ABJM 行列模型から 位相的弦理論へ Sanefumi Moriyama (NagoyaU/KMI) [Hatsuda+M+Okuyama 1207, 1211, 1301] [HMO+Marino 1306] [HMO+Honda 1306]
Lessons from '90s String Theory As Unified Theory String Theory, NOT JUST "A Theory Of Strings" Only Sensible After NonPerturbative Branes Various Perturbative String Theories Various Vacua Of "A Unique Theory" Further Unification By M theory
ABJM for NonPerturbative Strings ABJM Theory [N=6 Super Chern Simons Theory] N x M2 branes on C 4 /Z k ( CP 3 xs 1 as k ) To Understand Better M Theory (NonPerturbative Strings) Roles of ABJM Theory?
Yes Partition Function (Normalization = Free of Physical Poles) A Non Trivial Cancellation Mechanism Strings & Membranes: Poles Strings + Membranes: Cancel
Contents 1. Introduction 2. ABJM & Two Instanton Effects 3. Instantons from Numerics 4. Refined Topological Strings 5. Discussions
ABJM Matrix Model Partition Function of ABJM Theory Due to SUSY, Localized to Matrix Model Z(N) = N 1 =N 2 =N g s =2πi/k
't Hooft Expansion N Expansion with λ=n/k Fixed Genus Expansion F = N 2 F 0 (λ) + N 0 F 1 (λ) + N 2 F 2 (λ) + N 4 F 3 (λ) +... = N 2 [F pert 2 4 0 (λ) + # e 2π 2λ + # e 4π 2λ +...] + N 0 [F 1 pert (λ)+ # e 2π 2λ + # e 4π 2λ +...] +...... Sum Up To Airy Function! [Fuji Hirano M] Worldsheet Instanton e 2π 2λ = exp[ T F1 Area(CP 1 )] F String Wrapping CP 1 CP 3
M theory Expansion M theory Background: C 4 /Z k N with k Fixed Close To 't Hooft Limit But Not Exactly N M theory Regime k: Fixed k 't Hooft Regime λ=n/k Fixed but Large
Fermi Gas Formalism [Marino Putrov] Rewriting Partition Function Regarding as Fermi Gas with Density Matrix ρ Z(N) = (N!) 1 σ S(N) ( 1) σ i dq i q i ρ q σ(i) ρ = (2 cosh q/2) 1/2 (2 cosh p/2) 1 (2 cosh q/2) 1/2 [p,q] = i 2πk Introducing Grand Potential e J(μ) = dt(1 N=0 Z(N) e μn = det + e μ ρ) )
WKB (small k) expansion J(μ) = k 1 J 0 (μ) + k 1 J 1 (μ) + k 3 J 2 (μ) + k 5 J 3 (μ) +... = k 1 [ J pert 0 (μ) + (#μ 2 +#μ+#)e 2μ + (...) e 4μ +... ] + k [ J pert 1 (μ) + (#μ 2 +#μ+#)e 2μ + (...) e 4μ +... ] +...... = (#μ 3 +#μ+#) + (#μ 2 +#μ+#)e 2μ + (...)e 4μ +... Airy Function Reproduced [Marino Putrov, Honda et al] Membrane Instanton e 2μ e π 2Nk = exp[ T D2 Area(RP 3 )] D2 brane Wrapping RP 3 CP 3 [Drukker Marino Putrov]
