slides

Report
Exact Results for perturbative
partition functions of theories
with SU(2|4) symmetry
Shinji Shimasaki
(Kyoto University)
Based on the work in collaboration with
Y. Asano (Kyoto U.), G. Ishiki (YITP) and T. Okada(YITP)
JHEP1302, 148 (2013) (arXiv:1211.0364[hep-th])
and the work in progress
Introduction
Localization
Localization method is a powerful tool to
exactly compute some physical quantities
in quantum field theories.
i.e. Partition function, vev of Wilson loop in
super Yang-Mills (SYM) theories in 4d,
super Chern-Simons-matter theories in 3d,
SYM in 5d, …
M-theory(M2, M5-brane), AdS/CFT,…
In this talk, I’m going to talk about
localization for
SYM theories with SU(2|4) symmetry.
• gauge/gravity correspondence
for theories with SU(2|4) symmetry
• Little string theory ((IIA) NS5-brane)
Theories with SU(2|4) sym.
Consistent truncations of N=4 SYM on RxS3.
[Lin,Maldacena]
N=4 SYM on RxS3/Zk (4d)
“holonomy”
N=8 SYM on RxS2 (3d)
“monopole”
[Maldacena,Sheikh-Jabbari,Raamsdonk]
plane wave matrix model (1d)
(PWMM)
“fuzzy sphere”
[Berenstein,Maldacena,Nastase][Kim,Klose,Plefka]
 mass gap, many discrete vacua, SU(2|4) sym.(16 SUSY)
 gravity dual corresponding to each vacuum of each theory
is constructed (bubbling geometry in IIA SUGRA) [Lin,Maldacena]
 SYM on RxS2 and RxS3/Zk from PWMM
[Ishiki,SS,Takayama,Tsuchiya]
Theories with SU(2|4) sym.
Consistent truncations of N=4 SYM on RxS3.
N=4 SYM on RxS3/Zk (4d)
T-duality in
gauge theory
[Lin,Maldacena]
“holonomy”
[Taylor]
N=8 SYM on RxS2 (3d)
commutative limit
of fuzzy sphere
“monopole”
[Maldacena,Sheikh-Jabbari,Raamsdonk]
plane wave matrix model (1d)
(PWMM)
“fuzzy sphere”
[Berenstein,Maldacena,Nastase][Kim,Klose,Plefka]
 mass gap, many discrete vacua, SU(2|4) sym.(16 SUSY)
 gravity dual corresponding to each vacuum of each theory
is constructed (bubbling geometry in IIA SUGRA) [Lin,Maldacena]
 SYM on RxS2 and RxS3/Zk from PWMM
[Ishiki,SS,Takayama,Tsuchiya]
Our Results
Asano, Ishiki, Okada, SS
JHEP1302, 148 (2013)
• Using the localization method,
we compute the partition function of PWMM
up to instantons;
where
: vacuum configuration
characterized by
In the ’t Hooft limit, our result becomes exact.
•
is written as a matrix integral.
• We check that our result reproduces a one-loop
result of PWMM.
Our Results
Asano, Ishiki, Okada, SS
JHEP1302, 148 (2013)
• We also obtain the partition functions of N=8
SYM on RxS2 and N=4 SYM on RxS3/Zk from
that of PWMM by taking limits corresponding
to “commutative limit of fuzzy sphere” and
“T-duality in gauge theory”.
• We show that, in our computation, the partition
function of N=4 SYM on RxS3(N=4 SYM on
RxS3/Zk with k=1) is given by the gaussian
matrix model.
This is consistent with the known result of
N=4 SYM. [Pestun; Erickson,Semenoff,Zarembo; Drukker,Gross]
Application of our result
Work in progress; Asano, Ishiki, Okada, SS
• gauge/gravity correspondence for theories with
SU(2|4) symmetry
• Little string theory on RxS5
Plan of this talk
1. Introduction
2. Theories with SU(2|4) symmetry
3. Localization in PWMM
4. Exact results of theories
with SU(2|4) symmetry
5. Application of our result
6. Summary
Theories with
SU(2|4) symmetry
N=4 SYM on RxS3
: gauge field
: scalar field
(adjoint rep)
+ fermions
• vacuum
all fields=0
(Local Lorentz indices of RxS3)
N=4 SYM on RxS3
Hereafter we focus on the spatial part (S3) of the gauge fields.
Local Lorentz indices of S3
where
convention for S3
right inv. 1-form:
metric:
N=4 SYM on RxS3/Zk
Keep the modes with the periodicity
in N=4 SYM on RxS3.
