The Dynamics of D-branes with Dirac-Born-Infeld
and Chern-Simons/Wess-Zumino Actions
Boris Stoyanov
DARK MODULI INSTITUTE, Membrane Theory Research Department,
18 King William Street, London, EC4N 7BP, United Kingdom
E-mail: stoyanov@darkmodulinstitute.org
BRANE HEPLAB, Theoretical High Energy Physics Department,
East Road, Cambridge, CB1 1BH, United Kingdom
E-mail: stoyanov@braneheplab.org
SUGRA INSTITUTE, 125 Cambridge Park Drive, Suite 301,
Cambridge, 02140, Massachusetts, United States
E-mail: stoyanov@sugrainstitute.org
Abstract
We have explained, and shown by feature stringy examples, why a D-brane in superstring theory,
when treated as a fundamental dynamical object, can be described by a map ϕ from an Azumaya/matrix manifold XAz with a fundamental module with a connection (E, ∇) to the target
spacetime Y . In this sequel, we construct a non-Abelian Dirac-Born-Infeld action functional
(Φ,g,B)
SDBI (ϕ, ∇) for such pairs (ϕ, ∇) when the target spacetime Y is equipped with a background
(dilaton, metric, B)-field (Φ, g, B) from closed strings. We next develop a technical tool needed
(Φ,g,B)
to study variations of this action and apply it to derive the first variation δSDBI /δ(ϕ, ∇) of
(Φ,g,B)
SDBI
with respect to (ϕ, ∇). The equations of motion that govern the dynamics of D-branes
(ρ,h;Φ,g,B,C)
then follow. We introduce a new action Sstandard
for D-branes that is to D-branes as the
Polyakov action is to fundamental strings. This ‘standard action’ is abstractly a non-Abelian
gauged sigma model based on maps ϕ : (XAz , E; ∇) → Y from an Azumaya/matrix manifold
XAz with a fundamental module E with a connection ∇ to Y enhanced by the dilaton term, the
gauge-theory term, and the Chern-Simons/Wess-Zumino term that couples (ϕ, ∇) to RamondRamond field. In a special situation, this new theory merges the theory of harmonic maps and
a gauge theory, with a nilpotent type fuzzy extension. A complete action for a D-brane world(C)
volume must include also the Chern-Simons/Wess-Zumino term SCS/WZ (ϕ, ∇) that governs how
the D-brane world-volume couples with the Ramond-Ramond fields C on Y . The current notes
lay down a foundation toward the dynamics of D-branes along the line of this research project.
1
Introduction
We have explained, and shown by feature stringy examples, why a D-brane in superstring theory, when treated as a fundamental dynamical object, can be described by a map ϕ from an
Azumaya/matrix manifold XAz , served as the D-brane world-volume, with a fundamental module with a connection (E, ∇), served as the Chan-Paton bundle, to the target space-time Y . In
(Φ,g,B)
this sequel, we construct a non-Abelian Dirac-Born-Infeld action functional SDBI (ϕ, ∇) for such
pairs (ϕ, ∇) when the target spacetime Y is equipped with a background (dilaton, metric, B)-field
(Φ, g, B) from closed strings. We next develop a technical tool needed to study variations of this
(Φ,g,B)
(Φ,g,B)
action and apply it to derive the first variation δSDBI /δ(ϕ, ∇) of SDBI with respect to (ϕ, ∇).
The equations of motion that govern the dynamics of D-branes then follow. A complete action for
(C)
a D-brane world-volume must include also the Chern-Simons/Wess-Zumino term SCS/WZ (ϕ, ∇)
that governs how the D-brane world-volume couples with the Ramond-Ramond fields C on Y .
(C,B)
In the current notes, a version SCS/WZ (ϕ, ∇) of non-Abelian Chern-Simons/Wess-Zumino action
(Φ,g,B)
functional for (ϕ, ∇) that follows the same guide with which we construct SDBI (ϕ, ∇) is constructed for lower-dimensional D-branes (i.e. D(-1)-, D0-, D1-, D2-branes). Its first variation
(C,B)
δSCS/WZ (ϕ, ∇)/δ(ϕ, ∇) is derived and its contribution to the equations of motion for (ϕ, ∇) follows. For D-branes of dimension ≥ 3, an anomaly issue needs to be understood in the current
context. The current notes lay down a foundation toward the dynamics of D-branes along the
line of this D-project. Some highlights of the history of how the Born-Infeld action and the
Dirac-Born-Infeld action arise from open string theory and a list of issues one needs to resolve
to convert such an action to that for coincident D-branes are given in the research article. They
serve as a guide for the steps of our exceptional discussion.
(ρ,h;Φ,g,B,C)
for D-branes that is to D-branes as the (BrinkWe introduce a new action Sstandard
Di Vecchia-Howe/Deser-Zumino/) Polyakov action is to fundamental superstrings. This action
depends both on the (dilaton field ρ, metric h) on the underlying topology X of the D-brane
world-volume and on the background (dilaton field Φ, metric g, B-field B, Ramond-Ramond field
C) on the target space-time Y ; and is naturally a non-Abelian gauged sigma model — based on
maps ϕ : (XAz , E; ∇) → Y from an Azumaya/matrix manifold XAz with a fundamental module
E with a connection ∇ to Y — enhanced by the dilaton term that couples (ϕ, ∇) to (ρ, Φ),
the B-coupled gauge-theory term that couples ∇ to B, and the Chern-Simons/Wess-Zumino
(ρ,h;Φ,g,B,C)
. Before one can do so,
term that couples (ϕ, ∇) to (B, C) in our standard action Sstandard
one needs to resolve the built-in obstruction of pull-push of covariant tensors under a map from
a noncommutative manifold to a commutative manifold. Such issue already appeared in the
construction of the non-Abelian Dirac-Born-Infeld action. In this note, we give a hierarchy of
various admissible conditions on the pairs (ϕ, ∇) that are enough to resolve the issue while being
open-string compatible. This improves our understanding of admissible conditions beyond. With
the noncommutative analysis, we develop further in this note some covariant differential calculus
for such maps and use it to define the standard action for D-branes. After promoting the setting
to a family version, we work out the first variation and hence the corresponding equations of
motion for D-branes of the standard action and the second variation of the kinetic term for
maps and the dilaton term in this action. Compared with the non-Abelian Dirac-Born-Infeld
action constructed in the research article along the same line, the current standard action is
clearly much more manageable. Classically and mathematically and in the special case where
the background (Φ, B, C) on Y is set to vanish, this new theory is a merging of the theory of
harmonic maps and a gauge theory (free to choose either a Yang-Mills theory or other kinds
of applicable gauge theory) with a nilpotent type fuzzy extension. The current bosonic setting
is the first step toward fermionic D-branes and their quantization as fundamental dynamical
objects, in parallel to what happened for fundamental superstrings with inclusion of exceptional
type extremal brane systems.
2
2
The first variation of the Dirac-Born-Infeld action
Given an admissible Lorentzian map,
ϕ : (XAz , E; ∇) −→ (Y, g, B, Φ) ,
let T := (−ε, ε) ⊂ R1 and ϕt : (XAz , E; ∇t ) → (Y, g, B, Φ), t ∈ T , be a differentiable T -family of
admissible Lorentzian maps that deforms ϕ =: ϕ0 . In this subsection we derive in steps the first
variation
d
(Φ,g,B)
t
dt t=0 SDBI (ϕt , ∇ )
of the Dirac-Born-Infeld action. The derivation for the other two situations: (Y, g) Lorentzian
and ϕt spacelike, and (Y, g) Riemannian and ϕt Riemannian, are completely the same.
As the major part of the discussion is local and around 0 ∈ T , we will assume that ε is small
enough and set the computation over a small enough coordinate chart U ⊂ X (with coordinate
functions x = (x1 , · · · , xm ) so that E|U is trivializable and trivialized, and ϕt (U ) is contained
in a coordinate chart V ⊂ Y (with coordinate functions y = (y 1 , · · · , y n )). Recall from Sec. 3.2
that, over U ,
(Φ,g,B)
SDBI
|U (ϕt , ∇t )
Z
=
− Tm−1
 
p
Re Tr e−ϕt Φ − SymDet U (ϕt (g + B) + 2πα0 F∇t )
U
s
X
]
Re Tr e−ϕt (Φ) −SymDet
ϕ]t (Eij )Dµt ϕ]t (y i )Dνt ϕ]t (y j ) + 2πα0 [∇tµ , ∇tν ]
Z
= − Tm−1
U
i,j

µν
 dm x .
Here, we set the notation for the tensors and connections involved as follows:
· g + B = P (g + B ) dy ⊗ dy =: P E dy ⊗ dy , with g = g , B = −B ,
· ∇ = d + A = P (∂ + A ) dx is the connection on E| ,
· D = d + [A , · ] = P (∂ + [A , · ]) dx is the ∇ -induced connection on End (E|
· d x := dx ∧ · · · ∧ dx is compatible with the orientation on U ;
i,j
t
i
ij
t
t
m
ij
µ
t
µ
µ
t
i,j
i
ij
j
ij
µ
ji
ij
ji
U
µ
t
µ
µ
µ
1
j
t
C
U ),
m
and, for later use,
d
ϕ̇] (y i ) := dt
t=0
·(ϕ](yd)) :=
ϕ]T (y i ) ,
d
dt
t=0
d
Ȧµ := dt
(ϕ]T (y d )) ,
t=0
ATµ .
We assume further that the local chart U and ϕ > 0 are small enough so that the construction
over UT := U × (−ε, ε) in Sec. 4.1, with p ∈ U × {0} ⊂ UT , applies simultaneously to e−Φ and
Eij , i, j = 1, · · · , n, to give the local expression of ϕ]T (Φ) and ϕ]T (Eij ), i, j = 1 . . . , n, in terms
of elements in the polynomial ring over C ∞ (UT )
ϕ]T (Φ) , ϕ]T (Eij ) ∈
⊕sj=1 C ∞ (UT ) · Id E (j) [ ϕ]T (y 1 ), · · · , ϕ]T (y n ) ]
T
of multi-degree ≤ (r − 1, · · · , r − 1). Associated to these settings and with the notation from
Remark/Notataion 4.2.3.5, recall that
]
e−ϕT (Φ) = ϕ]T (e−Φ )
d
dt
−ϕ]T (Φ)
e
=
t=0
d
dt
t=0
= Re
−Φ
[0]|yd ϕ] (y)d ,
T
ϕ]T (e−Φ ) = R
e−Φ
[1]|yd · (ϕ] (y)d ) ;
T
and
ϕ]T (Eij ) = R Eij [0]|yd ϕ] (y)d ,
T
d
dt
t=0
ϕ]T (Eij )
= R
Eij
[1]|yd · (ϕ] (y)d ) ,
T
3
for i, j = 1, . . . , n. For simplicity of notation, it is understood that R Eij [1] is evaluated at t = 0
in the expression R Eij [1]|yd · (ϕ] (y)d ) ; and similarly for induced expressions that follow this.
T
The first variation of each ingredient in the Dirac-Born-Infeld action
]
(a) The first variation of e−ϕ (Φ) and ϕ] (Eij )
d
dt
−ϕ]T (Φ)
−Φ
= Re
e
t=0
=
n X
•
X
Then, it follows from Proposition 4.2.3.1 that
[1]|yd ·(ϕ] (yd ))
X
X
i0 =1 d=0 d, |d|=d ~
~
π ∈Ptn(1,d),
i(~π, d) =i0
−Φ
0
([∂y~πi0 ]R e [1](d) )L (ϕ] (y)) · ϕ̇] (y i ) · ([∂y~πi0 ]R e
n
X
=:
0
−Φ
−Φ
]
] i
e
L
R e [1](d,
~
π ) (ϕ (y)) · ϕ̇ (y ) · R
X
−Φ
[1](d) )R (ϕ] (y))
]
R
[1](d,
~
π ) (ϕ (y))
i0 =1 d, d, ~
π ; |d|=d, i(~π,d) =i0
and
d
dt
t=0
=
ϕ]T (Eij )
n X
•
X
= R Eij [1]|yd ·(ϕ] (yd ))
X
X
i0 =1 d=0 d, |d|=d ~
~
π ∈Ptn(1,d),
i(~π, d) =i0
0
([∂y~πi0 ]R Eij [1](d) )L (ϕ] (y)) · ϕ̇] (y i ) · ([∂y~πi0 ]R Eij [1](d) )R (ϕ] (y))
=:
n
X
0
L
]
] i
Eij
R
]
R Eij [1](d,
[1](d,
~
π ) (ϕ (y)) · ϕ̇ (y ) · R
~
π ) (ϕ (y)) .
X
i0 =1 d, d, ~
π ; |d|=d, i(~π,d) =i0
(b) The first variation of Dµ ϕ] (y i ) and Fµν
d
dt
t=0
DµT ϕ]T (y i )
=
By straightforward computation,
d
dt
∂µ ϕ]T (y i ) + [ATµ , ϕ]T (y i )]
t=0
] i
= Dµ ϕ̇ (y ) − [ϕ] (y i ), Ȧµ ]
and
d
dt
t=0
T
Fµν
=
d
dt
t=0
[∇Tµ , ∇Tν ]
= Dµ Ȧν − Dν Ȧµ .
The first variation of the Dirac-Born-Infeld action
With all the ingredients prepared, the computation of the first variation of SDBI (ϕ, ∇) is now
straightforward, though some of the expressions may look complicated due to noncommutativity.
We proceed in five steps.
Step (1) : Input from all the pieces
Let
M µν (t) :=
X ]
ϕt (Eij )Dµt ϕ]t (y i )Dνt ϕ]t (y j ) + 2πα0 [∇tµ , ∇tν ] ∈ C ∞ (End C (E|U ))
i,j
4
and M (t) := [M µν (t)]µν the m × m matrix with (µ, ν)-entry M µν (t). Then,
d
dt
SDBI (ϕt , ∇t )
d
dt
= −Tm−1
t=0
Z
t=0
p
]
Re Tr e−ϕt (Φ) −SymDet (M (t))
dm x
U
Re Tr
Z
= −Tm−1
U
d
dt
Tr
]
e−ϕt (Φ)
d
dt
]
e−ϕt (Φ)
p
−SymDet (M (t))
p
−SymDet (M (t))
t=0
q
]
Since (ϕ, ∇) is admissible, e−ϕ (Φ) and
Tr
]
e−ϕ
dm x ;
t=0
d
dt
p
−Φ
]
= Tr (R e [1]t=0 )|yI ·(ϕ] (yI )) · −SymDet (M (0)) + Tr e−ϕ (Φ)
(Φ)
d
dt
p
−SymDet (M (t))
t=0
−SymDet (M (t))
.
t=0
−SymDet (M (t)) commute. Thus,
−1
Tr
2
=
p
]
e−ϕ
(Φ)
p
−SymDet (M (0))
−1
·
d
dt
SymDet (M (t)) .
t=0
>
Denote by [ · ] the transpose of the matrix [ · ] and let


M (1) (t)
h
i>


>
>
···
M (t) = 
 = M (1) , · · · , M (m)
M (m) (t)
be the presentation of M (t) in terms of its row vectors and denote
for µ = 1, . . . , m. Then
d
dt
SymDet (M (t)) =
t=0
Denote
by
m
X
d
dt t=0
M (µ) (t) by Ṁ (µ) (0),
SymDet ([M (1) (0)> , · · · , M (µ−1) (0)> , Ṁ (µ) (0)> , M (µ+1) (0)> , · · · , M (m) (0)> ]> ) .
µ=1
d
dt t=0
M µν (t) by Ṁ µν (0), for µ, ν = 1, . . . , m. Then, the ν-th entry in Ṁ (µ) (0) is given
Ṁ µν (0)
=
X
R Eij [1]|yd ·(ϕ] (yd )) · Dµ ϕ] (y i )Dν ϕ] (y j )
i,j
+
X
+
X
ϕ] (Eij ) · (Dµ ϕ̇] (y i ) − [ϕ] (y i ), Ȧµ ]) · Dν ϕ] (y j )
i,j
ϕ] (Eij )Dµ ϕ] (y i ) · (Dν ϕ̇] (y j ) − [ϕ] (y j ), Ȧν ]) + 2πα0 (Dµ Ȧν − Dν Ȧµ ) .
i,j
With
M µν (0) =
X
ϕ] (Eij )Dµ ϕ] (y i )Dν ϕ] (y j ) + 2πα0 [∇µ , ∇ν ] ,
i,j
one has altogether:
d
d
t
dt t=0 SDBI (ϕt , ∇ ) = −Tm−1 dt
Z
= −Tm−1
U
Z
t=0
p
]
dm x
Re Tr e−ϕt (Φ) −SymDet (M (t))
U
p
−Φ
Re Tr R e [1]|yd ·(ϕ] (yd )) · −SymDet (M (0))
p
−1
]
1
− 2 e−ϕ (Φ) −SymDet (M (0))
m
X
·
SymDet ([M (1) (0)> , · · · , M (µ−1) (0)> , Ṁ (µ) (0)> , M (µ+1) (0)> , · · · , M (m) (0)> ]> ) dm x
µ=1
Z
= −Tm−1
U
p
−Φ
Re Tr (R e [1])|yd ·(ϕ] (yd )) · −SymDet (M (0))
p
−1
]
1
− 2 e−ϕ (Φ) −SymDet (M (0))
m
X
X
·
(−1)σ M 1 σ(1) (0) · · ·
µ=1 σ∈Symm
M (µ−1) σ(µ−1) (0)
Ṁ µ σ(µ) (0)
5
M (µ+1) σ(µ+1) (0)
···
M m σ(m) (0)
dm x .
Step (2) : Arrangement to boundary terms and the linear functional δSDBI (ϕ, ∇)/δ(ϕ, ∇) on
(ϕ̇] (y 1 ), · · · , ϕ̇] (y n ); Ȧ1 , · · · , Ȧm )
Summands from the first cluster
−Φ
R e [1]|yd ·(ϕ] (yd )) ·
q
−SymDet (M (0))
−Φ
contain only ϕ̇] (y i ), i = 1, . . . , n, from (R e [1])|y d ·(ϕ] (yd )) . Hence, it contributes solely to the
(Φ,g,B)
linear functional δSDBI (ϕ, ∇)/δ(ϕ, ∇) on (ϕ̇] (y 1 ), · · · , ϕ̇] (y n ); Ȧ1 , · · · , Ȧm ) and, hence, to the
equations of motion for (ϕ, ∇).
On the other hand, summands from the expansion of the second cluster
]
·
m
X
−1
q
− 21 e−ϕ (Φ)
−SymDet (M (0))
(−1)σ M 1 σ(1) (0)
X
· · · M (µ−1) σ(µ−1) (0)
Ṁ µ σ(µ) (0) M (µ+1) σ(µ+1) (0)
µ=1 σ∈Symm
···
M m σ(m) (0)
are of two types:
· One contains a factor in the list ϕ̇ (y ), i
]
i
= 1, . . . , n, Ȧµ , µ = 1, . . . , m from some
Ṁ µ0 ν 0 (0), µ , ν = 1, . . . , m. They contribute to the linear functional δSDBI (ϕ, ∇)/δ(ϕ, ∇)
on (ϕ̇] (y 1 ), · · · , ϕ̇] (y n ); Ȧ1 , · · · , Ȧm ) and, hence, to the equations of motion for (ϕ, ∇).
0
0
· The other contains a factor in the list D ϕ̇(y ), i
i
= 1, . . . , n, µ = 1, . . . , m, Dµ Ȧν ,
µ, ν = 1, . . . , m, from some Ṁ (0), µ , ν R= 1, . . . , m. After integration by parts, each
contributes a boundary term in an integral ∂U ( · · · ) and a term in the linear functional
δSDBI (ϕ, ∇)/δ(ϕ, ∇) on (ϕ̇] (y 1 ), · · · , ϕ̇] (y n ); Ȧ1 , · · · , Ȧm ). The latter contributes then to
the equations of motion for (ϕ, ∇).
µ0 ν 0
µ
0
0
We now proceed to study their details.
Step (3) : Details for the first cluster
For the first cluster,
−Φ
p
Tr R e [1]|yd ·(ϕ] (yd )) · −SymDet (M (0))
= Tr
n
X
X
Re
−Φ
0
[1](Ld, ~π) (ϕ] (y)) · ϕ̇] (y i ) · R e
−Φ
]
[1](R
d, ~π ) (ϕ (y))
p
· −SymDet (M (0))
0
i0 =1 d, d, ~
π ; |d|=d, i(~
π ,d) =i
= Tr
=: Tr
n X
X
i0 =1
0
d, d, ~
π ; |d|=d, i(~
π ,d) =i
n
X
1, (Φ,g,B)
NLi0
Re
0
−Φ
(ϕ, ∇) · ϕ̇] (y i )
]
[1](R
d, ~π ) (ϕ (y)) ·
p
−SymDet (M (0)) · R e
−Φ
[1](Ld, ~π) (ϕ] (y))
0
· ϕ̇] (y i )
.
i0 =1
Step (4) : Details for the second cluster
For the second cluster, we have
SymDet ([M (1) (0)> , · · · , M (µ−1) (0)> , Ṁ (µ) (0)> , M (µ+1) (0)> , · · · , M (m) (0)> ]> ) =
m
X
1 X
(−1)σ Det ([M (σ(1)) (0)> , · · · , M (σ(µ0 −1)) (0)> , Ṁ (µ) (0)> , M (σ(µ0 +1)) (0)> , · · · , M (σ(m)) (0)> ]> ) .
m! 0
µ =1 σ ∈ Symm
σ(µ0 ) = µ
6
]
Thus, denoting the factor − 21 e−ϕ (Φ)
Tr
p
−SymDet (M (0))
−1
by F2 (ϕ, ∇; Φ, g, B),
p
−1
]
1
− 2 e−ϕ (Φ) −SymDet (M (0))
m
X
·
SymDet ([M (1) (0)> , · · · , M (µ−1) (0)> , Ṁ (µ) (0)> , M (µ+1) (0)> , · · · , M (m) (0)> ]> )
µ=1
= Tr
= Tr
F2 (ϕ, ∇; Φ, g, B) ·
m
X
SymDet ([M (1) (0)> , · · · , M (µ−1) (0)> , Ṁ (µ) (0)> , M (µ+1) (0)> , · · · , M (m) (0)> ]> )
µ=1
m X
m
X
1
F
(ϕ,
∇;
Φ,
g,
B)
·
2
m!
0
X
µ=1 µ =1 σ ∈ Symm
σ(µ0 ) = µ
(−1)σ Det ([M (σ(1)) (0)> , · · · , M (σ(µ0 −1)) (0)> , Ṁ (µ) (0)> , M (σ(µ0 +1)) (0)> , · · · , M (σ(m)) (0)> ]> )
= Tr
m
m
1 X X
m!
0
X
(−1)σ
µ=1 µ =1 σ ∈ Symm
σ(µ0 ) = µ
· Det ([F2 (ϕ, ∇; Φ, g, B) M (σ(1)) (0)> , · · · , M (σ(µ0 −1)) (0)> , Ṁ (µ) (0)> , M (σ(µ0 +1)) (0)> , · · · , M (σ(m)) (0)> ]> )
= Tr
m
m
1 X X
m!
0
0
X
0
(−1)σ (−1)µ (m−µ )
µ=1 µ =1 σ ∈ Symm
σ(µ0 ) = µ
· Det ([M (σ(µ0 +1)) (0)> , · · · , M (σ(m)) (0)> , F2 (ϕ, ∇; Φ, g, B) M (σ(1)) (0)> , · · · , M (σ(µ0 −1)) (0)> , Ṁ (µ) (0)> ]> )
(by the invariance of trace under cyclic permutations) .
Note that Ṁ µν (0), µ, ν = 1, . . . , m, now appear uniformly as the last factor in the summands
from the expansion of Det ([ · · · ]> ) above. Let Minor(ϕ, ∇; Φ, g, B | µ0 , σ)µν be the (m, ν)-minor of
[M (σ(µ0 +1)) (0), · · · , M (σ(m)) (0), F2 (ϕ, ∇; Φ, g, B) M (σ(1)) (0), · · · , M (σ(µ0 −1)) (0), Ṁ (µ) (0)]> . Then:
= Tr
m
m
1 X X
m!
0
X
0
0
(−1)σ (−1)µ (m−µ ) ·
µ=1 µ =1 σ ∈ Symm
m
X
(−1)m+ν Minor(ϕ, ∇; Φ, g, B | µ0 , σ)µν Ṁ µν (0)
ν=1
σ(µ0 ) = µ
= Tr
m
m X
X
Ξ(ϕ, ∇; Φ, g, B)µν Ṁ µν (0)
,
µ=1 ν=1
where Ξ(ϕ, ∇; Φ, g, B)µν
m
1 X
:= m!
0
0
X
0
(−1)σ (−1)µ (m−µ )+m+ν Minor(ϕ, ∇; Φ, g, B | µ0 , σ)µν ,
µ =1 σ ∈ Symm
σ(µ0 ) = µ
= Tr
m
m X
X
Ξ(ϕ, ∇; Φ, g, B)µν
µ=1 ν=1
·
X
R Eij [1]|yd ·(ϕ] (yd )) · Dµ ϕ] (y i )Dν ϕ] (y j )
i,j
+
X
+
X
ϕ] (Eij ) · (Dµ ϕ̇] (y i ) − [ϕ] (y i ), Ȧµ ]) · Dν ϕ] (y j )
i,j
ϕ] (Eij )Dµ ϕ] (y i ) · (Dν ϕ̇] (y j ) − [ϕ] (y j ), Ȧν ]) + 2πα0 (Dµ Ȧν − Dν Ȧµ )
i,j
= (I) + (II) + (III) + (IV)
(defined in Step (4.1) – Step (4.4) below) .
Let us now study each of the four subclusters of the second cluster separately.
Step (4.1) : The subcluster (I)
(I)
:=
Tr
m X
m
X
µ=1 ν=1
Ξ(ϕ, ∇; Φ, g, B)µν ·
X
R Eij [1]|yd ·(ϕ] (yd )) · Dµ ϕ] (y i )Dν ϕ] (y j )
i,j
7
=
Tr
m X
m
X
Ξ(ϕ, ∇; Φ, g, B)µν
µ=1 ν=1
n
X X
·
=
Tr
i0 =1
]
R Eij [1](Ld, ~π) (ϕ] (y)) · ϕ̇] (y i ) · R Eij [1](R
d, ~π ) (ϕ (y))
m
X
m X
X
X
µ=1 ν=1 i,j
n
X
· Dµ ϕ] (y i )Dν ϕ] (y j )
]
] i
] j
R Eij [1](R
d, ~π ) (ϕ (y)) · Dµ ϕ (y )Dν ϕ (y )
d, d, ~
π ; |d|=d, i(~
π , d)
=i0
· Ξ(ϕ, ∇; Φ, g, B)µν · R Eij [1](Ld, ~π) (ϕ] (y))
=: Tr
0
i0 =1 d, d, ~
π ; |d|=d, i(~
π ,d) =i
i,j
n
X
0
X
2.I, (Φ,g,B)
NLi0
0
· ϕ̇] (y i )
0
(ϕ, ∇) · ϕ̇] (y i ) .
i0 =1
Step (4.2) : The subcluster (II)
This subcluster contributes also to boundary terms.
