An Uncertainty Principle for
Topological Sectors
JÄ
urg FrÄ
ohlich Fest
Zurich, July 2, 2007
Based on: hep-th/0605198, hep-th/0605200 with D. Freed and G. Segal
Introduction
Today I'll be talking about some interesting subtleties in the
quantization of Maxwell's theory and some of its generalizations.
There are many different generalizations of Maxwell's theory…
Abelian gauge theories: ¯eldstrengths are di®erential forms F 2 - ` (M ).
\ Generalized abelian gauge t heories:" { t he space of gauge-invariant ¯eld con¯gurat ions \ A =G" is a generalized di®erent ial cohomology group in t he sense of
Hopkins & Singer.
In part icular \ A =G" is an abelian group.
These kinds of theories arise naturally in supergravity and superstring theories,
and indeed play a key role in the theory of D-branes and in claims of moduli
stabilization in string theory made in the past few years.
Summary of the Results
1. The Hilbert space can be decomposed into electric and
magnetic flux sectors, but the operators that measure electric and
magnetic topological sectors don't commute and cannot be
simultaneously diagonalized.
This is surprising and nontrivial!
2. Group theoretic approach
Manifestly electric-magnetic dual formulation of the Hilbert space.
3. Group theoretic approach to the theory of a self-dual field.
4. In particular: the K-theory class of a RR field cannot be measured!
Generalized Maxwell Theory
Spacetime = M, with dim(M) =n
Gauge invariant fieldstrength is a differential form: F 2 - ` (M )
Free field action:
S = ¼R
R
2
M
F ¤F
If M = X £ R we have a Hilbert space H.
dF = 0
[F] is conserved
)
[F ] 2 H D` R (X )
Grade the Hilbert space by ``Flux sectors’’
more sophisticated…
H = ©m H m
`
m 2 H (X ; Z):
Example: Maxwell Theory on a
Lens Space
Degree l =2
3
M = S =Zk £ R
c1 (L ) 2 H (X ; Z) »
= Zk
2
Motivating Question:
Electric-Magnetic duality: For dim(M)=n, there is an equivalent formulation with
FD 2 -
n¡ `
(M )
RRD = ¹h
There must also be a grading by the (topological class of) electric flux:
H = ©eH e
e2 H
n¡ `
(X ; Z);
Can we simult aneously decompose H int o elect ric and magnet ic ° ux sect ors?
H
?
=
©e;m H e;m :
``Isn’t it obviously true?’’
For the periodic 1+1 scalar this is just
the decomposition into momentum + winding sectors!
R
Measure magnet ic ° ux:
Measure elect ric ° ux:
§1
R
§2
F where § 1 2 Z ` (X )
¤F where § 2 2 Z n ¡ ` (X )
Canonical moment um conjugat e t o A is ¦ » (¤F ) X .
Z
Z
Z
Z
[
F;
¤F ]
»
[
F;
¦ ]
§1
§2
§2
Z§ 1
Z
=
[ ! 1 F;
! 2¦ ]
X
X
R
= i ¹h X ! 1 d! 2 = 0
where !
i
are closed forms Poincar¶
e dual t o § i .
(1)
(2)
Fallacy in the argument
But! the above period integrals only measure the flux modulo torsion.
The topological sectors e and m are elements of integral cohomology groups.
These abelian groups in general have nontrivial torsion subgroups.
Recall: A ``torsion subgroup’’ of an abelian group is the subgroup
of elements of finite order:
0 ! H T` (X ; Z) ! H ` (X ; Z) ! H D` R (X )
Measuring flux sectors by periods misses our uncertainty
principle
What it is not:
1. We are not saying
2. We are also not saying
[Ei (x); B j (y)] 6
= 0
Z
[
§1
where
@
§ 1 = C1
Z
F;
¤F ] = L(C1 ; C2 )
§2
@
§ 2 = C2
3. Our uncertainty principle concerns
torsion classes of topological sectors.
Differential Cohomology
Correct Mathematical Framework: Deligne-Cheeger-Simons Cohomology
To a manifold M and degree l we associate
H· ` (M ).
D e¯nit ion: By \ generalized Maxwell t heory" we mean a ¯eld t heory such t hat
t he space of gauge inequivalent ¯elds is H· ` (M ) for some `.
Simplest example: Periodic Scalar in 1+1 dimensions: F = df
· 1 (S1 ) = M ap(S1 ; U(1)) = LU(1)
H
· ` (M ) in general
Next we want to get a picture of the space H
Structure of the Differential
Cohomology Group - I
Fieldstrength exact sequence:
F
° at
z
}|
{
0 ! H ` ¡ 1 (M ; R=Z) ! H· ` (M )
¡!
