Abstract
We give a concise summary of the impressive recent development unifying a number of different fundamental subjects. The quiver Nekrasov functions (generalized hypergeometric series) form a full basis for all conformal blocks of the Virasoro algebra and are sufficient to provide the same for some (special) conformal blocks of W-algebras. They can be described in terms of Seiberg-Witten theory, with the SW differential given by the 1-point resolvent in the DV phase of the quiver (discrete or conformal) matrix model (β-ensemble), \( dS = ydz + O\left( {{\varepsilon^2}} \right) = \sum\nolimits_p {{\varepsilon^{2p}}\rho_\beta^{\left( {p\left| 1 \right.} \right)}(z)} \), where ε and β are related to the LNS parameters ϵ1 and ϵ2. This provides explicit formulas for conformal blocks in terms of analytically continued contour integrals and resolves the old puzzle of the free-field description of generic conformal blocks through the Dotsenko-Fateev integrals. Most important, this completes the GKMMM description of SW theory in terms of integrability theory with the help of exact BS integrals, and provides an extended manifestation of the basic principle which states that the effective actions are the tau-functions of integrable hierarchies.
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References
N. Seiberg and E. Witten, Monopole condensation, and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys. B 426 (1994) 19 [Erratum ibid. B430 (1994) 485] [hep-th/9407087] [SPIRES].
N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD, Nucl. Phys. B 431 (1994) 484 [hep-th/9408099] [SPIRES].
A. Gorsky, I. Krichever, A. Marshakov, A. Mironov and A. Morozov, Integrability and Seiberg-Witten exact solution, Phys. Lett. B 355 (1995) 466 [hep-th/9505035] [SPIRES].
E.J. Martinec and N.P. Warner, Integrable systems and supersymmetric gauge theory, Nucl. Phys. B 459 (1996) 97 [hep-th/9509161] [SPIRES].
A. Hanany and Y. Oz, On the quantum moduli space of vacua of N = 2 supersymmetric SU(N c ) gauge theories, Nucl. Phys. B 452 (1995) 283 [hep-th/9505075] [SPIRES].
P.Argyres and A.Shapere, The vacuum structure of N=2 superQCD with classical gauge groups, Nucl. Phys. B 461 (1996) 437 [hep-th/9509175] [SPIRES].
J.Sonnenschein, S.Theisen and S.Yankielowicz, On the relation between the holomorphic prepotential and the quantum moduli in SUSY gauge theories, Phys. Lett. B 367 (1996) 145 [hep-th/9510129] [SPIRES].
J.Minahan and D.Nemeschansky, N=2 superYang-Mills and subgroups of SL(2, Z), Nucl. Phys. B 468 (1996) 72 [hep-th/9601059] [SPIRES].
N.Dorey, V.Khoze and M.Mattis, Multi instanton calculus in N=2 supersymmetric gauge theory. 2. Coupling to matter, Phys. Rev. D 54 (1996) 7832 [hep-th/9607202] [SPIRES].
N. Dorey, V.V. Khoze and M.P. Mattis, On N = 2 supersymmetric QCD with 4 flavors, Nucl. Phys. B 492 (1997) 607 [hep-th/9611016] [SPIRES].
R. Donagi and E. Witten, Supersymmetric Yang-Mills theory and integrable systems, Nucl. Phys. B 460 (1996) 299 [hep-th/9510101] [SPIRES].
E.Martinec, Integrable structures in supersymmetric gauge and string theory, Phys. Lett. B 367 (1996) 91 [hep-th/9510204] [SPIRES].
A.Gorsky, A.Marshakov, Towards effective topological gauge theories on spectral curves, Phys. Lett. B 374 (1996) 218 [hep-th/9510224] [SPIRES].
H.Itoyama and A.Morozov, Integrability and Seiberg-Witten theory: Curves and periods, Nucl. Phys. B 477 (1996) 855 [hep-th/9511126] [SPIRES].
H. Itoyama and A. Morozov, Prepotential and the Seiberg-Witten theory, Nucl. Phys. B 491 (1997) 529 [hep-th/9512161] [SPIRES].
H. Itoyama and A. Morozov, Integrability and Seiberg-Witten theory, hep-th/9601168 [SPIRES].
N. Nekrasov, Five dimensional gauge theories and relativistic integrable systems, Nucl. Phys. B 531 (1998) 323 [hep-th/9609219] [SPIRES].
H.W. Braden, A.Marshakov, A.Mironov and A.Morozov, Seiberg-Witten theory for a nontrivial compactification from five-dimensions to four-dimensions, Phys. Lett. B 448 (1999) 195 [hep-th/9812078] [SPIRES].
H.W. Braden, A. Marshakov, A. Mironov and A. Morozov, The Ruijsenaars-Schneider model in the context of Seiberg-Witten theory, Nucl. Phys. B 558 (1999) 371 [hep-th/9902205] [SPIRES].
H.W. Braden, A. Marshakov, A. Mironov and A. Morozov, WDVV equations for 6D Seiberg-Witten theory and bi-elliptic curves, hep-th/0606035 [SPIRES].
A. Gorsky, A. Marshakov, A. Mironov and A. Morozov, N = 2 supersymmetric QCD and integrable spin chains: rational case N f < 2N c , Phys. Lett. B 380 (1996) 75 [hep-th/9603140] [SPIRES].
A. Gorsky, A. Marshakov, A. Mironov and A. Morozov, A note on spectral curve for the periodic homogeneous XYZ-spin chain, hep-th/9604078 [SPIRES].