(II)
:=
Tr
m X
m
X
µ=1 ν=1
Ξ(ϕ, ∇; Φ, g, B)µν
X
X
·
ϕ] (Eij ) · Dµ ϕ̇] (y i ) · Dν ϕ] (y j ) +
ϕ] (Eij )Dµ ϕ] (y i ) · Dν ϕ̇] (y j )
i,j
=
Tr
m X
m X
X
Dµ ϕ] (y i ) · Ξ(ϕ, ∇; Φ, g, B)νµ ϕ] (Eji )
µ=1 ν=1 i,j
=
Tr
+ Ξ(ϕ, ∇; Φ, g, B)µν ϕ] (Eij )Dµ ϕ] (y i )
m X
m X
X
Dν
h
=
m
X
∂ν Tr
m
X
ϕ̇] (y j )
i
ϕ̇] (y j )
i
h
Dν Dµ ϕ] (y i ) · Ξ(ϕ, ∇; Φ, g, B)νµ ϕ] (Eji )
µ=1 ν=1 i=1
(−1)ν−1 ∂ν (BT
i
+ Ξ(ϕ, ∇; Φ, g, B)µν ϕ] (Eij )Dµ ϕ] (y i ) · ϕ̇] (y j )
+ Ξ(ϕ, ∇; Φ, g, B)µν ϕ] (Eij ) Dµ ϕ] (y i )
n
m X
n X
m X
X
j=1
Dµ ϕ] (y i ) Ξ(ϕ, ∇; Φ, g, B)νµ ϕ] (Eji )
µ=1 i,j
− Tr
· Dν ϕ̇] (y j )
h
Dν Dµ ϕ] (y i ) · Ξ(ϕ, ∇; Φ, g, B)νµ ϕ] (Eji )
µ=1 ν=1 i=1
m Xh X
ν=1
=:
+ Ξ(ϕ, ∇; Φ, g, B)µν ϕ] (Eij ) Dµ ϕ] (y i )
n
m X
n X
m X
X
j=1
Dµ ϕ] (y i ) Ξ(ϕ, ∇; Φ, g, B)νµ ϕ] (Eji )
ν=1 µ=1 i,j
− Tr
i,j
i
+ Ξ(ϕ, ∇; Φ, g, B)µν ϕ] (Eij )Dµ ϕ] (y i ) · ϕ̇] (y j )
2.II, (ϕ,∇;Φ,g,B)
(ϕ̇] (y)))
ν
+ Tr
ν=1
n
X
2.II, (Φ,g,B)
NLj
(ϕ, ∇) · ϕ̇] (y j ) .
j=1
Step (4.3) : The subcluster (III)
(III)
:=
Tr
−
m X
m
X
Ξ(ϕ, ∇; Φ, g, B)µν
µ=1 ν=1
·
X
ϕ] (Eij ) · [ϕ] (y i ), Ȧµ ] · Dν ϕ] (y j ) +
i,j
=
Tr
m X
m X
X
ν=1 µ=1 i,j
X
ϕ] (Eij ) Dµ ϕ] (y i ) · [ϕ] (y j ), Ȧν ]
i,j
Ξ(ϕ, ∇; Φ, g, B)µν ϕ] (Eij ) Dµ ϕ] (y i ) ϕ] (y j )
− ϕ] (y j ) Ξ(ϕ, ∇; Φ, g, B)µν ϕ] (Eij ) Dµ ϕ] (y i )
− Dµ ϕ] (y j ) Ξ(ϕ, ∇; Φ, g, B)νµ ϕ] (Eij ) ϕ] (y i )
+ ϕ] (y i ) Dµ ϕ] (y j ) Ξ(ϕ, ∇; Φ, g, B)νµ ϕ] (Eij )
=: Tr
m
X
(Φ,g,B)
NL2.III,
(ϕ, ∇) · Ȧν .
ν
ν=1
8
· Ȧν
Step (4.4) : The subcluster (IV)
This subcluster contributes also to boundary terms.
(IV)
:=
Tr 2πα
0
µ=1 ν=1
m X
m X
µ=1 ν=1
m X
m
X
Tr 2πα0
=
Ξ(ϕ, ∇; Φ, g, B)µν · (Dµ Ȧν − Dν Ȧµ )
Tr 2πα0
=
m X
m
X
Ξ(ϕ, ∇; Φ, g, B)µν − Ξ(ϕ, ∇; Φ, g, B)νµ · Dµ Ȧν
h
i
Ξ(ϕ, ∇; Φ, g, B)µν − Ξ(ϕ, ∇; Φ, g, B)νµ · Ȧν
Dµ
µ=1 ν=1
m X
m
X
0
h
i
Dµ Ξ(ϕ, ∇; Φ, g, B)µν − Ξ(ϕ, ∇; Φ, g, B)νµ · Ȧν
− Tr 2πα
ν=1
m
X
=
∂µ Tr 2πα0
µ=1
µ=1
m
X
Ξ(ϕ, ∇; Φ, g, B)µν − Ξ(ϕ, ∇; Φ, g, B)νµ · Ȧν
ν=1
− Tr 2πα0
m X
m
X
ν=1
m
X
=:
h
i
Dµ Ξ(ϕ, ∇; Φ, g, B)µν − Ξ(ϕ, ∇; Φ, g, B)νµ · Ȧν
µ=1
(−1)µ−1 ∂µ (BT
2.IV, (ϕ,∇;Φ,g,B)
(Ȧ))
µ
+ Tr
µ=1
m
X
(Φ,g,B)
NL2.IV,
(ϕ,
∇)
·
Ȧ
ν .
ν
ν=1
Step (5) : The final formula
In summary, with the notation introduced for the various nonlinear first-order and second-order
differential expressions on (ϕ, ∇) that depend on (Φ, g, B) and appear in the calculation (subject
to a relabelling of the dummy i0 index), one has
d
dt
SDBI (ϕt , ∇t )
= −Tm−1
t=0
Z
= −Tm−1
Re
U
m
X
d
dt
(−1)µ−1 ∂µ
Z
U
t=0
BT
p
]
Re Tr e−ϕt (Φ) −SymDet (M (t))
d mx
2.II, (ϕ,∇;Φ,g,B)
(ϕ̇] (y))
µ
+ BT
2.IV, (ϕ,∇;Φ,g,B)
(Ȧ)
µ
d mx
µ=1
n
X
1, (Φ,g,B)
2.I, (Φ,g,B)
2.II, (Φ,g,B)
Re Tr
(NLj
(ϕ, ∇) + NLj
(ϕ, ∇) + NLj
(ϕ, ∇)) · ϕ̇] (y j )
U
j=1
m
X
(Φ,g,B)
+
(NL2.III,
(ϕ, ∇) + NLν2.IV, (Φ,g,B) (ϕ, ∇)) · Ȧν d m x
ν
Z
−Tm−1
ν=1
Z
=: −Tm−1
(ϕ,∇;Φ,g,B)
Re (BT
(ϕ̇] (y), Ȧ))
∂U
Z
−Tm−1
U
n
m
X
X
(Φ,g,B);δϕ
Re Tr
NLj
(ϕ, ∇) · ϕ̇] (y j ) +
NL(Φ,g,B);δ∇
(ϕ, ∇) · Ȧν d m x .
ν
ν=1
j=1
Here,
BT
(ϕ,∇;Φ,g,B)
(ϕ̇] (y), Ȧ)
m X
(ϕ,∇;Φ,g,B)
:=
BT 2.II,
(ϕ̇] (y)) + BT
µ
2.IV, (ϕ,∇;Φ,g,B)
(Ȧ)
µ
dµ ∧ dxµ+1 · · · ∧ dxm ,
dx1 ∧ · · · ∧ dxµ−1 ∧ dx
µ=1
dµ meaning the removal of dxµ , is a complex-valued (m − 1)-form on U that depends
with the dx
linearly on (ẏ, Ȧ) and whose real part gives the total boundary term (up to the factor −Tm−1 )
(Φ,g,B)
of the first variation of SDBI (ϕ, ∇) with respect to (ϕ, ∇).
9
2.1
The equations of motion for D-branes
Remark 2.1.1 [ effect of Re ( · ) in action to equations of motion ] Due to the operation ‘Taking
the real part of ’ Re ( · ), to go from the the first variation formula to the expression for the
equations of motion there is a detail that depends
on how the space of pairs (ϕ, ∇) and its
√
−1θ
tangents (δϕ, δ∇) are parameterized; (cf. Re (e
z) = cos θ · Re (z) − sin θ · Im (z)).
(1) For the ϕ-part, first, caution that it is not that just because ϕ] (y i ), i = 1, . . . , n, take
values in a ring over C (i.e. C ∞ (End C (E))) that the space Map ((XAz , E), Y ) of all such
ϕ’s becomes a complex space. Indeed, due to the fact that all the eigenvalues of ϕ] (f ),
f ∈ C ∞ (Y ) are real (cf. [L-Y4: Sec. 3], D(11.1)), Map ((XAz , E), Y ) is intrinsically a real
space and there is no natural complex-space structure on it (even if exists) that can be made
compatible with the underlying
moduli problem since if δϕ is an unobstructed tangent to
√
Map ((XAz , E), Y ), then −1δϕ can never be an unobstructed tangent to Map ((XAz ), Y ).
So this part is good in the sense that if we fix a real presentation for ϕ’s in the study, then
Re (δSDBI /δϕ) gives the system of equations of motion for ϕ.
(2) For the ∇-part, if alone, the parameter space is complex in nature in our most general
setting. When E is Hermitian and ∇ is required to be compatible with the Hermitian
structure, the resulting parameter space becomes intrinsically real. In the latter case,
depending on the convention in presenting a unitary gauge theory (mathematicians vs.
(Φ,g,B)
(Φ,g,B)
physicists), one may take either Re (δSDBI /δ∇) or Im (δSDBI )/δ∇ as the system of
equations for ∇. However, this is not the full story as we imposed the admissible condition
∇• Aϕ ⊂ Aϕ on ∇. Details on writing the equations of motion will have to depend on how
we present this condition.
Not to let this additional detail to distract us in this first work in the D(13) subseries, we present
(Φ,g,B)
for the current notes the system of equations of motion that remove the effect of Re ( · ) in SDBI .
In other words, a true system of equations of motion will involve only a combination of what are
given below.
It follows from the study in Sec. 5.2 that the equations of motion for D-branes from the
Dirac-Born-Infeld action, with the D-brane world-volume modelled in the current context as an
admissible map
ϕ : (X Az , E; ∇) −→ (Y, Φ, g, B)
from an Azumaya/matrix manifold with a fundamental module with a connection (X Az , E; ∇)
to a space-time Y with massless background fields (Φ, g, B) from closed string excitations, are
given by the following system of second-order nonlinear partial differential equations on (ϕ, ∇):

 NL(Φ,g,B);δϕ (ϕ, ∇)
j

= 0 , for j = 1, . . . , n ;
NLν(Φ,g,B);δ∇ (ϕ, ∇) = 0 , for ν = 1, . . . , m .
Here, for the first subsystem,
(Φ,g,B);δϕ
NLj
1, (Φ,g,B)
(ϕ, ∇) = NLj
2.I, (Φ,g,B)
(ϕ, ∇) + NLj
2.II, (Φ,g,B)
(ϕ, ∇) + NLj
(ϕ, ∇)
with
1, (Φ,g,B)
NLj
(ϕ, ∇)
X
=
Re
−Φ
]
[1](R
d, ~π ) (ϕ (y)) ·
p
−Φ
−SymDet (M (0)) · R e [1](Ld, ~π) (ϕ] (y)) ,
d, d, ~
π ; |d|=d, i(~
π ,d) =j
2.I, (Φ,g,B)
NLj
(ϕ, ∇)
=
m X
m X
X
X
]
] i
] j
R Ei0 j0 [1](R
d, ~π ) (ϕ (y)) · Dµ ϕ (y )Dν ϕ (y )
µ=1 ν=1 i0 ,j 0 d, d, ~
π ; |d|=d, i(~
π ,d) =j
· Ξ(ϕ, ∇; Φ, g, B)µν · R Ei0 j0 [1](Ld, ~π) (ϕ] (y)) ,
2.II, (Φ,g,B)
NLj
(ϕ, ∇)
=
−
m X
m X
n
X
µ=1 ν=1
Dν Dµ ϕ] (y i ) · Ξ(ϕ, ∇; Φ, g, B)νµ ϕ] (Eji )
i=1
+ Ξ(ϕ, ∇; Φ, g, B)µν ϕ] (Eij )Dµ ϕ] (y i ) ;
10
and, for the second subsystem,
(Φ,g,B)
(ϕ, ∇)
(ϕ, ∇) = NLν2.III, (Φ,g,B) (ϕ, ∇) + NL2.IV,
NL(Φ,g,B);δ∇
ν
ν
with
(Φ,g,B)
NL2.III,
(ϕ, ∇)
ν
=
m X
X
Ξ(ϕ, ∇; Φ, g, B)µν ϕ] (Eij ) Dµ ϕ] (y i ) ϕ] (y j )
µ=1 i,j
− ϕ] (y j ) Ξ(ϕ, ∇; Φ, g, B)µν ϕ] (Eij ) Dµ ϕ] (y i )
− Dµ ϕ] (y j ) Ξ(ϕ, ∇; Φ, g, B)νµ ϕ] (Eij ) ϕ] (y i )
+ ϕ] (y i ) Dµ ϕ] (y j ) Ξ(ϕ, ∇; Φ, g, B)νµ ϕ] (Eij ) ,
(Φ,g,B)
NL2.IV,
(ϕ, ∇)
ν
=
2πα0
m
X
Dµ (Ξ(ϕ, ∇; Φ, g, B)νµ − Ξ(ϕ, ∇; Φ, g, B)µν ) .
µ=1
In both subsystems,
Ξ(ϕ, ∇; Φ, g, B)µν
=
m
1 X
m! 0
X
0
0
(−1)σ (−1)µ (m−µ )+m+ν Minor(ϕ, ∇; Φ, g, B | µ0 , σ)µν ,
µ =1 σ ∈ Symm
σ(µ0 ) = µ
where
Minor(ϕ, ∇; Φ, g, B | µ0 , σ)µν = the (m, ν)-minor of
[M (σ(µ0 +1)) (0)> , · · · , M (σ(m)) (0)> , F2 (ϕ, ∇; Φ, g, B) M (σ(1)) (0)> , · · · , M (σ(µ0 −1)) (0)> , Ṁ (µ) (0)> ]>
with
F2 (ϕ, ∇; Φ, g, B)
M ( • ) (0)
M µν (0)
p
−1
]
1
,
= − 2 e−ϕ (Φ) −SymDet (M (0))
= the • -th row vector of M (0) ,
X
0
0
= the (µ, ν)-entry of M (0) =
ϕ] (Ei0 j 0 ) Dµ ϕ] (y i ) Dν ϕ] (y j ) + 2πα0 [∇µ , ∇ν ] ,
i0 ,j 0
Ṁ µν (0)
=
X
i0 ,j 0
0
0
R Ei0 j0 [1]|yd ·(ϕ] (yd )) · Dµ ϕ] (y i )Dν ϕ] (y j )
+
X
+
X
0
0
0
ϕ] (Ei0 j 0 ) · (Dµ ϕ̇] (y i ) − [ϕ] (y i ), Ȧµ ]) · Dν ϕ] (y j )
i0 ,j 0
0
0
0
ϕ] (Ei0 j 0 )Dµ ϕ] (y i ) · (Dν ϕ̇] (y j ) − [ϕ] (y j ), Ȧν ]) + 2πα0 (Dµ Ȧν − Dν Ȧµ ) .
i0 ,j 0
Remark 2.1.2 [ origin/correction from anomaly equations for open strings ] From the stringtheory point of view, it is very important to understand further how such systems of differential
equations on the pair (ϕ, ∇) can arise from or be correced/improved by the anomaly-free conditions in open-string theory.
Remark 2.1.3 [the case of Hermitian/unitary D-branes] When in addition E is equipped with a
Hermitian structure and ϕ is Hermitian and ∇ is unitary, the Dirac-Born-Infeld action functional
(Φ,g,B)
S(ϕ,∇) and, hence, the resulting equations of motion can be simplified. The detail should be
studied further.
11
3
Remarks on the Chern-Simons/Wess-Zumino term
In view of Polchinski’s realization ([Po1]) that a D-brane world-volume can couple to a RamondRamond field in superstring theory (cf. Figure 6-0-1), the Chern-Simons/Wess-Zumino term
SCS/WZ for D-branes is also an indispensable part to understand the dynamics of D-branes.
(Φ,g,B)
With the same essence as for the construction of SDBI (ϕ, ∇), we construct in this section
the Chern-Simons/Wess-Zumino action SCS/WZ (ϕ, ∇) for lower-dimensional D-branes, in which
cases anomaly issues do not occur, derive their first variation formula and, hence, obtain their
contribution to the equations of motions for D-branes.
To begin, with anomalies taken into account, the coupling of a simple embedded D-brane
f : X ,→ Y
with the Ramond-Ramond field C on Y (with a B-field background B), is encoded in the ChernSimons/Wess-Zumino action for D-branes, which takes the form
(C,B)
SCS/WZ (f, ∇)
= Tm−1
Z
∗
f C ∧e
2πα0 F∇ +f ∗ B
∧
q
X
,
Â(X)/Â(NX/Y )
(m)
where
· m = dim X, T
the D(m − 1)-brane tension, Â( · ) the Â-class of the bundle in question,
the normal bundle of X in Y along f ,
m−1
NX/Y
· (···)
(m)
is the degree-m component of a differential form ( · · · ) on X.
The fact that the over coupling strength is identical with the D-brane tension Tm−1 is a consequence of supersymmetry.
With the lesson already learned from studying the Dirac-Born-Infeld action, formally the
Chern-Simons/Wess-Zumino action generalizes to the case of coincident D-brane in our setting
ϕ : (XAz , E; ∇) −→ Y ,
as
(C,B)
SCS/WZ (f, ∇)
formally
=
Z
Tm−1
Re
Tr
0
B
ϕ C ∧ e2πα F∇ +ϕ
X
∧
q
Â(XAz )/Â(NXAz /Y )
.
(m)
One now has to resolve in addition the following issues:
q
(8) the anomaly factor “ Â(XAz )/Â(NXAz /Y ) ”, which presumably is an End C (E)-valued differential form on X;
(9) wedging of of End C (E)-valued
differential forms on X :
q
2πα0 F∇ +ϕ B
ϕ C ∧e
∧ Â(X)/Â(NXAz /Y ) .
3.1
Resolution of issues in the Chern-Simons/Wess-Zumino term
We address in this subsection the resolution of Issue (9) in a way that is compatible with how
we treat/interpret the Dirac-Born-Infled action in Sec. 3. This gives us a version of the Chern(C,B)
Simons/Wess-Zumino term SCS/WZ for D-branes of dimension −1, 0, 1, and 2 that matches the
(Φ,g,B)
Dirac-Born-Infeld action SDBI
constructed.
12
From determinant function to wedge product of differential forms
For an ordinary differentiable manifold M , the wedge product of differential forms is determined
by the wedge product of a collection of 1-forms and the latter is set by the determinat function
through the following rule
(ω 1 ∧ · · · ∧ ω s )(e1 ∧ · · · ∧ es ) = Det (ω i (ej )) .
Here, e1 , · · · , es are vector fields on M , ω 1 , · · · , ω s are 1-forms on M , e1 ∧ · · · ∧es := σ∈Syms (−1)σ eσ(1) ⊗
· · · ⊗ eσ(s) , and (ω i (ej )) is the s × s matrix with the (i, j)-entry ω i (ej ). When ω 1 , · · · , ω s are
enhanced to 1-forms with value in a noncommutative ring R, the original determinant function
Det ( · ) needs to be enhanced/generalized as well to a determinant function for matrices with
entries in R since now ω j (ei ) ∈ R, for i, j = 1, . . . , s.
Recall that in the study of non-Abelian Dirac-Born-Infeld action for the pair (ϕ, ∇), we ran
into the need for such a generalization, too, and introduced the notion of symmetrized determinant SymDet ; cf. Definition 3.1.3.6. There, we propose an Ansatz that this is the determinant
function for the construction of the non-Abelian Dirac-Born-Infeld action, cf. Ansatz 3.1.3.11. It
is very natural to suggest that the same notion of determinant function is applied to both the
Dirac-Born-Infeld term and the Chern/Simons/Wess-Zumino term in the full action for D-branes:
P
Ansatz 3.1.1 [wedge product in the Chern-Simons/Wess-Zumino action] We interpret
the wedge products that appear in the formal expresion for the Chern-Simons/Wess-Zumino term
(C,B)
SCS/WZ through the symmetrized determinant that applies to the above defining identities for
wedge product; namely, we require that
(ω 1 ∧ · · · ∧ ω s )(e1 ∧ · · · ∧ es ) = SymDet (ω i (ej ))
for End C (E)-valued 1-forms ω 1 , · · · , ω s on X. Denote this generalized wedge product by ∧.
Example 3.1.2 [C(1) ∧ F ∧ F ] Let C(1) = µ Cµ dxµ and F =
End C (E)-valued 1-form and 2-form respectively, then
P
X
C(1) ∧ F ∧ F =
(Cµ
0
0
P
µ0 ,ν 0
0
Fµ0 ν 0 dxµ ∧ dxν be an
0
00
00
Fµ00 ν 00 ) dxµ ∧ dxµ ∧ dxν ∧ dxµ ∧ dxν ,
Fµ0 ν 0
µ, µ0 , ν 0 , µ00 , ν 00
where, recall that, Cµ
Fµ0 ν 0
Fµ00 ν 00 is the symmetrized product of the triple (Cµ , Fµ0 ν 0 , Fµ00 ν 00 ).
Remark 3.1.3 [ on the ring (C ∞ ( • T ∗ X ⊗R End C (E)), +, ∧) ] (Cf. Remark 3.1.3.10.) Properties
V
of ∧ follow from properties of
on C ∞ (End C (E)) and properties of ∧ on C ∞ ( • T ∗ X) . In
particular, for example, C(1) ∧ F ∧ F is directly defined for the triple (C(1) , F, F ) of End C (E)valued differential forms on X, rather than through a train of applications of a binary operation.
V
The three elements in 5 T ∗ X ⊗R End C (E)
V
C(1) ∧ F ∧ F ,
(C(1) ∧ F ) ∧ F ,
V•
in general are all different. The ring (C ∞ (
but not associative.
C(1) ∧ (F ∧ F )
T ∗ X⊗R End C (E)), +, ∧) is Z2 -graded, Z2 -commutative,
13
Lemma 3.1.4 [ ϕ , ∧, and ∧ ] Let ϕ : (XAz , E, ∇) → Y be an admissible map and ζ1 , · · · , ζk
differential forms on Y . Then
ϕ ζ1 ∧ · · · ∧ ϕ ζk = ϕ (ζ1 ∧ · · · ∧ ζk ) .
Recall the surrogate Xϕ of XAz specified by ϕ and the built-in maps
Xϕ [rr]− fϕ [d]− πϕ Y X
.
Since the function-ring Aϕ := C ∞ (X)hIm ϕ] i of Xϕ is commutative, for differential forms ζ10 , · · · , ζk0
on Xϕ ,
πϕ ∗ ζ10 ∧ · · · ∧ πϕ ∗ ζk0 = πϕ ∗ ζ10 ∧ · · · ∧ πϕ ∗ ζk0 = πϕ ∗ (ζ10 ∧ · · · ∧ ζk0 ) .
It follows that
ϕ ζ1 ∧ · · · ∧ ϕ ζk = πϕ ∗ (fϕ∗ ζ1 ) ∧ · · · ∧ πϕ ∗ (fϕ∗ ζk )
= πϕ ∗ (fϕ∗ ζ1 ∧ · · · ∧ fϕ∗ ζk ) = πϕ ∗ (fϕ∗ (ζ1 ∧ · · · ∧ ζk )) = ϕ (ζ1 ∧ · · · ∧ ζk ) .
The Chern-Simons/Wess-Zumino action for lower dimensional D-branes
q
For a simple D-brane world-volume f : X ,→ Y , the anomaly factor Â(X)/Â(NX/Y ) = 1, for
dim X = m ≤ 3. This may not hold for ϕ since ϕ(XAz ) can have fuzzy/nilpotent structure of
nilpotency ≤ r (the rank of E as a complex vector bundle on X), which can be large even when
the dimension m of X is small. However, if one formally assume that the same is true, then
for lower dimensional D-branes (i.e. D(−1)-, D0-, D1-, D2-branes), one has: (Assuming that
P
B = i,j Bij dy i ⊗ dy j , Bji = −Bij )
· For D(−1)-brane world-point (m = 0) :
(C
)
(0)
SCS/WZ
(ϕ) = T−1 · Tr (ϕ C(0) ) = T−1 · Tr (ϕ] (C(0) )) .
· For D-particle world-line (m = 1) : Assume that C
(C
)
(1)
SCS/WZ
(ϕ, ∇) = T0
Z
X
Tr (ϕ C(1) )
locally
=
(1)
T0
=
Z
Pn
i=1
Tr
Ci dy i locally; then
n
X
U
ϕ] (Ci ) · Dx ϕ] (y i ) dx .
i=1
Here, Dx := D∂/∂x .
· For D-string world-sheet (m = 2) : Assume that C
(2)
=
Pn
i,j=1
Cij dy i ⊗ dy j locally,
with Cij = −Cji ; then
(C
,C
(0)
(2)
SCS/WZ
=
Z
(ϕ, ∇) = T1
Re (Tr (ϕ C(2) + ϕ (C(0) B) + 2πα0 ϕ] (C(0) ) F∇ ))
X
Z
T1
Re (Tr (ϕ (C(2) + C(0) B) + πα0 ϕ] (C(0) )F∇ + πα0 F∇ ϕ] (C(0) )))
,B)
X
locally
=
Z
T1
U
n
X
Re Tr
ϕ] (Cij + C(0) Bij ) Dx1 ϕ] (y i ) Dx2 ϕ] (y j )
i,j=1
+ πα0 ϕ] (C(0) ) [∇x1 , ∇x2 ] + πα0 [∇x1 , ∇x2 ] ϕ] (C(0) )
d2x .
Here, Dx1 := D∂/∂x1 , Dx2 := D∂/∂x2 and ∇x1 := ∇∂/∂x1 , ∇x2 := ∇∂/∂x2 .
