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 of 11d Membrane-theory uplifting 10d type IIA string theory [Townsend '95] [Witten '95] 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] Matrix model of M-theory [Banks-Fischer-Shenker-Susskind '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 [Schwarz '04] Intersecting M2-M5 brane solution by using 3-bracket [Basu-Harvey '04] Maximal SUSY CS theory by using 3-bracket [Bagger-Lambert '06,'07] [Gutavsson '07] time
Superconformal CSM theory Feature Abelian moduli space # of SUSY Name N=8 Lie 3 bracket BLG model [BLG '06, 07] Gaiotto-Witten model Linear quiver N=4 [Gaiotto-Witten '08] Orbifold [Fuji-Terashima-Yamazaki '08] Linear & HLLLP model circular quiver [Hosomichi-Lee-Lee-Lee-Park ’08] circular quiver ABJM model N=6 [Aharony-Bergmann-Jafferis-Maldacena ’08] Orbifold N=4 [Benna-Klebanov-Klose-Smedback ’08] circular quiver [Imamura-Kimura ’08] time
Superconformal CSM theory Feature Abelian moduli space # of SUSY Name N=8 Lie 3 bracket BLG model [BLG '06, 07] Gaiotto-Witten model Linear quiver N=4 [Gaiotto-Witten '08] Orbifold [Fuji-Terashima-Yamazaki '08] Linear & HLLLP model circular quiver [Hosomichi-Lee-Lee-Lee-Park ’08] circular quiver ABJM model N=6 [Aharony-Bergmann-Jafferis-Maldacena ’08] Orbifold N=4 [Benna-Klebanov-Klose-Smedback ’08] circular quiver [Imamura-Kimura ’08] SUSY localization on S 3 N≧2 [Kapustin-Willet-Yaakov ’09] time
Superconformal CSM theory Feature Abelian moduli space # of SUSY Name N=8 Lie 3 bracket BLG model [BLG '06, 07] Gaiotto-Witten model Linear quiver N=4 [Gaiotto-Witten '08] Today! Orbifold [Fuji-Terashima-Yamazaki '08] Linear & HLLLP model circular quiver [Hosomichi-Lee-Lee-Lee-Park ’08] circular quiver ABJM model N=6 [Aharony-Bergmann-Jafferis-Maldacena ’08] Orbifold N=4 [Benna-Klebanov-Klose-Smedback ’08] circular quiver [Imamura-Kimura ’08] SUSY localization on S 3 N≧2 [Kapustin-Willet-Yaakov ’09] time
Plan 1. Introduction ✓ 2. Minimal N=4 CSM theory 3. Exact partition function 4. Level/rank duality 5. All order 't Hooft expansion 6. Summary
U(N 1 ) k x U(N 2 ) -k N=4 CSM theory [Gaiotto-Witten '08] ① Linear quiver gauge theory ② Type IIB brane configuration [Hanany-Witten '98] NS5 3 NS5 1 (1,k)5 N 1 D3 N 2 D3
U(N 1 ) k x U(N 2 ) -k N=4 CSM theory [Gaiotto-Witten '08] ① Linear quiver gauge theory 3d N=4 U(N 1 ) k vector multiplet 3d N=4 U(N 2 ) -k vector multiplet 3d N=4 bifundamental hypermultiplet ② Type IIB brane configuration [Hanany-Witten '98] NS5 3 NS5 1 (1,k)5 N 1 D3 N 2 D3
U(N 1 ) k x U(N 2 ) -k N=4 CSM theory [Gaiotto-Witten '08] ③ Action (Euclidean) SUSY mass deformation [HLLLP '08] cf. [SY '13]
U(N 1 ) k x U(N 2 ) -k N=4 CSM theory [Gaiotto-Witten '08] ③ Action (Euclidean) SUSY mass deformation [HLLLP '08] cf. [SY '13] Global symmetry (I) R-symmetry (II) Parity (massless case)
Plan 1. Introduction ✓ 2. Minimal N=4 CSM theory ✓ 3. Exact partition function 4. Level/rank duality 5. All order 't Hooft expansion 6. Summary
Exact partition function ⇒ Supersymmetric localization [Kapustin-Willet-Yaakov ’09] [Jafferis ’10] [Hama-Hosomichi-Lee ’10] cf.
