AGT 関係式とその一般化に向けて
(Towards the generalization of AGT relation)
高エネルギー加速器研究機構 (KEK)
素粒子原子核研究所 (IPNS)
柴 正太郎 (Shotaro Shiba)
S. Kanno, Y. Matsuo, S.S. and Y. Tachikawa, Phys. Rev. D81 (2010) 046004.
S. Kanno, Y. Matsuo and S.S., work in progress.
Introduction
What is the multiple M-branes’ system like? (The largest motivation of my research)
• The system of single M-brane in 11-dim spacetime is understood, at least classically.
• However, at this time, we have too little information on the multiple M-branes’ system.
• Now I hope to understand more on M-theory by studying the internal degrees of
freedom which the multiple branes’ systems must always have.
 D-branes’ case : internal d.o.f ~ N2
• The superstrings ending on a D-brane compose the internal d.o.f.
• It is well known that this system is described by DBI action with gauge symmetry of
Lie algebra U(N), which is reduced to Yang-Mills theory in the low-energy limit.
2
 M2-branes’ case : internal d.o.f. ~ N3/2
• The proposition of BLG model is the important breakthrough.
[Bagger-Lambert ’07] [Gustavsson ’07]
• We can derive the internal d.o.f. of order N3/2 naturally and successfully, using the finite
representation of Lie 3-algebra which is the gauge symmetry algebra of BLG model.
[Chu-Ho-Matsuo-SS ’08]
• However, at this moment, we don’t know at all what compose these d.o.f.
Subject of today’s seminar
 M5-branes’ case : internal d.o.f. ~ N3
The near horizon geometry of M-branes is
AdS x S, so we can use AdS/CFT discussion.
Then this internal d.o.f. corresponds to the
entropy of AdS blackhole. (~ area of horizon)
Based on the recent research of AGT relation and its generalization, not a few researchers
now hope that
[Alday-Gaiotto-Tachikawa ’09] [Wyllard ’09] etc.
• Toda fields on 2-dim Riemann surface (or Seiberg-Witten curve [Seiberg-Witten ’94])
• W-algebra which is the symmetry algebra of Toda field theory
bring us some new understanding on the multiple M5-branes’ internal d.o.f !
3
Intersecting M5-branes’ system makes 4-dim spacetime and 2-dim surface.
• From the condition of 11-dim supergravity (i.e. intersection rule), the intersection
surface of two bundles of M5-branes at right angles must be 3-dim space.
• In this 3-dim space (i.e. 4-dim spacetime), N=2 gauge theory lives. (We see this next.)
In this time, M5-branes keep only ½ x ½ SUSYs.
• The remaining part of M5-branes becomes 2-dim surface (complex 1-dim curve).
• Since it is believed that M5-branes’ worldvolume theory is conformal (from AdS/CFT),
if 4-dim gauge theory is conformal, the theory on this 2-dim surface (called as the
Seiberg-Witten curve) must also be conformal field theory.
This is Seiberg-Witten system. [Seiberg-Witten ’94]
?
bundle of M5-branes
0,1,2,3
4,5
6,10
4
Seiberg-Witten curve determines the field contents of 4-dim gauge theory.
• Now we compactify 1-dim space out of 11-dim spacetime, and go to the D4-NS5 system
in superstring theory, since we have very little knowledge on M5-brane.
• In string theory, (vibration modes) of F1-strings describe the gauge and matter fields.
• The fields of this gauge theory are composed by F1-strings moving in 4-dim spacetime.
4,5
D4-brane (M5-brane)
flavor brane
color brane
flavor brane
(length = infinite) (length ~ 1/coupling)
[Seiberg-Witten ’94]
6, 10
D6-brane
7,8,9
more generally…
antifund. gauge bifund.
fund.
F1-string
gluons / quarks
NS5-brane (M5-brane)
increasing
(from Hanany-Witten’s discussion)
increasing
• In general, gauge group is SU(d1) x SU(d2) x … x SU(N) x … x SU(N) x … x SU(d’2) x SU(d’1).
This theory is conformal, when # of D6-branes is
.
5
A kind of ‘deformations’ makes clear the structure of Seiberg-Witten curve.
• To see the structure of Seiberg-Witten curve, now we move each D4-brane for
longitudinal direction of NS5-branes to each distance.
• After this ‘deformation’, the gauge fields get VEV’s, and the matter fields get masses.
(This means, of course, that the gauge theory is no longer conformal.)
• In general cases, the Seiberg-Witten curve is described in terms of a polynomial as
~ direction of D4
~ direction of NS5
Note that
 The coefficient of y N is 1. : normalization which causes the divergence of
!