14
· For
D-membrane world-volume (m = 3) :
P
Assume that C(1) = ni=1 Ci dy i and C(3) =
n
i
j
k
i,j,k=1 Cijk dy ⊗ dy ⊗ dy locally, with Cijk alternating with respect to ijk; then
(C
,C
(1)
(3)
SCS/WZ
,B)
locally
=
Z
(ϕ, ∇) = T2
Z
T2
U
Re (Tr (ϕ C(3) + ϕ (C(1) ∧ B) + 2πα0 ϕ C(1) ∧ F∇ ))
X
n
X
Re Tr
ϕ] (Cijk + Ci Bjk + Cj Bki + Ck Bij ) Dx1 ϕ] (y i ) Dx2 ϕ] (y j ) Dx3 ϕ] (y k )
i,j,k=1
n
X
X
+ πα0
P
(−1)(λµν)
ϕ] (Ci ) Dxλ (ϕ] (y i )) [∇xµ , ∇xν ] + [∇xµ , ∇xν ] ϕ] (Ci ) Dxλ ϕ] (y i )
(λµν)∈Sym3 i=1
The technical issue of anomaly is the focus of another work. For the moment, we will take the
above as our working Anzatz for the Chern-Simons/Wess-Zumino action for lower-dimensional
D-branes.
3.2
The first variation and the contribution to the equations of motion
Under the same setup as in Sec. 5.2, we derive in this subsection the first variation of the
(C,B)
Chern-Simons/Wess-Zumino action SCS/WZ for lower-dimensional D-brane world-volumes. The
additional contribution to the equations of motion for such lower-dimensional D-branes due to
(C,B)
the additional term SCS/WZ in the total action for D-brane world-volume would then follow.
3.2.1
D(−1)-brane world-point (m = 0)
(C
)
(0)
For a D(−1)-brane world-point, dim X = 0, ∇ = 0, and SCS/WZ
(ϕ) = T−1 · Tr (ϕ] (C(0) )). It
follows that
d
d
d
(C(0) )
SCS/WZ
(ϕT ) = T−1
Tr (ϕ]T (C(0) )) = T−1 Tr
ϕ]T (C(0) )
dt t=0
dt t=0
dt t=0
= T−1 Tr
n X
X
j=1
n
X
=: T−1 Tr
]
C(0)
R C(0) [1](R
[1](Ld, ~π) (ϕ] (y))
d, ~π ) (ϕ (y)) · R
· ϕ̇] (y j )
d, d, ~
π
(C(0) );δϕ
NLj
(ϕ) · ϕ̇] (y j )
.
j=1
Here, the following identities are employed:
d
dt
t=0
=
ϕ]T (C(0) )
n X
•
X
= R C(0) [1]|yd ·(ϕ] (yd ))
X
X
j=1 d=0 d, |d|=d ~
~
π ∈Ptn(1,d),
i(~π, d) =j
([∂y~πj ]R C(0) [1](d) )L (ϕ] (y)) · ϕ̇] (y j ) · ([∂y~πj ]R C(0) [1](d) )R (ϕ] (y))
=:
n
X
X
L
]
] j
C(0)
R
]
R C(0) [1](d,
[1](d,
~
π ) (ϕ (y)) .
~
π ) (ϕ (y)) · ϕ̇ (y ) · R
j=1 d, d, ~
π
(Φ,g,B)
(Φ,g,B)
In this case, SDBI (ϕ) = 0 always and the full action SDBI
The full system of equations of motion is thus
(Φ,g,B,C(0) );δϕ
NLj
(C(0) );δϕ
(ϕ) := NLj
(C
)
(ϕ) = 0 ,
j = 1, . . . , n , for D(−1)-brane. Such world-points give rise to instantons in space-time.
15
(C
)
(0)
(0)
+ SCS/WZ
is simply SCS/WZ
.
d3x .
3.2.2
D-particle world-line (m = 1)
For a D-particle world-line, dim X = 1 and
(C
)
(1)
SCS/WZ
(ϕ, ∇) = T0
Z
Tr
n
X
U
ϕ] (Ci ) · Dx ϕ] (y i ) dx
i=1
locally over X. It follows that
d
dt
(C(1) )
n
X
Z
T
SCS/WZ (ϕT , ∇ ) = T0
Tr
U
t=0
n
X
= T0 Tr
ϕ] (Ci ) ϕ̇] (y i )
i=1
Z
+ T0
Tr
U
= T0 BT
(ϕ;C(1) )
i=1
∂U
n
X
ϕ̇] (Ci ) · Dx ϕ] (y i ) + ϕ] (Ci ) · (Dx ϕ̇] (y i ) − [ϕi (])(y i ), Ȧx ]) dx
ϕ̇] (Ci ) · Dx ϕ] (y i ) − Dx ϕ] (Ci ) · ϕ̇] (y i ) − ϕ] (Ci ) · [ϕi (])(y i ), Ȧx ] dx
i=1
Z
]
(ϕ̇ (y))|∂U + T0
Tr
n
X
U
(C(1) );δϕ
NLj
(C(1) );δ∇
(ϕ, ∇) · ϕ̇] (y j ) + NLx
(ϕ) · Ȧx dx ,
j=1
where
BT
(ϕ;C(1) )
(ϕ̇] (y))
= Tr
n
X
ϕ] (Ci ) ϕ̇] (y i ) ,
i=1
(C );δϕ
NLj (1)
(ϕ, ∇)
= − Dx ϕ] (Cj ) +
n
X
X
]
] i
Ci
L
]
RCi [1]R
π ) (ϕ (y)) ,
(d,~
π ) (ϕ (y)) · Dx ϕ (y ) · R [1](d,~
i=1 d,d,~
π ; |d|=d,i(~
π ,d) =j
(C(1) );δ∇
NLx
(ϕ)
=
n
X
[ϕ] (y i ) , ϕ] (Ci )] = 0 .
i=1
(Φ,g,B)
The full action SDBI
D-particle moving in Y :
(Φ,g,B,C(1) );δϕ
NLj
(C
)
(1)
(ϕ, ∇) + SCS/WZ
(ϕ, ∇) gives the system of equations of motion for a
(Φ,g,B);δϕ
(ϕ, ∇) := NLj
(Φ,g,B,C(1) );δ∇
NLx
(ϕ, ∇)
(C(1) );δϕ
(ϕ, ∇) + NLj
(ϕ, ∇) = 0 ,
:= NLx(Φ,g,B);δ∇ (ϕ, ∇)
= 0,
j = 1, . . . , n .
For the current case, the curvature F∇ of ∇ is zero and the above system may still involves
Ax but not its differentials with respect to x. I.e. it is a system of differential equations on ϕ
but non-differential equations on ∇. ∇ is thus non-dynamical, as is anticipated. Thus, after a
re-trivialization fo the fundamental module E on X, one may assume that Ax ≡ 0 and the above
system is reduced to a system
NL(Φ,g,B,C(1) );δϕ (ϕ) = 0 ,
j = 1, . . . , n ,
of second-order nonlinear differential equations that involve ϕ alone.
3.2.3
D-string world-sheet (m = 2)
Denote
C̆(2) := C(2) + C(0) B =
(Cij + C(0) Bij ) dy i ⊗ dy j =
X
X
ij
i,j
C̆ij dy i ⊗ dy j
in local coordinates of Y . Then, for a D-string world-sheet, dim X = 2 and
(C
,C
(0)
(2)
SCS/WZ
,B)
Z
(ϕ, ∇)
=
T1
U
n
X
Re Tr
ϕ] (C̆ij ) D1 ϕ] (y i ) D2 ϕ] (y j )
i,j=1
+ πα0 ϕ] (C(0) ) F12 + πα0 F12 ϕ] (C(0) )
16
d2x
locally over X. (Here, D1 := D∂/∂x1 , D2 := D∂/∂x2 , and F12 := [∇x1 , ∇x2 ] is the curvature of ∇.)
It follows then from a straightforward computation that
d
dt
(C
,C
(0)
(2)
SCS/WZ
,B)
(ϕT , ∇T )
t=0
n X
Re Tr
ϕ̇] (C̆ij ) D1 ϕ] (y i ) D2 ϕ] (y j ) + ϕ] (C̆ij ) (D1 ϕ̇] (y i ) − [ϕ] (y i ), Ȧ1 ]) D2 ϕ] (y j )
U
i,j=1
+ ϕ] (C̆ij ) D1 ϕ] (y i ) (D2 ϕ̇] (y j ) − [ϕ] (y j ), Ȧ2 ])
Z
= T1
+ πα0 ϕ̇] (C(0) ) F12 + πα0 ϕ] (C(0) ) (D1 Ȧ2 − D2 Ȧ1 )
+ πα0 (D1 Ȧ2 − D2 Ȧ1 ) ϕ] (C(0) ) + πα0 F12 ϕ̇] (C(0) )
Z
= T1
Re (BT
(ϕ,∇;C(0) ,C(2) ,B)
d2x
(ϕ̇] (y), Ȧ))
∂U
n
2
X
X
(C ,C ,B);δϕ
(C );δ∇
Re Tr
NLj (0) (2)
(ϕ, ∇) · ϕ̇] (y j ) +
NLν (0)
(ϕ, ∇) · Ȧν d2x ,
Z
+ T1
U
ν=1
j=1
where
· the boundary term is given by
BT
(ϕ,∇;C(0) ,C(2) ,B)
= Tr
n X
j=1
(ϕ̇] (y), Ȧ) , a 1-form on U ,
n
X
D2 ϕ] (y i ) ϕ] (C̆ji ) · ϕ̇] (y j ) + 2πα0 ϕ] (C(0) ) · Ȧ2 dx2
i=1
− Tr
n X
n
X
j=1
ϕ] (C̆ij ) D1 ϕ] (y i )
· ϕ̇] (y j ) − 2πα0 ϕ] (C(0) ) · Ȧ1
dx1 ,
i=1
· the subsystem associated to variations of ϕ :
(C(0) ,C(2) ,B);δϕ
NLj
(ϕ, ∇)
n
X
=
X
0
0
]
] i
] j
C̆i0 j 0
]
RC̆i0 j0 [1]R
[1]L
(d,~
π ) (ϕ (y)) D1 ϕ (y ) D2 ϕ (y ) R
(d,~
π ) (ϕ (y))
i0 ,j 0 =1 d,d,~
π ; |d|=d,i(~
π ,d) =j
−
n X
D2 ϕ] (y i ) D1 ϕ] (C̆ji ) + D2 ϕ] (C̆ij ) D1 ϕ] (y i ) + [F12 , ϕ] (y i )] · ϕ] (C̆ji )
i=1
+ 2πα0
X
]
C(0)
]
RC(0) [1]R
[1]L
(d,~
π ) (ϕ (y)) F12 R
(d,~
π ) (ϕ (y)) ,
d,d,~
π ; |d|=d,i(~
π ,d) =j
· the subsystem associated to variations of ∇ :
(C(0) );δ∇
NL1
(C(0) );δ∇
(ϕ, ∇) = 2πα0 D2 ϕ] (C(0) ) ,
NL2
(ϕ, ∇) = − 2πα0 D1 ϕ] (C(0) ) .
Note that, as a consequence of Leibniz rule or integration by parts, there are at first
summands
(C(0) );δ∇
−D2 ϕ] (y j ) ϕ] (C̆ij ) ϕ] (y i ) + ϕ] (y i ) D2 ϕ] (y j ) ϕ] (C̆ij )
in NL1
−ϕ] (C̆ij ) D1 ϕ] (y i ) ϕ] (y j ) + ϕ] (y j ) ϕ] (C̆ij ) D1 ϕ] (y i )
(ϕ, ∇) ,
(C );δ∇
NL2 (0) (ϕ, ∇) ,
in
respectively. However, they vanish for (ϕ, ∇) admissible. Thus, the 2-forms C(2) and B has
(C
,C
(0) (2)
no consequence to the variation of SCS/WZ
,B)
with respect to ∇. This is anticipated since
(C
,C
(0) (2)
there is no coupling term between C(2) , B and ∇ in SCS/WZ
(C
,B)
.
,C
(0) (2)
The contribution of the Chern-Simon/Wess-Zumino term SCS/WZ
motion for a D-string follows immediately.
17
,B)
to the equations of
3.2.4
D-membrane world-volume (m = 3)
Denote
C(3) + C(1) ∧ B
C̆(3) :=
(Cijk + Ci Bjk + Cj Bki + Ck Bij ) dy i ⊗ dy j ⊗ dy k =
X
=
X
i,j,k
C̆ijk dy i ⊗ dy j ⊗ dy k
i,j,k
in local coordinates of Y . Then, for D-membrane world-volume, dim X = 3 and
(C
,C
(1)
(3)
SCS/WZ
,B)
Z
= T2
(ϕ, ∇)
n
X
Re Tr
ϕ] (C̆ijk ) D1 ϕ] (y i ) D2 ϕ] (y j ) D3 ϕ] (y k )
U
i,j,k=1
+ 2πα
0
n
X
X
(−1)(λµν) ϕ] (Ci ) Dλ ϕ] (y i ) Fµν
d3x
(λµν)∈Sym3 i=1
locally over X. (Here, Fµν := [∇xµ , ∇xν ] is the curvature of ∇.) It follows then from a straightforward computation that
d
dt
(C
,C
(1)
(3)
SCS/WZ
,B)
(ϕT , ∇T )
t=0
n
X
Re Tr
ϕ̇] (C̆ijk ) D1 ϕ] (y i ) D2 ϕ] (y j ) D3 ϕ] (y k )
Z
= T2
U
i,j,k=1
+ ϕ] (C̆ijk · (D1 ϕ̇] (y i ) − [ϕ] (y i ), Ȧ1 ]) · D2 ϕ] (y j ) D3 ϕ] (y k )
+ ϕ] (C̆ijk ) D1 ϕ] (y i ) · (D2 ϕ̇] (y j ) − [ϕ] (y j ), Ȧ2 ]) · D3 ϕ] (y k )
+ ϕ] (C̆ijk ) D1 ϕ] (y i ) D2 ϕ] (y j ) · (D3 ϕ̇] (y k ) − [ϕ] (y k ), Ȧ3 ])
n
X
X
+ 2πα0
(−1)(λµν) ϕ̇] (Ci ) Dλ ϕ] (y i ) Fµν
(λµν)∈Sym3 i=1
+ ϕ] (Ci ) · (Dλ ϕ̇] (y i ) − [ϕ] (y i ), Ȧλ ]) · Fµν + ϕ] (Ci ) Dλ ϕ] (y i ) · (Dµ Ȧν − Dν Ȧµ )
Z
= T2
Re (BT
(ϕ,∇;C(1) ,C(3) ,B)
(ϕ̇] (y), Ȧ))
∂U
n
3
X
X
(C ,C ,B);δϕ
(C );δ∇
Re Tr
NLj (1) (3)
(ϕ, ∇) · ϕ̇] (y j ) +
NLν (1)
(ϕ, ∇) · Ȧν d3x ,
Z
+ T2
U
ν=1
j=1
where
· the boundary term is given by
BT
(ϕ,∇;C(1) ,C(3) ,B)
(ϕ̇] (y), Ȧ) , a 2-form on U ,
n X
n
X
= Tr
D2 ϕ] (y i ) D3 ϕ] (y k ) ϕ] (C̆jik ) + 4πα0 F23 ϕ] (Cj ) · ϕ̇] (y j )
j=1
i,k=1
+ 4πα0
n
X
ϕ] (Ci ) D3 ϕ] (y i )
· Ȧ2 − 4πα0
i=1
− Tr
D3 ϕ] (y k ) ϕ] (C̆ijk ) D1 ϕ] (y i ) − 4πα0 F13 ϕ] (Cj )
n
X
ϕ] (Ci ) D3 ϕ] (y i )
· Ȧ1 + 4πα0
i=1
n X
n
X
j=1
· Ȧ3
d2 ∧ dx3
· ϕ̇] (y j )
i,k=1
− 4πα0
+ Tr
ϕ] (Ci ) D2 ϕ] (y i )
i=1
n X
n
X
j=1
n
X
n
X
ϕ] (Ci ) D1 ϕ] (y i )
· Ȧ3
d1 ∧ dx3
i=1
ϕ] (C̆ikj ) D1 ϕ] (y i ) D2 ϕ] (y k ) + 4πα0 F12 ϕ] (Cj )
· ϕ̇] (y j )
i,k=1
+ 4πα0
n
X
ϕ] (Ci ) D2 ϕ] (y i )
· Ȧ1 − 4πα0
i=1
n
X
i=1
18
ϕCi D1 ϕ] (y i )
· Ȧ2
dx1 ∧ dx2 ,
d3x
· the subsystem associated to variations of ϕ :
(C(1) ,C(3) ,B);δϕ
NLj
=
(ϕ, ∇)
n
X
0
X
0
0
]
] i
] j
] k
C̆i0 j 0 k0
]
RC̆i0 j0 k0 [1]R
[1]L
(d,~
π ) (ϕ (y)) D1 ϕ (y ) D2 ϕ (y ) D3 ϕ (y )R
(d,~
π ) (ϕ (y))
i0 ,j 0 ,k0 =1 d,d,~
π ; |d|=d,i(~
π ,d) =j
n X
−
D2 ϕ] (y i ) D3 ϕ] (y k ) D1 ϕ] (C̆jik ) + D3 ϕ] (y k ) D2 ϕ] (C̆ijk ) D1 ϕ] (y i )
i,k=1
+ D3 ϕ] (C̆ikj ) D1 ϕ] (y i ) D2 ϕ] (y k ) + [F12 , ϕ] (y i )] · D3 ϕ] (y k ) ϕ] (C̆jik )
+ [F23 , ϕ] (y k )] · ϕ] (C̆ijk ) D1 ϕ] (y i ) + [F31 , ϕ] (y i )] · D2 ϕ] (y k ) ϕ] (C̆ikj )
+ 2πα
0
n
X
X
X
]
] i
Ci
L
]
(−1)(λµν) RCi [1]R
(d,~
π ) (ϕ (y)) Dλ ϕ (y ) Fµν R [1](d,~
π ) (ϕ (y))
i=1 (λµν)∈Sym3 d,d,~
π ;|d|=d,i(~
π ,d) =j
X
− 2πα0
(−1)(λµν)
Fµν Dλ ϕ] (Cj ) + Dλ Fµν ϕ] (Cj )
,
(λµν)∈Sym3
· the subsystem associated to variations of ∇ :
(C(1) );δ∇
NLλ
(ϕ, ∇)
n
X
= 2πα0
[ϕ] (y i ) , Fµν ϕ] (Ci )]
i=1
+ 4πα0
n X
Dµ ϕ] (Ci ) Dν ϕ] (y i ) − Dν ϕ] (Ci ) Dµ ϕ] (y i ) + ϕ] (Ci ) · [Fµν , ϕ] (y i )]
,
i=1
where (λµν) = (123), (231), (312) .
Note that, as a consequence of Leibniz rule or integration by parts, there are at first
summands
n
X
[ϕ] (y i ), D2 ϕ] (y j )D3 ϕ] (y k )ϕ] (C̆ijk )]
i,j,k=1
n
X
[ϕ] (y j ), D3 ϕ] (y k )ϕ] (C̆ijk )D1 ϕ] (y i )]
i,j,k=1
n
X
[ϕ] (y k ), ϕ] (C̆ijk )D1 ϕ] (y i )D2 ϕ] (y j )]
(C(1) );δ∇
(ϕ, ∇) ,
(C(1) );δ∇
(ϕ, ∇) ,
(C(1) );δ∇
(ϕ, ∇)
in NL1
in NL2
in NL3
i,j,k=1
respectively. However, they vanish for (ϕ, ∇) admissible. Thus, the 3-forms C(3) and
(C
,C
(1) (3)
C(1) ∧ B have no consequence to the variation of SCS/WZ
,B)
with respect to ∇. This is
(C
,C
(1) (3)
anticipated since there is no coupling term between C(3) , C(1) ∧ B and ∇ in SCS/WZ
(C
,C
(1) (3)
The contribution of the Chern-Simon/Wess-Zumino term SCS/WZ
motion for a D-membrane follows immediately.
,B)
,B)
.
to the equations of
Remark 3.2.4.1 [contribution only to first-order terms in EOM ] As observed from these exam(C,B)
ples, for lower dimensional D-branes, the Chern-Simons/Wess-Zumino term SCS/WZ in the action
contributes an additional set of first-order nonlinear differential-expression terms to the system
of equations of motion fo D-branes. In particular, they preserve the signature of the original
(Φ,g,B)
system from the Dirac-Born-Infeld term SDBI
in the action.
19
The current notes lay down some foundation toward the dynamics of D-branes along the
line of our D-project. Solutions to the system of equations of motion from the total action
(Φ,g,B)
(C,B)
SDBI (ϕ, ∇) + SCS/WZ (ϕ, ∇) for a D-brane world-volume should be thought of as an Azumaya/matrix version of minimal submanifolds or harmonic maps, twisted/bent, on one hand, by
the (dynamical) gauge field ∇ on the domain manifold X with a (noncommutative) endomorphism/matrix function-ring and, on the other hand, by the background field (Φ, g, B, C), created
by closed (super)strings, on the target space(-time) Y . Further details, issues, and examples are
the focus of the sequels.
4
The standard action for D-branes
We introduce in this section the standard action, which is to D-branes as the (Brink-Di VecchiaHowe/Deser-Zumino/) Polyakov action is to fundamental superstrings. Abstractly, it is an enhanced non-Abelian gauged sigma model based on maps ϕ : (XAz , E; ∇) → Y .
The gauge-symmetry group C ∞ (Aut C (E))
Let Aut C (E) be the automorphism bundle of the complex vector bundle E (of rank r) over E.
Aut C (E) ⊂ End C (E) canonically as the bundle of invertible endomorphisms; it is a principal
GLr (C)-bundle over X. The set
Ggauge := C ∞ (Aut C (E))
of smooth sections of Aut C (E) forms an infinite-dimensional Lie group and acts on the space of
pairs (ϕ, ∇) as a gauge-symmetry group:
g 0 ∈ Ggauge :
0
0
0
(ϕ, ∇ = d + A) 7−→ (gϕ , g ∇ = d + gA)
:= (g 0 ϕg 0 −1 , d − (dg 0 )g 0 −1 + g 0 Ag 0 −1 ) .
The induced action of Ggauge on other basic objects are listed in the lemma below:
Lemma 4.1 [induced action of Ggauge on other basic objects] (All the Ggauge -actions are
denoted by a representation ρgauge of Ggauge , if in need.)
Az
(01 ) on OX
:
Az
ρgauge (g 0 )(m) = g 0 mg 0 −1 for m ∈ OX
.
(02 ) on induced connections :
D = d + [A, · ] 7−→
D := d + [ gA , · ].
g0
0
ρgauge (g 0 )(ω ⊗ m) = ω ⊗ (g 0 mg 0 −1 ) =: g 0 (ω ⊗ m)g 0 −1 .
Az
:
(1) on T ∗ X C ⊗OXC OX
(2) for ϕ∗ T∗ Y :
ϕ ∗ T∗ Y
g0 ∗
−→
ϕ T∗ Y
0
m ⊗ v 7−→ (g mg
0 −1
) ⊗ v =: g 0 (m ⊗ v)g 0 −1 .
.
(3) for T ∗ X ⊗OXϕ∗ T∗ Y :
T ∗ X ⊗OXϕ∗ T∗ Y
ω⊗m⊗v
0
T ∗ X ⊗OX gϕ∗ T∗ Y
−→
7−→ ω ⊗ (g 0 mg 0 −1 ) ⊗ v =: g 0 (ω ⊗ m ⊗ v)g 0 −1 .
.
20
0
g0
D gϕ = g 0 Dϕ g 0 −1 .
(4) for covariant differential :
Dϕ 7−→
(5) for pull-push :
(gϕ) α = g 0 ϕ α g 0 −1 .
0
The proof is elementary. Let us demonstrate Item (2) as an example.
Az
⊗ϕ] ,OY T∗ Y ,
For m ⊗ v ∈ ϕ∗ T∗ Y := OX
−1
ρgauge (g 0 )(m ⊗ v) = ρgauge (g 0 )(m) ⊗ v = (g 0 mg 0 ) ⊗ v
since Ggauge acts on T∗ Y trivially (i.e. by by the identity map Id Y ). The only issue is: Where
does (g 0 mg 0 −1 ) ⊗ v now live? To answer this, note that, for f ∈ C ∞ (Y ), on one hand
−1
ρgauge (g 0 )(m ⊗ f v) = ρgauge (g 0 )(mϕ] (f ) ⊗ v) = (g 0 mϕ] (f )g 0 ) ⊗ v ,
while on the other hand
−1
ρgauge (g 0 )(m ⊗ f v) = (g 0 mg 0 ) ⊗ f v ,
It follows that
−1
(g 0 mg 0 ) ⊗ f v
−1
= (g 0 mϕ] (f )g 0 ) ⊗ v = (g 0 mg 0
−1
−1
· g 0 ϕ] (f )g 0 ) ⊗ v = (g 0 mg 0
−1
0
]
· gϕ (f )) ⊗ v .
0
Which says that our section (g 0 mg 0 −1 ) ⊗ v now lives in gϕ∗ T∗ Y .
The standard action for D-branes
Fix a (dilaton field ρ, metric h) on the underlying smooth manifold X (of dimension m) of the
Azumaya/matrix manifold with a fundamental module (XAz , E). Fix a background (dilaton field
Φ, metric g , B-field B, Ramond-Ramond field C) on the target space(-time) Y (of dimension
n). Here, h and g can be either Riemannian or Lorentzian.
Definition 4.2 [standard action = enhanced non-Abelian gauged sigma model] With
(ρ,h;Φ,g,B,C)
the given background fields (ρ, h) on X and (Φ, g, B, C) on Y , the standard action Sstandard
(ϕ, ∇)
for (∗1 )-admissible pairs (ϕ, ∇) is defined to be the functional
(ρ,h;Φ,g,B,C)
Sstandard
(ρ,h;Φ,g,B,C)
(ϕ, ∇) := SnAGSM+
(ϕ, ∇)
(h;B)
(ρ,h;Φ,g)
(C,B)
:= Smap:kinetic+ (ϕ, ∇) + Sgauge /YM (ϕ, ∇) + SCS/WZ (ϕ, ∇)
with the enhanced kinetic term for maps
1
(ρ,h;Φ,g)
Smap:kinetic+ (ϕ, ∇) := 2 Tm−1
Z
X
Re Tr hDϕ , Dϕi(h,g) vol h +
Z
X
Re Tr hdρ, ϕ dΦih vol h ,
the gauge/Yang-Mills term
1
(h;B)
Sgauge /YM (ϕ, ∇) := − 2
Z
X
Re Tr k2πα0 F∇ + ϕ Bk2h vol h
and the Chern-Simons/Wess-Zumino term
(if (ϕ, ∇) is furthermore (∗2 )-admissible, cf. Remark 2.1.13)
(C,B)
SCS/WZ (ϕ, ∇)
formally
=
Z
Tm−1
0
B
Re Tr ϕ C ∧ e2πα F∇ +ϕ
X
Here,
21
∧
q
Â(XAz )/Â(NXAz /Y )
(m)
.