Exact partition function ⇒ Supersymmetric localization [Kapustin-Willet-Yaakov ’09] [Jafferis ’10] [Hama-Hosomichi-Lee ’10] 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! ➡ ② Take weak coupling limit. ③ Path integral is exactly performed by gaussian integration (WKB approximation). Tree + 1 loop exact!!
Exact partition function ⇒ Supersymmetric localization [Kapustin-Willet-Yaakov ’09] [Jafferis ’10] [Hama-Hosomichi-Lee ’10] 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! ➡ ② Take weak coupling limit. ③ Path integral is exactly performed by gaussian integration (WKB approximation). Tree + 1 loop exact!! Result ➡ NOTE: In ABJ(M) case, the denominator is squared. GW matrix model is less convergent.
Comment on convergence 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]
Comment on convergence 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] 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”
Comment on convergence 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] Gaiotto-Witten's classification ① (The dimension of ∀monopole operators) > ½. → “Good” ⇔ The partition function is absolutely convergent. ② (The dimension of ∃monopole operator) = ½. → “Ugly” ⇔ The partition function is marginally divergent. ③ (The dimension of ∃monopole operator) < ½. → “Bad” ⇔ The partition function is absolutely divergent.
Comment on convergence 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] Gaiotto-Witten's classification ① (The dimension of ∀monopole operators) > ½. → “Good” ⇔ The partition function is absolutely convergent. ② (The dimension of ∃monopole operator) = ½. → “Ugly” ⇔ The partition function is marginally divergent. ③ (The dimension of ∃monopole operator) < ½. → “Bad” ⇔ The partition function is absolutely divergent. ➡ We regularize the theory by giving a suitable imaginary part for each CS level.
Complete factorization [Nosaka-SY '17]
Complete factorization [Nosaka-SY '17] ➡ Regime: where NOTE: In massless case (ζ=0), there are poles @ k+N 2 -N 1 odd.
Plan 1. Introduction ✓ 2. Minimal N=4 CSM theory ✓ ✓ 3. Exact partition function 4. Level/rank duality 5. All order 't Hooft expansion 6. Summary
Level rank duality ⇒ Self dual for minimal N=4 CSM theory
Level rank duality ⇒ Self dual for minimal N=4 CSM theory ① From partition function (I) Level/rank duality for pure CS theory ⇆ k is the renormalized coupling, k=k B +N, where level-rank duality exchanges k B N.
Level rank duality ⇒ Self dual for minimal N=4 CSM theory ① From partition function (I) Level/rank duality for pure CS theory ⇆ k is the renormalized coupling, k=k B +N, where level-rank duality exchanges k B N. (II) Level/rank duality for matter partition function [Nosaka-SY '17]
Level rank duality ⇒ Self dual for minimal N=4 CSM theory ① From partition function (I) Level/rank duality for pure CS theory ⇆ k is the renormalized coupling, k=k B +N, where level-rank duality exchanges k B N. (II) Level/rank duality for matter partition function [Nosaka-SY '17] Contribution of ∃decoupled sector in the dual theory!? ➡ NOTE: A similar prefactor appears for Seiberg-like duality realtion [Yaakov '13] between “good” theory and “bad” one.
Level rank duality ② From brane realization [Nosaka-SY '17] NS5 3 NS5 1 (1,k)5 N 1 D3 N 2 D3
Level rank duality ② From brane realization [Nosaka-SY '17] NS5 3 NS5 1 (1,k)5 N 1 D3 N 2 D3 Hanany-Witten transition. [Hanany-Witten '98] ➡ (5-brane movement with Linking # unchanged.) cf. [Giveon-Kutsov '08] (1,k)5 NS5 1 NS5 3 -N 2 +k D3 -N 1 +k D3
Plan 1. Introduction ✓ 2. Minimal N=4 CSM theory ✓ ✓ 3. Exact partition function 4. Level/rank duality ✓ 5. All order 't Hooft expansion 6. Summary
Free energy in 't Hooft expansion [Nosaka-SY '17] For simplicity we consider massless case: ζ=0 Def.
Free energy in 't Hooft expansion [Nosaka-SY '17] For simplicity we consider massless case: ζ=0 Def. The 't Hooft expansion
Free energy in 't Hooft expansion [Nosaka-SY '17] For simplicity we consider massless case: ζ=0 Def. The 't Hooft expansion All order result
Free energy in vector model limit [Nosaka-SY '17] ➡ The large M scaling behavior of the free energy in the vector model limit Claim can be deduced from the planar result.
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