The uses of the refined matrix model recursion

We study matrix models in the beta ensemble by building on the refined recursion relation proposed by Chekhov and Eynard. We present explicit results for the first beta-deformed corrections in the one-cut and the two-cut cases, as well as two applications to supersymmetric gauge theories: the calculation of superpotentials in N=1 gauge theories, and the calculation of vevs of surface operators in superconformal N=2 theories and their Liouville duals. Finally, we study the beta deformation of the Chern-Simons matrix model. Our results indicate that this model does not provide an appropriate description of the Omega-deformed topological string on the resolved conifold, and therefore that the beta-deformation might provide a different generalization of topological string theory in toric Calabi-Yau backgrounds.

[1]  Loop equations, matrix models, and Script N = 1 supersymmetric gauge theories , 2003, hep-th/0304138.

[2]  Perturbative computation of glueball superpotentials for SO(N) and USp(N) , 2002, hep-th/0211261.

[3]  L. Alday,et al.  Loop and surface operators in $ \mathcal{N} = 2 $ gauge theory and Liouville modular geometry , 2009, 0909.0945.

[4]  M. F. ATIYAHf.  THE UNITY OF MATHEMATICS , 2006 .

[5]  Andrei Okounkov,et al.  Seiberg-Witten theory and random partitions , 2003, hep-th/0306238.

[6]  A. Morozov,et al.  The matrix model version of AGT conjecture and CIV-DV prepotential , 2010, 1004.2917.

[7]  B. Eynard,et al.  Topological expansion of the Bethe ansatz, and non-commutative algebraic geometry , 2008, 0809.3367.

[8]  S. Pasquetti,et al.  A& B model approaches to surface operators and Toda thoeries , 2010, 1004.2025.

[9]  L. Alday,et al.  Effective superpotentials via Konishi anomaly , 2003, hep-th/0304119.

[10]  Nikita A. Nekrasov.  Seiberg-Witten prepotential from instanton counting , 2002 .

[11]  R. Schiappa,et al.  Ja n 20 10 An A r threesome : Matrix models , 2 d CFTs and 4 d N = 2 gauge theories , 2010 .

[12]  The refined topological vertex , 2007, hep-th/0701156.

[13]  M. Mariño.  Les Houches lectures on matrix models and topological strings , 2004, hep-th/0410165.

[14]  B. Eynard,et al.  Hermitian matrix model free energy: Feynman graph technique for all genera , 2005, hep-th/0504116.

[15]  G. Parisi,et al.  Planar diagrams , 1978 .

[16]  M. Atiyah.  The Unity of Mathematics , 1978 .

[17]  Liouville Field Theory — A decade after the revolution , 2004, hep-th/0402009.

[18]  B. Eynard,et al.  Topological expansion of the Bethe ansatz, and quantum algebraic geometry , 2009, 0911.1664.

[19]  A. Brini,et al.  Exact Results for Topological Strings on Resolved Y p,q Singularities , 2008, 0804.2598.

[20]  B. Eynard,et al.  Invariants of algebraic curves and topological expansion , 2007, math-ph/0702045.

[21]  R. Gopakumar,et al.  On the Gauge Theory/Geometry Correspondence , 1998, hep-th/9811131.

[22]  M. Tierz.  SOFT MATRIX MODELS AND CHERN–SIMONS PARTITION FUNCTIONS , 2002, hep-th/0212128.

[23]  Chern-Simons Theory, Matrix Integrals, and Perturbative Three-Manifold Invariants , 2002, hep-th/0207096.

[24]  S. Gukov,et al.  Vortex Counting and Lagrangian 3-Manifolds , 2010, 1006.0977.

[25]  M. Mariño.  Open string amplitudes and large order behavior in topological string theory , 2006, hep-th/0612127.

[26]  B. Eynard,et al.  Topological expansion for the 1-hermitian matrix model correlation functions , 2004 .