(0) On Re
Note that while eigenvalues of ϕ] (f ) are all real ([L-Y5: Sec. 3.1] (D(11.1))) for
f ∈ OY , the eigenvalues of Dξ ϕ] (f ), ξ ∈ T∗ X, may not be so under the (∗1 )-Admissible
(ρ,h;Φ,g,B,C)
Condition. Thus, Tr (· · · · · ·) in the integrand of terms in Sstandard
(ϕ, ∇) are in general
C-valued and we take the real part Re Tr (· · · · · ·) of it.
(ρ,h;Φ,g)
(1) The enhanced kinetic term for maps
kinetic energy
The first summand of Smap:kinetic+ defines the
1
(h;g)
E ∇ (ϕ) := Smap:kinetic (ϕ, ∇) := 2 Tm−1
Z
X
Re Tr hDϕ , Dϕi(h,g) vol h
of the map ϕ for a given ∇ and, hence, will be called the kinetic term for maps in the
(ρ,h;Φ,g,B,C)
standard action Sstandard
(ϕ, ∇). When the metric g on Y is Lorentzian, then depending
on the convention of its signature (−, +, · · · +) vs. (+, −, · · · −), one needs to add an
overall minus − vs. plus + sign. In this note, for simplicity of presentation, we choose
h and g to be both Riemannian (i.e. for Euclideanized/Wick-rotated D-branes and spacetime).
· The world-volume X
Az
of D-brane is m-dimensional; Tm−1 is the tension of (m − 1)dimensional D-branes. Like the tension of the fundamental string, it is a fixed constant of
nature.
· The second summand of S
(ρ,h;Φ,g)
map:kinetic+
(ρ,h;Φ)
Sdilaton (ϕ)
:=
Z
X
Re Tr hdρ, ϕ dΦih vol h ,
(ρ,h;Φ,g,B,C)
will be called the dilaton term of the standard action Sstandard
(ϕ, ∇).
Note that if let U be small enough and fix a local trivialization of E|U . and assume that
∇ = d + A with respect to this local trivialization. Then D = d + [A, · ] and, over U with
an orthonormal frame (eµ )µ ,
Tr hdρ, ϕ dΦih =
X
Tr (dρ(eµ )Deµ ϕ] (Φ))
µ
=
X
Tr dρ(eµ ) eµ ϕ] (Φ) + [A(eµ ), ϕ] (Φ)]
µ
=
X
Tr dρ(eµ )(eµ ϕ] (Φ)) .
µ
Thus, while ϕ dΦ depends on the connection ∇, the integrand (Tr hdρ, ϕ dΦih ) vol h does
not. This justifies the dilaton term as a functional of ϕ alone.
In contrast, over U with the above setting, Tr hDϕ, Dϕi(h,g) contains summand
XX
µ
Tr [A(eµ ), ϕ] (y i )] [A(eµ ), ϕ] (y j )] ϕ] (gij ) ,
i,j
which does not vanish in general. Thus, Tr hDϕ, Dϕi(h,g) does depend on the pair (ϕ, ∇).
(h;B)
(2) The gauge/Yang-Mills term Sgauge /YM (ϕ, ∇)
to the tension of a fundamental string.
·F
α0 is the Regge slope; 2πα0 is the inverse
Az
is the curvature tensor of the connection ∇ on E; 2πα0 F∇ + ϕ B is an OX
-valued
2-tensor on X; and
∇
k2πα0 F∇ + ϕ Bk2h := h2πα0 F∇ + ϕ B , 2πα0 F∇ + ϕ Bih
from Sec. 3.2.1. Up to the shift by ϕ B, this is a norm-squared of the field strength of the
(h;B)
gauge field, and hence the name Yang-Mills term. Note that in Sgauge /YM (ϕ, ∇), ∇ couples
with ϕ only through the background B-field B. When B = 0, this is simply a functional
(h)
Sgauge /YM (∇) of ∇ alone.
22
· In the current bosonic case, the Yang-Mills functional for the gauge term S
(h;B)
gauge /YM (ϕ, ∇)
can be replaced any other standard action functional, e.g. Chern-Simons functional, in
gauge theories.
(C,B)
(3) The Chern-Simons/Wess-Zumino term SCS/WZ (ϕ, ∇)
The coupling constant of
Ramond-Ramond fields with D-branes is taken to be equal to the D-brane tension Tm−1 .
This choice is adopted from the situation of the Dirac-Born-Infeld action. However, in the
(C,B)
current bosonic case, one may take a different constant. As given here, SCS/WZ (ϕ, ∇) is only
q
formal; the anomaly factor
in the current situation.
· The wedge product of O
Â(XAz )/Â(NXAz /Y ) in its integrand remains to be understood
Az
X -valued
differential forms was discussed in [L-Y8: Sec.6.1] (D(13.1)).
An Ansatz was proposed there in accordance with the notion of ‘symmetrized determinant’
Az
for an OX
-valued 2-tensor on X in the construction of the non-Abelian Dirac-Born-Infeld
action ibidem. Here, we no longer have a direct guide from the construction of the kinetic
(h;g)
term Smap:kinetic (ϕ, ∇) for maps as to how to define such wedge products. However, just
like Polyakov string should be thought of as being equivalent to Nambu-Goto string (at
least at the classical level) but technically more robust, here we would think that ‘standard
D-branes’ should be equivalent to ‘Dirac-Born-Infeld D-branes’ (at least classically) and,
hence, will take the same Ansatz:
Ansatz [wedge product in the Chern-Simons/Wess-Zumino action] We interpret the wedge products that appear in the formal expression for the Chern(C,B)
Simons/Wess-Zumino term SCS/WZ (ϕ, ∇) through the symmetrized determinant that
applies to the above defining identities for wedge product; namely, we require that
(ω 1 ∧ · · · ∧ ω s )(e1 ∧ · · · ∧ es ) = SymDet (ω i (ej ))
Az
for OX
-valued 1-forms ω 1 , · · · , ω s on X. Denote this generalized wedge product by ∧.
Then, for lower-dimensional D-branes m = 0, 1, 2, 3, it is reasonable to assume that the
(C,B)
anomaly factor is 1 (i.e. no anomaly) and SCS/WZ (ϕ, ∇) can be written out precisely.
Locally in terms of a local frame (eµ )µ on an open set U ⊂ X and a coordinate
P
(y , · · · , y n ) on a local chart of Y , one has: (Assuming that B = i,j Bij dy i ⊗ dy j , Bji =
−Bij .)
1
· For D(−1)-brane world-point (m = 0) :
(C
)
(0)
SCS/WZ
(ϕ) = T−1 · Tr (ϕ C(0) ) = T−1 · Tr (ϕ] (C(0) )) .
· For D-particle world-line (m = 1) : Assume that C
Z
(C(1) )
SCS/WZ (ϕ) = T0
X
Re (Tr (ϕ C(1) ))
locally
=
(1)
Z
T0
=
Re Tr
U
Pn
i=1
n
X
Ci dy i locally; then
ϕ] (Ci ) · De1 ϕ] (y i )
de1 .
i=1
(ρ,h;Φ)
Note that as in the case of the dilaton term Sdilaton (ϕ), this is a functional of ϕ alone.
· For D-string world-sheet (m = 2) : Assume that C
(2)
=
Pn
i,j=1
Cij dy i ⊗ dy j locally,
with Cij = −Cji ; then
(C
,C(2) ,B)
(0)
SCS/WZ
(ϕ, ∇) = T1
X
Z
=
Z
T1
X
Re (Tr (ϕ C(2) + ϕ (C(0) B) + 2πα0 ϕ] (C(0) )
F∇ ))
Re (Tr (ϕ (C(2) + C(0) B) + πα0 ϕ] (C(0) )F∇ + πα0 F∇ ϕ] (C(0) )))
23
Z
locally
=
n
X
T1
Re Tr
U
ϕ] (Cij + C(0) Bij ) De1 ϕ] (y i ) De2 ϕ] (y j )
i,j=1
0 ]
+ πα ϕ (C(0) ) F∇ (e1 , e2 ) + πα0 F∇ (e1 , e2 ) ϕ] (C(0) )
Z
=
T1
n
X
Re Tr
U
e1 ∧ e2
ϕ] (Cij + C(0) Bij ) De1 ϕ] (y i ) De2 ϕ] (y j )
i,j=1
+ 2 πα0 ϕ] (C(0) ) F∇ (e1 , e2 )
e1 ∧ e2 .
Here, the last identity comes from the effect of the trace map Tr .
· For
D-membrane world-volume (m = 3) : Assume that C
P
n
i,j,k=1
(C
Cijk dy i ⊗ dy j ⊗ dy k locally, with Cijk
,C(3) ,B)
(1)
SCS/WZ
locally
=
(ϕ, ∇) = T2
Z
T2
Re Tr
X
n
X
U
Z
ϕ] (Cijk + Ci Bjk + Cj Bki + Ck Bij )
· De1 ϕ] (y i ) De2 ϕ] (y j ) De3 ϕ] (y k )
n
X
X
(−1)(λµν)
ϕ] (Ci ) Deλ (ϕ] (y i )) F∇ (eµ , eν )
(λµν)∈Sym3 i=1
Z
Re Tr
T2
U
+ 2πα0
n
X
+ F∇ (eµ , eν ) ϕ] (Ci ) Deλ ϕ] (y i )
e1 ∧ e2 ∧ e3
ϕ] (Cijk + Ci Bjk + Cj Bki + Ck Bij )
i,j,k=1
X
P
Re (Tr (ϕ C(3) + ϕ (C(1) ∧ B) + 2πα0 ϕ C(1) ∧ F∇ ))
i,j,k=1
+ πα0
=
= ni=1 Ci dy i and C(3) =
alternating with respect to ijk; then
(1)
n
X
· De1 ϕ] (y i ) De2 ϕ] (y j ) De3 ϕ] (y k )
(−1)(λµν)
ϕ] (Ci ) Deλ (ϕ] (y i )) F∇ (eµ , eν )
e1 ∧ e2 ∧ e3 .
(λµν)∈Sym3 i=1
Here, the last identity comes from the effect of the trace map Tr .
Their partial study was done in [L-Y8 : Sec. 6.2] (D(13.1)).
(4) The background B-field
in the part
The coupling of (ϕ, ∇) with the background B-field B on Y
(h;B)
(C,B)
Sgauge /YM (ϕ, ∇) + SCS/WZ (ϕ, ∇)
of the standard action means that we have to adjust the fundamental module E on X
by a compatible “twisting” governed by ϕ and B. With this “twisting”, E now lives on
a gerb over X. See [L-Y2] (D(5)) for details and further references. However, since the
study of the variational problems in this note is mainly local and focuses on the enhanced
(ρ,h;Φ,g)
kinetic term for maps Smap:kinetic+ , we’ll ignore this twisting for the current note to keep
the language and expressions simple.
Remark 4.3 [other effects from B-field and Ramond-Ramond field ] There are other effects to
D-branes beyond just mentioned above from the background B-field and Ramond-Ramond field
that have not yet been taken into account in this project so far; e.g. [H-M1], [H-M2], and [H-Y].
They can influence the action for D-branes as well. Such additional effects should be investigated
in the future.
Theorem 4.4 [well-defined gauge-symmetry-invariant action] Except the anomaly factor
in the Chen-Simons/Wess-Zumino term, which is yet to be understood, the standard action
(ρ,h;Φ,g,B,C)
(C,B)
Sstandard
(ϕ, ∇) as given in Definition 4.2 for (∗1 )-admissible pairs (ϕ, ∇) (and SCS/WZ (ϕ, ∇)
for (∗2 )-admissible (ϕ, ∇)) is well-defined. Assume that the anomaly factor in the Chen-Simons/
24
Az
under a gauge symmetry, then
Wess-Zumino term transforms also by conjugation as for OX
(ρ,h;Φ,g,B,C)
Sstandard
(ϕ, ∇) is invariant under gauge symmetries:
(ρ,h;Φ,g,B,C)
Sstandard
(ρ,h;Φ,g,B,C) g 0
0
( ϕ, g ∇)
(ϕ, ∇) = Sstandard
for g 0 ∈ Ggauge := C ∞ (Aut C (E)).
For the kinetic term for maps
1
(h;g)
Smap:kinetic (ϕ, ∇) := 2 Tm−1
Z
Re Tr hDϕ , Dϕi(h,g) vol h ,
X
that it is well-defined follows Lemma 3.2.2.4. Under a gauge transformation g 0 ∈ Ggauge :=
C ∞ (Aut C (E)) and in terms of local coordinates (x1 , · · · , xm ) on X and (y 1 , · · · , y n ) on Y ,
g0
0
D gϕ =
X
dxµ ⊗
X
µ
g0
D
i
∂
g0 ]
∂
X
ϕ ( ∂y i ) ⊗g0ϕ ∂y i =
∂
∂xµ
dxµ ⊗
X
µ
g0 D
i
∂
∂xµ
∂
ϕ] ( ∂y i ) g 0
−1
∂
⊗g0ϕ ∂y i .
Thus,
0
0
0
0
h gD gϕ , gD gϕi(h,g)
=
XX
hµν ⊗ g 0 D
µ,ν i,j
=
XX
hµν · g 0 D
∂
∂xµ
µ,ν i,j
= g0
XX
hµν · D
µ,ν i,j
= g hDϕ , Dϕi(h,g) g 0
0
0
∂
ϕ] ( ∂y i ) g 0
∂
ϕ] ( ∂y i ) g 0
∂
ϕ] ( ∂y i ) · D
∂
∂xµ
0
0
∂
∂xµ
−1
−1
−1
∂
∂xν
· g0 D
∂
∂xν
· g0 D
∂
∂xν
∂
ϕ] ( ∂y j ) g 0
∂
ϕ] ( ∂y j ) g 0
∂
ϕ] ( ∂y j ) · ϕ] (gij ) g 0
−1
−1
⊗g0ϕ gij
· g 0 ϕ] (gij )g 0
−1
−1
.
0
It follows that Tr h gD gϕ , gD gϕi(h,g) = Tr hDϕ , Dϕi(h,g) and, hence,
0
(h;g)
0
(h;g)
Smap:kinetic ( gϕ, g ∇) = Smap:kinetic (ϕ, ∇) .
(ρ,h;Φ,g,B,C)
The other terms in Sstandard
(ϕ, ∇) do not involve a partially-defined inner product and
hence are all defined. That the integrand inside Tr all transform by conjugation under a gauge
Az
follows Lemma 4.1.
symmetry as for OX
This proves the theorem.
Remark 4.5 [gauge-fixing condition] As in any gauge field theory (e.g. [P-S]), understanding how
(ρ,h;Φ,g,B,C)
to fix the gauge is an important part of understanding Sstandard
(ϕ, ∇).
The standard action as an enhanced non-Abelian gauged sigma model
Recall that, in an updated language and in a form for easy comparison, a sigma model (σ-model,
SM) on a (Riemannian or Lorentzian) manifold (Y, g) (of dimension n) is a field theory on a
(Riemannian or Lorentzian) manifold (X, h) (of some dimension m) with
· Field :
differentiable maps
· Action functional :
(h,g)
Ssigma model
1
(f ) := ± 2
Z
1
:= ± 2
Z
X
f :X →Y,
hdf , df i(g,h) vol h
m
n
X
X
X µ,ν=1 i,j=1
1
= ±2
Z
X
kf ∗ gk2h vol h
q
∂f i
∂f j
hµν (x)gij (f (x)) ∂xµ (x) ∂xν (x) |det h(x)| d m x ,
25
in terms of local coordinates x = (x1 , · · · , xm ) on X and y = (y 1 , · · · , y n ) on Y ; cf. [GM-L] and
see e.g. [C-T] for modern update and further references. (The ± sign depends on the signature
of the metric.) At the classical level, this is a theory of harmonic maps; cf. [E-L], [E-S], [Ma],
[Sm].
Back to our situation. To begin with, the kinetic term
1
(h;g)
Smap:kinetic (ϕ, ∇) := 2 Tm−1
(ρ,h;Φ,g,B,C)
qualifies the standard action Sstandard
· Field :
Z
X
Re (Tr hDϕ , Dϕi(h,g) ) vol h
(ϕ, ∇) to be regarded as a sigma model, now based on
(∗1 )-admissible differentiable maps ϕ : (XAz , E; ∇) → Y .
(ρ,h;Φ,g,B,C)
The fact that Sstandard
(ϕ, ∇) is invariant under the gauge symmetry group
∞
Ggauge := C (Aut C (E)) and that the latter is non-Abelian justify that this sigma model is indeed
a non-Abelian gauged sigma model (nAGSM). However, compared with, for example, the wellstudied d = 2, N = (2, 2) (Abelian) gauged linear sigma model, e.g. [H-V] and [Wi1], the gauge
(ρ,h;Φ,g,B,C)
symmetry of Sstandard
(ϕ, ∇) does not arise from gauging a global group-action on the target
(ρ,h;Φ,g,B,C)
space Y . (For this reason, one may call Sstandard
(ϕ, ∇) a sigma model with non-Abelian gauge
symmetry as well.) For D-branes, its additional coupling to the background Ramond-Ramond
(C,B)
field C on Y is essential ([Po1]) and, hence, the Chern-Simons/Wess-Zumino term SCS/WZ (ϕ, ∇).
Also, we like our dynamical field (ϕ, ∇) coupled to the background dilaton field Φ on Y as well
so that the essence of the other important action — the Dirac-Born-Infeld action — for D-branes
(ρ,h;Φ)
can be retained as much as we can. This motivates the dilaton term Sdilaton (ϕ). In summary,
(ρ,h;Φ,g,B,C)
Sstandard
(ϕ, ∇) :=
(ρ,h;Φ,g,B)
SnAGSM
(C,B)
(ρ,h;Φ,g,B,C)
=: SnAGSM+
(ρ,h;Φ)
(ϕ, ∇) + SCS/WZ (ϕ, ∇) + Sdilaton (ϕ)
(ϕ, ∇) ,
which explains the name enhanced non-Abelian gauged sigma model (nAGSM+ ).
5
Admissible family of admissible pairs (ϕT , ∇T )
In this section we introduce the notion of one-parameter admissible families of admissible pairs
and rephrase the basic settings and results in Sec. 3.2 in a relative format for such a family. Some
curvature tensor computations are given for later use. The natural generalization (without work)
to two-parameter admissible families of admissible pairs is remarked in the last theme of the
section. This prepares us for the study of the variational problem of the enhanced kinetic term
(ρ,h;Φ,g)
(ρ,h;Φ,g,B,C)
for maps Smap:kinetic+ (ϕ, ∇) in the standard action Sstandard
(ϕ, ∇) for D-branes.
Basic setup and the notion of admissible families of admissible pairs (ϕT , ∇T )
Let T = (−ε, ε) ⊂ R1 , with coordinate t and ε > 0 small, be the one-parameter space and
∂t := ∂/∂t and dt be respectively the tangent vector field and the 1-form determined by the
coordinate t on T . Let (X, E) be a manifold X of dimension m with a complex vector bundle E
of rank r. Recall the structure sheaf OX of X and the OX -module E from E.
Consider the following families of objects over T :
·X
:= X ×T , with the structure sheaf OXT and regarded as the constant family of manifolds
over T determined by X. XT is equipped with the built-in projection maps prX : XT → X
and prT : XT × T → T . For U ⊂ X an open set, we will denote by UT the corresponding
open set U × T ⊂ X × T over T .
T
26
·TX
∗ T := the tangent bundle of XT and T∗ XT := the tangent sheaf of XT ;
T ∗ XT := the cotangent bundle of XT and T ∗ XT := the cotangent sheaf of XT ;
T∗ (XT /T ) := the relative tangent bundle of XT over T and
T∗ (XT /T ) := the relative tangent sheaf of XT over T ;
T ∗ (XT /T ) := the relative cotangent bundle of XT over T and
T ∗ (XT /T ) := the relative cotangent sheaf of XT over T .
When X is endowed with a (Riemannain or Lorentzian) metric h, h induces canonically
an inner-product structure on fibers of T∗ (XT /T ) and its dual, T ∗ (XT /T ), over T . These
induced inner-product structure will be denoted by h · , ·ih .
·E
:= pr∗X E the pull-back vector bundle of E to XT , regarded as the constant T -family of
vector bundles over X determined by E; and ET := pr∗X E the corresponding OXT -module,
regarded as the constant T -family of OX -modules determined by E.
The projection map prX : XT → X induces a projection map prE : ET → E between the
total space of bundles in question. T∗ ET (resp. T∗ ET ) denotes the tangent space (resp. the
tangent sheaf) of the total space of ET .
T
· (X
Az
T , ET )
Az
:= (XT , OX
:= End OXC (ET ), ET ), regarded as the constant T -family of AzuT
T
maya/matrix manifolds with a fundamental module determined by (XAz , E). There is a
trace map
Az
Tr : OX
−→ OXCT
T
as OXT -modules, which takes Id ET to r.
and take the following notational conventions:
· Through the product structure X
T = X × T , a vector field ξ (resp. 1-form ω) on X and
the vector field ∂t on T lift canonically to a vector field (resp. 1-form) on XT , which will
still be denoted by ξ (resp. ω) and ∂t respectively.
· For referral, the restriction of X
Az
T , XT ,
ET , · · ·T to over t ∈ T will be denoted Xt , XAz
t , Et ,
· · ·t respectively.
Definition 5.1 [connection/covariant derivation trivially flat over T ] A connection ∇T
on ET (equivalently, connection/covariant derivative ∇T on ET ) is said to be trivially flat over T
if the horizontal lifting of ∂t to T∗ ET lies in the kernel of the map prE ∗ : T∗ ET → T∗ E. For such
a ∇T , we will denote the covariant derivative ∇T∂t simply by ∂t . The curvature tensor of ∇T will
be denoted by F∇T .
Note that any connection on ET is flat over T and hence, due to the topology of T , can be
made trivially flat over T after a bundle-isomorphism. Thus the notion of ‘trivially flat’ is only a
notational convenience for our variational problem, not a true constraint. However, caution that
while ∇T is always flat over T , its restriction ∇t to Xt varies as t varies in T . Thus, in general,
F∇T (∂t , · ) 6= 0.
Definition 5.2 [admissible family of admissible pairs (ϕT , ∇T )] A T-family of maps with
varying connections from (XAz , E) to Y is a pair (ϕT , ∇T ), where
ϕT : (XAz , ET ) −→ Y
is a map from (XAz
T , ET ) to Y defined contravariantly by a ring-homomorphism
ϕ]T : C ∞ (Y ) −→ C ∞ (End C (ET ))
27
over R ⊂ C and ∇T is a connection on ET that is trivially flat over T . ϕ]T induces a homomorphism
Az
OY −→ OX
T
between equivalence classes of gluing systems of rings, which will still be denoted by ϕ]T .
Az
Let AϕT ⊂ OX
= OXT hIm ϕ]T i. Then (ϕT , ∇T ) is said to be a (∗i )-admissible T -family
T
of (∗j )-admissible pairs if (ϕT , ∇T ) satisfies Admissible Condition (∗i ) along T and Admissible
Condition (∗j ) along X, for i, j = 1, 2, 3.
Example 5.3 [(∗2 )-admissible T -family of (∗1 )-admissible pairs] A (∗2 )-admissible T family of (∗1 )-admissible pairs (ϕT , ∇T ) is a T -family of maps ϕT with a varying connection
∇T trivially flat over T such that
(∗2 ) : ∂t Comm (AϕT ) ⊂ Comm (AϕT )
(∗1 ) : ∇Tξ AϕT ⊂ Comm (AϕT )
and
Az
for all ξ ∈ T∗ (XT /T ). Here, Comm (AϕT ) is the commutant of AϕT in OX
.
T
Three basic OXT -modules with induced structures
Let X be endowed with a (Riemannian or Lorentzian) metric h and Y be endowed with a
(Riemannian or Lorentzian) metric g. Denote the canonically induced inner-product structure
from h and g on whatever bundle applicable by h · , · ih and h · , · ig respectively. Denote the
induced connection on T∗ (XT /T ) and T ∗ (XT /T ) by ∇h and the Levi-Civita connection on T∗ Y
by ∇g . The associated Riemann curvature tensor is denoted by Rh and Rg respectively.
Let (ϕT , ∇T ) be a (∗1 )-admissible T -family of (∗1 )-admissible pairs. The basic OXCT -modules
with induced structures from the setting, as in Sec. 3.2, are listed below to fix notations.
Az
: the noncommutative structure sheaf on XT
(0) OX
T
· The induced connection D from ∇ , which is also trivially flat over T ,
· An O -valued, O -bilinear (nonsymmetric) inner product from the multiplication
T
Az
XT
Az
in OXT ;
an OXC -valued,
T
C
X
OXC -bilinear (symmetric) inner product after the post-composition with
Tr .
· Both inner products are covariantly constant with respect to D
T
and one has the
Leizniz rules
DT (m1T m2T ) = (DT m1T )m2T + m1T DT m2T ;
dTr (m1T m2T ) = Tr DT (m1T m2T )
= Tr ((DT m1T )m2T ) + Tr (m1T DT m2T ) .
Az
Az
(1) T ∗ (XT /T ) ⊗OXT OX
: OX
-valued relative 1-forms on XT /T
T
T
· The induced connection ∇
:= ∇ ⊗ Id + Id ⊗ D , trivially flat over T .
· An O -valued, O -bilinear (nonsymmetric) inner product h · , · i ;
T,(h,DT )
an
Az
XT
OXC -valued,
C
XT
OXC -bilinear
h
T
(symmetric) inner product Tr h · , · ih .
h
· Both inner products are covariantly constant with respect to ∇
T,(h,DT )
and one has
the Leibniz rules
· · i = h∇
·, · i
dTr h · , · i = Tr (D h · , · i )
for ·, · ∈ T (X /T ) ⊗
O .
DT h ,
0
0
∗
T
O XT
h
0
T
h
0
0
T,(h,DT )
h
h
·
+ h , ∇T,(h,D
T)
·i ,
·, · i
T,(h,DT )
= Tr h ∇
Az
XT
28
0
g
0
h
·
+ Tr h , ∇T,(h,D
T)
·i
0
h
Az
Az
⊗ϕ] , OY T∗ Y : OX
(2) ϕ∗T T∗ Y := OX
-valued derivations on OY
T
T
T
· The induced connection ∇
T,(ϕT ,g)
:= DT ⊗ Id + Id ·
Pn
i=1
DT ϕ]T (y i ) ⊗ ∇g ∂ (in local
∂y i
expression), trivially flat over T .
· A partially defined O
a partially defined
h · , · ig ;
· , · ig .