- `Z (M ) ! 0
Characteristic class exact sequence:
A
0! |
¯el dst r engt h
`¡ 1
(M )={z
`¡ 1
Z (M
Topologi call y t r ivi al
Connected!
) ! H· ` (M )
}
char :cl ass
¡!
H ` (M ; Z)
| {z }
Topol ogi cal sect or
! 0
Structure of the Differential
Cohomology Group - II
The space of differential characters has the form:
·` = T£ ¡ £ V
H
T: Connected torus of topologically trivial ° at ¯elds:
W ` ¡ 1(M ) = H ` ¡ 1 (M ; Z) - R=Z
¡ : Discrete (possibly in¯nite) abelian group of topological sectors: H ` (M ; Z).
V: In¯nite-dimensional vector space of \ oscillator modes." V »
= Imdy .
W` ¡
1
H ` (M ; Z)
V
Example: Loop Group of U(1)
Configuration space of a periodic scalar field on a circle:
· 1 (S1 ) = M ap(S1 ; U(1)) = LU(1)
H
Topological class = Winding number:
Flat fields = Torus
Vector Space:
T
of constant maps:
w 2 H 1 (S1; Z) »
= Z
H 0 (S1 ; R=Z) »
= R=Z
V = - 0 =R Loops admitting a logarithm.
H· 1 (S1 ) = T £ Z £ V
This corresponds to the explicit decomposition:
·
X
' (¾) = exp 2¼i Á0 + 2¼i w¾+
n6
=0
¸
Án 2¼i n ¾
e
n
The space of flat fields
H ` ¡ 1 (X ; R=Z)
Is a compact abelian group
… it is not necessarily connected!
Connected component of identity:
Group of components:
Example:
»
= [F i nite gr oup] £ [Tor us]
W ` ¡ 1 (M ) = H ` ¡ 1 (M ; Z) - R=Z
= H T` (M ; Z)
H 1 (M ; Z) »
= H 2 (M ; Z) »
= Zk
H 1 (M ; R=Z) »
= Zk
` = 2, M = S3 =Zk
Discrete Wilson lines, defining topologically nontrivial flat gauge fields.
Entirely determined by holonomy:
Âr (° ) = e2¼i r =k = ! kr
r 2 Zk
Poincare-Pontryagin Duality
M is compact, oriented, dim(M) = n
There is a very subtle PERFECT PAIRING on differential cohomology:
H· ` (M ) £ H· n + 1¡ ` (M ) ! R=Z
Z
H·
h[A· 1 ]; [A· 2 ]i :=
[A· 1 ] ¤ [A· 2 ]
M
On topologically trivial fields:
Z
h[A 1 ]; [A 2 ]i =
A 1 dA 2
M
mod Z
Example:Cocycle of the Loop Group
· 1 (S1 ) = LU(1):
Recall H
h' 1 ; ' 2 i = ??
' = exp(2¼i Á)
Á(s + 1) = Á(s) + w
Z
h' 1 ; ' 2 i =
0
1
Á: R ! R
w 2 Z is the winding number.
2
dÁ
Á1
ds ¡ w1 Á2 (0)
ds
mod Z
Not e! This is (twice!) t he cocycle which de¯nes t he basic cent ral ext ension
of L U(1).
Hamiltonian Formulation of
Generalized Maxwell Theory
Spacet ime: M = X £ R.
· 2 H· ` (M ).
Generalized Maxwell ¯elds: [A]
Z
S = ¼R 2
F ¤F
M
· ` (X ))
Hilbert space: H = L 2 ( H
This breaks manifest electric-magnetic duality.
There is a better way to characterize the Hilbert space.
But for the moment we use this formulation.
Defining Magnetic Flux Sectors
Return to our question:
Can we simultaneously decompose H into electric and
magnetic flux sectors?
H
Since
?
=
· ` (X ))
H»
= L 2(H
©e;m H e;m :
and since
H· ` (X ) has components,
Definition: A state y of definite topological class of magnetic
flux has its wavefunction supported on the component labelled by
m 2 H ` (X ; Z):
Electric flux can be defined similarly in the dual frame,
H»
= L 2 ( H· n ¡ ` )
but…
Defining Electric Flux Sectors
We need to understand more deeply the grading by electric flux
· ` (X )):
in the polarization H »
= L 2(H
Diagonalizing (¤F ) X means diagonalizing ¦ ,
¦ is the generator of translations.
