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Summing Up All Genus Free Energy of ABJM Matrix Model Sanefumi Moriyama (Nagoya U) JHEP [arXiv:1106.4631] with H.Fuji and S.Hirano

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M is for Mother IIA IIB I Het-SO(32) Het-E8xE8 5 Consistent String Theories in 10D

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M is for Mother IIA IIB I Het-SO(32) Het-E8xE8 5 Consistent String Theories in 10D T-duality S-dualityOrientifold 5 Vacua of A Unique String Theory

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M is for Mother M (11D) IIA IIB I Het-SO(32) Het-E8xE8 M2 / M5 10D & Compactifications Non- Perturbative Strong Coupling Limit

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M is for Mystery 11D Supergrav as LowEnergyTheory Supergrav Sol for M2- & M5-branes Near Horizon Geometry Superconformal Symmetry osp(8|4) Action for Single M2-/M5-brane DOF N 3/2 /N 3 for N M2-/M5-branes Little was known before ABJM

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cf. D-branes DOF N 2 for D-branes MatrixDescribed by

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M is for Membrane Non-Abelian M2-brane theory by ABJM [Aharony-Bergman-Jafferis-Maldacena] Partition Function Localized to CS Matrix [Kapustin-Willett-Yaakov, Hama-Hosomichi-Lee] Planar N 3/2 Behavior Reproduced [Drukker-Marino-Putrov] Today: All Genus Sum A breakthrough by ABJM

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Summary of Strategy & Result Chern-Simons Theory = Open A-model Duality: Closed B-model Holomorphic Anomaly Eq (Recurrence Eq) Modified Bessel Differential Eq Solution Implications?

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Contents 1.Introduction 2.ABJM Theory 3.Localization 4.Planar Limit 5.All Genus Sum 6.Discussions

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Contents 1.Introduction 2.ABJM Theory 3.Localization 4.Planar Limit 5.All Genus Sum 6.Discussions

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(N 1 +N 2 )/2 M2 with N 1 -N 2 fractional M2 on C 4 / Z k ABJ(M) theory (N 1 =N 2 ) U(N 2 ) -k U(N 1 ) k Gauge Field Bifundamental Matter Fields N=6 Chern-Simons-matter Theory Superpotential A 1, A 2 B 1, B 2

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N=2 Chern-Simons Theory Dim Red of 4D N=1 N=2 Vector Multiplet V N=2 Chiral Multiplet Φ L gauge (V) =... L matter (Φ) =... (Topological & Auxiliary)

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N=3 Chern-Simons-matter Theory Field Contents of N=4 N=4 Vector Multiplet (V, φ) N=4 HyperMultiplet (Φ i, Φ i † ) Adj Rep Conj Rep (For General Gauge Groups & Representations) (CS Term: N=3 / No Kinetic Term for φ)

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N=6 Chern-Simons-matter Theory Application to U(N 1 ) x U(N 2 ) U(N 2 ) -k U(N 1 ) k A 1, A 2 B 1, B 2

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N=6 Chern-Simons-matter Theory so(3) = su(2) diag ⊂ su(2) A x su(2) B ⊂ su(4) = so(6) A 1, A 2 B 1, B 2 su(2) R-Symmetry

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Brane Construction NS5 0 1 2 3 4 5 (1,k)5 0 1 2 [3,7] θ [4,8] θ [5,9] θ tanθ = k D3 0 1 2 6 N 1 x D3 N 2 x D3 (1,k)5 NS5 Preserving N=6 SUSY (Expected to be N=6 Chern-Simons-matter Theory)

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Brane Construction T 6 -duality KKm (6dual) + k D6 KKm (6dual) (N 1 +N 2 )/2 x D2 (N 1 -N 2 ) x fractional D2 KKm (6dual, k x 10) KKm (6dual) (N 1 +N 2 )/2 x M2 (N 1 -N 2 ) x fractional M2 C 4 / Z k M-lift

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Contents 1.Introduction 2.ABJM Theory 3.Localization 4.Planar Limit 5.All Genus Sum 6.Discussions

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Localization Grassmann-odd Symmetry Grassmann-odd Quantity Integration Localized to