Az
C
XT -valued, OX -bilinear (nonsymmetric) inner product
OXC -valued, OXC -bilinear (symmetric) inner product Tr h
· Both inner products, when defined, are covariantly constant with respect to ∇
T,(ϕT ,g)
and one has the Leibniz rules
DT h – , –0 ig = h ∇T,(ϕT ,g) – , –0 ig + h – , ∇T,(ϕT ,g) –0 ig ,
dTr h – , –0 ig = Tr (DT h – , –0 ig ) = Tr h ∇T,(ϕT ,g) – , –0 ig + Tr h – , ∇T,(ϕT ,g) –0 ig ,
whenever all h –00 , –000 ig and Tr h –00 , –000 ig involved are defined.
Az
-valued relative 1-form)-valued derivations on OY
(3) T ∗ (XT /T ) ⊗OXT ϕ∗T T∗ Y : (OX
T
This is a combination of the construction in Item (1) and in Item (2).
· The induced connection
∇T,(h,ϕT ,g) = ∇h ⊗ Id ⊗ Id + Id ⊗ DT ⊗ Id + Id ⊗ Id ·
n
X
DT ϕ]T (y i ) ⊗ ∇g ∂
∂y i
i=1
(in local expression), trivially flat over T .
· A partially defined O
a partially defined
· , · i(h,g);
· , · i(h,g).
C
Az
XT -valued, OX -bilinear (nonsymmetric) inner product h
OXC -valued, OXC -bilinear (symmetric) inner product Tr h
· Both inner products, when defined, are covariantly constant with respect to ∇
T,(h,ϕT ,g)
and one has the Leibniz rules
DT h ∼ , ∼0 i(h,g) = h ∇T,(h,ϕT ,g) ∼ , ∼0 i(h,g) + h ∼ , ∇T,(h,ϕT ,g) ∼0 i(h,g) ,
dTr h ∼ , ∼0 i(h,g) = Tr (DT h ∼ , ∼0 i(h,g) )
= Tr h ∇T,(h,ϕT ,g) ∼ , ∼0 i(h,g) + Tr h ∼ , ∇T,(h,ϕT ,g) ∼0 i(h,g) ,
whenever the h ∼00 , ∼000 i(h,g) and Tr h ∼00 , ∼000 i(h,g) involved are defined.
Curvature tensors with ∂t and other order-switching formulae
Let (ϕT , ∇T ) be a (∗1 )-admissible T -family of (∗1 )-admissible pairs. A very basic step in (particularly the second) variational problem involves passing ∂t over a differential operator on Xt ’s. In
general, a curvature term appears whenever such passing occurs. In this theme, we collect and
prove such formulae we need.
First, passing ∂t over a differential operator usually means the appearance of a curvature term
by the very definition of a curvature tensor:
Lemma 5.4 [curvature tensor with ∂t ] Let (ϕT , ∇T ) be a (∗1 )-admissible T -family of (∗1 )admissible pairs. Let ξ be a vector field on an open set U ⊂ X small enough so that ϕT (UTAz )
is contained in a coordinate chart on Y , with coordinates (y 1 , · · · , y n ). The standard lifting of ξ
to UT is denoted also by ξ. Note that, by construction, [∂t , ξ] = 0 and all our connection ∇· in
Az
Theme ‘Three basic OX
-modules with induced structures’ are trivially flat; hence, F∇· (∂t , ξ) =
∂t ∇·ξ − ∇·ξ ∂t . One has the following curvature expressions with ∂t on the basic OXT -modules:
(Below we adopt the convention that the Riemann curvature tensor from a metric is denoted by
R while the curvature tensor of a connection in all other bundle situations is denoted by F .)
29
(01 ) For sections ωT of T ∗ (XT /T ) : R∇h (∂t , ξ) ωT = ∂t ∇hξ ωT − ∇hξ ∂t ωT = 0.
Az
(02 ) For sections mT of OX
: FDT (∂t , ξ) mT = ∂t DξT mT − DξT ∂t mT = [(∂t ∇T )(ξ), mT ] .
T
As a consequence of this, if (ϕT , ∇T ) is furthermore a (∗2 )-admissible T -family of (∗2 )admissible pairs, then
Az
)
(∂t ∇T )(ξ) ∈ Inn ϕ(∗1 ) (OX
i.e.
[(∂t ∇T )(ξ), AϕT ] ⊂ Comm (AϕT ) .
Az
(1) For sections ωT ⊗ mT of T ∗ (XT /T ) ⊗OXT OX
:
T
F∇T,(h,DT ) (∂t , ξ) (ωT ⊗ mT )
T,(h,DT )
= ∂t ∇ξ
T,(h,DT )
(ωT ⊗ mT ) − ∇ξ
∂t (ωT ⊗ mT ) = ωT ⊗ [(∂t ∇T )(ξ), mT ] .
Az
(2) For sections mT ⊗ v of ϕ∗T T∗ Y := OX
⊗ϕ] ,OY T∗ Y :
T
T
(v on the coordinate chart of Y above, with coordinates (y 1 , · · · , y n ))
T,(ϕT ,g)
F∇T,(ϕT ,g) (∂t , ξ)(mT ⊗ v) = ∂t ∇ξ
= [(∂t ∇T )(ξ), mT ] ⊗ v + mT
+ mT
n
X
T,(ϕT ,g)
(mT ⊗ v) − ∇ξ
∂t (mT ⊗ v)
n
X
[(∂t ∇T )(ξ), ϕ]T (y i )] ⊗ ∇g ∂ v
∂y i
i=1
DξT ϕ]T (y i ) ∂t ϕ]T (y j )
!
⊗∇
g
i,j=1
∂
∂y j
∇
g
∂
∂y i
v −
∂t ϕ]T (y j ) DξT ϕ]T (y i )
⊗∇
g
∂
∂y i
∇
g
∂
∂y j
v .
If (ϕT , ∇T ) is furthermore a (∗2 )-admissible T -family of (∗1 )-admissible pairs, then the
last term has a Y -coordinate-free form
The last term = mT
X
∂t ϕ]T (y j ) Dξ ϕ]T (y i ) ⊗ Rg ( ∂y∂ j , ∂y∂ i ) v = mT ((ϕT Rg )(∂t , ξ))v .
i,j
Az
⊗ϕ] ,OY T∗ Y :
(3) For sections ωT ⊗ mT ⊗ v of T ∗ (XT /T ) ⊗ ϕ∗T T∗ Y := T ∗ (XT /T ) ⊗OXT OX
T
T
1
n
(v on the coordinate chart of Y above, with coordinates (y , · · · , y ) )
F∇T,(h,ϕT ,g) (∂t , ξ)(ωT ⊗ mT ⊗ v)
T,(h,ϕT ,g)
= ∂t ∇ξ
(ωT ⊗ mT ⊗ v) − ∇T,(h,ϕT ,g) ∂t (ωT ⊗ mT ⊗ v)
= ωT ⊗ F∇T,(ϕT ,g) (∂t , ξ)(mT ⊗ v) .
Statement (01 ) follows from the fact that XT is a constant family over T . Statement (02 ),
First Part, follows from a computation with respect to an induced local trivialization of ET from
a local trivialization of E
∂t DξT mT
= ∂t (ξmT + [A∇T (ξ), mT ])
= ξ∂t mT + [∂t A∇T (ξ), mT ] + [A∇T (ξ), ∂t mT ] = DξT ∂t mT + [(∂t ∇T )(ξ), mT ] .
For Second Part, if (ϕT , ∇T ) is furthermore a (∗2 )-admissible T-family of (∗2 )-admissible pairs,
then for f1 , f2 ∈ OY , by First Part and the (∗2 )-Admissible Condition,
[[(∂t ∇T )(ξ), ϕ]T (f1 )], ϕ]T (f2 )] = [∂t DξT ϕ]T (f1 ), ϕ]T (f2 )] − [DξT ∂t ϕ]T (f1 ), ϕ]T (f2 )] = 0 .
Az
Which says that (∂t ∇T )(ξ) ∈ Inn ϕ(∗T1 ) (OX
).
T
Statement (1) is a consequence of Statement (01 ) and Statement (02 ). Statement (3) is a consequence of Statement (01 ) and a property of the induced connection on a tensor product of
30
OXCT -modules with a connection. Let us carry out Statement (2) as a demonstration of the
covariant differential calculus involved.
Let mT ⊗ v ∈ ϕ∗T T∗ Y . Then, by Statement (02 ),
T,(ϕT ,g)
∂t ∇ξ
(mT ⊗ v) = ∂t DξT mT ⊗ v + mT
= (DξT ∂t mT + [(∂t ∇T )(ξ), mT ]) ⊗ v +
+ (∂t mT )
X
DξT ϕ]T (y i ) ⊗ ∇g ∂ v +
∂y i
i
X
DξT ϕ]T (y i ) ⊗ ∇g ∂ v
i
∂y
i
X
(DξT mT )
∂t ϕ]T (y i ) ⊗ ∇g ∂ v
∂y i
i
X
mT
(DξT ∂t ϕ]T (y i ) + [(∂t ∇T )(ξ), ϕ]T (y i )]) ⊗
i
X
+ mT
DξT ϕ]T (y i )∂t ϕ]T (y j ) ⊗ ∇g ∂ ∇g ∂ v
∂y i
∂y j
i,j
∇g ∂ v
∂y i
while
T,(ϕT ,g)
∇ξ
T,(ϕT ,g)
∂t (mT ⊗ v) = ∇ξ
∂t mT ⊗ v + mT
X
∂t ϕ]T (y i ) ⊗ ∇g ∂ v
∂y i
i
= DξT ∂t mT ⊗ v + (∂t mT )
X
DξT ϕ]T (y i ) ⊗ ∇g ∂ v + (DξT mT )
∂y i
i
+ mT
X
DξT ∂t ϕ]T (y i ) ⊗ ∇g ∂ v + mT
∂y i
i
X
X
∂t ϕ]T (y i ) ⊗ ∇
i
∂
∂y i
v
∂t ϕ]T (y i )DξT ϕ]T (y j ) ⊗ ∇g ∂ ∇g ∂ v .
∂y j
i,j
∂y i
Thus,
F∇T,(ϕT ,g) (∂t , ξ)(mT ⊗ v) = (∂t ∇T,(ϕT ,g) − ∇T,(ϕT ,g) ∂t )(mT ⊗ v)
= [(∂t ∇T )(ξ), mT ] ⊗ v + mT
[(∂t ∇T )(ξ), ϕ]T (y i )] ⊗ ∇g ∂ v
X
∂y i
i
+ mT
X
DξT ϕ]T (y i )∂t ϕ]T (y j ) ⊗ ∇g ∂ ∇g ∂ v − ∂t ϕ]T (y j )DξT ϕ]T (y i ) ⊗ ∇g ∂ ∇g ∂ v
∂y j
i,j
∂y i
∂y i
∂y j
as claimed, after a relabeling of i, j.
If (ϕT , ∇T ) is furthermore a (∗2 )-admissible T -family of (∗1 )-admissible pairs, then DξT ϕ]T (y i )
and ∂t ϕ]T (y j ) commute since [DξT ϕ]T (y i ), ϕ]T (y j )] = 0 by the (∗1 )-Admissible Condition along X
and, hence,
0 = ∂t [DξT ϕ]T (y i ), ϕ]T (y j )]
= [∂t DξT ϕ]T (y i ), ϕ]T (y j )] + [DξT ϕ]T (y i ), ∂t ϕ]T (y j )] = [DξT ϕ]T (y i ), ∂t ϕ]T (y j )]
by the (∗1 )-Admissible Condition along X and the (∗2 )-Admissible Condition along T . The last
summand of F∇T,(ϕT ,g) (∂t , ξ)(mT ⊗ v) is then equal to
mT
X
∂t ϕ]T (y j )DξT ϕ]T ⊗ ∇g ∂ ∇g ∂ − ∇g ∂ ∇g ∂
∂y j
i,j
∂y i
∂y i
∂y j
v = mT ((ϕT Rg )(∂t , ξ)) v .
This proves the lemma.
The following lemma addresses the issue of passing ∂t over the covariant differential DϕT of
ϕT . Though such passing is not a curvature issue in the conventional sense, it does carry a taste
of curvature calculations.
Lemma 5.5 [∂t DT ϕT versus ∇T,(ϕT , g) ∂t ϕT ] Let (ϕT , ∇T ) be a (∗1 )-admissible T -family of (∗1 )admissible pairs. With the above notation and convention, let ξ be a vector field on X. Then,
for a chart of Y with coordinates (y 1 , · · · , y n ), one has
∂t DξT ϕT
=
T,(ϕ , g)
∇ξ T ∂t ϕT
− (ad ⊗ ∇
g
)∂t ϕT DξT ϕT
+
n
X
[(∂t ∇T )(ξ), ϕ]T (y i )] ⊗
i=1
31
∂
∂y i
.
Here, only as a compact notation,
(ad ⊗ ∇
g
)∂t ϕT DξT ϕT
n
X
:=
∂
j
∂y
∂y i
[∂t ϕ] (y i ), DξT ϕ] (y j )] ⊗ ∇g ∂
i,j=1
n
X
= −
[DξT ϕ] (y j ), ∂t ϕ] (y i )] ⊗ ∇g ∂
∂y j
i,j=1
∂
=: − (ad ⊗ ∇g )DξT ϕT ∂t ϕT .
∂y i
If (ϕT , ∇T ) is furthermore a (∗2 )-admissible T -family of (∗2 )-admissible pairs, then the last
term has a Y -coordinate-free expression
ad (∂t ∇T )(ξ) ϕT .
Under the given setting and by Lemma 5.4 (02 ),
∂t DξT ϕT = ∂t
X
i
=
X
i
∂
DξT ϕ]T (y i ) ⊗ ∂y i
∂
(DξT ∂t ϕ]T (y i ) + [(∂t ∇T )(ξ), ϕ]T (y i )]) ⊗ ∂y i +
X
i,j
∂
DξT ϕ]T (y i )∂t ϕ]T (y j ) ⊗ ∇g ∂ ∂y i
∂y j
while
T,(ϕT ,g)
∇ξ
T,(ϕT ,g)
∂t ϕT = ∇ξ
X
i
=
X
DξT ∂t ϕ]T (y i )
i
∂
∂t ϕ]T (y i ) ⊗ ∂y i
X
∂
∂
⊗ ∂y i +
∂t ϕ]T (y i )DξT ϕ]T (y j ) ⊗ ∇g ∂ ∂y i .
j
∂y
i,j
Thus,
T,(ϕT ,g)
∂t DξT ϕ − ∇ξ
=
X
i
∂t ϕT
∂
[(∂t ∇T )(ξ), ϕ]T (y i )] ⊗ ∂y i
+
X
i,j
∂
DξT ϕ]T (y i )∂t ϕ]T (y j ) ⊗ ∇g ∂ ∂y i −
∂y j
Either apply the identity ∇g ∂
∂y j
or apply the identity ∇g ∂
∂y j
X
i,j
∂
∂y i
∂
∂y i
= ∇g ∂
∂y i
= ∇g ∂
∂y i
∂
∂y j
∂
∂y j
∂
=
[DξT ϕ]T (y i ) ,
i,j
= −
X
i,j
∂t ϕ]T (y j )]
i,j
∂
∂t ϕ]T (y i )DξT ϕ]T (y j ) ⊗ ∇g ∂ ∂y i
∂y j
to the second term and relabeling i, j of the third,
to the third term and relabeling i, j of the second,
DξT ϕ]T (y i )∂t ϕ]T (y j ) ⊗ ∇g ∂ ∂y i −
∂y j
X
X
⊗∇
X
i,j
g
∂
∂y i
∂
∂t ϕ]T (y i )DξT ϕ]T (y j ) ⊗ ∇g ∂ ∂y i
∂y j
∂
∂y j
:= (ad ⊗ ∇g )DT ϕ ∂t ϕT
:= − (ad ⊗ ∇g )∂t ϕT DξT ϕT
ξ
∂
[∂t ϕ]T (y i ) , DξT ϕ]T (y j )] ⊗ ∇g ∂ ∂y j
∂y i
.
This proves the First Statement in Lemma.
The Second Statement in Lemma is a consequence of Corollary 3.1.10 and Lemma 5.4 (02 ).
This proves the lemma.
Before continuing the discussion, we introduce a notion that is needed in the next lemma.
32
1
,•
Definition 5.6 [half-torsion tensor Tor ∇2 g ] Recall the torsion tensor Tor ∇0 of a connection
∇0 on Y
Tor ∇0 (v1 , v2 ) := ∇0v1 v2 − ∇0v2 v1 − [v1 , v2 ]
for v1 , v2 ∈ T∗ Y . For the Levi-Civita connection ∇g associated to a metric g on Y , Tor ∇g ≡ 0
by construction. Thus, in this case, for a Φ ∈ C ∞ (Y ),
(∇gv1 v2 − v1 v2 )Φ = (∇gv2 v1 − v2 v1 )Φ
for v1 , v2 ∈ T∗ Y . This defines a symmetric 2-tensor on Y
1
,Φ
: T∗ Y ×Y T∗ Y
(v1 , v2 )
Tor ∇2 g
−→
OY
,
g
7−→ (∇v1 v2 − v1 v2 ) Φ
called the half-torsion tensor of (the torsion-free connection) ∇g associated to Φ ∈ C ∞ (Y ).
Az
The following lemma addresses the issue of passing ∂t over ‘evaluation of an OX
-valued
T
∞
derivation on C (Y )’, and another similar situation:
T,(ϕ , g)
Lemma 5.7 [∂t ((DξT ϕT )Φ) versus (∂t DξT ϕT )Φ ; DξT ((∂t ϕT )Φ) versus (∇ξ T ∂t ϕT )Φ]
Let (ϕT , ∇T ) be a (∗1 )-admissible T -family of (∗1 )-admissible pairs. Continue the notation and
Az
Az
,
⊗ϕ] ,OY OY ' OX
convention in Lemma 5.4. Under the canonical isomorphism OX
T
T
T
∂t ((DξT ϕT )Φ)
= (∂t DξT ϕT )Φ +
n
X
∂ ∂
∂
DξT ϕ]T (y i ) ∂t ϕ]T (y j ) ⊗ ∂y
∇g ∂ ∂y
i ∂y j Φ −
j Φ
∂y i
i,j=1
1
,Φ
= (∂t DξT ϕT )Φ − (ϕT Tor ∇2 g )(ξ, ∂t ) ;
and
DξT ((∂t ϕT )Φ)
T,(ϕT , g)
= (∇ξ
n
X
∂t ϕT )Φ +
i,j=1
T,(ϕT , g)
= (∇ξ
∂ ∂
∂
∇g ∂ ∂y
∂t ϕ]T (y i ) DξT ϕ]T (y j ) ⊗ ∂y
i ∂y j Φ −
j Φ
∂y i
1
,Φ
∂t ϕT )Φ − (ϕT Tor ∇2 g )(∂t , ξ) .
For the first identity,
∂t ((DξT ϕT )Φ) = ∂t
X
i
X
=
∂t DξT ϕ]T
i
∂
DξT ϕ]T (y i ) ⊗ ∂y i Φ
X
∂ ∂
∂
⊗ ∂y i Φ +
DξT ϕ]T (y i )∂t ϕ]T (y j ) ⊗ ∂y j ∂y i Φ
i,j
while
(∂t DξT ϕT )Φ =
∂t
X
i
=
X
i
∂
DξT ϕ]T (y i ) ⊗ ∂y i Φ
∂t DξT ϕ]T (y i )
∂
⊗ ∂y i +
X
i,j
33
∂
DξT ϕ]T (y i )∂t ϕ]T (y j ) ⊗ ∇g ∂ ∂y i Φ .
∂y j
Thus,
∂t ((DξT ϕT )Φ) − (∂t DξT ϕT )Φ
=
X
=
X
i,j
i,j
∂
∂
∂
∂
∂
∂
DξT ϕ]T (y i )∂t ϕ]T (y j ) ⊗ ∂y j ∂y i − ∇g ∂ ∂y i Φ
∂y j
1
,Φ
2
DξT ϕ]T (y i )∂t ϕ]T (y j ) ⊗ ∂y i ∂y j − ∇g ∂ ∂y j Φ = − (ϕT Tor ∇
g )(ξ, ∂t )
∂y i
and the first identity follows.
For the second identity,
DξT ((∂t ϕT )Φ) = DξT
X
i
=
X
DξT ∂t ϕ]T (y i )
i
∂
∂t ϕ]T (y i ) ⊗ ∂y i Φ
∂
⊗ ∂y i Φ +
X
i,j
∂
∂
∂t ϕ]T (y i )DξT ϕ]T (y j ) ⊗ ∂y j ∂y i Φ
while
T,(ϕT , g)
(∇ξ
∂t ϕT )Φ =
T,(ϕT ,g) X
∇ξ
i
=
X
DξT ∂t ϕ]T (y i )
∂
∂t ϕ]T (y i ) ⊗ ∂y i Φ
X
∂
∂
∂t ϕ]T (y i )DξT ϕ]T (y j ) ⊗ ∇g ∂ ∂y i Φ .
⊗ ∂y i +
∂y j
i,j
i
Thus,
T,(ϕT , g)
DξT ((∂t ϕT )Φ) − (∇ξ
=
X
=
X
i,j
i,j
∂t ϕT )Φ
∂
∂
∂
∂
∂
∂
∂t ϕ]T (y i )DξT ϕ]T (y j ) ⊗ ∂y j ∂y i − ∇g ∂ ∂y i Φ
∂y j
1
,Φ
2
∂t ϕ]T (y i )DξT ϕ]T (y j ) ⊗ ∂y i ∂y j − ∇g ∂ ∂y j Φ = − (ϕT Tor ∇
g )(∂t , ξ)
∂y i
and the second identity follows.
This proves the lemma.
Remark 5.8 [for (∗2 )-admissible family of (∗1 )-admissible pairs ] If (ϕT , ∇T ) is furthermore
a (∗2 )-admissible T -family of (∗1 )-admissible pairs, then, as in the proof of Lemma 5.4 (2),
DξT ϕ]T (y i ) and ∂t ϕ]T (y j ) commute for all i, j. In this case,
1
1
,Φ
,Φ
(ϕ Tor ∇2 g )(ξ, ∂t ) = (ϕ Tor ∇2 g )(∂t , ξ) .
Two-parameter admissible families of admissible pairs
Let T = (−ε, ε)2 ⊂ R2 , ε > 0 small, be a two-parameter space with coordinates (s, t). The
setting and results above for one-parameter admissible families of admissible pairs generalizes
without work to two-parameter admissible of admissible pairs. In particular,
Definition 5.9 [two-parameter admissible family of admissible pairs] A (∗2 )-admissible
T
T -family of (∗1 )-admissible maps is a (∗1 )-admissible map ϕT : (XAz
T , ET ; ∇ ) → Y , where ET is
trivially flat over T , such that ∂s Comm AϕT ⊂ Comm (AϕT ) and ∂t Comm AϕT ⊂ Comm (AϕT ).
The following is a consequence of the proof of Lemma 3.2.2.5:
34
Lemma 5.10 [symmetry property of Tr hF∇T,(ϕT ,g) (∂s , ξ2 )∂t ϕT , DξT4 ϕT i] Let
T
ϕT : (XAz
T , ET ; ∇ ) → Y be a (∗2 )-admissible T -family of (∗1 )-admissible maps. Let ξ2 , ξ4 ∈ T∗ X
and denote the same for their respective lifting to T∗ (XT /T ). Then,
Tr hF∇T,(ϕT ,g) (∂s , ξ2 )∂t ϕT , DξT4 ϕT ig = − Tr h∂t ϕT , F∇T,(ϕT ,g) (∂s , ξ2 )Dξ4 ϕT ig
= − Tr hF∇T,(ϕT ,g) (∂s , ξ2 )DξT4 ϕT , ∂t ϕT ig = Tr hF∇T,(ϕT ,g) (ξ2 , ∂s )DξT4 ϕT , ∂t ϕT ig .
Let ξ be ξ2 or ξ4 . Since ∂s Comm (AϕT ) ⊂ Comm (AϕT ) and and ∂ξ AϕT ⊂ Comm (AϕT ), both
∂s DξT ϕT and ∂s ∂t ϕT lie in Comm (AϕT ) ⊗ϕ] ,OY T∗ Y . Locally explicitly,
∂s DξT ϕT =
X
∂s ∂t ϕT =
X
∂s DξT ϕ]T (y i ) ⊗
i
∂s ∂t ϕ]T (y i ) ⊗
i
X
∂
∂
+
DξT ϕ]T (y i ) ∂s ϕ]T (y j ) ⊗ ∇g ∂
i
i;
∂y
∂y
j
∂y
i,j
X
∂
∂
+
∂t ϕ]T (y i ) ∂s ϕ]T (y j ) ⊗ ∇g ∂
i
i.
∂y
∂y
j
∂y
i,j
Now follow the proof of Lemma 3.2.2.5, but under only the (∗1 )-Admissible Condition on
(ϕT , ∇T ), to convert Tr hF∇T,(ϕT ,g) (∂s , ξ2 )∂t ϕT , DξT4 ϕT ig to Tr h∂t ϕT , F∇T,(ϕT ,g) (∂s , ξ2 )Dξ4 ϕT ig .
Since Tr h – , –0 ig is defined as long as one of –, –0 is in Comm (AϕT ) ⊗ϕ] ,OY T∗ Y , one realizes
T
that all the terms that appear in the process via the Leibniz rule are defined except
T,(ϕT ,g)
− Tr h∇ξ2
T,(ϕT ,g)
∂t ϕT , ∂s DξT4 ϕT ig + Tr h∂s ∂t ϕT , ∇ξ2
DξT4 ϕT ig .
Under the additional (∗2 )-Admissible Condition on (ϕT , ∇T ) along T , both ∂s DξT4 ϕT and ∂s ∂t ϕT
now lie in Comm (AϕT ) ⊗ϕ] ,OY T∗ Y ; and the above two exceptional terms become defined.
The lemma follows.
6
The first variation of the enhanced kinetic term for
maps and ......
Let (ϕ, ∇) be a (∗1 )-admissible pair. Recall the setup in Sec. 5. Let T = (−ε, ε) ⊂ R1 , for some
ε > 0 small, and (ϕT , ∇T ) be a (∗1 )-admissible T -family of (∗1 )-admissible pairs that deforms
(ϕ, ∇) = (ϕT , ∇T )|t=0 . We derive in Sec. 6.1 and Sec. 6.2 the first variation formula of the newly
introduced enhanced kinetic term for maps
(ρ,h;Φ,g)
Smap:kinetic+ (ϕ, ∇)
Z
Z
1
Re Tr hDϕ , Dϕi(h,g) vol h +
Re Tr hdρ, ϕ dΦih vol h
:= Tm−1
2
X
X
in the standard action for D-branes. As the ‘taking the real part’ operation Re (· · · · · ·) is a
OX -linear operation and can always be added back in the end, we will consider
(ρ,h;Φ,g)
Smap:kinetic+ (ϕ, ∇)C :=
Z
Z
1
Tm−1
Tr hDϕ , Dϕi(h,g) vol h +
Tr hdρ, ϕ dΦih vol h
2
X
X
so that we don’t have to carry Re around.