Definition: A state y of definite topological class of electric
flux is an eigenstate under translation by flat fields
8Áf 2 H ` ¡ 1 (X ; R=Z);
µ
Z
· f ) = exp 2¼i
Ã( A· + Á
¶
H
eÁf
·
Ã( A)
X
The topological classes of electric flux are labelled by
e 2 H n ¡ ` (X ; Z):
Noncommuting Fluxes
So: the decomposition wrt electric flux is diagonalizing the group
H ` ¡ 1 (X ; R=Z)
of translations by flat fields:
Suppose à is in a state of de¯nite magnetic ° ux m 2 H ` (X ; Z):
· ` (X ) m .
the support of the wavefunction à is in the topological sector H
Such a state cannot be in an eigenstate of translations by flat characters, if
there are any flat, but topologically nontrivial fields.
Translation by such a nontrivial flat field must translate the support of y to
a different magnetic flux sector.
Therefore, one cannot in general measure both electric and magnetic flux.
m
Group Theoretic Approach
Let S be any (locally compact) abelian group (with a measure)
H = L 2 (S) is a represent at ion of S: 8s0 2 S
Let
S^
(Ts0 Ã)(s) := Ã(s + s0 ):
be the Pontryagin dual group of characters of S
^ 8Â 2 S
^
H = L 2 (S) is also a represent at ion of S:
(M Â Ã)(s) := Â(s)Ã(s)
But!
Ts0 M Â = Â(s0)M Â Ts0 :
So H = L 2 (S): is a representation of the Heisenberg group central extension:
^ ! S£ S
^! 1
1 ! U(1) ! Heis(S £ S)
Heisenberg Groups
~ of
T heor em A Let G be a t opological abelian group. Cent ral ext ensions, G,
G by U(1) are in one-one correspondence wit h cont inuous bimult iplicat ive maps
s : G £ G ! U(1) which are alt ernat ing (and hence skew).
• s is alternating:
s(x,x) = 1
•s is skew: s(x,y) = s(y,x) -1
•s
is bimultiplicative:
s(x1 + x2; y) = s(x1; y)s(x2; y)
&
s(x; y1 + y2) = s(x; y1)s(x; y2)
~ then s(x; y) = x~y~x~¡ 1 y~¡ 1 .
If x 2 G lifts to x~ 2 G
~ is a Heisenberg group.
D e¯nit ion: If s is nondegenerate then G
~ is a Heisenberg group then
T heor em B : (Stone-von Neuman theorem). If G
~ where U(1) acts canonically is unique up to isomorphism.
the unitary irrep of G
Heisenberg group for generalized
Maxwell theory
^ = H· n ¡ ` (X ):
If S = H· ` (X ), then PP duality ) S
¡ `
¢
n
¡
`
~ := Heis H· (X ) £ H·
G
(X )
via the group commutator:
·
¸
¡
¢
¡
¢
D
D
D
D
·
·
·
·
·
·
·
·
s ([A 1 ]; [A 1 ]); ([A 2 ]; [A 2 ]) = exp 2¼i h[A 2 ]; [A 1 ]i ¡ h[A 1 ]; [A 2 ]i :
The Hilbert space of the generalized Maxwell theory
~
is the unique irrep of the Heisenberg group G
N.B! This formulation of the Hilbert space is manifestly
electric-magnetic dual.
Flux Sectors from Group Theory
H ` ¡ 1 (X ; R=Z)
Electric flux sectors diagonalize the flat fields
Magnetic flux sectors diagonalize dual flat fields
H n¡
These groups separately lift to commutative subgroups of
`¡ 1
(X ; R=Z)
~ := Heis( H· ` £ H· n ¡ ` ).
G
However they do not commute with each other!
UE (´ e) := translation operator by ´ e 2 H ` ¡ 1 (X ; R=Z)
UM (´ m ) := translation operator by ´ m 2 H n ¡ ` ¡ 1 (X ; R=Z)
µ
¶
Z
[Ue(´ e); Um (´ m )] = T(´ e; ´ m ) = exp 2¼i
´ e ¯´ m
X
T: torsion pairing, ¯= Bockstein: ¯(´ m ) 2 H Tn ¡ ` (X ; Z).
The New Uncertainty Relation
µ
¶
Z
[Ue(´ e); Um (´ m )] = T(´ e; ´ m ) = exp 2¼i
´ e ¯´ m
X
T
torsion pairing,
b = Bockstein.
Translations by W ` ¡ 1 (X ) and by W n ¡
`¡ 1
(X ) commute
H = ©e¹ ;m¹ H e¹ ;m¹ :
However: The pairing does not
commute on the subgroups of all flat fields.