 The y N-1 term doesn’t exist. : suitable shift of coordinates
6
Contents
1. Introduction
(pp.2-6)
2. Gaiotto’s discussion
3. AGT relation
(pp.8-10)
(pp.11-17)
4. Towards proof of AGT relation
(pp.18-22)
5. Towards generalized AGT relation
6. Conclusion
(pp.23-29)
(p.30)
7
Gaiotto’s discussion
Seiberg-Witten curve may be described by 2-dim conformal field theory.
When we recognize the intersecting point of D4-branes and NS5-branes as ‘punctures’,
2-dim conformal field theory can be defined on Seiberg-Witten curve.
NS5-branes
0
∞
0
[Gaiotto ’09]
∞
deformation to
2-dim sphere
multiple D4-branes
6
…
d3 – d2
d2 – d1
d1
…
10 (compactified)
…
…
…
…
…
4,5
…
…
…
…
d’3 – d’2
d’2 – d’1
d’1
(All Young tableaux are composed by N boxes.)
For gauge group : SU(d1) x SU(d2) x … x SU(N) x … x SU(N) x … x SU(d’2) x SU(d’1)
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What is the breakthrough provided by Gaiotto’s discussion?
• Therefore, 4-dim gauge theory relates to 2-dim theory at the following points :
 gauge group
 coupling const.
type of punctures at z=0 and ∞ (which are classified with Young tableaux)
length between neighboring punctures
• For example, when we infinitely lengthen a distance between punctures (i.e. take a weak
coupling limit), the following transformation occurs :
S-dual
…
SU(N)
…
…
…
…
SU(N)
SU(N)
…
• Also, he strongly suggested that the larger class of 4-dim gauge theories than those
described by brane configurations in string theory can be recognized as the 2-dim
compactification of multiple M5-branes’ system. For example, famous(?) TN theory.
9
What is the breakthrough provided by Gaiotto’s discussion?
• TN theory is obtained as S-dual of SU(N) quiver gauge theory, as follows :
TN
…
…
interchange
lengthen
…
…
…
…
…
…
…
…
In other words,
…
SU(N)
SU(N)
SU(N)
…
SU(N)
SU(N)
SU(N)
U(1)
U(1)
U(1)
U(1)
SU(N)
SU(N)
SU(N)
SU(N-1)
U(1)
…
U(1)
SU(3)
SU(2)
U(1)
U(1)
• However, in the following, we concentrate on the systems of brane configuration,
i.e. the cases where 4-dim theory is a quiver gauge theory.
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AGT relation
AGT relation reveals the concrete correspondence between partition function of 4-dim
SU(2) quiver gauge theory and correlation function of 2-dim Liouville theory.
1. The partition function of 4-dim gauge theory
 Action (Besides the classical part…)
 1-loop correction : more than 1-loop is cancelled, because of N=2 supersymmetry.
 instanton correction : Nekrasov’s calculation with Young tableaux
 Parameters
(Sorry, they are different from Gaiotto’s ones!)
 coupling constants
 masses of fundamental / antifund. / bifund. fields and VEV’s of gauge fields
link
 Nekrasov’s deformation parameters : background of graviphoton
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1-loop part of partition function of 4-dim quiver gauge theory
We can obtain it of the analytic form :
gauge
VEV
where
antifund.
bifund.
mass
mass
fund.
mass
deformation parameters
< Case of SU(N) x SU(N’) >
: 1-loop part can be written in terms of double Gamma function!
12
Instanton part of partition function of 4-dim quiver gauge theory
We obtain it of the expansion form of instanton number :
where
Young tableau
: coupling const. and
< Case of instanton # = 1 >
+
where
(fractions of simple polynomials)
13
2. The correlation function of 2-dim field theory
• We put the (primary) vertex operators
at punctures, and consider the
correlation functions of them:
• In general, the following expansion is valid:
primaries
descendants
For the case of Virasoro algebra,
, and e.g. for level-2,
: Shapovalov matrix
• It means that all correlation functions consist of 3-point function and propagator, and
the intermediate states (i.e. descendant fields) can be classified by Young tableaux.
 Parameters (They correspond to parameters of 4-dim gauge theory!)
 position of punctures
 momentum
of vertex operators for internal / external lines
 central charge of the field theory
14
Correlation function of 2-dim conformal field theory
We obtain it of the factorization form of 3-point functions and propagators :
 3-point function
where
highest weight
~ simple punc.
 propagator (2-point function) : inverse Shapovalov matrix
15
AGT relation : SU(2) gauge theory  Liouville theory !
[Alday-Gaiotto-Tachikawa ’09]
 4-dim theory : SU(2) quiver gauge theory
 2-dim theory : Liouville (SU(2) Toda) field theory
In this case, the 4-dim theory’s partition function Z and the 2-dim theory’s correlation
function correspond each other :
Gauge theory
coupling const.
