perturbative partition functions of theories with SU24 symmetry Shinji Shimasaki Kyoto University JHEP1302 148 2013 arXiv12110364 hepth Based on the work in collaboration w ID: 427192
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Slide1
Exact Results for perturbative partition functions of theories with SU(2|4) symmetry
Shinji Shimasaki
(Kyoto University)
JHEP1302, 148 (2013) (arXiv:1211.0364[hep-th])
Based on the work in collaboration with Y. Asano (Kyoto U.), G. Ishiki (YITP) and T. Okada(YITP)
and the work in progressSlide2
IntroductionSlide3
Localization method is a powerful tool to exactly compute some physical quantities in quantum field theories.
Localizationsuper Yang-Mills (SYM) theories in 4d,super
Chern-Simons-matter theories in 3d,SYM in 5d, …M-theory(M2, M5-brane),
AdS/CFT,…
i.e. Partition function, vev of Wilson loop inSlide4
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)Slide5
Theories with SU(2|4) sym.
mass gap, many discrete vacua, SU(2|4) sym.(16 SUSY) SYM on RxS2 and RxS
3/Zk from PWMM[
Ishiki,SS,Takayama,Tsuchiya]
gravity dual corresponding to each vacuum of each theory is constructed (bubbling geometry in IIA SUGRA)[Lin,Maldacena]
N=4 SYM on RxS
3
/
Z
k
(4d)
Consistent truncations of N=4 SYM on RxS
3
.
(
PWMM
)
[
Lin,Maldacena
]
[
Maldacena
,
Sheikh-Jabbari
,
Raamsdonk
]
N=8 SYM on RxS2 (3d)
plane wave matrix model (1d)
[
Berenstein,Maldacena,Nastase
][Kim,Klose,Plefka]
“holonomy”
“monopole”
“fuzzy sphere”Slide6
Theories with SU(2|4) sym.
N=4 SYM on RxS3/
Zk (4d)Consistent truncations of N=4 SYM on RxS
3.(
PWMM)[Lin,Maldacena]
[
Maldacena
,
Sheikh-Jabbari
,
Raamsdonk
]
N=8 SYM on RxS
2
(3d)
plane wave matrix model (1d)
“
holonomy
”
“monopole”
“fuzzy sphere”
T-duality in
gauge theory
[Taylor]
commutative limit
of fuzzy sphere
[
Berenstein,Maldacena,Nastase
][
Kim,Klose,Plefka
]
m
ass gap, many discrete vacua, SU(2|4) sym.(16 SUSY)
SYM on RxS2 and RxS
3
/
Z
k
from PWMM
[
Ishiki,SS,Takayama,Tsuchiya
]
gravity dual corresponding to each vacuum of each theory
is constructed (bubbling geometry in IIA SUGRA)
[
Lin,Maldacena
]Slide7
Our Results
Using the localization method, we compute the partition function of PWMM up to instantons;
We check that our result reproduces a one-loop result of PWMM.where
: vacuum configuration
characterized by
In the ’t
Hooft
limit, our result becomes exact.
is written as a matrix integral.
Asano,
Ishiki
, Okada, SS
JHEP1302, 148 (2013)Slide8
Our Results
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]
We also obtain the partition functions of
N=8
SYM on RxS
2
and
N=4 SYM on RxS
3
/
Z
k
from
that of
PWMM
by taking limits corresponding
to
“commutative limit of fuzzy sphere”
and
“T-duality in gauge theory”.
Asano, Ishiki, Okada, SS
JHEP1302, 148 (2013)Slide9
Application of our result
gauge/gravity correspondence for theories with SU(2|4) symmetryWork in progress; Asano, Ishiki, Okada, SS
Little string theory on RxS5Slide10
Plan of this talk
1. Introduction2. Theories with SU(2|4) symmetry3. Localization in PWMM4. Exact results of
theories with SU(2|4) symmetry5. Application of our result
6. SummarySlide11
Theories with SU(2|4) symmetrySlide12
N=4 SYM on RxS
3
(Local Lorentz indices of RxS
3
)
vacuum
all fields=0
: gauge field
: scalar field
(
adjoint
rep)
+ fermionsSlide13
N=4 SYM on RxS
3
convention for S3
right inv. 1-form:
metric:
Local Lorentz indices of S
3
Hereafter we focus on the spatial part (S
3
) of the gauge fields.
whereSlide14
vacuum
“
holonomy”
Angular momentum op.
on S
2
Keep the modes with the periodicity
in N=4 SYM on RxS
3
.