Summary J(μ) ) = J pert (μ) + J WS (μ) + J MB (μ) +... J pert (μ) = C μ 3 /3 + B μ + A e 2π 2λ e 4μ/k J WS (μ) = m=1 d (m) k e m x 4μ/k J MB (μ) = l=1 (a (l) 2 + b (l) + c (l) e l x 2μ k μ k μ k ) WS Instantons from Topological Strings d k (m) = g d m ( 1) m/d N g d/d (2 Sin[2πd/k]) 2g 2 N g d: Gopakumar Vafa Invariant on F 0 =P 1 xp 1
Contents 1. Introduction 2. ABJM & Two Instanton Effects 3. Instantons from Numerics 4. Refined Topological Strings 5. Discussions
Partition Function Z(N) with Fixed k Example: (For k=1) ) Z(1) = 1/4 Z(2) = 1/16π Z(3) = (π 3)/2 6 π Z(4) = ( π 2 +10)/2 10 π 2 Z(5) ( ) = ( 9π 2 +20π+26)/2 12 π 2 Z(6) = (36π 3 121π 2 +78)/2 14 3 2 π 3 Z(7) = ( 75π 3 +193π 2 +174π 126)/2126)/2 16 3π 3 Z(8) = (1053π 4 2016π 3 4148π 2 +876)/2 21 3 2 π 4 Z(9) = (5517π 4 13480π 3 15348π 2 +8880π+4140)/2 23 3 2 π 4
Exact Values From Numerical Studies Define Approximate Grand Potential J(μ) = log[ Nmax Z(N) e μn N=0 ( ) ] Compare with Expectation J(μ)=(#μ )(# 3 +#μ+#)+ #) m 4 #) l 2 m=1 #e mx4μ/k + l=1 (#μ 2 +#μ+#)e lx2μ Read off Exact Values of Coefficients
Results (Schematically) J 4μ 8μ 12μ k=1 (μ) = [#μ 2 +#μ+#]e + [#μ 2 +#μ+#]e + [#μ 2 +#μ+#]e +... J k=2 (μ) = [#μ 2 +#μ+#]e 2μ + [#μ 2 +#μ+#]e 4μ + [#μ 2 +#μ+#]e 6μ +... J k=3 (μ) = [#]e 4μ/3 + [#]e 8μ/3 + [#μ 2 +#μ+#]e 4μ +... J k=4 (μ) = [#]e [] μ + [#μ μ 2 +#μ+#]eμ 2μ + [#]e [] 3μ +...... J 2μ/3 4μ/3 2μ k=6 (μ) = [#]e + [#]e + [#μ 2 +#μ+#]e +... WS(1) WS(2) WS(3)
Comparison with WS Instanton (Top String) J 4μ 8μ 12μ k=1 (μ) = [#μ 2 +#μ+#]e + [#μ 2 +#μ+#]e + [#μ 2 +#μ+#]e +... J k=2 (μ) = [#μ 2 +#μ+#]e 2μ + [#μ 2 +#μ+#]e 4μ + [#μ 2 +#μ+#]e 6μ +... J k=3 (μ) = [#]e 4μ/3 + [#]e 8μ/3 + [#μ 2 +#μ+#]e 4μ +... J k=4 (μ) = [#]e [] μ + [#μ μ 2 +#μ+#]eμ 2μ + [#]e [] 3μ +...... J 2μ/3 4μ/3 2μ k=6 (μ) = [#]e + [#]e + [#μ 2 +#μ+#]e +... WS(1) WS(2) WS(3) : Match : Divergent : Not Match
Divergences Cancelled by MB Instanton J 4μ 8μ 12μ k=1 (μ) = [#μ 2 +#μ+#]e + [#μ 2 +#μ+#]e + [#μ 2 +#μ+#]e +... J k=2 (μ) = [#μ 2 +#μ+#]e 2μ + [#μ 2 +#μ+#]e 4μ + [#μ 2 +#μ+#]e 6μ +... J k=3 (μ) = [#]e 4μ/3 + [#]e 8μ/3 + [#μ 2 +#μ+#]e 4μ +... J k=4 (μ) = [#]e [] μ + [#μ μ 2 +#μ+#]eμ 2μ + [#]e [] 3μ +... k=4... MB(2) J 2μ/3 4μ/3 2μ k=6 (μ) = [#]e + [#]e + [#μ 2 +#μ+#]e +... WS(1) WS(2) WS(3) MB(1)
1 Membrane Instanton Vanishing in k=odd Canceling Divergence @ k=2m Matching the WKB data a k (1) = 4(π 2 k) 1 Cos[πk/2] b k (1) = 2π 1 Cot[πk/2] Cos[πk/2] k [ / ] [ / ] c k (1) =...