The first variation of the gauge/Yang-Mills term is analogous to that in the ordinary YangMills theory and the first variation of the Chern-Simons/Wess-Zumino term is an update from
[L-Y8: Sec. 6] (D(13.1)). Both are given in Sec. 6.3 under the stronger (∗2 )-Admissible Condition.
35
6.1
The first variation of the kinetic term for maps
t
Recall the (complexified) kinetic energy E ∇ (ϕt )C of ϕt for a given ∇t , t ∈ T := (−ε, ε),
t
(h;g)
E ∇ (ϕt )C := Smap:kinetic (ϕt , ∇t )C :=
Z
1
Tm−1
Tr hDt ϕt , Dt ϕt i(h,g) vol h .
2
X
As t varies, with a slight abuse of notation, denote the resulting function of t by
T
(h;g)
E ∇ (ϕT )C := Smap:kinetic (ϕT , ∇T )C :=
Z
1
Tm−1
Tr hDT ϕT , DT ϕT i(h,g) vol h ,
2
X
with the understanding that all expressions are taken on Xt with t varying in T .
Let U ⊂ X be an open set with an orthonormal frame (eµ )µ=1, ··· , m . Let (eµ )µ=1, ··· , m be the
dual co-frame. Assume that U is small enough so that ϕT (UTAz ) is contained in a coordinate chart
of Y , with coordinates (y 1 , · · · , y n ). Then, over U ,
d ∇T
E (ϕT )C
dt
1
Tm−1
∂t Tr hDT ϕT , DT ϕT i(h,g) vol h
2
U
Z
1
Tm−1
Tr ∂t hDT ϕT , DT ϕT i(h,g) vol h
2
U
Z
=
=
=
1
Tm−1
2
Z
Tr ∂t
U
Z
= Tm−1
Tr
U
Tr
U
hDeTµ ϕT , DeTµ ϕT ig vol h
µ=1
m
X
h∂t DeTµ ϕT , DeTµ ϕT ig vol h
µ=1
Z
= Tm−1
m
X
X
T ,g)
h∇eT,(ϕ
∂t ϕT , DeTµ ϕT ig vol h
µ
µ
Z
Tr
+ Tm−1
U
X
µ
+ Tm−1
h(ad ⊗∇g )DeT
µ ϕT
Z X X
n
h
U µ
∂t ϕT , DeTµ ϕT ig vol h
[(∂t ∇T )(eµ ), ϕ]T (y i )] ⊗
∂
∂y i
, DeTµ ϕT ig vol h
i=1
= (I.1) + (I.2) + (I.3) .
(I.1) = Tm−1
Z X
U µ
= Tm−1
Z X
U µ
T ,g)
Tr DeTµ h∂t ϕT , DeTµ ϕT ig − h∂t ϕT , ∇T,(ϕ
DeTµ ϕT ig vol h
eµ
eµ Tr h∂t ϕT , DeTµ ϕT ig vol h + Tm−1
Z
Tr h∂t ϕT , −
U
T,(ϕT ,g) T
Deµ ϕT ig
µ ∇eµ
P
vol h
= (I.1.1) + (I.1.2) .
Summand (I.1.1) suggests a boundary term. To really extract the boundary term from it,
consider the T -family of C-valued 1-forms on U
T
T
α(I,
∂t ϕT ) := Tr h∂t ϕT , D ϕT ig ,
which depends C ∞ (U )C -linearly on ∂t ϕT . Let
T
ξ(I,
∂t ϕT ) :=
m X
Tr h∂t ϕT , DeTµ ϕT ig eµ
µ=1
36
T
be the T -family of dual C-valued vector fields of α(I,
∂t ϕT ) on U with respect to the metric h.
T
∞
C
Note that ξ(I, ∂t ϕT ) depends C (U ) -linearly on ∂t ϕT as well. Then
(I.1.1) = Tm−1
Z X
U
Z X
= Tm−1
U
µ
T
eµ hξ(I,
∂t ϕT ) , eµ ih vol h
µ
T
h∇heµ ξ(I,
∂t ϕT ) , eµ ih vol h + Tm−1
Z
U
T
hξ(I,
∂ t ϕT ) ,
P
µ
∇heµ eµ ih vol h .
The first term is equal to
Tm−1
Z
(− div
U
T
ξ(I,
∂t ϕT ) ) vol h
= Tm−1
Z
U
d iξT
(I, ∂t ϕT )
vol h = Tm−1
Z
∂U
iξT
(I, ∂t ϕT )
vol h ,
which is the sought-for boundary term, whose integrand satisfies the requirement that it be
C ∞ (U )C -linear on ∂t ϕT . The second term is equal to
Tm−1
Z
T
Tr h∂t ϕT , DP
µ
U
ϕT ig
∇h
eµ e µ
vol h
by construction, which is C ∞ (U )C -linear in ∂t ϕT and hence in a final form.
The integrand of Summand (I.1.2) is already C ∞ (U )C -linear in ∂t ϕT and hence in a final
form.
Summand (I.2) can be re-written as
Z
(I.2) = − Tm−1
Tr
h(ad ⊗∇g )∂t ϕT DeTµ ϕT , DeTµ ϕT ig vol h .
X
U
µ
Thus, its integrand is already C ∞ (U )C -linear in ∂t ϕT and hence in a final form.
Finally, since the built-in inclusion OUC ⊂ OUAz identifies OUC with the center of OUAz , Summand
(I.3) is C ∞ (U )C -linear and hence in its final fom.
Altogether, we almost complete the calculation except the issue of whether all the inner
products Tr h , ig that appear in the procedure are truly defined. For this, one notices that
wherever such an inner product appears above, at least one of its arguments is either ∂t ϕT or
DeTµ ϕT , for some µ. It follows from Lemma 3.2.2.4 that they are indeed defined.
In summary,
· ·
Proposition 6.1.1 [first variation of kinetic term for maps]
admissible T -family of (∗1 )-admissible pairs. Then,
Z
d 1
d ∇T
E (ϕT )C =
Tm−1 Tr hDT ϕT , DT ϕT i(h,g) vol h
dt
dt 2
U
= Tm−1
Z
∂U
+ Tm−1
iξT
(I, ∂t ϕT )
Z
U
+ Tm−1
∇h
eµ e µ
µ=1
Z
vol h
T
m
Tr h∂t ϕT , (DP
− Tm−1
Let (ϕT , ∇T ) be a (∗1 )-
Tr
U
Z X
m X
n
h
U µ=1 i=1
m
X
−
Pm
T,(ϕT ,g) T
Deµ ) ϕT ig
µ=1 ∇eµ
vol h
h(ad ⊗∇g )∂t ϕT DeTµ ϕT , DeTµ ϕT ig vol h
µ=1
[(∂t ∇T )(eµ ), ϕ]T (y i )] ⊗
∂
∂y i
, DeTµ ϕT ig vol h .
m
T
T
Here, the first summand is the boundary term with ξ(I,
µ=1 (Tr h∂t ϕT , Deµ ϕT ig )eµ
∂t ϕT ) :=
C ∞ (U )C -linear in ∂t ϕT ; the integrand of the second and the third terms are C ∞ (U )C -linear in
P
37
∂t ϕT and their real part contribute first-order and second-order terms to the equations of motion
for (ϕ, ∇); the integrand of the last term is C ∞ (U )C -linear in ∂t ∇T and its real part contributes
terms, first order in ϕ but zeroth order in the connection 1-from of ∇, to the equations of mo(ρ,h;Φ,g,B,C)
tion for (ϕ, ∇) in addition to those from the first variation of the rest part of Sstandard
(ϕ, ∇).
These lower-order terms contribute to the equations of motion for (ϕ, ∇) but do not change the
signature of the system.
Remark 6.1.2 [for (∗2 )-admissible T -family of (∗2 )-admissible pairs] If furtheremore (ϕ, ∇) is
(∗2 )-admissible and (ϕT , ∇T ) is a (∗2 )-admissible T -family of (∗2 )-admissible pairs that deforms
(ϕ, ∇), then the third summand of the first variation formula in Proposition 6.1.1 vanishes and
the fourth/last summand has a Y -coordinate-free form
Tm−1
Z X
m
U µ=1
had (∂t ∇T )(eµ ) ϕT , DeTµ ϕT ig vol h .
In this case, the first variation with respect to ϕ alone (i.e. setting ∂t ∇T = 0), cf. the first two
summands, takes the form of a direct formal generalization of the first variation formula in the
study of harmonic maps; e.g. [E-L], [E-S], [Ma], [Sm].
6.2
The first variation of the dilaton term
(ρ,h;Φ,g,B,C)
(ϕ, ∇)C .
We now turn to the (complexified) dilaton term in Sstandard
Let ϕT : (XAz , ET ; ∇T ) → Y be a (∗1 )-admissible T -family of (∗1 )-admissible pairs. Then,
over an open set U ⊂ X,
(ρ,h;Φ)
Sdilaton (ϕT )C =
=
Z
U
Z
Tr hdρ , ϕT dΦih vol h
Tr
U
=
m
X
Z
=
Tr
U
Z
Tr
U
m
X
∂Φ i=1
X
Tr
!
vol h
∂y i
dρ(eµ ) ((DeTµ ϕT )Φ)
vol h .
µ
dρ(eµ ) ∂t (DeTµ ϕT )Φ vol h
µ=1
X
DeTµ ϕ]T (y i ) ϕ]T
!
Z
U
d (ρ,h;Φ)
S
(ϕT )C =
dt dilaton
dρ(eµ )
µ=1
n
X
dρ(eµ ) (∂t DeTµ ϕT )Φ vol h
µ
Z
+
Tr
U
X
n
X
dρ(eµ )
µ
DeTµ ϕ]T (y i )∂t ϕ]T (y j )
⊗
∂ ∂
Φ
∂y j ∂y i
− ∇
i,j=1
g
∂
∂y j
∂
∂y i
!
Φ vol h
= (II.1) + (II.2) .
The integrand of Summand (II.2) is C ∞ (U )C -linear in ∂t ϕT and hence in a final form.
Z
(II.1) =
Tr
U
X
T , g)
dρ(eµ ) (∇eT,(ϕ
∂t ϕT )Φ vol h
µ
µ
−
Z
Tr
U
X
dρ(eµ ) ((ad ⊗ ∇g )∂t ϕT DeTµ ϕT )Φ vol h
µ
Z
+
Tr
U
X
dρ(eµ )
µ
X
i
= (II.1.1) + (II.1.2) + (II.1.3) .
38
[(∂t ∇T )(eµ ) , ϕ]T (y i )] ⊗
∂
∂y i
Φ vol h
Both Summand (II.1.2) and Summand (II.1.3) vanish since
Tr ([a, b]c) = 0
if [b, c] = 0
for r × r matrices a, b, c.
Z
(II.1.1) =
Tr
X
U
dρ(eµ ) DeTµ ((∂t ϕT )Φ)) vol h
µ
−
Z
Tr
U
X
dρ(eµ )
µ
n
X
∂t ϕ]T (y i ) DeTµ ϕ]T (y j )
⊗
∂ ∂
Φ
∂y i ∂y j
−
i,j=1
∇
g
∂
∂y i
∂
∂y j
!
Φ vol h
= (II.1.1.1) + (II.1.1.2) .
The integrand of Summand (II.1.1.2) is C ∞ (U )C -linear in ∂t ϕT and hence in a final form. It can
be combined with Summand (II.2) to give
(II.1.1.2) + (II.2)
Z
= −
Tr
U
X
dρ(eµ )
µ
n
X
[∂t ϕ]T (y i ) ,
i,j=1
DeTµ ϕ]T (y j )]
∂ ∂
Φ
∂y i ∂y j
⊗
−
∇
g
∂
∂y i
∂
∂y j
!
Φ vol h ,
which again vanishes due to Tr .
(II.1.1.1) =
Z X
U µ
=
Z X
U µ
=
Z X
U µ
dρ(eµ )Tr DeTµ ((∂t ϕT )Φ) vol h
dρ(eµ ) eµ Tr ((∂t ϕT )Φ) vol h
eµ dρ(eµ ) Tr ((∂t ϕT )Φ) vol h −
Z X
U
eµ dρ(eµ ) Tr ((∂t ϕT )Φ) vol h
µ
= (II.1.1.1.1) + (II.1.1.1.2)
The integrand of Summand (II.1.1.1.2) is C ∞ (U )C -linear in ∂t ϕT and hence in a final form. To
extract the boundary term from Summand (II.1.1.1.1), consider the T -family of C-valued 1-forms
on U
T
α(II,∂
:= dρ Tr ((∂t ϕT )Φ) ,
t ϕT )
which depends C ∞ (U )C -linearly on ∂t ϕT . Let
T
ξ(II,∂
=
t ϕT )
m X
dρ(eµ ) Tr ((∂t ϕT )Φ) eµ
µ=1
T
be the T-family of dual C-valued vector fields of α(II,∂
on U with respect to the metric h.
t ϕT )
T
∞
C
Note that ξ(II,∂t ϕT ) depends C (U ) -linearly on ∂t ϕT as well. Then
(II.1.1.1.1) =
Z X
U µ
=
Z X
U µ
T
eµ hξ(II,∂
, eµ ih vol h
t ϕT )
T
h∇heµ ξ(II,∂
, eµ ih vol h +
t ϕT )
Z
U
T
hξ(II,∂
,
t ϕT )
h
µ ∇eµ eµ ih vol h
P
The first term is equal to
Z
U
(− div
T
ξ(II,
∂t ϕT ) ) vol h
=
Z
U
d iξT
(II, ∂t ϕT )
39
vol h =
Z
∂U
iξT
(II, ∂t ϕT )
vol h ,
which is the sought-for boundary term, whose integrand satisfies the requirement that it be
C ∞ (U )C -linear in ∂t ϕT . The second term is equal to
Z
dρ(
h
µ ∇eµ eµ ) Tr ((∂t ϕT )Φ) vol h
P
U
by construction, which is C ∞ (U )C -linear in ∂t ϕT and hence in a final form.
In summary,
Proposition 6.2.1 [first variation of dilaton term] Let (ϕT , ∇T ) be a (∗1 )-admissible T family of (∗1 )-admissible pairs. Then,
d (ρ,h;Φ)
d
Sdilaton (ϕT )C =
dt
dt
Z
U
Tr hdρ , ϕT dΦih vol h
Z
=
∂U
+
iξT
(II, ∂t ϕT )
Z dρ(
vol h
Pm
h
µ=1 ∇eµ eµ ) −
U
Pm
µ=1 eµ dρ(eµ ) Tr ((∂t ϕT )Φ) vol h .
T
Here, the first summand is the boundary term with ξ(II,∂
:= m
µ=1 (dρ(eµ ) Tr ((∂t ϕT )Φ))eµ
t ϕT )
∞
C
∞
C (U ) -linear in ∂t ϕT ; the integrand of the second summand C (U )C -linear in ∂t ϕT and they
contribute additional zeroth-order terms to the equations of motion for (ϕ, ∇). In particular,
while the dilaton term of the standard action modifies the equations of motion for (ϕ, ∇), it does
not change the signature of the system.
P
6.3
The first variation of the gauge/Yang-Mills term and the ChernSimons/ Wess-Zumino term
Az
To make sure that differential forms on Y of rank ≥ 2 are pull-pushed to (OX
-valued-)differential
Az
forms on X (cf. Lemma 2.1.11), we assume in this subsection that ϕT : (XT , ET ; ∇T ) → Y is a
(∗2 )-family of (∗2 )-admissible maps. (Note that as the gauge/Yang-Mills term is defined through
a norm-squared, (∗1 )-admissible family of (∗1 )-admissible (ϕT , ∇T ) is enough for the derivation of
the first variation formula of the gauge/Yang-Mills term but the result will be slightly messier.)
6.3.1
The first variation of the gauge/Yang-Mills term
Let (e1 , · · · , em ) be an orthonormal frame on U . Then, over U ,
1
(h;B)
Sgauge /YM (ϕT , ∇T )C := − 2 Tr k2πα0 F∇T + ϕT Bk2h vol h
U
Z
2
X
1
= − 2 Tr
(2πα0 F∇T + ϕT B)(eµ , eν ) vol h .
U
µ,ν
Z
Applying the following basic identities:
∂t F∇T (eµ , eν ) = DeTµ ((∂t ∇T )(eν )) − DeTν ((∂t ∇T )(eµ )) − (∂t ∇T )([eµ , eν ]) ,
∂t ((ϕT B)(eµ , eν )) =
X
∂t (ϕ]T (Bij ))DeTµ ϕ]T (y i )DeTν ϕ]T (y j )
i,j
+
X ]
ϕT (Bij ) DeTµ ∂t ϕ]T (y i ) + [(∂t ∇T )(eµ ), ϕ]T (y i )] DeTν ϕ]T (y j )
i,j
+
X ]
ϕT (Bij )DeTµ ϕ]T (y i ) DeTν ∂t ϕ]T (y j ) + [(∂t ∇T )(eν ), ϕ]T (y j )] .
i,j
40
and proceeding similarly to Sec. 6.1, one has the following results.
d (h;B)
1
S
(ϕT , ∇T )C = −
2
dt gauge /YM
= −
Z
Tr
X
Tr
X
U
= −
Z
Tr ∂t
U
X
2
(2πα0 F∇T + ϕT B)(eµ , eν ) vol h
µ,ν
∂t (2πα0 F∇T + ϕT B)(eµ , eν ) · (2πα0 F∇T + ϕT B)(eµ , eν ) vol h
µ,ν
Z
U
2πα0 ∂t (F∇T (eµ , eν )) · (2πα0 F∇T + ϕT B)(eµ , eν ) vol h
µ,ν
Z
−
Tr
U
X
∂t (ϕT B(eµ , eν )) · (2πα0 F∇T + ϕT B)(eµ , eν ) vol h
µ,ν
= (III.1) + (III.2) .
(III.1) :=
−
Z
X
Tr
U
=
2πα0 ∂t (F∇T (eµ , eν )) · (2πα0 F∇T + ϕT B)(eµ , eν ) vol h
µ,ν
− 2πα0
Z
X
Tr
U
DeTµ ((∂t ∇T )(eν )) − DeTν ((∂t ∇T )(eµ )) − (∂t ∇T )([eµ , eν ])
µ,ν
· (2πα0 F∇T + ϕT B)(eµ , eν ) vol h
=
− 4πα0
Z
∂U
− 4πα0
iξT
(III,∂t ∇T )
Z
Tr
vol h
X
(∂t ∇T )(eν ) ·
U
(2πα0 F∇T + ϕT B)(
h
µ ∇eµ eµ , eν )
P
ν
−
X
DeTµ ((2πα0 F∇T + ϕT B)(eµ , eν ))
µ
1X ν
e ([eµ , eλ ])(2πα0 F∇T + ϕT B)(eµ , eλ ) vol h .
−2
µ,λ
Here,
T
ξ(III,∂
:=
T
t∇ )
Tr (∂t ∇T )(eν ) · (2πα0 F∇T + ϕT B)(eµ , eν ) eµ
X
∈ T∗ (UT /T )C
µ,ν
is OUC -linear in ∂t ∇T ; and the second summand contributes to the equations of motion for (ϕ, ∇).
The latter are standard terms from non-Abelian Yang-Mills theory with additional terms from
ϕ B.
(III.2) :=
−
Z
Tr
U
=
−
∂t (ϕT B(eµ , eν )) · (2πα0 F∇T + ϕT B)(eµ , eν ) vol h
µ,ν
Z
Tr
U
X
XX
µ,ν
∂t (ϕ]T (Bij ))DeTµ ϕ]T (y i )DeTν ϕ]T (y j )
i,j
+
X ]
+
X ]
ϕT (Bij ) DeTµ ∂t ϕ]T (y i ) + [(∂t ∇T )(eµ ), ϕ]T (y i )] DeTν ϕ]T (y j )
i,j
ϕT (Bij )DeTµ ϕ]T (y i )
DeTν ∂t ϕ]T (y j )
+ [(∂t ∇
T
i,j
· (2πα0 F∇T + ϕT B)(eµ , eν ) vol h
=
(III.2.1) + + ((III.2.2.1) + (III.2.2.2)) + (III.2.3.1) + (III.2.3.2)
in the order of the appearance of the five summands after the expansion.
(III.2.1) :=
−
Z
Tr
U
XX
∂t (ϕ]T (Bij ))DeTµ ϕ]T (y i )DeTν ϕ]T (y j )
µ,ν i,j
41
)(eν ), ϕ]T (y j )]
· (2πα0 F∇T + ϕT B)(eµ , eν ) vol h
Z
−
:=
XX
Tr
U
((∂t ϕT )Bij )DeTµ ϕ]T (y i )DeTν ϕ]T (y j )
µ,ν i,j
· (2πα0 F∇T + ϕT B)(eµ , eν ) vol h
has an integrand OUC -linear in ∂t ϕT and hence in a final form.
(III.2.2.1) + (III.2.3.1)
:= −
Z
XX ]
ϕT (Bij )DeTµ ∂t ϕ]T (y i )DeTν ϕ]T (y j )
Tr
U
µ,ν
i,j
X ]
ϕT (Bij )DeTµ ϕ]T (y i )DeTν ∂t ϕ]T (y j )
+
i,j
· (2πα F∇T + ϕT B)(eµ , eν ) vol h
Z
= −2
XX
Tr
U
0
DeTµ ∂t ϕ]T (y i ) ϕ]T (Bij ) DeTν ϕ]T (y j )
µ,ν i,j
·((2πα0 F∇T + ϕT B)(eµ , eν ))vol h
Z
= −2
iξT
(III,∂t ϕT )
Z∂U
−2
XX
Tr
U
vol h
∂t ϕ]T (y i )
ν i,j
ϕ]T (Bij ) DeTν ϕ]T (y j ) · ((2πα0 F∇T + ϕT B)(
h
µ ∇eµ eµ , eν ))
P
−
X
DeTµ
ϕ]T (Bij ) DeTν ϕ]T (y j )
0
· ((2πα F∇T +
ϕT B)(eµ , eν ))
vol h .
µ
Here,
T
ξ(III,∂
:=
t ϕT )
XXX
µ
ν
∂t ϕ]T (y i ) ϕ]T (Bij ) DeTν ϕ]T (y j ) · ((2πα0 F∇T + ϕT B)(eµ , eν )) eµ
i,j
in T∗ (UT /T )C is OUC -linear in ∂t ϕT ; and the second summand contributes to
(ρ,h;Φ,g,B,C)
(ϕ, ∇)/δϕ-part of the equations of motion for (ϕ, ∇).
δSstandard
(III.2.2.2) + (III.2.3.2)
:= −
Z
Tr
U
XX ]
ϕT (Bij )[(∂t ∇T )(eµ ), ϕ]T (y i )]DeTν ϕ]T (y j )
µ,ν
i,j
+
ϕT (Bij )DeTµ ϕ]T (y i )[(∂t ∇T )(eν ), ϕ]T (y j )]
X ]
i,j
· (2πα0 F∇T + ϕT B)(eµ , eν ) vol h .
has an integrand OUC -linear in ∂t ∇T and hence in a final form.
In summary,
Proposition 6.3.1.1 [first variation of gauge/Yang-Mills term] Let (ϕT , ∇T ) be a (∗2 )admissible family of (∗2 )-admissible pairs. Then
d (h;B)
1 d
T C
dt Sgauge /YM (ϕT , ∇ ) = − 2 dt
= − 4πα
0
− 4πα0
Z
Z∂U
iξT
(III,∂t ∇T )
Tr
U
X
Z
vol h − 2
(∂t ∇T )(eν ) ·
U
Z
Tr k2πα0 F∇T + ϕT Bk2h vol h
∂U
iξ T
(III,∂t ϕT )
vol h
(2πα0 F∇T + ϕT B)(
h
µ ∇eµ eµ , eν )
P
ν
42
−
X
DeTµ ((2πα0 F∇T + ϕT B)(eµ , eν ))
µ
1X ν
e ([eµ , eλ ])(2πα0 F∇T + ϕT B)(eµ , eλ ) vol h .
−2
µ,λ
−
Z
Tr
U
XX ]
ϕT (Bij )[(∂t ∇T )(eµ ), ϕ]T (y i )]DeTν ϕ]T (y j )
µ,ν
i,j
+
X ]
ϕT (Bij )DeTµ ϕ]T (y i )[(∂t ∇T )(eν ), ϕ]T (y j )]
i,j
· (2πα F∇T + ϕT B)(eµ , eν ) vol h
−
Z
Tr
U
XX
0
((∂t ϕT )Bij )DeTµ ϕ]T (y i )DeTν ϕ]T (y j )
µ,ν i,j
· (2πα0 F∇T + ϕT B)(eµ , eν ) vol h
−2
Z
Tr
U
XX
ν
∂t ϕ]T (y i )
i,j
ϕ]T (Bij ) DeTν ϕ]T (y j ) · ((2πα0 F∇T + ϕT B)(
h
µ ∇eµ eµ , eν ))
P
−
X
DeTµ
ϕ]T (Bij ) DeTν ϕ]T (y j )
0
· ((2πα F∇T +
ϕT B)(eµ , eν ))
vol h .
µ
Here,
Tr (∂t ∇T )(eν ) · (2πα0 F∇T + ϕT B)(eµ , eν ) eµ ,
T
ξ(III,∂
:=
T
t∇ )
X
T
ξ(III,∂
:=
t ϕT )
XXX
µ,ν
µ
ν
∂t ϕ]T (y i ) ϕ]T (Bij ) DeTν ϕ]T (y j ) · ((2πα0 F∇T + ϕT B)(eµ , eν )) eµ
i,j
in T∗ (UT /T )C , with the first OUC -linear in ∂t ∇T and the second OUC -linear in ∂t ϕT .
6.3.2
The first variation of the Chern-Simons/Wess-Zumino term for lower dimensional D-branes
This is an update of [L-Y8: Sec.6.2] (D(13.1)) in the current setting. Let ϕT : (XAz , ET ; ∇T ) →
Y be an (∗2 )-family of (∗2 )-admissible maps. We work out the first variation of the Chern(C,B)
Simons/Wess-Zumino term SCS/WZ (ϕ, ∇) for the cases where m := dim X = 0, 1, 2, 3. As the
details involve no identities or techniques that have not yet been used in Sec. 6.1, Sec. 6.2, and/or
Sec. 6.3.1, we only summarize the final results below.
6.3.2.1
D(−1)-brane world-point (m = 0)
(C
)
(0)
For a D(−1)-brane world-point, dim X = 0, ∇ = 0, and SCS/WZ
(ϕT ) = T−1 · Tr (ϕ]T (C(0) )). It
follows that
d (C(0) )
]
dt SCS/WZ (ϕT ) = T−1 Tr ∂t (ϕT (C(0) )) = T−1 Tr ((∂t ϕT )C(0) ) .