• vacuum
“holonomy”
N=8 SYM on RxS2
Angular momentum op.
on S2
N=8 SYM on RxS2
In the second line we rewrite
and the scalar field on S2 as
• vacuum
in terms of the gauge fields
.
“Dirac monopole” monopole charge
plane wave matrix model
plane wave matrix model
• vacuum
“fuzzy sphere”
: spin
rep. matrix
Relations among theories
with SU(2|4) symmetry
N=4 SYM on RxS3/Zk (4d)
T-duality in
gauge theory
[Taylor]
N=8 SYM on RxS2 (3d)
commutative limit
of fuzzy sphere
Plane wave matrix model (1d)
N=8 SYM on RxS2 from PWMM
N=4 SYM on RxS3/Zk (4d)
N=8 SYM on RxS2 (3d)
commutative limit
of fuzzy sphere
Plane wave matrix model (1d)
N=8 SYM on RxS2 from PWMM
 PWMM around the following fuzzy sphere vacuum
with
fixed
 N=8 SYM on RxS2 around the following monopole vacuum
N=8 SYM on RxS2
around a monopole vacuum
• monopole vacuum
• Expand the fields around a monopole vacuum
• Decompose fields into blocks according to the block
structure of the vacuum
(s,t) block
matrix
N=8 SYM on RxS2
around a monopole vacuum
: Angular momentum op. in the presence
of a monopole with charge
PWMM around a fuzzy sphere
vacuum
• fuzzy sphere vacuum
• Expand the fields around a fuzzy sphere vacuum
• Decompose fields into blocks according to the block
structure of the vacuum
(s,t) block
matrix
PWMM around a fuzzy sphere
vacuum
 N=8 SYM on RxS2 around a monopole vacuum
: Angular momentum op. in the presence
of a monopole with charge
 PWMM around a fuzzy sphere vacuum
Spherical harmonics
 monopole spherical harmonics (basis of sections of a line bundle on S2)
[Wu,Yang]
 fuzzy spherical harmonics
(basis of
rectangular matrix )
[Grosse,Klimcik,Presnajder; Baez,Balachandran,Ydri,Vaidya; Dasgupta,Sheikh-Jabbari,Raamsdonk;…]
with
fixed
Mode expansion
 N=8 SYM on RxS2
Expand
in terms of the monopole spherical harmonics
 PWMM
Expand
in terms of the fuzzy spherical harmonics
N=8 SYM on RxS2 from PWMM
 N=8 SYM on RxS2 around a monopole vacuum
 PWMM around a fuzzy sphere vacuum
N=8 SYM on RxS2 from PWMM
 N=8 SYM on RxS2 around a monopole vacuum
 PWMM around a fuzzy sphere vacuum
In the limit in which
with
PWMM coincides with N=8 SYM on RxS2.
fixed
N=4 SYM on RxS3/Zk
from N=8 SYM on RxS2
N=4 SYM on RxS3/Zk (4d)
T-duality in
gauge theory
[Taylor]
N=8 SYM on RxS2 (3d)
Plane wave matrix model (1d)
N=4 SYM on RxS3/Zk
from N=8 SYM on RxS2
 N=8 SYM on RxS2 around the following monopole vacuum
with
Identification among blocks of fluctuations (orbifolding)
 (an infinite copies of) N=4 SYM on RxS3/Zk around
the trivial vacuum
N=4 SYM on RxS3/Zk
from N=8 SYM on RxS2
N=4 SYM on RxS3/Zk
(S3/Zk : nontrivial S1 bundle over S2)
KK expand along S1 (locally)
N=8 SYM on RxS2 with infinite number of KK modes
• These KK mode are sections of line bundle on S2
and regarded as fluctuations around a monopole
background in N=8 SYM on RxS2.
(monopole charge = KK momentum)
• N=4 SYM on RxS3/Zk can be obtained by expanding
N=8 SYM on RxS2 around an appropriate monopole
background so that all the KK modes are reproduced.
N=4 SYM on RxS3/Zk
from N=8 SYM on RxS2
This is achieved in the following way.
Extension of Taylor’s T-duality to that on nontrivial fiber bundle
[Ishiki,SS,Takayama,Tsuchiya]
• Expand N=8 SYM on RxS2 around the following monopole vacuum
with
• Make the identification among blocks of fluctuations (orbifolding)
• Then, we obtain (an infinite copies of) N=4 U(N) SYM on RxS3/Zk.