Liouville theory
position of punctures
VEV of gauge fields
momentum of internal lines
mass of matter fields
momentum of external lines
1-loop part
DOZZ factors
instanton part
conformal blocks
deformation parameters
Liouville parameters
central charge :
16
Natural expectation : SU(N>2) gauge theory  SU(N) Toda theory… !?
 4-dim theory : SU(N) quiver gauge theory
[Wyllard ’09]
[Kanno-Matsuo-SS-Tachikawa ’09]
 2-dim theory : SU(N) Toda field theory
• Similarly, we want to study on correspondence between partition function of 4-dim
theory and correlation function of 2-dim theory :
• This discussion is somewhat complicated, since in these cases, punctures are classified
with more than one kinds of Young tableaux (which composed by N boxes) :
< full-type >
< simple-type >
< other types >
…
…
…
…
… …
…
(cf. In SU(2) case, all these Young tableaux become ones of the same type
.)
17
Towards proof of AGT relation
(or background physics)
6-dim :
Multiple M5-branes’ worldvolume theory
Contradiction? of
compactification and
coupling constant…
4-dim :
2-dim :
Correspondence of
worldvolume anomaly
and central charge
SU(N) quiver gauge theory
[Alday-Benini-Tachikawa ’09]
SU(N) Toda field theory
<concrete calculations>
Conformal blocks, Dotsenko-Fateev
integral, Selberg integral, …
[Mironov-Morozov-Shakirov-… ’09, ’10]
0-dim :
Dijkgraaf-Vafa matrix model
~ ‘quantization’ of Seiberg-Witten curve?
18
Existence of Toda fields? : multiple M5-branes’ worldvolume anomaly
First, we remember how the anomaly is cancelled in the single M5-brane’s case.
For example, [Berman ’07] for a review.
 worldvolume fields : bosons (5 d.o.f.) / fermions (8 d.o.f.) / self-dual 2-form field (3 d.o.f.)
 inflow mechanism (interaction term in the 11-dim supergravity action at 1-loop level in l p) :
 Chern-Simons interaction (which needs careful treatment because of presence of M5-branes) :
Therefore, when we naively consider, in the case of (multiple) N M5-branes’ case,
xN
x
N3
Cancellation doesn’t work!! (T_T)
It is believed that this is an indication of some extra fields on M5-branes’ worldvolume :
19
Existence of Toda fields? : multiple M5-branes’ worldvolume anomaly
• This story is related to AGT relation, if we compactify M5-branes’ worldvolume on 4-dim
[Alday-Benini-Tachikawa ’09]
space X4. We define 2-dim anomaly by integrating I8 over X4:
• On the spacetime symmetry, we consider the following situation:
TW
NW
• We twist R5 over X4 so that N=2 supersymmetry on X4 remains. In this case, N=(0,2)
supersymmetry with U(1) R-symmetry remains on
. The general form of anomaly is
F : external U(1) bundle
coupling to U(1)R symmetry
• Especially, in the case of
with Nekrasov’s deformation
,
This is precisely the same as central charge of Toda theory!
(from AGT relation)
20
Toda theory is quantum theory of SW curve? : Dijkgraaf-Vafa matrix model
• We consider 4-dim and 2-dim system in type IIB string theory.
[Dijkgraaf-Vafa ’09]
 4-dim : Topological strings on Calabi-Yau 3-fold
 2-dim : Seiberg-Witten curve embedded in Calabi-Yau 3-fold
• Dijkgraaf-Vafa matrix model may provide a bridge between them.
 matrix model is powerful tool of description of topological B-model strings.
 matrix model is also related to Liouville and Toda systems (, as we will see concisely).
• Concretely, the partition function of 4-dim theory and the correlation function of 2-dim
theory may be connected via the partition function of matrix model :
where
,
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Toda theory is quantum theory of SW curve? : Dijkgraaf-Vafa matrix model
• It is known that the free fermion system (
) can describe the system of creation and
annihilation of D-branes which are extended, for example, as
• To define this system, we ‘quantize’ Seiberg-Witten curve as
, so the following
chiral path integral must be given naturally :
• On the other hand, it is known that x classically act on fermions as
• To sum up, in ‘quantum’ theory, x may be represented as
• This means that an additional term is given in chiral path integral :
When we bosonize the fermions, this additional term is nothing but the Toda potential !
22
Towards generalized AGT relation
• In the previous section, we saw some evidence(?) that Toda fields live on SeibergWitten curve or multiple M5-branes’ worldvolume.
• Now let us return the discussion on generalization of AGT relation. To do this, we need
to consider…
 momentum
of Toda fields in vertex operators
:
Again, in SU(N>2) case, we need to determine the form of vertex operators which corresponds to
each kind of punctures (classified with Young tableaux).
 how to calculate the conformal blocks of W-algebra: 3pt functions and propagators
 correspondence between parameters of SU(N) quiver gauge theory and those of
SU(N) Toda field theory
23
What is SU(N) Toda field theory? : some extension of Liouville field theory
• In this theory, there are energy-momentum tensor
and higher spin fields
as Noether currents.