6.3.2.2
D-particle world-line (m = 1)
43
For a D-particle world-line, dim X = 1. Let e1 be the orthonormal frame on an open set U ⊂ X;
e1 its dual co-frame. Then, over U ,
(C(1) )
SCS/WZ (ϕT )
Z
C
= T0
U
Tr ϕT C(1)
Z
= T0
Tr
U
n
X
ϕ]T (Ci ) · DeT1 ϕ]T (y i ) e1 .
i=1
It follows that
X
d (C(1) )
∂t ϕ]T (y i )ϕ]T (Ci ) |∂U
SCS/WZ (ϕ) = T0 Tr
dt
i
Z
− T0
Tr
U
6.3.2.3
X
∂t ϕ]T (y i )DeT1 ϕ]T (Ci )
1
Z
e + T0
i
Tr
X
U
De1 ϕ]T (y i ) · (∂t ϕT )Ci e1 .
i
D-string world-sheet (m = 2)
Denote
C̆(2) := C(2) + C(0) B =
1
(Cij + C(0) Bij ) dy i ⊗ dy j =
X
X
ij
i,j
C̆ij dy i ⊗ dy j
n
in a local coordinate (y , · · · , y ) of Y . For a D-string world-sheet, dim X = 2. Let (e1 , e2 ) be
an orthonormal frame on an open set U ⊂ X; (e1 , e2 ) its dual co-frame. Then, over U ,
(C
,C(2) ,B)
(0)
SCS/WZ
(ϕT , ∇T )C
n
X
Z
= T1
Tr
U
ϕ]T (C̆ij ) DeT1 ϕ]T (y i ) DeT2 ϕ]T (y j )
i,j=1
n
X
Z
Tr
= T1
U
+ πα0 ϕ]T (C(0) ) F∇T (e1 , e2 ) + πα0 F∇T (e1 , e2 ) ϕ]T (C(0) ) e1 ∧ e2
ϕ]T (C̆ij ) DeT1 ϕ]T (y i ) DeT2 ϕ]T (y j ) + 2 πα0 ϕ]T (C(0) ) F∇T (e1 , e2 ) e1 ∧ e2 .
i,j=1
It follows that
d (C(0) ,C(2) ,B)
(ϕT , ∇T )C
dt SCS/WZ
Z
n
X
= T1
Tr ∂t
ϕ]T (C̆ij ) DeT1 ϕ]T (y i ) DeT2 ϕ]T (y j ) + 2 πα0 ϕ]T (C(0) ) F∇T (e1 , e2 ) e1 ∧ e2
U
i,j=1
Z
= T1
1
(IV,∂t ϕT )
Z
+ T1
Tr
Z
iξT
∂U
X
n
U
0
(e ∧ e ) + 2πα T1
iξT
∂U
2
(IV,∂t ∇T )
(e1 ∧ e2 )
∂t ϕ] (y i ) DeT2 ϕ]T (y j )ϕ]T (C̆ij )e1 − DeT1 ϕ]T (y j )ϕ]T (C̆ij )e2 (∇he1 e1 + ∇he2 e2 ) e1 ∧ e2
i,j=1
Z Z
− T1
Tr
X
U
i,j
Z
+ T1
1
∂t ϕ]T (y i ) DeT1 DeT2 ϕ]T (y j ) · ϕ]T (C̆ij ) − DeT2 DeT1 ϕ]T (y j ) · ϕ]T (C̆ij )
e ∧ e2
Tr
X
U
∂t ϕ]T (C̆ij )DeT1 ϕ]T (y i )DeT2 ϕ]T (y j ) e1 ∧ e2
i,j
Z
+ 2πα0 T1
Tr ∂t ϕ]T (C(0) ) · F∇T (e1 , e2 ) e1 ∧ e2
U
Z
h
]
0
T
1
T
2
h
T
+ 2πα T1
Tr ϕT (C(0) ) (∂t ∇ )(e2 ) e − (∂t ∇ )(e1 ) e (∇e1 e1 + ∇e2 e2 ) − (∂t ∇ )([e1 , e2 ])
e1 ∧ e2
U
Z
0
− 2πα T1
Tr DeT1 ϕ]T (C(0) ) · (∂t ∇T )(e2 ) − DeT2 ϕ]T (C(0) ) · (∂t ∇T )(e1 ) e1 ∧ e2 .
U
Here,
P
]
]
]
]
i
T ]
j
i
T ]
j
i,j ∂t ϕT (y )De2 ϕT (y )ϕT (C̆ij )) e1 − Tr ( i,j ∂t ϕT (y )De1 ϕT (y )ϕT (C̆ij )) e2 ,
Tr (ϕ]T (C(0) ) · (∂t ∇T )(e2 )) e1 − Tr (ϕ]T (C(0) ) · (∂t ∇T )(e1 )) e2
T
ξ(IV,∂
:= Tr (
t ϕT )
P
T
ξ(IV,∂
:=
T
t∇ )
in T∗ (UT /T )C , with the first OUC -linear in ∂t ϕT and the second OUC -linear in ∂t ∇T .
44
6.3.2.4
D-membrane world-volume (m = 3)
Denote
C(3) + C(1) ∧ B
C̆(3) :=
(Cijk + Ci Bjk + Cj Bki + Ck Bij ) dy i ⊗ dy j ⊗ dy k =
X
=
i,j,k
X
C̆ijk dy i ⊗ dy j ⊗ dy k
i,j,k
in a local coordinate (y 1 , · · · , y n ) of Y . For D-membrane world-volume, dim X = 3. Let
(e1 , e2 , e3 ) be an orthonormal frame on an open set U ⊂ X; (e1 , e2 , e3 ) its dual co-frame. Then,
over U ,
(C
,C(3) ,B)
(1)
SCS/WZ
(ϕT , ∇T )C
Z
= T2
Tr
n
X
U
ϕ] (C̆ijk ) DeT1 ϕ]T (y i ) DeT2 ϕ]T (y j ) DeT3 ϕ]T (y k )
i,j,k=1
+ 2πα
0
n
X
X
(−1)(λµν) (ϕ]T (Ci ) DλT ϕ]T (y i ) F∇T (eµ , eν ) ) e1 ∧ e2 ∧ e3
(λµν)∈Sym3 i=1
It follows that
d (C(1) ,C(3) ,B)
(ϕT , ∇T )C
dt SCS/WZ
Z
n
X
= T2
Tr ∂t
ϕ] (C̆ijk ) DeT1 ϕ]T (y i ) DeT2 ϕ]T (y j ) DeT3 ϕ]T (y k )
U
i,j,k=1
n
X
X
+ 2πα0
(−1)(λµν) ϕ]T (Ci ) DeTλ ϕ]T (y i ) F∇T (eµ , eν )
e1 ∧ e2 ∧ e3
(λµν)∈Sym3 i=1
Z
= T2
1
3
Z
(IV,∂t ϕT ;C̆(3) )
∂U
(IV,∂t ∇
∂U
+ T2
0
(e ∧ e ∧ e ) + 4πα T2
iξT
(e1 ∧ e2 ∧ e3 )
(IV,∂t ϕT ;C(1) )
∂U
Z
+ 2πα0 T2
iξT
(e1 ∧ e2 ∧ e3 )
T
iξT
2
Z Tr
U
X
)
ϕ]T (C̆ijk )∂t ϕ]T (y i )DeT2 ϕ]T (y j )DeT3 ϕ]T (y k ) e1
i,j,k
X
− Tr
ϕ]T (C̆ijk )∂t ϕ]T (y i )DeT1 ϕ]T (y j )DeT3 ϕ]T (y k ) e2
i,j,k
− Tr
X
ϕ]T (C̆ijk )∂t ϕ]T (y i )DeT2 ϕ]T (y j )DeT1 ϕ]T (y k )
e
3
P3
( µ=1 ∇heµ eµ ) e1 ∧ e2 ∧ e3
i,j,k
Z
− T2
X
Tr
U
∂t ϕ]T (y i ) DeT1 ϕ]T (C̆ijk )DeT2 ϕ]T (y j )DeT3 ϕ]T (y k ) − DeT2 ϕ]T (C̆ijk )DeT1 ϕ]T (y j )DeT3 ϕ]T (y k )
i,j,k
− DeT3 ϕ]T (C̆ijk )DeT2 ϕ]T (y j )DeT1 ϕ]T (y k ) e1 ∧ e2 ∧ e3
Z
+ T2
X
Tr
U
+ 4πα0 T2
∂t ϕ]T (C̆ijk )DeT1 ϕ]T (y i )DeT2 ϕ]T (y j )DeT3 ϕ]T (y k ) e1 ∧ e2 ∧ e3
i,j,k
Z Tr
X
U
ϕ]T (Ci )∂t ϕ]T (y i )F∇T (e2 , e3 ) e1
i
X
− Tr
ϕ]T (Ci )∂t ϕ]T (y i )F∇T (e1 , e3 ) e2
i
+ Tr
X
P
3
ϕ]T (Ci )∂t ϕ]T (y i )F∇T (e1 , e2 ) e3 ( µ=1 ∇heµ eµ ) e1 ∧ e2 ∧ e3
i
− 4πα0 T2
Z
Tr
U
X
∂t ϕ]T (y i ) DeT1 ϕ]T (Ci )F∇T (e2 , e3 ) − DeT2 ϕ]T (Ci )F∇T (e1 , e3 )
i
+ DeT3 ϕ]T (Ci )F∇T (e1 , e2 ) e1 ∧ e2 ∧ e3
45
+ 2πα0 T2
Z
U
+ 2πα0 T2
X
X
(λµν)∈Sym3
i
Tr
Z Tr
X
U
(−1)(λµν) ∂t ϕ]T (Ci ) DeTλ ϕ]T (y i ) F∇T (eµ , eν ) e1 ∧ e2 ∧ e3
ϕ]T (Ci ) DeT3 ϕ]T (y i )(∂t ∇T )(e2 ) − DeT2 ϕ]T (y i )(∂t ∇T )(e3 ) e1
i
+ Tr
X
X
ϕ]T (Ci ) DeT3 ϕ]T (y i )(∂t ∇T )(e1 ) + DeT1 ϕ]T (y i )(∂t ∇T )(e3 ) e2
i
+ Tr
P
3
ϕ]T (Ci ) DeT1 ϕ]T (y i )(∂t ∇T )(e2 ) − DeT2 ϕ]T (y i )(∂t ∇T )(e1 ) e3 ( µ=1 ∇heµ eµ )
i
+ 2πα0 T2
Z
X
X
(λµν)∈Sym3
i
Tr
U
(−1)(λµν)
e1 ∧ e2 ∧ e3
ϕ]T (Ci ) (∂t ∇T )(eλ ), ϕ]T (y i ) F∇T (eµ , eν )
− ϕ]T (Ci ) DeTλ ϕ]T (y i ) (∂t ϕT )([eµ , eν ]) e1 ∧ e2 ∧ e3 .
Here,
T
ξ(IV,∂
:=
t ϕT ;C̆(3) )
Tr
X
ϕ]T (C̆ijk )∂t ϕ]T (y i )DeT2 ϕ]T (y j )DeT3 ϕ]T (y k ) e1
i,j,k
X
− Tr
ϕ]T (C̆ijk )∂t ϕ]T (y i )DeT1 ϕ]T (y j )DeT3 ϕ]T (y k ) e2
i,j,k
− Tr
X
ϕ]T (C̆ijk )∂t ϕ]T (y i )DeT2 ϕ]T (y j )DeT1 ϕ]T (y k ) e3 ,
i,j,k
T
ξ(IV,∂
t ϕT ;C(1) )
:=
Tr
X
ϕ]T (Ci )∂t ϕ]T (y i )F∇T (e2 , e3 )
e1
i
X
X
ϕ]T (Ci )∂t ϕ]T (y i )F∇T (e1 , e2 ) e3 ,
− Tr
ϕ]T (Ci )∂t ϕ]T (y i )F∇T (e1 , e3 ) e2 + Tr
i
i
T
ξ(IV,∂
T
t∇ )
:=
Tr
X
ϕ]T (Ci )
DeT3 ϕ]T (y i )(∂t ∇T )(e2 )
−
DeT2 ϕ]T (y i )(∂t ∇T )(e3 )
e1
i
X
+ Tr
ϕ]T (Ci ) DeT3 ϕ]T (y i )(∂t ∇T )(e1 ) + DeT1 ϕ]T (y i )(∂t ∇T )(e3 ) e2
i
+ Tr
X
ϕ]T (Ci ) DeT1 ϕ]T (y i )(∂t ∇T )(e2 ) − DeT2 ϕ]T (y i )(∂t ∇T )(e1 ) e3
i
in T∗ (UT /T )C , with the first two OUC -linear in ∂t ϕT and the third OUC -linear in ∂t ∇T .
7
The second variation of the enhanced kinetic term for
maps
Let T = (−ε, ε)2 ⊂ R2 , with coordinate (s, t), and (ϕT , ∇T ) be an (∗2 )-admissible family of
(∗1 )-admissible pairs, with (ϕ(0,0) , ∇(0,0) ) = (ϕ, ∇). Assume further that
Dξ ∂s AϕT ⊂ Comm (AϕT )
for all ξ ∈ T∗ (XT /T ) .
We work out in this section the second variation formula of the enhanced kinetic term
(ρ,h;Φ,g)
(ρ,h;Φ,g,B,C)
Smap:kinetic+ (ϕ, ∇) in the standard action Sstandard
(ϕ, ∇).
7.1
The second variation of the kinetic term for maps
Recall
E
∇T
(ϕT ) :=
(h;g)
Smap:kinetic (ϕT , ∇T )
Z
1
:= Tm−1
Tr hDT ϕT , DT ϕT i(h,g) vol h ,
2
X
46
with the understanding that all expressions are taken on X(s,t) with (s, t) varying in T .
Let U ⊂ X be an open set with an orthonormal frame (eµ )µ=1, ··· , m . Let (eµ )µ=1, ··· , m be the
dual co-frame. Assume that U is small enough so that ϕT (UTAz ) is contained in a coordinate chart
of Y , with coordinates (y 1 , · · · , y n ). Then, as in Sec. 6.1, over U ,
∂ ∇T
E (ϕT )
∂t
m D
X
Z
=
Tm−1
Tr
U
T ,g)
∇eT,(ϕ
∂t ϕT , DeTµ ϕT
µ
E
g
µ=1
Z
+ Tm−1
Tr
U
m D
X
vol h
(ad ⊗∇g )DeT
ϕ ∂t ϕT
µ T
µ=1
Z X
m DX
n
+ Tm−1
U µ=1
, DeTµ ϕT
E
g
vol h
[(∂t ∇T )(eµ ), ϕ]T (y i )] ⊗
, DeTµ ϕT
∂
∂y i
E
i=1
=: (I 2 .1) + (I 2 .2) + (I 2 .3) ;
and
∂2
∂ 2
∂ 2
∂ 2
T
E ∇ (ϕT ) =
(I .1) +
(I .2) +
(I .3) .
∂s ∂t
∂s
∂s
∂s
Which we now compute term by term.
The term
∂
∂s
(I 2 .1)
m D
E
X
∂ Z
∂ 2
T
T ,g)
vol h
∇eT,(ϕ
∂
ϕ
,
D
ϕ
(I .1) = Tm−1
Tr
t
T
T
e
µ
µ
g
∂s
∂s U
µ=1
= Tm−1
= Tm−1
Z
Tr
X
U
Z
D
T ,g)
∂s ∇T,(ϕ
∂t ϕT , DeTµ ϕT
eµ
µ
Tr
XD
U
T ,g)
∂t ϕT , DeTµ ϕT
∂s ∇T,(ϕ
eµ
µ
+ Tm−1
Z
Tr
U
E
g
E
g
vol h
vol h
XD
T ,g)
∇eT,(ϕ
∂t ϕT , ∂s DeTµ ϕT
µ
µ
E
g
vol h
= (I 2 .1.1) + (I 2 .1.2) .
(a) Term (I 2 .1.1)
2
(I .1.1) :=
=
Tm−1
Tm−1
Z
T ,g)
∂t ϕT , DeTµ ϕT
∂s ∇eT,(ϕ
µ
U
Z
E
g
µ
XD
T ,g)
∇T,(ϕ
∂s ∂t ϕT , DeTµ ϕT
eµ
Tr
U
+ Tm−1
=
XD
Tr
µ
Z
Tr
U
XD
E
g
vol h
vol h
F∇T,(ϕT ,g) (∂s , eµ ) ∂t ϕT , DeTµ ϕT
µ
E
g
vol h
(I 2 .1.1.1) + (I 2 .1.1.2) .
For Term (I 2 .1.1.1), as in Sec. 6.1 for Summand (I.1.1), consider the 1-form on UT /T
T
T
α(I
2 ,∂ ∂ ϕ ) := Tr h∂s ∂t ϕT , D ϕT ig
s t T
and let
ξ(IT 2 ,∂s ∂t ϕT )
:=
n
X
Tr h∂s ∂t ϕT , DeTµ ϕT ig eµ
µ=1
47
g
vol h
be its dual on UT /T with respect to h. Then,
2
(I .1.1.1) = Tm−1
Z
∂U
iξT 2
(I ,∂s ∂t ϕT )
Z
+ Tm−1
vol h
D
U
T
Tr ∂s ∂t ϕT , (DP
−
∇h
eµ eµ
µ
P
T,(ϕT ,g) T
Deµ )ϕT
µ ∇eµ
E
g
vol h .
For Term (I 2 .1.1.2), recall Lemma 3.2.2.5. Then,
2
(I .1.1.2) = − Tm−1
Z
D
U
= − Tm−1
Z
Tr
X
U
g
vol h
[F∇ (∂s , eµ ) , h∂t ϕT , DeTµ ϕT ig ] vol h
µ
D
U
E
F∇T,(ϕT ,g) (∂s , eµ )DeTµ ϕT
µ
+ Tm−1
Z
X
Tr ∂t ϕT ,
X
Tr ∂t ϕT ,
E
F∇T,(ϕT ,g) (∂s , eµ )DeTµ ϕT
g
µ
vol h .
Here,
T ,g)
T ,g)
F∇T,(ϕT ,g) (∂s , eµ ) DeTµ ϕT = (∂s ∇T,(ϕ
− ∇eT,(ϕ
∂s )
eµ
µ
=
n
X
i=1
+
∂
[(∂s ∇T )(eµ ), Deµ ϕ]T (y i )] ⊗ ∂y i +
n
X
n X
Deµ ϕ]T (y i )
i=1
n
X
Pn
]
i
i=1 Deµ ϕT (y )
Deµ ϕ]T (y i )
i=1
n
X
∂
⊗ ∂y i
∂
[(∂s ∇T )(eµ ), ϕ]T (y j )] ⊗ ∇g ∂ ∂y i
∂y j
j=1
DeTµ ϕ]T (y j ) ∂s ϕ]T (y k )
j,k=1
⊗∇
∂
g
∂
∂y k
∇g ∂ ∂y i
∂y j
∂
− ∂s ϕ]T (y k ) DeTµ ϕ]T (y j ) ⊗ ∇g ∂ ∇g ∂ ∂y i
∂y j
∂y k
explicitly.
(b) Term (I 2 .1.2)
(I 2 .1.2) :=
=
Tm−1
Tm−1
Z
Tr
U
Z
+ Tm−1
T ,g)
∂t ϕT , ∂s DeTµ ϕT
∇T,(ϕ
eµ
µ
Tr
U
+ Tm−1
XD
XD
Tr
U
Z
U
vol h
E
g
vol h
XD
µ
Tr
g
T ,g)
T ,g)
∇T,(ϕ
∂t ϕT , ∇eT,(ϕ
∂s ϕT
eµ
µ
µ
Z
E
T ,g)
∇T,(ϕ
∂t ϕT , (ad ⊗ ∇g )DeTµ ϕT ∂s ϕT
eµ
XD
T ,g)
∇T,(ϕ
∂t ϕT ,
eµ
µ
E
g
vol h
E
Pn
]
∂
T
i
[(∂
∇
)(e
),
ϕ
(y
)]
⊗
s
µ
i
T
i=1
∂y g vol h .
As in Sec. 6.1, consider the 1-forms on UT /T ,
T
α(I
= Tr h∂t ϕT , ∇T,(ϕT ,g) ∂s ϕT ig ,
2 ,∂ ϕ ,∇T,(ϕT ,g) )
t T
T
α(I
= Tr h∂t ϕT , (ad ⊗ ∇g )DT ϕT ∂s ϕT ig ,
2 ,∂ ϕ ,DT ϕ )
t T
T
T
α(I
= Tr h∂t ϕT ,
2 ,∂ ϕ ,∂ ∇T )
t T s
Pn
]
∂
T
i
i=1 [∂s ∇ , ϕT (y )] ⊗ ∂y i ig
and let
ξ(IT 2 ,∂t ϕT ,∇T,(ϕT ,g) ) =
X
ξ(IT 2 ,∂t ϕT ,DT ϕT )
=
X
=
X
ξ(IT 2 ,∂t ϕT ,∂s ∇T )
T ,g)
Tr h∂t ϕT , ∇eT,(ϕ
∂s ϕT ig eµ ,
µ
µ
µ
µ
Tr h∂t ϕT , (ad ⊗ ∇g )DeTµ ϕT ∂s ϕT ig eµ ,
Tr h∂t ϕT ,
Pn
]
∂
T
i
i=1 [(∂s ∇ )(eµ ), ϕT (y )] ⊗ ∂y i ig eµ
48
be their respective dual on UT /T with respect to h. Then,
Z
2
(I .1.2) = Tm−1
∂U
i ξT
(I2 ,∂t ϕT ,∇T,(ϕT ,g) )
Z
+ Tm−1
ZU
+ Tm−1
U
D
+ ξT 2
(I ,∂t ϕT ,DT ϕT )
T,(ϕT ,g)
∇h e
µ eµ µ
Tr ∂t ϕT , ∇P
D
D
Tr ∂t ϕT ,
∇h
eµ eµ
∂s ϕT
E
g
vol h
ϕT
T,(ϕT ,g)
(ad ⊗ ∇g )DeT
µ ∇eµ
P
ϕT
µ
∂s ϕT
Pn ]
T P
h
i
i=1 [(∂s ∇ )( µ ∇eµ eµ ), ϕT (y )]
P T,(ϕ ,g)
T
−
∂
∂s
T,(ϕT ,g) T,(ϕT ,g)
∇eµ
µ ∇eµ
µ
U
The term
vol h
−
+ Tm−1
(I ,∂t ϕT ,∂s ∇T )
P
−
Tr ∂t ϕT , (ad ⊗ ∇g )DP
T
Z
+ ξT 2
µ ∇eµ
T
E
g
vol h
∂
E
[(∂s ∇ )(eµ ), ϕ]T (y i )] ⊗ ∂y i
g
vol h .
(I 2 .2)
m D
E
X
∂ 2
∂ Z
Tr
(ad ⊗∇g )DeTµ ϕT ∂t ϕT , DeTµ ϕT vol h
(I .2) = Tm−1
g
∂s
∂s U
µ=1
= Tm−1
Z
Tr
U
XD
∂s ((ad ⊗ ∇g )Deµ ϕT ∂t ϕT ) , DeTµ ϕT
µ
+ Tm−1
Z
Tr
U
XD
E
g
vol h
(ad ⊗ ∇g )Deµ ϕT ∂t ϕT , ∂s DeTµ ϕT
µ
E
vol h
g
= (I 2 .2.1) + (I 2 .2.2) .
(a) Term (I 2 .2.1)
(I 2 .2.1) = Tm−1
Z
Tr
U
XD
Z
Tr
U
Tr
+ Tm−1
XDX
µ
Z
U
E
µ
µ
+ Tm−1
(ad ⊗ ∇g )DeT (∂s ∂t ϕT ) , DeTµ ϕT
vol h
[DeTµ ∂s ϕ]T (y j ), ∂t ϕ]T (y i )] ⊗ ∇g ∂
∂y j
i,j
XD X µ
g
E
∂
, DeTµ ϕT vol h
i
∂y
g
∂
∂
∂
[DeTµ ϕ]T (y j ), ∂t ϕ]T (y i )]∂s ϕ]T (y k ) ⊗ Rg ( ∂y k , ∂y j ) ∂y i
i,j,k
∂
− ∂t ϕ]T (y i )[DeTµ ϕ]T (y j ), ∂s ϕ]T (y k )] ⊗ ∇g ∂ ∇g ∂ ∂y i ,
∂y j
∂y k
DeTµ ϕT
Z
+ Tm−1
Tr
U
XDX
µ
i,j
∂
[ad (∂s ∇T )(eµ ) ϕ]T (y j ), ∂t ϕ]T (y i )] ⊗ ∇g ∂ ∂y i , DeTµ ϕT
∂y j
E
g
E
g
vol h
vol h .
The integrand of the first summand captures a related part in the system of equations of motion
for (ϕ, ∇). The integrand of the second summand is tensorial in ∂t ϕT and first-order differential
operatorial in ∂s ϕT . The integrand of the third summand is tensorial in both ∂t ϕT and ∂s ϕT .
The integrand of the fourth summand is tensorial in ∂t ϕT and ∂s ∇T .
(b) Term (I 2 .2.2)
(I 2 .2.2) = Tm−1
Z
Tr
U
XD
(ad ⊗ ∇g )Deµ ϕT ∂t ϕT , ∂s DeTµ ϕT
µ
49
E
g
vol h
Z
= Tm−1
XD
T ,g)
∂s ϕT
(ad ⊗ ∇g )Deµ ϕT ∂t ϕT , ∇eT,(ϕ
µ
Tr
U
E
− Tm−1
Z
Tr
XD
E
(ad ⊗ ∇g )Deµ ϕT ∂t ϕT , (ad ⊗ ∇g )∂s ϕT DeTµ ϕT
U
Tr
XD
(ad ⊗ ∇g )Deµ ϕT ∂t ϕT ,
U
X
µ
i
vol h
g
µ
Z
+ Tm−1
vol h
g
µ
∂
[(∂s ∇T )(eµ ), ϕ]T (y i )] ⊗ ∂y i
E
g
vol h .
The integrand of the first summand is tensorial in ∂t ϕT and first-order differential operatorial in
∂s ϕT . The integrand of the second summand is tensorial in both ∂t ϕT and ∂s ϕT . The integrand
of the third summand is tensorial in ∂t ϕT and ∂s ∇T .
The term
∂
∂s
(I 2 .3)
m DX
n
∂ Z X
∂ 2
(I .3) = Tm−1
[(∂t ∇T )(eµ ), ϕ]T (y i )] ⊗
∂s
∂s U µ=1 i=1
= Tm−1
Z X
m DX
n
U µ=1
+ Tm−1
Z
∂s [(∂t ∇T )(eµ ), ϕ]T (y i )] ⊗
i=1
m
X
U µ=1
2
n
DX
∂
∂y i
, DeTµ ϕT
∂
∂y i
[(∂t ∇T )(eµ ), ϕ]T (y i )] ⊗
E
g
, DeTµ ϕT
∂
∂y i
vol h
E
g
vol h
, ∂s DeTµ ϕT
E
i=1
g
vol h
= (I 2 .3.1) + (I .3.2) .