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Grassmann-odd Symmetry δ: Chiral SUSY Grassmann-odd Quantity V: δV = S YM Localized to F μν = 0 & D μ σ = 0 (σ = A 4 ) Pure Gauge A μ = 0 & Constant σ Classical Action: Quadratic If Rescaled by t 1-Loop Exact Application to ABJM

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Gauge Field Matter Field... = cosh -2 Application to ABJM

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Finally, Localized to Matrix Model

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Contents 1.Introduction 2.ABJM Theory 3.Localization 4.Planar Limit 5.All Genus Sum 6.Discussions

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ABJM Matrix Model

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If, instead Lens Space Matrix Model

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If, further simplified Chern-Simons Matrix Model

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The simplest one Gaussian Matrix Model Study from Gaussian MM to ABJM MM

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Matrix Model Eigenvalue Density Wanted Definition - Resolvent - Planar Limit Eigenvalues

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Nice Properties of Resolvent 1. Behavior 2.Dispersion Relation 3.EOM ・・・ Discontinuity Eq Integration Force

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Integration Contour z Partition Func F 0 (λ)Resolvent ω 0 (z)

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Chern-Simons Matrix Model

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Resolvent Asymptotic Behavior as Discontinuity Eq from EOM

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Chern-Simons Matrix Model A Regular Function [Halmagyi-Yasnov] Asymptotic Behavior Determined!

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Lens Space Matrix Model

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Same Tech as Chern-Simons Matrix Model Two Cuts Instead Lens Space Matrix Model sinh cosh

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ABJM Matrix Model

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Analytic Cont from Lens Space Matrix Model Result (for ABJM Slice: λ = λ 1 = λ 2 )

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Result Neglecting Worldsheet Instanton N 3/2 Reproduced and More

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Interpretation Charge Shift [Aharony-Hashimoto-Hirano-Ouyang] from Euler Coupling Match in Planar Case

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Furthermore, Non-Planar Prediction Renormalization of 't Hooft coupling Or in terms of

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Contents 1.Introduction 2.ABJM Theory 3.Localization 4.Planar Limit 5.All Genus Sum 6.Discussions

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Strategy Chern-Simons Theory on Lens Space S 3 /Z 2 (String Completion) Open Top A-model on T*(S 3 /Z 2 ) (Large N Duality) Closed Top A on Hirzebruch Surface F 0 = P 1 x P 1 (Mirror Symmetry) Closed Top B on Spectral Curve u v = H(x,y) Holomorphic Anomaly Equation!!

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Holomorphic Anomaly Eq General Case [Bershadsky-Cecotti-Ooguri-Vafa] Determined by F 0 : 2 nd & 3 rd Derivatives

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Application to ABJM One Modulus = Ordinary Differential Eq 2 Cuts = Torus

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Derivative 2 nd Derivative 3 rd Derivative (Yukawa Coupling) Elliptic Int Torus Modulus K: α-cycle K': β-cycle τ 1

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Ansatz & Reduction Ansatz for Partition Function Eisenstein Series Reduced HAE [Drukker-Marino-Putrov] Covariant Derivative

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Higher Genus Behavior

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Resum 1 Discrepancy Discussed Later

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Resum 2 After Partial Sum Recursion Relation (cf. SW theory [Huang-Klemm]) Translated into Differential Eq Solved by Modified Bessel Functions

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Airy Function Result Result [Fuji-Hirano-M] Airy Function

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Contents 1.Introduction 2.ABJM Theory 3.Localization 4.Planar Limit 5.All Genus Sum 6.Discussions

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Discrepancy? SU(N) x SU(N) instead of U(N) x U(N)? Another Superconformal Theory by ABJM SU(N) instead of U(N) for D3-branes

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Airy Function Modified Bessel Functions Everywhere as for Loop Operator VEV [Moore-Seiberg-Staudacher, Drukker-Gross] Cubic Integral Reminiscent of Chern-Simons Airy Function in Other Setups of M theory "Wave Function of the Universe" [Ooguri-Vafa-Verlinde] Corrections to Quantum Gravity? Max SUSY ・・・ No Corrections?

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Future Directions [work in progress] Relation to Superalgebra U(N|N) Airy Function Beyond ABJM Slice Other Analysis (Surgery?) for WS Instanton Good Variables for ABJM Theory Three-Algebra? [Bagger-Lambert, Gutsvasson]

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