How About? (l, m) Bound State? e (m x 4μ/k + l x 2μ) Ex: e 3μ Effects in k=4 Sector From Both (0,3) & (1,1) Contribution frombound States J(μ) = J pert (μ) + J MB (μ) + J WS (μ) + J bnd (μ) J bnd (μ) = m=1 l=1 # e (m x 4μ/k + l x 2μ)
Bound States J(μ) = J pert (μ) + J MB (μ) + J WS (μ) + J bnd (μ) Bound States Incorporated by μ μ eff J WS (μ)+ J bnd (μ) = J WS (μ eff ) μ eff = μ + l=1 a (l) k e l x 2μ All in Effective Chemical Potential J(μ) = J pert (μ eff ) + J' MB (μ eff ) + J WS (μ eff ) J' MB (μ (b~ +c~ eff ) = l=1 (l) k μ eff (l) k ) e l x 2μeff Membrane Instanton in Linear Functions c~ k (l) = k 2 k [b~ k (l) /k] Only a k (l) & b k (l) NonTrivial
Summary Exact Values From Numerical Study Poles Cancellation Requirement Explicit First Few Membrane Instantons Bound States (Neither fromgenus OR WKB) No Other Contributions
Contents 1. Introduction 2. ABJM & Two Instanton Effects 3. Instantons from Numerics 4. Refined Topological Strings 5. Discussions
Refined Topological Strings a (l) k & b (l) k A & B Periods of Q Spectral Curve! Classical Spectral Curve 1 + e x + e p + z 1 e x + z 2 e p = 0 Quantum Spectral Curve ( 1+e x +z 1 e x )Ψ(x)+Ψ(x+ )+z 2 Ψ(x )=0 Quantization Refined Topological String of Spectral Curve in Nekrasov Shatashivili Limit [Mironov Morozov, Aganagic Cheng Dijkgraaf Krefl Vafa]
All Explicitly In Topological Strings J(μ)=J pert (μ eff )+J WS (μ eff )+J' MB (μ eff ) J pert (μ)=cμ 3 /3+Bμ+A J WS (μ eff )=F T top (T eff 1,T eff 2,λ) J' MB (μ eff )=(2πi) 1 λ [λf NS (T 1 eff /λ,t 2 eff /λ,1/λ)] F(T 1,T 2,τ 1,τ 2 ): Free Energy of Refined Top Strings T 1,T 2 : Kahler Moduli τ 1,τ 2 : Coupling Constants T eff 1 =4μ eff /k iπ T eff 2 =4μ eff /k+iπ λ=2/k Topological Limit F top (T 1,T 2,τ)=lim τ1 τ,τ2 τ F(T 1,T 2,τ 1,τ 2 ) NS Limit F NS (T 1,T 2,τ)=lim τ1 τ,τ2 0 2πiτ 2 F(T 1,T 2,τ 1,τ 2 )
Triple Sine Function Triple Sine Function Defined frommultiple Zeta Function All Poles Cancelled Formally log S 3 (z;ω)= z + z + Sign Modifications Pole Cancellation Still Holds! z /ω ω 1,ω 3 /ω 2 ω 2,ω 1 /ω 3 ω 3,ω 2 1
Relation to Triple Sine Function F 1 top + (2πi) 1 λ [ λ F NS ] = lim τ1 τ,τ2 τ d j N d j log S 3 ω z z z 3 /ω ω 1 ω 2 ω 1,ω 3 /ω 2 ω 2,ω 1 /ω 3 ω 3,ω 2 1 F F top NS Before The Limit? Ellipsoid S 3? Squashed S 3? Similar to M5 Partition Function? Duality???
Contents 1. Introduction 2. ABJM & Two Instanton Effects 3. Instantons from Numerics 4. Refined Topological Strings 5. Discussions
Summary Instanton Bound States of WS & MB Neither from Genus Exp OR WKB Exp An Explicit Formula for ABJM Matrix Model Perturbative: Cubic Function WS Instanton: Top String MB Instanton: Refined Top String in NS Limit Partition Function (& Wilson Loop)
Punch Lines M theory / Nonperturbative Strings String Theory, NOT JUST A Theory Of Strings Only Sensible After NonPerturbative Branes ABJM Matrix Model Poles from WS & MB Cancel Only After Sum Cancellation Mechanism Raison D'Etre DEtrefor M theory First Few Membrane Instantons Infinite i Cancellation in General lformula
Thank you for your attention.