[27]  A large N duality via a geometric transition , 2001, hep-th/0103067.

[28]  H. Itoyama,et al.  Method of generating q-expansion coefficients for conformal block and N=2 Nekrasov function by β-deformed matrix model , 2010, 1003.2929.

[29]  E. Martinec,et al.  Unoriented strings and matrix ensembles , 1990 .

[30]  Min-xin Huang,et al.  Direct integration for general Omega backgrounds , 2010, 1009.1126.

[31]  C. Vafa,et al.  Toda Theories, Matrix Models, Topological Strings, and N=2 Gauge Systems , 2009, 0909.2453.

[32]  B. Eynard,et al.  Algebraic methods in random matrices and enumerative geometry , 2008, 0811.3531.

[33]  P. Di Francesco,et al.  2D gravity and random matrices , 1993 .

[34]  L. Alday,et al.  Liouville Correlation Functions from Four-Dimensional Gauge Theories , 2009, 0906.3219.

[35]  A. Mironov,et al.  CONFORMAL BLOCKS AS DOTSENKO–FATEEV INTEGRAL DISCRIMINANTS , 2010 .

[36]  Multicritical points of unoriented random surfaces , 1991 .

[37]  D. Gaiotto.  Preprint Typeset in Jhep Style -hyper Version N = 2 Dualities , 2022 .

[38]  On low rank classical groups in string theory, gauge theory and matrix models , 2003, hep-th/0311181.

[39]  Matrix eigenvalue model: Feynman graph technique for all genera , 2006, math-ph/0604014.

[40]  S. Majumdar,et al.  Large deviations of the maximal eigenvalue of random matrices , 2010, 1009.1945.

[41]  B. Eynard.  All genus correlation functions for the hermitian 1-matrix model , 2004, hep-th/0407261.

[42]  A. Mironov,et al.  Matrix model conjecture for exact BS periods and Nekrasov functions , 2009, 0911.5721.

[43]  M. Mariño,et al.  Multi-instantons and multicuts , 2008, 0809.2619.

[44]  M. Aganagic,et al.  Topological Strings and (Almost) Modular Forms , 2006, hep-th/0607100.

[45]  Stavros Garoufalidis,et al.  Universality and asymptotics of graph counting problems in non-orientable surfaces , 2008, J. Comb. Theory A.

[46]  On the matter of the Dijkgraaf-Vafa conjecture , 2003, hep-th/0303104.

[47]  L. Chekhov.  Logarithmic potential beta-ensembles and Feynman graphs , 2010, 1009.5940.

[48]  P. Sułkowski.  Matrix models for β-ensembles from Nekrasov partition functions , 2009, 0912.5476.

[49]  C. Vafa,et al.  Matrix models, topological strings, and supersymmetric gauge theories , 2002, hep-th/0206255.

[50]  D. Krefl,et al.  Extended Holomorphic Anomaly in Gauge Theory , 2010, 1007.0263.

[51]  V. Bouchard,et al.  Remodeling the B-Model , 2007, 0709.1453.

[52]  A. Klemm,et al.  Direct integration and non-perturbative effects in matrix models , 2010, 1002.3846.

[53]  V. Bouchard,et al.  Topological Open Strings on Orbifolds , 2008, 0807.0597.

[54]  C. Vafa,et al.  A Perturbative Window into Non-Perturbative Physics , 2002, hep-th/0208048.

[55]  Duality of Orthogonal and Symplectic Matrix Integrals and Quaternionic Feynman Graphs , 2002, math-ph/0206011.

[56]  M. Spreafico.  On the Barnes double zeta and Gamma functions , 2009 .

[57]  T. Eguchi,et al.  Penner type matrix model and Seiberg-Witten theory , 2009, 0911.4797.

[58]  Matrix model as a mirror of Chern-Simons theory , 2002, hep-th/0211098.

[59]  Holomorphic anomaly and matrix models , 2007, hep-th/0702110.