exact results for perturbative partition functions of theories with su(2|4) symmetry
DESCRIPTION
Exact Results for perturbative partition functions of theories with SU(2|4) symmetry. Shinji Shimasaki. (Kyoto University). Based on the work in collaboration w ith Y. Asano (Kyoto U.), G. Ishiki (YITP) and T. Okada(YITP). JHEP1302, 148 (2013) ( arXiv:1211.0364[ hep-th ]). - PowerPoint PPT PresentationTRANSCRIPT
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Exact Results for perturbative partition functions of theories
with SU(2|4) symmetryShinji 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 progress
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Introduction
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Localization method is a powerful tool to exactly compute some physical quantities in quantum field theories.
Localization
super 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 in
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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)
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Theories with SU(2|4) sym.
mass gap, many discrete vacua, SU(2|4) sym.(16 SUSY)
SYM on RxS2 and RxS3/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 RxS3/Zk (4d)
Consistent truncations of N=4 SYM on RxS3.
(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”
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Theories with SU(2|4) sym.
N=4 SYM on RxS3/Zk (4d)
Consistent truncations of N=4 SYM on RxS3.
(PWMM)
[Lin,Maldacena]
[Maldacena,Sheikh-Jabbari,Raamsdonk] N=8 SYM on RxS2 (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]
mass gap, many discrete vacua, SU(2|4) sym.(16 SUSY)
SYM on RxS2 and RxS3/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]
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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, SSJHEP1302, 148 (2013)
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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 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”.
Asano, Ishiki, Okada, SSJHEP1302, 148 (2013)
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Application of our result
• gauge/gravity correspondence for theories with SU(2|4) symmetry
Work in progress; Asano, Ishiki, Okada, SS
• Little string theory on RxS5
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Plan of this talk1. Introduction2. Theories with SU(2|4) symmetry3. Localization in PWMM4. Exact results of theories with SU(2|4) symmetry5. Application of our result6. Summary
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Theories with SU(2|4) symmetry
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N=4 SYM on RxS3
(Local Lorentz indices of RxS3)
• vacuum all fields=0
: gauge field: scalar field (adjoint rep)
+ fermions
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N=4 SYM on RxS3
convention for S3
right inv. 1-form:
metric:
Local Lorentz indices of S3
Hereafter we focus on the spatial part (S3) of the gauge fields.
where
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• vacuum“holonomy”
Angular momentum op. on S2
Keep the modes with the periodicityin N=4 SYM on RxS3.
N=4 SYM on RxS3/Zk
N=8 SYM on RxS2
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• vacuum “Dirac monopole”
In the second line we rewrite in terms of the gauge fieldsand the scalar field on S2 as .
plane wave matrix model
monopole charge
N=8 SYM on RxS2
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• vacuum “fuzzy sphere”
: spin rep. matrix
plane wave matrix model
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N=4 SYM on RxS3/Zk (4d)
N=8 SYM on RxS2 (3d)
Plane wave matrix model (1d)commutative limit of fuzzy sphere
Relations among theorieswith SU(2|4) symmetry
T-duality in gauge theory [Taylor]
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N=4 SYM on RxS3/Zk (4d)
N=8 SYM on RxS2 (3d)
Plane wave matrix model (1d)commutative limit of fuzzy sphere
N=8 SYM on RxS2 from PWMM
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PWMM around the following fuzzy sphere vacuum
N=8 SYM on RxS2 from PWMM
N=8 SYM on RxS2 around the following monopole vacuum
fixedwith
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N=8 SYM on RxS2 around a monopole vacuum
matrix
• Decompose fields into blocks according to the block structure of the vacuum
• monopole vacuum
(s,t) block
• Expand the fields around a monopole vacuum
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: Angular momentum op. in the presence of a monopole with charge
N=8 SYM on RxS2 around a monopole vacuum
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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 vacuum
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PWMM around a fuzzy spherevacuum
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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
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Spherical harmonics monopole spherical harmonics
fuzzy spherical harmonics
(basis of sections of a line bundle on S2)
(basis of rectangular matrix )
with fixed
[Grosse,Klimcik,Presnajder; Baez,Balachandran,Ydri,Vaidya; Dasgupta,Sheikh-Jabbari,Raamsdonk;…]
[Wu,Yang]
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Mode expansion N=8 SYM on RxS2
PWMM
Expand in terms of the monopole spherical harmonics
Expand in terms of the fuzzy spherical harmonics
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N=8 SYM on RxS2 from PWMM
PWMM around a fuzzy sphere vacuum
N=8 SYM on RxS2 around a monopole vacuum
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N=8 SYM on RxS2 from PWMM
PWMM around a fuzzy sphere vacuum
N=8 SYM on RxS2 around a monopole vacuum
fixed
In the limit in which
with
PWMM coincides with N=8 SYM on RxS2.