(a) Term (I 2 .3.1)
2
(I .3.1) = Tm−1
= Tm−1
Z X
m DX
n
U µ=1 i=1
Z X
m DX
n
U µ=1
Z
U µ=1
∂
[(∂s ∂t ∇T )(eµ ), ϕ]T (y i )] ⊗ ∂y i , DeTµ ϕT
i=1
m DX
n X
+ Tm−1
∂
∂s [(∂t ∇T )(eµ ), ϕ]T (y i )] ⊗ ∂y i , DeTµ ϕT
i=1
E
g
E
g
vol h
vol h
∂
[(∂t ∇T )(eµ ), ∂s ϕ]T (y i )] ⊗ ∂y i
+ [(∂t ∇T )(eµ ), ϕ]T (y i )]
∂
g
]
j
j ∂s ϕT (y ) ⊗ ∇ ∂ ∂y i
∂y j
P
, DeTµ ϕT
E
g
vol h .
The integrand of the first summand captures a related part in the system of equations of motion
for (ϕ, ∇). The integrand of the second summand is tensorial in ∂s ϕT and ∂t ∇T .
(b) Term (I 2 .3.2)
(I 2 .3.2) = Tm−1
= Tm−1
Z X
m DX
n
U µ=1 i=1
Z X
m DX
n
U µ=1
− Tm−1
+ Tm−1
Z
i=1
m
XD
U µ=1
Z X
m
∂
[(∂t ∇T )(eµ ), ϕ]T (y i )] ⊗ ∂y i , ∂s DeTµ ϕT
∂
E
g
vol h
T ,g)
[(∂t ∇T )(eµ ), ϕ]T (y i )] ⊗ ∂y i , ∇T,(ϕ
∂s ϕT
eµ
∂
Pn
E
g
vol h
]
T
i
g
T
i=1 [(∂t ∇ )(eµ ), ϕT (y )] ⊗ ∂y i , (ad ⊗ ∇ )∂s ϕT Deµ ϕT
n D
X
U µ=1 i,j=1
∂
E
g
vol h
∂
[(∂t ∇T )(eµ ), ϕ]T (y i )] ⊗ ∂y i , [(∂s ∇T )(eµ ), ϕ]T (y j )] ⊗ ∂y j
E
g
vol h .
The integrand of the first summand is tensorial in ∂t ∇T and first-order differential operatorial
in ∂s ϕT . The integrand of the second summand is tensorial in ∂s ϕT and ∂t ∇T . The integrand of
the third summand is tensorial in both ∂t ∇T and ∂s ∇T .
50
Finally, recall Lemma 3.2.2.4 and note that with the additional assumption at the beginning
of this section, all the inner products Tr h , ig that appear in the calculation above are defined.
· ·
In summary,
Proposition 7.1.1 [second variation of kinetic term for maps] Let (ϕT , ∇T ) be a (∗2 )admissible T -family of (∗1 )-admissible pairs with the additional assumption that
Dξ ∂s AϕT ⊂ Comm (AϕT ) for all ξ ∈ T∗ (XT /T ). Then,
Z
∂ ∂ 1
∂ ∂ ∇T
C
T
T
E
(ϕ
)
=
T
Tr
hD
ϕ
,
D
ϕ
i
vol
T
T
T (h,g)
h
∂s ∂t
∂s ∂t 2 m−1 U
Z
= Tm−1
iξ T 2
vol h
∂U (I ,∂s ∂t ϕT )
Z
+ Tm−1
i ξT
vol h
+ ξT 2
+ ξT 2
(I ,∂t ϕT ,DT ϕT )
(I ,∂t ϕT ,∂s ∇T )
(I2 ,∂t ϕT ,∇T,(ϕT ,g) )
Z ∂U D
E
P T,(ϕT ,g) T T
+ Tm−1
Tr ∂s ∂t ϕT , DP
−
∇
D ϕT vol h
h
U
µ
Z
+ Tm−1
U
+ Tm−1
µ
∇eµ eµ
eµ
eµ
g
E
XD
Tr
(ad ⊗ ∇g )DeTµ (∂s ∂t ϕT ) , DeTµ ϕT vol h
g
µ
Z X
m DX
n
E
∂
(∂s ∂t ∇T )(eµ ), ϕ]T (y i ) ⊗ ∂y i , DeTµ ϕT vol h
g
U
µ=1
i=1
Z
E
D
P
T,(ϕ ,g)
T ,g)
T ,g)
+ Tm−1
Tr ∂t ϕT , ∇P T∇h e − µ ∇T,(ϕ
∇eT,(ϕ
∂s ϕT vol h
e
µ
µ
eµ µ
g
µ
U
Z
D
T
+ Tm−1
Tr ∂t ϕT , (ad ⊗ ∇g )DP
ϕT
U
µ
−
∇h
eµ eµ
E
T,(ϕT ,g)
g
T
∇
(ad
⊗
∇
)
∂
ϕ
vol h
s T
µ eµ
Deµ ϕT
P
g
Z
+ Tm−1
U
XDX
DeTµ ∂s ϕ]T (y j ), ∂t ϕ]T (y i ) ⊗ ∇g ∂
Tr
µ
∂y j
i,j
E
∂
T
,
D
ϕ
vol h
T
∂y i eµ
g
Z
X D X ∂
∂ ∂
+ Tm−1
Tr
DeTµ ϕ]T (y j ), ∂t ϕ]T (y i ) ∂s ϕ]T (y k ) ⊗ Rg ∂y k , ∂y j ∂y i
U
µ
i,j,k
∂
− ∂t ϕ]T (y i ) DeTµ ϕ]T (y j ), ∂s ϕ]T (y k ) ⊗ ∇g ∂ ∇g ∂ ∂y i ,
∂y j
∂y k
E
DeTµ ϕT vol h
g
Z
E
XD
T ,g)
+ Tm−1
Tr
(ad ⊗ ∇g )Deµ ϕT ∂t ϕT , ∇T,(ϕ
∂s ϕT vol h
eµ
U
Z
− Tm−1
U
Z
− Tm−1
U
g
µ
E
XD
Tr
(ad ⊗ ∇g )Deµ ϕT ∂t ϕT , (ad ⊗ ∇g )∂s ϕT DeTµ ϕT vol h
g
µ
D
E
X
Tr ∂t ϕT ,
F∇T ,(ϕT ,g) (∂s , eµ )DeTµ ϕT vol h
g
µ
Z
D
Pn P
Tr ∂t ϕT , i=1 (∂s ∇T )( µ ∇heµ eµ ), ϕ]T (y i )
U
E
P
∂
T ,g)
− µ ∇T,(ϕ
(∂s ∇T )(eµ ), ϕ]T (y i ) ⊗ ∂y i vol h .
eµ
g
Z
E
XDX
∂
+ Tm−1
Tr
ad (∂s ∇T )(eµ ) ϕ]T (y j ), ∂t ϕ]T (y i ) ⊗ ∇g ∂ ∂y i , DeTµ ϕT vol h
+ Tm−1
U
µ
∂y j
i,j
g
Z
+ Tm−1
E
XD
X
∂
Tr
(ad ⊗ ∇g )Deµ ϕT ∂t ϕT ,
(∂s ∇T )(eµ ), ϕ]T (y i ) ⊗ ∂y i vol h
g
U
µ
+ Tm−1
i
Z X
m DX
n ∂
(∂t ∇T )(eµ ), ∂s ϕ]T (y i ) ⊗ ∂y i
U
µ=1
i=1
E
P
∂
+ (∂t ∇T )(eµ ), ϕ]T (y i ) j ∂s ϕ]T (y j ) ⊗ ∇g ∂ ∂y i , DeTµ ϕT vol h
g
∂y j
51
+ Tm−1
Z X
m DX
n
E
∂
T ,g)
∂s ϕT vol h
(∂t ∇T )(eµ ), ϕ]T (y i ) ⊗ ∂y i , ∇T,(ϕ
eµ
g
U
− Tm−1
Z X
m D
E
Pn ∂
]
g
T
T
i
vol h
i=1 (∂t ∇ )(eµ ), ϕT (y ) ⊗ ∂y i , (ad ⊗ ∇ )∂s ϕT Deµ ϕT
g
U
µ=1
i=1
µ=1
Z X
m X
n D
E
∂
∂ (∂t ∇T )(eµ ), ϕ]T (y i ) ⊗ ∂y i , (∂s ∇T )(eµ ), ϕ]T (y j ) ⊗ ∂y j vol h .
g
U
+ Tm−1
µ=1 i,j=1
Here,
n
X
T
ξ(I
:=
2 ,∂ ∂ ϕ )
s t T
Tr h∂s ∂t ϕT , DeTµ ϕT ig eµ ,
µ=1
=
X
T
ξ(I
2 ,∂ ϕ ,DT ϕ )
t T
T
=
X
T
ξ(I
2 ,∂ ϕ ,∂ ∇T )
t T s
=
T
ξ(I
2 ,∂ ϕ ,∇T,(ϕT ,g) )
t T
T ,g)
Tr h∂t ϕT , ∇eT,(ϕ
∂s ϕT ig eµ ,
µ
µ
Tr h∂t ϕT , (ad ⊗ ∇g )DeT
µ ϕT
µ
X
∂s ϕT ig eµ ,
Pn
∂
]
T
i
i=1 [(∂s ∇ )(eµ ), ϕT (y )] ⊗ ∂y i ig eµ
Tr h∂t ϕT ,
µ
and
T ,g)
T ,g)
F∇T,(ϕT ,g) (∂s , eµ ) DeTµ ϕT = (∂s ∇T,(ϕ
− ∇T,(ϕ
∂s )
eµ
eµ
=
n
X
i=1
+
n
X
∂
[(∂s ∇T )(eµ ), Deµ ϕ]T (y i )] ⊗ ∂y i +
n
X
n X
Deµ ϕ]T (y i )
i=1
j,k=1
Pn
]
i
i=1 Deµ ϕT (y )
Deµ ϕ]T (y i )
n
X
∂
⊗ ∂y i
∂
[(∂s ∇T )(eµ ), ϕ]T (y j )] ⊗ ∇g ∂ ∂y i
∂y j
j=1
i=1
∂
DeTµ ϕ]T (y j ) ∂s ϕ]T (y k ) ⊗ ∇g ∂ ∇g ∂ ∂y i
∂y j
∂y k
∂
− ∂s ϕ]T (y k ) DeTµ ϕ]T (y j ) ⊗ ∇g ∂ ∇g ∂ ∂y i
∂y j
∂y k
The summands
+ Tm−1
Z
D
T
Tr ∂s ∂t ϕT , (DP
∇h e
µ eµ µ
UZ
+ Tm−1
Tr
E
T,(ϕT ,g) T
Deµ )ϕT
µ ∇eµ
g
P
XD
U
(ad ⊗ ∇g )DeTµ (∂s ∂t ϕT ) , DeTµ ϕT
µ
Z X
m DX
n
+ Tm−1
−
U µ=1
i=1
E
g
vol h
vol h
E
∂
T
[(∂s ∂t ∇T )(eµ ), ϕ]T (y i )] ⊗ ∂y
vol h
ϕ
,
D
i
eµ T
g
will vanish when imposing the equations of motion for (ϕ, ∇).
If (ϕT , ∇T ) is furthermore a (∗2 )-admissible T -family of (∗2 )-admissible pairs, Then, the above
expression reduces to
∂ ∂ ∇T
C
∂s ∂t E (ϕT )
Z
∂ ∂
Z
1
= ∂s ∂t 2 Tm−1
= Tm−1
∂U
iξT
(I2 ,∂s ∂t ϕT )
U
Tr hDT ϕT , DT ϕT i(h,g) vol h
vol h
Z
+ Tm−1
∂U
Z
+ Tm−1
i ξT
(I2 ,∂t ϕT ,∇T,(ϕT ,g) )
D
∇h
eµ e µ
µ
Z X
m DX
n
U µ=1
Z
+ Tm−1
U
(I ,∂t ϕT ,DT ϕT )
T
Tr ∂s ∂t ϕT , (DP
U
+ Tm−1
+ ξT 2
D
i=1
−
+ ξT 2
(I ,∂t ϕT ,∂s ∇T )
E
T,(ϕT ,g) T
Deµ )ϕT
µ ∇eµ
g
P
∂
[(∂s ∂t ∇T )(eµ ), ϕ]T (y i )] ⊗ ∂y i , DeTµ ϕT
T,(ϕT ,g)
∇h
eµ e µ
µ
Tr ∂t ϕT , ∇P
−
vol h
E
g
vol h
vol h
T,(ϕT ,g) T,(ϕT ,g)
∇eµ
∂s ϕT
µ ∇eµ
P
52
E
g
vol h
Z
+ Tm−1
U
D
Tr ∂t ϕT , (ad ⊗ ∇g )DP
T
µ
−
− Tm−1
Z
D
Tr ∂t ϕT ,
U
Z
Tr ∂t ϕT ,
U
T,(ϕT ,g)
(ad
µ ∇eµ
Z X
m DX
n U µ=1
i=1
⊗ ∇g )DeT
P
F∇T,(ϕT ,g) (∂s , eµ )DeTµ ϕT
µ ∇eµ
E
g
∂s ϕT
E
g
vol h
vol h
∂
[(∂s ∇ )(eµ ), ϕ]T (y i )] ⊗ ∂y i
T
E
g
vol h .
∂
[(∂t ∇T )(eµ ), ∂s ϕ]T (y i )] ⊗ ∂y i
+ [(∂t ∇T )(eµ ), ϕ]T (y i )]
]
j
j ∂s ϕT (y )
P
Z X
m DX
n
∂
⊗ ∇g ∂ ∂y i
∂y j
∂
, DeTµ ϕT
E
g
vol h
E
T ,g)
∂s ϕT
[(∂t ∇T )(eµ ), ϕ]T (y i )] ⊗ ∂y i , ∇T,(ϕ
eµ
vol h
g
U µ=1 i=1
Z X
m D
E
Pn
∂
]
T
i
g
T
− Tm−1
vol h
i=1 [(∂t ∇ )(eµ ), ϕT (y )] ⊗ ∂y i , (ad ⊗ ∇ )∂s ϕT Deµ ϕT
g
U µ=1
Z X
m X
n D
E
∂
∂
Tm−1
[(∂t ∇T )(eµ ), ϕ]T (y i )] ⊗ ∂y i , [(∂s ∇T )(eµ ), ϕ]T (y j )] ⊗ ∂y j vol h .
g
U µ=1 i,j=1
+ Tm−1
+
ϕ
µ T
Pn ]
T P
h
i
i=1 [(∂s ∇ )( µ ∇eµ eµ ), ϕT (y )]
P T,(ϕ ,g)
T
−
+ Tm−1
ϕT
µ
D
+ Tm−1
X
∇h
eµ eµ
If one further imposes the equations of motion on (ϕ, ∇), then the expression reduces further
to
∂ ∂ ∇T
C
∂s ∂t E (ϕT )
Z
∂ ∂
Z
1
= ∂s ∂t 2 Tm−1
= Tm−1
∂U
iξT
(I2 ,∂s ∂t ϕT )
U
Tr hDT ϕT , DT ϕT i(h,g) vol h
vol h
Z
+ Tm−1
∂U
Z
i ξT
(I2 ,∂t ϕT ,∇T,(ϕT ,g) )
D
+ Tm−1
U
+ ξT 2
(I ,∂t ϕT ,DT ϕT )
T,(ϕT ,g)
∇h
eµ e µ
µ
Tr ∂t ϕT , ∇P
Z
+ Tm−1
U
D
−
D
Tr ∂t ϕT ,
X
Tr ∂t ϕT ,
U
Z X
m DX
n U µ=1
i=1
T,(ϕT ,g)
(ad ⊗ ∇g )DeT
µ ∇eµ
P
µ ϕT
µ ∇eµ
vol h
E
g
∂s ϕT
E
g
vol h
vol h
∂
[(∂s ∇ )(eµ ), ϕ]T (y i )] ⊗ ∂y i
T
E
g
vol h .
∂
[(∂t ∇T )(eµ ), ∂s ϕ]T (y i )] ⊗ ∂y i
]
j
j ∂s ϕT (y )
P
+
g
Pn ]
i
T P
h
i=1 [(∂s ∇ )( µ ∇eµ eµ ), ϕT (y )]
P T,(ϕ ,g)
T
+ [(∂t ∇T )(eµ ), ϕ]T (y i )]
+ Tm−1
E
ϕT
F∇T,(ϕT ,g) (∂s , eµ )DeTµ ϕT
−
+ Tm−1
∇h
eµ eµ
µ
D
+ Tm−1
T,(ϕT ,g) T,(ϕT ,g)
∇eµ
∂s ϕT
µ ∇eµ
µ
U
Z
vol h
−
− Tm−1
(I ,∂t ϕT ,∂s ∇T )
P
Tr ∂t ϕT , (ad ⊗ ∇g )DP
T
Z
+ ξT 2
Z X
m DX
n
∂
∂
⊗ ∇g ∂ ∂y i
∂y j
T ,g)
[(∂t ∇T )(eµ ), ϕ]T (y i )] ⊗ ∂y i , ∇T,(ϕ
∂s ϕT
eµ
E
, DeTµ ϕT
E
g
vol h
vol h
g
U µ=1 i=1
Z X
m D
E
Pn
∂
]
T
i
g
T
vol h
− Tm−1
i=1 [(∂t ∇ )(eµ ), ϕT (y )] ⊗ ∂y i , (ad ⊗ ∇ )∂s ϕT Deµ ϕT
g
U µ=1
Z X
m X
n D
E
∂
∂
Tm−1
[(∂t ∇T )(eµ ), ϕ]T (y i )] ⊗ ∂y i , [(∂s ∇T )(eµ ), ϕ]T (y j )] ⊗ ∂y j vol h .
g
U µ=1 i,j=1
53
7.2
The second variation of the dilaton term
We now work out the second variation of the (complexified) dilaton term
(ρ,h;Φ)
Sdilaton (ϕT )C
Z
=
Tr hdρ , ϕT dΦih vol h
U
Z
=
Tr
X
U
dρ(eµ ) ((DeTµ ϕT )Φ) vol h .
µ
for an (∗1 )-admissible family of (∗1 )-admissible pairs (ϕT , ∇T ), T := (−ε, ε)2 ⊂ R2 with coordinate (s, t).
It follows from Sec. 6.2 that, due to the effect of the trace map Tr ,
Z
m
X
∂ (ρ,h;Φ)
C
Sdilaton (ϕT ) =
Tr
dρ(eµ ) ∂t ((DeTµ ϕT )Φ)vol h
∂t
U
µ=1
Z
=
X
Tr
U
dρ(eµ )DeTµ ((∂t ϕT )Φ)vol h .
µ
Thus, due to the effect of the trace map Tr again,
∂ ∂ (ρ,h;Φ)
Sdilaton (ϕT )C =
∂s ∂t
Z
=
Tr
X
Tr
X
U
X
Tr
U
dρ(eµ )∂s DeTµ ((∂t ϕT )Φ)vol h
µ
dρ(eµ )DeTµ ∂s ((∂t ϕT )Φ)vol h
µ
Z
=
Z
U
dρ(eµ )DeTµ ((∂s ∂t ϕT )Φ)vol h
µ
Z
+
Tr
U
2
X
dρ(eµ )DeTµ
X
µ
i,j
∂
∂
∂
∂t ϕ]T (y i )∂s ϕ]T (y j ) ⊗ ∂y i ∂y j − ∇g ∂ ∂y j Φ vol h
∂y i
2
= (II .1) + (II .2) .
For Summand (II 2 .1), repeating the same argument in Sec. 6.1 for Summand (I.1.1), one
concludes that
Z
2
(II .1) =
∂U
iξT
(II2 ,∂s ∂t ϕT )
vol h
Z X
+
U
∇heµ eµ −
µ
X
eµ dρ(eµ ) Tr ((∂s ∂t ϕT )Φ)vol h ,
µ
where
T
ξ(II
2 ,∂ ∂ ϕ ) :=
s t T
X
∈ T∗ (UT /T ) .
dρ(eµ )Tr ((∂s ∂t ϕT )Φ) eµ
µ
(ρ,h;Φ)
The second summand of Summand (II 2 .1) above is the term that captures the Sdilaton (ϕ)contribution to the system of equations of motion for (ϕ, ∇).
P
g
∂ ∂
∂
With ∂s ∂t ϕT replaced by i,j ∂t ϕ]T (y i )∂s ϕ]T (y j ) ⊗ ( ∂y
i ∂y j − ∇ ∂ ∂y j )Φ, one has similarly
∂y i
(II 2 .2) =
Z
∂U
iξT
(II2 ,∂t ϕT ,∂s ϕT )
+
Z X
U
vol h
∇heµ eµ −
µ
X
eµ dρ(eµ ) ·
µ
Tr
X
∂t ϕ]T (y i )∂s ϕ]T (y j )
i,j
g
∂
∂ ∂
⊗ ∂y
i ∂y j − ∇ ∂ ∂y j Φ vol h ,
i
∂y
where
T
ξ(II
2 ,∂ ϕ ,∂ ϕ ) :=
t T s T
X
µ
dρ(eµ )Tr
X
i,j
g
∂ ∂
∂
∂t ϕ]T (y i )∂s ϕ]T (y j ) ⊗ ∂y
eµ
i ∂y j − ∇ ∂ ∂y j Φ
i
∂y
54
in T∗ (UT /T ). The second summand of Summand (II 2 .2) above contributes to the zeroth order
(ρ,h;Φ,g,B,C)
terms of the differential operator on (∂s ϕT , ∂t ϕT ) from the second variation of Sstandard
(ϕ, ∇).
In summary,
(ρ,h;Φ)
Proposition 7.2.1 [second variation of Sdilaton (ϕ)C ] For the (complexified) dilaton term
Z
(ρ,h;Φ)
Sdilaton (ϕ)C :=
U
Tr hdρ , ϕ dΦih vol h ,
its second variation for a (∗1 )-admissible family of (∗1 )-admissible pairs (ϕT , ∇T ), T := (−ε, ε)2 ⊂
R2 with coordinate (s, t), is given by
∂ ∂ (ρ,h;Φ)
Sdilaton (ϕT )C
∂s ∂t Z
=
∂U
+
iξ T
(II2 ,∂s ∂ ϕT )
Z Xt
U
+
+ξ T
(II2 ,∂t ϕT ∂s ϕT )
∇heµ eµ −
X
∇heµ eµ −
X
µ
eµ dρ(eµ ) Tr ((∂s ∂t ϕT )Φ)vol h
µ
Z X
U
vol h
µ
eµ dρ(eµ ) ·
µ
Tr
X
∂t ϕ]T (y i )∂s ϕ]T (y j )
i,j
∂
∂
∂
⊗ ∂y i ∂y j − ∇g ∂ ∂y j Φ vol h ,
∂y i
where
T
:=
ξ(II
2 ,∂ ∂ ϕ )
s t T
X
T
ξ(II
:=
2 ,∂ ϕ ,∂ ϕ )
t T s T
X
dρ(eµ )Tr ((∂s ∂t ϕT )Φ) eµ
µ
dρ(eµ )Tr
X
µ
i,j
∂
∂
∂
∂t ϕ]T (y i )∂s ϕ]T (y j ) ⊗ ∂y i ∂y j − ∇g ∂ ∂y j Φ
∂y i
eµ
in T∗ (UT /T ). The integral
Z X
U
µ
∇heµ eµ −
X
eµ dρ(eµ ) Tr ((∂s ∂t ϕT )Φ)vol h
µ
would vanish when imposing the equations of motion of (ϕ, ∇) after the combination with other
(ρ,h;Φ,g,B,C)
(ϕ, ∇)C .
Equations-of-Motion capturing parts from the second variation of other terms in Sstandard
8
Conclusion
The current notes lay down some foundation toward the dynamics of D-branes along the line of
(Φ,g,B)
our D-project. Solutions to the system of equations of motion from the total action SDBI (ϕ, ∇)+
(C,B)
SCS/WZ (ϕ, ∇) for a D-brane world-volume should be thought of as an Azumaya/matrix version of
minimal submanifolds or harmonic maps, twisted/bent, on one hand, by the (dynamical) gauge
field ∇ on the domain manifold X with a (noncommutative) endomorphism/matrix function-ring
and, on the other hand, by the background field (Φ, g, B, C), created by closed (super)strings,
on the target space(-time) Y . Further details, issues, and examples are the focus of the sequels.
At the classical level Polyakov superstring or its generalization, a sigma model, is a theory of
harmonic maps on the mathematical side. We construct all the building blocks to generalize
the existing theory of harmonic maps to a theory of maps ϕ : (XAz , E; ∇) → Y , which describe
D-branes. It will turn out that both the connection ∇ and the Admissible Condition (∗1 ) chosen
are needed to build up a mathematically sound theory for such maps ϕ. We introduce a new
(ρ,h;Φ,g,B,C)
action Sstandard
for D-branes that is to D-branes as the Polyakov action is to fundamental
55
superstrings. This ‘standard action’ is abstractly a non-Abelian gauged sigma model based on
maps ϕ : (XAz , E; ∇) → Y from an Azumaya/matrix manifold XAz with a fundamental module E
with a connection ∇ to Y enhanced by the dilaton term, the gauge-theory term, and the ChernSimons/Wess-Zumino term that couples (ϕ, ∇) to Ramond-Ramond field. In a special situation,
this new theory merges the theory of harmonic maps and a gauge theory, with a nilpotent type
fuzzy extension. With the analysis developed for such maps and an improved understanding of
the hierarchy of various admissible conditions on the pairs (ϕ, ∇) and how they resolve the builtin obstruction to pull-push of covariant tensors under a map from a noncommutative manifold
to a commutative manifold, we develop further in this note some covariant differential calculus
needed and apply them to work out the first variation and hence the corresponding equations of
motion for D-branes of the standard action and the second variation of the kinetic term for maps
and the dilaton term in this action. Compared with the non-Abelian Dirac-Born-Infeld action
constructed in the article along the same line, the current note brings the Nambu-Goto-stringto-Polyakov-string analogue to D-branes. The current bosonic setting is the first step toward
the dynamics of fermionic D-branes and their quantization as fundamental dynamical objects,
in parallel to what happened to the theory of fundamental superstrings. It’s by now a history
that as the built-in structure of a string is far richer than that for a point, a physical theory
that takes strings as fundamental objects has brought us to where a physical theory that takes
only point-particles as fundamental objects cannot reach. Now that a D-brane carries even more
built-in structures, are these even-richer-than-string structures all just in vain? Or is a physical
theory that takes D-branes as fundamental objects going to lead us to somewhere beyond that
from string theories? Besides a theory in its own right, a theory that takes D-branes as fundamental objects has deep connection with other themes outside. In particular, at low dimensions,
that there should be the following connections are “obvious” but most details to realize these
connections remain far from reach at the moment.
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