N=4 SYM on RxS
3
/
Z
k
N=8 SYM on RxS
2Slide15
vacuum
“Dirac monopole”
I
n the second line we rewrite in terms of the gauge fields
and the scalar field on S
2
as .
plane wave matrix model
monopole charge
N=8 SYM on RxS
2Slide16
vacuum
“fuzzy sphere”
: spin rep. matrix
plane wave matrix modelSlide17
N=4 SYM on RxS
3/Zk (4d)
N=8 SYM on RxS2 (3d) Plane wave matrix model (1d)
commutative limit
of fuzzy sphere
Relations among theories
with SU(2|4) symmetry
T-duality in
gauge theory
[Taylor]Slide18
N=4 SYM on RxS
3/Zk (4d)
N=8 SYM on RxS2 (3d) Plane wave matrix model (1d)
commutative limit
of fuzzy sphere
N=8 SYM on RxS
2
from PWMMSlide19
PWMM around the following fuzzy sphere vacuum
N=8 SYM on RxS
2
from PWMM
N=8 SYM on RxS
2
around the following monopole vacuum
fixed
withSlide20
N=8 SYM on RxS2
around a monopole vacuummatrix
Decompose fields into blocks according to the block structure of the vacuum
m
onopole vacuum
(
s,t
) block
Expand the fields around a monopole vacuumSlide21
: Angular momentum op. in the presence
of
a
monopole
with charge
N=8 SYM on RxS
2
around a monopole vacuumSlide22
PWMM around a fuzzy spherevacuum
fuzzy sphere vacuum
Decompose fields into blocks according to the block
structure of the vacuum
matrix
(
s,t
) block
Expand the fields around a fuzzy sphere vacuumSlide23
PWMM around a fuzzy sphere
vacuumSlide24
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 Slide25
Spherical harmonics
monopole spherical harmonics
fuzzy spherical harmonics
(basis of sections of a line bundle on S
2
)
(basis of rectangular matrix )
with
fixed
[
Grosse,Klimcik,Presnajder
;
Baez,Balachandran,Ydri,Vaidya
;
Dasgupta,Sheikh-Jabbari,Raamsdonk
;…]
[
Wu,Yang
]Slide26
Mode expansion N=8 SYM on RxS2
PWMM
Expand in terms of the
monopole spherical harmonics
Expand in terms of the fuzzy spherical harmonicsSlide27
N=8 SYM on RxS2 from PWMM
PWMM around a fuzzy sphere vacuum
N=8 SYM on RxS2 around a monopole vacuumSlide28
N=8 SYM on RxS
2
from PWMM PWMM around a fuzzy sphere vacuum
N=8 SYM on RxS
2 around a monopole vacuum
fixed
In the limit in which
with
PWMM
coincides with
N=8 SYM on RxS
2
.Slide29
N=4 SYM on RxS
3/Zk (4d)
N=8 SYM on RxS2 (3d) Plane wave matrix model (1d)
T-duality in
gauge theory
[Taylor]
N=4 SYM on RxS
3
/
Z
k
from N=8 SYM on RxS
2Slide30
N=8 SYM on RxS2 around the following monopole vacuum
Identification among blocks of fluctuations (
orbifolding
)
with
(an infinite copies of)
N=4 SYM on RxS
3
/
Z
k
around
the trivial vacuum
N=4 SYM on RxS
3
/
Z
k
from N=8 SYM on RxS
2Slide31
N=4 SYM on RxS3/
Zk from N=8 SYM on RxS
2(S3/
Zk : nontrivial S1 bundle over S2)
KK expand along S1 (locally)
N=8 SYM on RxS
2
with infinite number of KK modes
These KK mode are sections of line bundle on S
2
and regarded as fluctuations around a monopole
background in N=8 SYM on RxS
2
.