• The symmetry algebra of this theory is called W-algebra.
• For the simplest example, in the case of N=3, the generators are defined as
And, their commutation relation is as follows:
For simplicity, we ignore
Toda potential (interaction)
at this present stage.
which can be regarded as the extension of Virasoro algebra, and where
,
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As usual, we compose the primary, descendant, and null fields.
• The primary fields are defined as
acting
/
, so the descendant fields are composed by
on the primary fields as uppering / lowering operators.
• First, we define the highest weight state as usual :
Then we act lowering operators on this state, and obtain various descendant fields as
• However, (special) linear combinations of descendant fields accidentally satisfy the
highest weight condition. Such states are called null states. For example, the null states in
level-1 descendant fields are
• As we will see next, we found the fact that this null state in W-algebra is closely related
to the singular behavior of Seiberg-Witten curve near the punctures. That is, Toda fields
whose existence is predicted by AGT relation may describe the form (or behavior) of
Seiberg-Witten curve.
25
The singular behavior of SW curve is related to the null fields of W-algebra.
[Kanno-Matsuo-SS-Tachikawa ’09]
• As we saw, Seiberg-Witten curve is generally represented as
~ direction of D4
and Laurent expansion near z=z0 of the coefficient function
~ direction of NS5
is generally
• This form is similar to Laurent expansion of W-current (i.e. definition of W-generators)
• Also, the coefficients satisfy the similar equation, except the full-type puncture’s case
null condition
This correspondence becomes exact, when we take some ‘classical’ limit :
(which is related to Dijkgraaf-Vafa’s discussion on free fermion’s system?)
• This fact strongly suggest that vertex operators corresponding non-full-type punctures
must be the primary fields which has null states in their descendant fields.
26
The punctures on SW curve corresponds to the ‘degenerate’ fields!
• If we believe this suggestion, we can conjecture the form of
momentum
of Toda field
[Kanno-Matsuo-SS-Tachikawa ’09]
in vertex operators
, which
corresponds to each kind of punctures.
• To find the form of vertex operators which have the level-1 null state, it is useful to
define the screening operator (a special type of vertex operator)
• We can easily show that the state
satisfies the highest weight
condition, since the screening operator commutes with all the W-generators.
(Note that the screening operator itself has non-zero momentum.)
• This state doesn’t vanish, if the momentum
satisfies
for some j. In this case, the vertex operator has a null state at level
.
27
The punctures on SW curve corresponds to the ‘degenerate’ fields!
• Therefore, when we write the simple root as
the condition of level-1 null state becomes
(as usual),
for some j.
• It means that the general form of mometum of Toda fields satisfying this null state
condition is
.
Note that this form naturally corresponds to Young tableaux
.
• More generally, the null state condition can be written as
(The factors
are abbreviated, since they are only the images under Weyl transformation.)
• Moreover, from physical state condition (i.e. energy-momentum is real), we need to
choose
Here,
, instead of naive generalization of Liouville case
.
is the same form of β,
is Weyl vector, and
.
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Our plans of current and future research on generalized AGT relation
 Case of SU(3) quiver gauge theory
 SU(3) : already checked successfully.
[Wyllard ’09] [Mironov-Morozov ’09]
 SU(3) x … x SU(3) : We checked 1-loop part, and now calculate instanton part.
 SU(3) x SU(2) : We check it now, but correspondence seems very complicated!
 Case of SU(4) quiver gauge theory
• In this case, there are punctures which are not full-type nor simple-type.
• So we must discuss in order to check our conjucture (of the simplest example).
• The calculation is complicated because of W4 algebra, but is mostly streightforward.
 Case of SU(∞) quiver gauge theory
• In this case, we consider the system of infinitely many M5-branes, which may relate to
AdS dual system of 11-dim supergravity.
• AdS dual system is already discussed using LLM’s droplet ansatz, which is also governed
by Toda equation.
[Gaiotto-Maldacena ’09]
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Conclusion

It is well known that Seiberg-Witten system can be regarded as the multiple M5branes’ system. This system is composed by intersecting M5-branes, and can be
described by (direct sum? of) 4-dim quiver gauge theory and 2-dim conformal
field theory on Seiberg-Witten curve.

Recently, it was strongly suggested that the partition function of 4-dim theory
and the correlation function of 2-dim theory closely correspond to each other. In
particular, this correspondence requires that Toda (or Liouville) field should live
in 2-dim theory on Seiberg-Witten curve.

We showed that the singular behavior of SW curve near punctures corresponds
to the composition of null states in W-algebra. Also, we conjectured the
momentum of vertex operators corresponding each kind of punctures.

Again, we expect that this subject brings us new understanding on M5-branes!
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