It descends to the ``torsion pairing'' or ``link pairing'':
H T` (X ) £ H Tn ¡ ` (X ) ! R=Z
This is a perfect pairing, so it is maximally noncommutative on torsion.
Maxwell theory on a Lens space
S3=Zk £ R
H 1 (L k ; R=Z) »
= H 2 (L k ; Z) = Zk
Acting on the Hilbert space the flat fields generate a Heisenberg group
extension
0 ! Zk ! Heis(Zk £ Zk ) ! Zk £ Zk ! 0
This has unique irrep P = clock operator, Q = shift operator
PQ = e2¼i =k QP
1 X 2¼i em =k
e
jmi
States of definite electric and magnetic flux jei = p
k m
This example already appeared in string theory in Gukov, Rangamani, and Witten,
hep-th/9811048. They studied AdS5xS5/Z3 and in order to match nonperturbative
states concluded that in the presence of a D3 brane one cannot simultaneously
measure D1 and F1 number.
Remarks- Principles of Quantum
Mechanics
The pairing of topologically nontrivial flat fields has no tunable
parameter. Therefore, this is a quantum effect which does not
become small in the semiclassical or large volume limit.
In general, there are no clear rules for quantization of disconnected
phase spaces. We are extending the standard rules of quantum
mechanics by demanding electric-magnetic duality.
This becomes even more striking in the theory of the self-dual field.
An Experimental Test
Since our remark applies to Maxwell theory: Can we test it experimentally?
Discouraging fact: No region in
R3
has torsion in its cohomology
With A. Kitaev and K. Walker we noted that using arrays of Josephson
Junctions, in particular a device called a ``superconducting mirror,’’
we can ``trick’’ the Maxwell field into behaving as if
it were in a space with torsion.
To exponentially good accuracy the groundstates of the electromagnetic
field are an irreducible representation of Heis(Zn £ Zn )
See arXiv:0706.3410 for more details.
Self-dual fields
Now suppose dimM = 4k + 2, and ` = 2k + 1.
We can impose a self-duality condition F = *F.
This is a very subtle quantum theory: Differential cohomology is the ideal tool.
For the non-self-dual field we represent
· ` (X ) £ H· ` (X ))
Heis( H
Proposal: For the self-dual field we represent:
Heis( H· ` (X ))
Attempt to define this Heisenberg group via
st rial ([A· 1 ]; [A· 2 ]) = exp 2¼i h[A· 1 ]; [A· 2 ]i :
It is skew and and nondegenerate, but not
alternating!
R
· [A])
· = (¡ 1)
st r i al ([A];
X
º 2k m
Gomi 2005
Z 2 -graded Heisenberg groups
Theorem A’: Skew bimultiplicative maps classify
Z 2 -graded Heisenberg groups.
Z
· = 0
²([A])
Z2
if
grading in our case:
Z
· = 1
²([A])
Theorem B’: A
º 2k m = 0 mod 2
Z 2-graded Heisenberg group has a
if
unqiue
º 2k m = 1 mod 2
Z2
-graded
irreducible representation.
This defines the Hilbert space of the self-dual field
Example: Self-dual scalar: k = 0. By bosonizat ion à = ei Á .
The Z 2 -grading is just fermion number!
Relation to Nonselfdual Field
.
Remark: One can show that the nonself-dual field at a
special radius, R2 = 2¹h, decomposes into
H n sd »
= ©®H sd;® - H asd;®
The subscript ® is a sum over \ generalized spin structures" -
a torsor for 2-torsion points in H 2k + 1(X ; Z) - R=Z.
For the self dual scalar ® labels R and N S sectors.
Thus, self-dual theory generalizes VOA theory
String Theory Applications
1. RR field of type II string theory is valued in differential K-theory
K· (X )
· (X ))
RR field is self dual, hence Hilbert space = unique irrep of Heis( K
Hence – the full K theory class is not measurable!
2. B-fields
3. AdS/CFT Correspondence
Open Problems
1. What happens when X is noncompact?
2. Complete the quantum theory of the free self-dual field.
3. If one cannot measure the complete K-theory class of
RR flux what about the D-brane charge?
a.) If no, we need to make an important conceptual revision of
the standard picture of a D-brane
b.) If yes, then flux-sectors and D-brane charges are
classified by different groups
tension with AdS/CFT and
geometric transitions.
4. What is the physical meaning of the fermionic sectors in
the RR Hilbert space?
5. How is this compatible with noncommutativity of 7-form
Page charges in M-theory?