Complete factorization in minimal N=4 Chern-Simons matter theory 7. - - PowerPoint PPT Presentation
Complete factorization in minimal N=4 Chern-Simons matter theory 7. Aug. 2017 @ YITP Strings and Fields 2017 Shuichi Yokoyama Yukawa Institute for Theoretical Physics Ref. T.Nosaka-SY arXiv:1706.07234 From M-theory to CS theory Existence
Yukawa Institute for Theoretical Physics
Ref.
arXiv:1706.07234 T.Nosaka-SY
Strings and Fields 2017
[Townsend '95] [Witten '95]
Existence of 11d Membrane-theory uplifting 10d type IIA string theory
[Basu-Harvey '04]
Matrix model of M-theory
[Banks-Fischer-Shenker-Susskind '97]
World-volume action for a (super) M2-brane
[Townsend '96] [Duff '96]
World-volume action for a (super) M5-brane
[Pasti-Sorokin-Tonin '97] [Aganagic-Park-Popescu-Schwarz '97]
Implication of relation between M2-branes and (SUSY) CS theory
[Kitao-Ohta-Ohta '98] [Bergman-Hanany-Karch-Kol '99]
A no-go theorem on maximal SUSY CS theory Intersecting M2-M5 brane solution by using 3-bracket
[Schwarz '04] [Bagger-Lambert '06,'07] [Gutavsson '07]
Maximal SUSY CS theory by using 3-bracket time
Name
[Hosomichi-Lee-Lee-Lee-Park ’08]
# of SUSY
[Benna-Klebanov-Klose-Smedback ’08] [Aharony-Bergmann-Jafferis-Maldacena ’08]
time BLG model
[BLG '06, 07]
Feature Lie 3 bracket Abelian moduli space
[Gaiotto-Witten '08]
Gaiotto-Witten model
Linear quiver
[Fuji-Terashima-Yamazaki '08]
Orbifold Linear & circular quiver
ABJM model
circular quiver
HLLLP model
Orbifold
[Imamura-Kimura ’08]
circular quiver
Name
[Hosomichi-Lee-Lee-Lee-Park ’08]
# of SUSY
[Benna-Klebanov-Klose-Smedback ’08] [Aharony-Bergmann-Jafferis-Maldacena ’08]
time BLG model
[BLG '06, 07]
Feature Lie 3 bracket Abelian moduli space
[Gaiotto-Witten '08]
Gaiotto-Witten model
Linear quiver
[Fuji-Terashima-Yamazaki '08]
Orbifold Linear & circular quiver
ABJM model
circular quiver
HLLLP model
Orbifold
[Imamura-Kimura ’08]
circular quiver
SUSY localization on S3
[Kapustin-Willet-Yaakov ’09]
Name
[Hosomichi-Lee-Lee-Lee-Park ’08]
# of SUSY
[Benna-Klebanov-Klose-Smedback ’08] [Aharony-Bergmann-Jafferis-Maldacena ’08]
time BLG model
[BLG '06, 07]
Feature Lie 3 bracket Abelian moduli space
[Gaiotto-Witten '08]
Gaiotto-Witten model
Linear quiver
[Fuji-Terashima-Yamazaki '08]
Orbifold Linear & circular quiver
ABJM model
circular quiver
HLLLP model
Orbifold
[Imamura-Kimura ’08]
circular quiver
Today!
[Kapustin-Willet-Yaakov ’09]
SUSY localization on S3
① Linear quiver gauge theory ② Type IIB brane configuration
[Gaiotto-Witten '08] [Hanany-Witten '98]
N1 D3 NS51 N2 D3 NS53 (1,k)5
① Linear quiver gauge theory ② Type IIB brane configuration
[Gaiotto-Witten '08] [Hanany-Witten '98]
N1 D3 NS51 N2 D3 NS53 (1,k)5 3d N=4 U(N1)k vector multiplet 3d N=4 U(N2)-k vector multiplet 3d N=4 bifundamental hypermultiplet
[Gaiotto-Witten '08] SUSY mass deformation [HLLLP '08]
[Gaiotto-Witten '08] SUSY mass deformation [HLLLP '08]
Global symmetry (I) R-symmetry (II) Parity (massless case)
[Kapustin-Willet-Yaakov ’09] [Jafferis ’10] [Hama-Hosomichi-Lee ’10]
⇒ Supersymmetric localization
cf.
① Add Q-exact term which gives the free kinetic term for each SUSY multiplet. NOTE: Q-exact deformation does not change the partition function!
[Kapustin-Willet-Yaakov ’09] [Jafferis ’10] [Hama-Hosomichi-Lee ’10]
⇒ Supersymmetric localization
cf.
② Take weak coupling limit. ③ Path integral is exactly performed by gaussian integration (WKB approximation). Tree + 1 loop exact!!
① Add Q-exact term which gives the free kinetic term for each SUSY multiplet. Result NOTE: Q-exact deformation does not change the partition function!
[Kapustin-Willet-Yaakov ’09] [Jafferis ’10] [Hama-Hosomichi-Lee ’10]
⇒ Supersymmetric localization
cf.
② Take weak coupling limit. ③ Path integral is exactly performed by gaussian integration (WKB approximation). Tree + 1 loop exact!! NOTE: In ABJ(M) case, the denominator is squared. GW matrix model is less convergent. ➡
[Kapustin-Willett-Yaakov '10]
The condition of convergence of the partition function is related to a classification by Gaiotto-Witten of N=4 linear quiver gauge theories without Chern-Simons coupling constant.