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N=4 SYM on RxS3/Zk (4d)
N=8 SYM on RxS2 (3d)
Plane wave matrix model (1d)
T-duality in gauge theory [Taylor]
N=4 SYM on RxS3/Zk from N=8 SYM on RxS2
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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 RxS3/Zk around the trivial vacuum
N=4 SYM on RxS3/Zk from N=8 SYM on RxS2
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N=4 SYM on RxS3/Zk from N=8 SYM on RxS2
(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
• 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.
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This is achieved in the following way.
• Expand N=8 SYM on RxS2 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 RxS3/Zk.
Extension of Taylor’s T-duality to that on nontrivial fiber bundle [Ishiki,SS,Takayama,Tsuchiya]
N=4 SYM on RxS3/Zk from N=8 SYM on RxS2
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Plan of this talk1. Introduction2. Theories with SU(2|4) symmetry3. Localization in PWMM4. Exact results of theories with SU(2|4) symmetry5. Application of our result6. Summary
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Localization in PWMM
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Localization
Suppose that is a symmetry
and there is a function such that
Define
is independent of
[Witten; Nekrasov; Pestun; Kapustin et.al.;…]
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one-loop integral around the saddle points
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We perform the localization in PWMM following Pestun,
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Plane Wave Matrix Model
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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]
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const. matrix
where
Saddle point
We choose
Saddle point
In , and are vanishing.
is a constant matrix commuting with :
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Saddle points are characterized by reducible representations of SU(2), , and constant matrices
1-loop around a saddle point with integral of
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The solutions to the saddle point equations we showed are the solutions when is finite.
In , some terms in the saddle point equationsautomatically 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]
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Plan of this talk1. Introduction2. Theories with SU(2|4) symmetry3. Localization in PWMM4. Exact results of theories with SU(2|4) symmetry5. Application of our result6. Summary
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Exact results of theories with SU(2|4) symmetry
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Partition function of PWMM
Contribution from the classical action
Partition function of PWMM with is given by
whereEigenvalues of
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Partition function of PWMMTrivial vacuum
(cf.) partition function of 6d IIB matrix model[Kazakov-Kostov-Nekrasov][Kitazawa-Mizoguchi-Saito]
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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
fixedwith
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Partition function of N=8 SYM on RxS2
trivial vacuum
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Partition function of N=4 SYM on RxS3/Zk
such thatand impose orbifolding condition .
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
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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 modelConsistent with the result of N=4 SYM
[Pestun; Erickson,Semenoff,Zarembo; Drukker,Gross]
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Application of our result
• gauge/gravity duality for N=8 SYM on RxS2 around the trivial vacuum
• NS5-brane limit
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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.
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NS5-brane limitBased 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 RxS5
(# of NS5 = )
with and fixed
In this limit, instantons are suppressed.So, we can check this conjecture by using our result.
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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 .
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NS5-brane limit
is nicely fitted by with for various !
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Summary
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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.
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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