(monopole charge = KK momentum)
N=4 SYM on RxS
3
/
Z
k
N=4 SYM on RxS
3
/
Z
k
can be obtained by expanding
N=8 SYM on RxS
2 around an appropriate monopole background so that all the KK modes are reproduced.Slide32
This is achieved in the following way.
Expand N=8 SYM on RxS
2
around the following monopole vacuum
Make the identification among blocks of fluctuations (
orbifolding
)
with
Then, we obtain (an infinite copies of) N=4 U(N) SYM on RxS
3
/
Z
k
.
Extension of Taylor’s T-duality to that on nontrivial fiber bundle
[
Ishiki
,SS,Takayama,Tsuchiya
]
N=4 SYM on RxS
3
/
Z
k
from N=8 SYM on RxS
2Slide33
Plan of this talk
1. Introduction2. Theories with SU(2|4) symmetry
3. Localization in PWMM4. Exact results of theories w
ith SU(2|4) symmetry5. Application of our result6
. SummarySlide34
Localization in PWMMSlide35
Localization
Suppose that is a symmetry
and there is a function such that
Define
is independent of
[
Witten;
Nekrasov
;
Pestun
; Kapustin et.al.;…
]Slide36
one-loop integral around the saddle pointsSlide37
We perform the localization in PWMM following Pestun,Slide38
Plane Wave Matrix ModelSlide39
Off-shell SUSY in PWMM
SUSY algebra is
closed
if there exist
spinors
which satisfy
Indeed, such exist
: invariant under the off-shell SUSY.
:Killing vector
[
Berkovits
]Slide40
c
onst. matrix
where
Saddle
point
We choose
Saddle point
In , and are vanishing.
is a constant matrix commuting with : Slide41
Saddle points are characterized by reducible representations of SU(2), , and constant matrices
1-loop around a saddle point with integral ofSlide42
The solutions to the saddle point equations we showed are the solutions when is finite.
In , some terms in the saddle point equations
automatically vanish.
In this case, the saddle point equations for remainingterms are reduced to (anti-)self-dual equations.
(mass deformed Nahm equation)
In addition to these, one should also take into account
the
instanton
configurations localizing at .
Here we neglect the
instantons
.
Instanton
[
Yee,Yi;Lin;Bachas,Hoppe,Piolin
]Slide43
Plan of this talk
1. Introduction2. Theories with SU(2|4) symmetry
3. Localization in PWMM4. Exact results of theories w
ith SU(2|4) symmetry5. Application of our result6
. SummarySlide44
Exact results of theories with SU(2|4) symmetrySlide45
Partition function of PWMM
Contribution from
the classical action
Partition function of PWMM w
ith is given by
where
Eigenvalues
ofSlide46
Partition function of PWMM
Trivial vacuum
(cf.) partition function of 6d IIB matrix model[Kazakov-Kostov-Nekrasov]
[Kitazawa-Mizoguchi-Saito
]Slide47
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
fixed
withSlide48
Partition function of
N=8 SYM on RxS2
trivial vacuumSlide49
Partition function of
N=4 SYM on RxS
3/Z
k
such thatand impose orbifolding condition .
In order to obtain the partition function of N=4 SYM on
RxS
3
/
Z
k
around the trivial background from that of
N=8 SYM on RxS
2
, we take Slide50
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
]Slide51
Application of our result gauge/gravity duality
for N=8 SYM on RxS2 around the trivial vacuum
NS5-brane limitSlide52
Gauge/gravity duality for N=8 SYM on RxS2
around the trivial vacuumPartition 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
.Slide53
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 giveslittle string theory (IIA NS5-brane theory) on RxS5.
Expand PWMM around and take the limit in which
and
Little string theory on RxS
5
(# of
NS5 = )
with
and
fixed
In this limit,
instantons
are suppressed.
So, we can check this conjecture by using our result.Slide54
If this conjecture is true,the vev of an operator can be expanded as
NS5-brane limit
We checked this numerically in the case where
and
for various .Slide55
NS5-brane limit
is nicely fitted by with for various ! Slide56
SummarySlide57
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 RxS
5
.Slide58
Future work take into account instantons
N=8 SYM on RxS2 ABJM on RxS2?
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
can we say something about NS5-brane?
meaning of Q-closed operator in the gravity
dual
M-theory on 11d plane wave geometry