[Kapustin-Willett-Yaakov '10]
The condition of convergence of the partition function is related to a classification by Gaiotto-Witten of N=4 linear quiver gauge theories without Chern-Simons coupling constant. Gaiotto-Witten's classification ① (The dimension of ∀monopole operators) > ½. → “Good” ② (The dimension of ∃monopole operator) = ½. → “Ugly” ③ (The dimension of ∃monopole operator) < ½. → “Bad”
[Kapustin-Willett-Yaakov '10]
The condition of convergence of the partition function is related to a classification by Gaiotto-Witten of N=4 linear quiver gauge theories without Chern-Simons coupling constant. Gaiotto-Witten's classification ① (The dimension of ∀monopole operators) > ½. → “Good” ② (The dimension of ∃monopole operator) = ½. → “Ugly” ③ (The dimension of ∃monopole operator) < ½. → “Bad” ⇔ The partition function is absolutely divergent. ⇔ The partition function is marginally divergent. ⇔ The partition function is absolutely convergent.
[Kapustin-Willett-Yaakov '10]
The condition of convergence of the partition function is related to a classification by Gaiotto-Witten of N=4 linear quiver gauge theories without Chern-Simons coupling constant. Gaiotto-Witten's classification ① (The dimension of ∀monopole operators) > ½. → “Good” ② (The dimension of ∃monopole operator) = ½. → “Ugly” ③ (The dimension of ∃monopole operator) < ½. → “Bad” ⇔ The partition function is absolutely divergent. ⇔ The partition function is marginally divergent. ⇔ The partition function is absolutely convergent. ➡ We regularize the theory by giving a suitable imaginary part for each CS level.
[Nosaka-SY '17]
[Nosaka-SY '17]
where
NOTE: In massless case (ζ=0), there are poles @ k+N2-N1 odd. Regime:
⇒ Self dual for minimal N=4 CSM theory
① From partition function
(I) Level/rank duality for pure CS theory
⇒ Self dual for minimal N=4 CSM theory
k is the renormalized coupling, k=kB+N, where level-rank duality exchanges kB N. ⇆
① From partition function
[Nosaka-SY '17]
(I) Level/rank duality for pure CS theory (II) Level/rank duality for matter partition function
⇒ Self dual for minimal N=4 CSM theory
k is the renormalized coupling, k=kB+N, where level-rank duality exchanges kB N. ⇆
① From partition function
[Nosaka-SY '17]
(I) Level/rank duality for pure CS theory (II) Level/rank duality for matter partition function
⇒ Self dual for minimal N=4 CSM theory
[Yaakov '13]
NOTE: A similar prefactor appears for Seiberg-like duality realtion between “good” theory and “bad” one. k is the renormalized coupling, k=kB+N, where level-rank duality exchanges kB N. ⇆ Contribution of ∃decoupled sector in the dual theory!?
② From brane realization
N1 D3 NS51 N2 D3 NS53 (1,k)5 [Nosaka-SY '17]
② From brane realization
N1 D3 NS51 N2 D3 NS53 (1,k)5 NS53 NS51 (1,k)5
Hanany-Witten transition. (5-brane movement with Linking # unchanged.)
[Hanany-Witten '98]
[Nosaka-SY '17]
[Nosaka-SY '17]
Def.
For simplicity we consider massless case: ζ=0
[Nosaka-SY '17]
Def.
For simplicity we consider massless case: ζ=0
The 't Hooft expansion
[Nosaka-SY '17]
Def.
For simplicity we consider massless case: ζ=0
The 't Hooft expansion All order result
[Nosaka-SY '17]
The large M scaling behavior of the free energy in the vector model limit can be deduced from the planar result. Claim ➡
[Nosaka-SY '17]
The large M scaling behavior of the free energy in the vector model limit can be deduced from the planar result. Claim ➡ Suppose the planar free energy is expanded in terms of λ1 as ➡ ➡ Proof QED
[Nosaka-SY '17]
The large M scaling behavior of the free energy in the vector model limit can be deduced from the planar result. Claim ➡ Suppose the planar free energy is expanded in terms of λ1 as ➡ ➡ Proof QED In the current case, ➡ Transition between (confined) phase (~M) and deconfined one (~M2) occurs in a smooth way.
・ We have argued the level/rank duality from partition function by verifying its invariance under the duality transformation, and from type IIB brane realization by moving 5-branes taking into account the Hanany-Witten effect. ・ By performing the integration explicitly (with a suitable regularization) the partition function completely factorized into that of pure CS theory for two gauge groups and an analogous contribution from the hypermultiplet (“Complete factorization”).
・ We have investigated a minimal N=4 CSM theory from its S3 partition function, which reduces to a matrix model via SUSY localization. ・ We have presented the all order 't Hooft expansion of the free energy, and commented on the connection to the higher-spin theory.
・ Topological string theory interpretation of matter partition function?
・ Physical meaning of poles in partition function?
[Nosaka-SY, working in progress]
・ Purely algebraic and representational origin of the partition function?
New toplogical object? Generalizing classification of modular transformation matrices? ・ Generalization to more general N=4 linear quiver gauge theory? Relation to ABJ(M) matrix model?
[Nosaka-SY, working in progress]
・ Physical meaning of prefactor? New massless d.o.f from decoupled sector?
・ Topological string theory interpretation of matter partition function?
・ Physical meaning of poles in partition function?
[Nosaka-SY, working in progress]
・ Purely algebraic and representational origin of the partition function?
New toplogical object? Generalizing classification of modular transformation matrices? ・ Generalization to more general N=4 linear quiver gauge theory? Relation to ABJ(M) matrix model?
[Nosaka-SY, working in progress]
・ Physical meaning of prefactor? New massless d.o.f from decoupled sector?