exact results of theories with su(2|4) symmetry and … · 2013. 5. 13. · super yang-mills (sym)...
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Exact Results of theories with SU(2|4) symmetry
and gauge/gravity correspondence
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 progress
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Introduction
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Localization method is a powerful tool to exactly compute vev of some particular operators 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 on RxS5 (IIA NS5-brane)
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Theories with SU(2|4) sym.
mass gap, many discrete vacua, SU(2|4) sym.(16 SUSY)
SU(2|4) theories from PWMM [Ishiki,SS,Takayama,Tsuchiya]
gravity duals corresponding to each vacuum of each theory are constructed (bubbling geometry in IIA SUGRA) [Lin,Maldacena]
N=4 SYM on RxS3/Zk (4d)
Consistent truncations of N=4 SYM on RxS3.
(BMN)
[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.
(BMN)
[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)
SU(2|4) theories from PWMM [Ishiki,SS,Takayama,Tsuchiya]
gravity duals corresponding to each vacuum of each theory are 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 checked our result by comparing with perturbative computation.
where : vacuum configuration characterized by
In the ’t Hooft limit, our result becomes exact.
• is written as a matrix integral.
Asano, Ishiki, Okada, SS
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Our Results
Partition func. = gaussian matrix model
• We checked our result in the k=1 case of N=4 SYM on RxS3/Zk, N=4 SYM on RxS
3, by comparing with the known result of N=4 SYM. [Pestun; Erickson,Semenoff,Zarembo; Drukker,Gross]
• 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”.
Asano, Ishiki, Okada, SS
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Applications of our results
• gauge/gravity correspondence for theories with SU(2|4) symmetry
Work in progress; Asano, Ishiki, Okada, SS
• Little string theory on RxS5
We will discuss the gauge/gravity correspondence for N=8 SYM on RxS2 around the trivial vacuum.
This theory is also one of theories with SU(2|4) symmetry.
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Plan of this talk
1. Introduction
2. Theories with SU(2|4) symmetry
3. gauge/gravity correspondence for theories
with SU(2|4) symmetry
4. Localization in PWMM
5. Exact results for theories
with SU(2|4) symmetry
6. Application of our result
7. Summary
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Theories with SU(2|4) symmetry
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Theories with SU(2|4) sym.
N=4 SYM on RxS3
convention for S3
right inv. 1-from:
metric:
Local Lorentz indices of spatial directions
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• vacuum “holonomy”
N=4 SYM on RxS3/Zk
Angular momentum op. on S2
Keep the modes with the periodicity in N=4 SYM on RxS3.
Theories with SU(2|4) sym.
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N=8 SYM on RxS2
• vacuum “monopole”
In the second line we rewrite in terms of the gauge fields and the scalar field as .
Plane wave matrix model
monopole charge
Theories with SU(2|4) sym.
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Plane wave matrix model (PWMM)
• vacuum “fuzzy sphere”
: spin rep. matrix
Theories with SU(2|4) sym.
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Plan of this talk
1. Introduction
2. Theories with SU(2|4) symmetry
3. gauge/gravity correspondence for theories
with SU(2|4) symmetry
4. Localization in PWMM
5. Exact results for theories
with SU(2|4) symmetry
6. Application of our result
7. Summary
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gauge/gravity correspondence for theories
with SU(2|4) symmetry
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general solutions with 16 SUSY and RxSO(3)xSO(6) isometry
black region surrounded by white region
Lin-Lunin-Maldacena geometry
black region : shrinks
white region : shrinks
white region surrounded by black region
M2 flux
M5 flux
• 11d SUGRA
translational invariance along 10d IIA SUGRA
“bubbling geometry”
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Lin-Maldacena geometry
• 10d IIA SUGRA
black region surrounded by white region
white region surrounded by black region
D2 flux
NS5 flux
general solutions with 16 SUSY and RxSO(3)xSO(6) isometry
black region : shrinks
white region : shrinks
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Lin-Maldacena geometry
• 10d IIA SUGRA
identify
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vacua of SYM on RxS2
vacua of SYM on RxS3/Zk
vacua of PWMM
Lin-Maldacena geometry
• 10d IIA SUGRA
identify
D2 flux
NS5 flux
D2 flux
NS5 flux
D2 flux
NS5 flux
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Lin-Maldacena geometry
• 10d IIA SUGRA
vacua of little string theory on RxS5
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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]
commutative limit of fuzzy sphere
The relation among theories with SU(2|4) symmetry can be seen from the gravity duals.
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D2 flux
NS5 flux
D2 flux
NS5 flux
• PWMM around • N=8 SYM on RxS2 around the trivial vacuum
N=8 SYM on RxS2 from PWMM
In field theory language, this limit corresponds to the commutative limit of fuzzy sphere.
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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
fixed with
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• N=8 SYM on RxS2 around
In field theory language, this procedure corresponds to the Taylor’s T–duality.
with • N=4 SYM on RxS3/Zk around the trivial vauum
identify periodic
extract one period
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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Little string theory on RxS5 from PWMM ?
D2 flux
NS5 flux
D2 flux
NS5 flux
• PWMM around • Little string theory on RxS5 around the trivial vacuum
I will discuss this later..
[Ling, Mohazab, Shieh, Anders, Raamsdonk]
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Plan of this talk
1. Introduction
2. Theories with SU(2|4) symmetry
3. gauge/gravity correspondence for theories
with SU(2|4) symmetry
4. Localization in PWMM
5. Exact results for theories
with SU(2|4) symmetry
6. Application of our result
7. 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 by constructing equivariant cohomology 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.
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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 equations automatically vanish.
In this case, the saddle point equations for remaining terms 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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: gauge transformation with parameter ;
: U(1) transformation (diagonal U(1) subgroup of SO(3)xSO(6)R)
Change of variables of fermion
Rewrite in the basis
SUSY transformation is rewritten as
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In order to perform gauge-fixing simultaneously, we define and add ghosts
Field contents
SUSY transformation
SUSY+BRST
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: linear diff. op. depend on
1-loop around saddle point
Relevant part of the 1-loop computation
Fluctuations are vanishing at infinity
Expand around the saddle point
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1-loop around saddle point
: functional space of which vanishes at infinity
In the second equality, we used the fact that there is cancellation between and
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Plan of this talk
1. Introduction
2. Theories with SU(2|4) symmetry
3. gauge/gravity correspondence for theories
with SU(2|4) symmetry
4. Localization in PWMM
5. Exact results for theories
with SU(2|4) symmetry
6. Application of our result
7. Summary
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Exact results for 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
where
Eigenvalues of
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Partition function of PWMM
Trivial vacuum
(cf.) partition function of 6d IIB matrix model
[Moore-Nekrasov-Shatashvili][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 limit in which
fixed with
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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 that
and 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 except for the Vandermonde determinant completely cancel out.
Gaussian matrix model
Consistent with the results for 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 trivial vacuum • NS5-brane limit
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Gauge/gravity duality for N=8 SYM on RxS2 around trivial vacuum
Partition function of N=8 SYM on RxS2 around trivial vacuum
This can be solved in the large-N and the large ’t Hooft coupling limit;
The dependence of and is consistent with the gravity dual obtained by Lin and Maldacena.
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NS5-brane limit
Based on the gauge/gravity duality by Lin-Maldacena, Ling, Mohazab, Shieh, Anders and Raamsdonk proposed a double scaling limit in which little string theory (IIA NS5 -brane theory) on RxS5 is obtained from PWMM.
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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where
Q-exact term
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