Plan of this talk
1. Introduction
2. Theories with SU(2|4) symmetry
3. Localization in PWMM
4. Exact results of theories
with SU(2|4) symmetry
5. Application of our result
6. Summary
Localization in
PWMM
Localization
Suppose that
[Witten; Nekrasov; Pestun; Kapustin et.al.;…]
is a symmetry
and there is a function
Define
is independent of
such that
one-loop integral around the saddle points
We perform the localization in PWMM
following Pestun,
Plane Wave Matrix Model
Off-shell SUSY in PWMM
SUSY algebra is closed if there exist spinors
Indeed, such
[Berkovits]
: invariant under the off-shell SUSY.
•
•
exist
which satisfy
:Killing vector
Saddle point
We choose
Saddle point
where
is a constant matrix commuting with
:
const. matrix
In
,
and
are vanishing.
Saddle points are characterized by reducible
representations of SU(2),
, and
constant matrices
1-loop around a saddle point
with integral of
Instanton
The solutions to the saddle point equations we showed
are the solutions when
is finite.
In addition to these, one should also take into account
the instanton configurations localizing at
.
In
, some terms in the saddle point equations
automatically vanish.
In this case, the saddle point equations for remaining
terms are reduced to (anti-)self-dual equations.
(mass deformed Nahm equation)
[Yee,Yi;Lin;Bachas,Hoppe,Piolin]
Here we neglect the instantons.
Plan of this talk
1. Introduction
2. Theories with SU(2|4) symmetry
3. Localization in PWMM
4. Exact results of theories
with SU(2|4) symmetry
5. Application of our result
6. Summary
Exact results of
theories with
SU(2|4) symmetry
Partition function of PWMM
Partition function of PWMM
Eigenvalues of
where
with
is given by
Contribution from
the classical action
Partition function of PWMM
Trivial vacuum
(cf.) partition function of 6d IIB matrix model
[Kazakov-Kostov-Nekrasov]
[Kitazawa-Mizoguchi-Saito]
Partition function of
N=8 SYM on RxS2
In order to obtain the partition function of N=8 SYM on RxS2
from that of PWMM, we take the commutative limit of
fuzzy sphere, in which
with
fixed
Partition function of
N=8 SYM on RxS2
trivial vacuum
Partition function of
N=4 SYM on RxS3/Zk
In order to obtain the partition function of N=4 SYM on
RxS3/Zk around the trivial background from that of
N=8 SYM on RxS2, we take
such that
and impose orbifolding condition
.
Partition function of
N=4 SYM on RxS3/Zk
When
, N=4 SYM on RxS3,
the measure factors completely cancel out except for
the Vandermonde determinant.
Gaussian matrix model
Consistent with the result of N=4 SYM
[Pestun; Erickson,Semenoff,Zarembo; Drukker,Gross]
Application of
our result
• gauge/gravity duality for N=8 SYM
on RxS2 around the trivial vacuum
• NS5-brane limit
Gauge/gravity duality for N=8 SYM
on RxS2 around the trivial vacuum
Partition function of N=8 SYM on RxS2 around the trivial vacuum
This can be solved in the large-N and the large ’t Hooft
coupling limit;
The
and
dependences are consistent with
the gravity dual obtained by Lin and Maldacena.
NS5-brane limit
Based on the gauge/gravity duality by Lin-Maldacena,
Ling, Mohazab, Shieh, Anders and Raamsdonk proposed
a double scaling limit of PWMM which gives
little string theory (IIA NS5-brane theory) on RxS5.
Expand PWMM around
and take the limit in which
and
with
and
fixed
Little string theory on RxS5
(# of NS5 =
)
In this limit, instantons are suppressed.
So, we can check this conjecture by using our result.
NS5-brane limit
If this conjecture is true,
the vev of an operator
can be expanded as
We checked this numerically in the case where
and
for various
.
NS5-brane limit
is nicely fitted by
with
for various
!
Summary
Summary
• Using the localization method,
we compute the partition function of PWMM
up to instantons.
• We also obtain the partition function of
N=8 SYM on RxS2 and N=4 SYM on RxS3/Zk
from that of PWMM by taking limits corresponding
to “commutative limit of fuzzy sphere” and
“T-duality in gauge theory”.
• We may obtain some nontrivial evidence for
the gauge/gravity duality for theories with
SU(2|4) symmetry and the little string theory
on RxS5.
Future work
 take into account instantons
• N=8 SYM on RxS2
ABJM on RxS2?
• M-theory on 11d plane wave geometry
• What is the meaning of the full partition function
in the gravity(string) dual?
geometry change?
baby universe? (cf) Dijkgraaf-Gopakumar-Ooguri-Vafa
 precise check of the gauge/gravity duality
• meaning of Q-closed operator
in the gravity dual
 can we say something about NS5-brane?

similar documents