A.Gorsky, S.Gukov and A.Mironov, Multiscale N = 2 SUSY field theories, integrable systems and their stringy/brane origin. 1, Nucl. Phys. B 517 (1998) 409 [hep-th/9707120] [SPIRES].
A.Gorsky and A.Mironov, Integrable many body systems and gauge theories, [hep-th/0011197] [SPIRES].
E. D’Hoker, I.M. Krichever and D.H. Phong, The effective prepotential of N = 2 supersymmetric SU(N c ) gauge theories, Nucl. Phys. B 489 (1997) 179 [hep-th/9609041] [SPIRES].
E. D’Hoker, I.M. Krichever and D.H. Phong, The renormalization group equation in N = 2 supersymmetric gauge theories, Nucl. Phys. B 494 (1997) 89 [hep-th/9610156] [SPIRES].
A.Gorsky, A.Marshakov, A.Mironov and A.Morozov, RG equations from Whitham hierarchy, Nucl. Phys. B 527 (1998) 690 [hep-th/9802007] [SPIRES].
J.Edelstein, M.Gomez-Reino, M.Marino and J.Mas, N=2 supersymmetric gauge theories with massive hypermultiplets and the Whitham hierarchy, Nucl. Phys. B 574 (2000) 587 [hep-th/9911115] [SPIRES].
J.Edelstein, M.Gomez-Reino and J.Mas, Instanton corrections in N=2 supersymmetric theories with classical gauge groups and fundamental matter hypermultiplets, Nucl. Phys. B 561 (1999) 273 [hep-th/9904087] [SPIRES].
A.Marshakov, Seiberg-Witten theory and integrable systems, World Scientific, Singapore (1999).
A. Losev, N. Nekrasov and S.L. Shatashvili, Issues in topological gauge theory, Nucl. Phys. B 534 (1998) 549 [hep-th/9711108] [SPIRES].
A. Lossev, N. Nekrasov and S.L. Shatashvili, Testing Seiberg-Witten solution, hep-th/9801061 [SPIRES].
A. Lossev, N. Nekrasov and S.L. Shatashvili, Testing Seiberg-Witten solution, hep-th/9801061 [SPIRES].
G.W. Moore, N. Nekrasov and S. Shatashvili, D-particle bound states and generalized instantons, Commun. Math. Phys. 209 (2000) 77 [hep-th/9803265] [SPIRES].
N. Dorey, T.J. Hollowood, V.V. Khoze and M.P. Mattis, The calculus of many instantons, Phys. Rept. 371 (2002) 231 [hep-th/0206063] [SPIRES].
N.A. Nekrasov and S.L. Shatashvili, Supersymmetric vacua and Bethe ansatz, Nucl. Phys. Proc. Suppl. 192–193 (2009) 91 [arXiv:0901.4744] [SPIRES].
N.A. Nekrasov and S.L. Shatashvili, Quantum integrability and supersymmetric vacua, Prog. Theor. Phys. Suppl. 177 (2009) 105 [arXiv:0901.4748] [SPIRES].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161] [SPIRES].
R.Flume and R.Pogossian, An Algorithm for the microscopic evaluation of the coefficients of the Seiberg-Witten prepotential, Int. J. Mod. Phys. A 18 (2003) 2541 [hep-th/0208176] [SPIRES].
H. Nakajima and K. Yoshioka, Instanton counting on blowup. I, math/0306198 [SPIRES].
H. Nakajima, K. Yoshioka, Lectures on instanton counting, math/0311058 [SPIRES].
S.Shadchin, Status report on the instanton counting, SIGMA 2 (2006) 008 [hep-th/0601167] [SPIRES].
S. Shadchin, On certain aspects of string theory/gauge theory correspondence, hep-th/0502180 [SPIRES].
D. Bellisai, F. Fucito, A. Tanzini and G. Travaglini, Multi-instantons, supersymmetry and topological field theories, Phys. Lett. B 480 (2000) 365 [hep-th/0002110] [SPIRES].
U.Bruzzo, F.Fucito, A.Tanzini, G.Travaglini, On the multi - instanton measure for superYang-Mills theories, Nucl. Phys. B 611 (2001) 205 [hep-th/0008225] [SPIRES].
U.Bruzzo, F.Fucito, J.Morales and A.Tanzini, Multiinstanton calculus and equivariant cohomology, JHEP 05 (2003) 054 [hep-th/0211108] [SPIRES].
U.Bruzzo and F.Fucito, Superlocalization formulas and supersymmetric Yang-Mills theories, Nucl. Phys. B 678 (2004) 638 [math-ph/0310036] [SPIRES].
N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, hep-th/0306238 [SPIRES].
F.Fucito, J.Morales and R.Pogossian, Instantons on quivers and orientifolds, JHEP 10 (2004) 037 [hep-th/040890] [SPIRES].
L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville correlation functions from four-dimensional gauge theories, arXiv:0906.3219 [SPIRES].
N. Wyllard, A N−1 conformal Toda field theory correlation functions from conformal N = 2 SU(N) quiver gauge theories, JHEP 11 (2009) 002 [arXiv:0907.2189] [SPIRES].
N.Drukker, D.Morrison and T.Okuda, Loop operators and S-duality from curves on Riemann surfaces JHEP 09 (2009) 031 [arXiv:0907.2593] [SPIRES].
A. Marshakov, A. Mironov and A. Morozov, On combinatorial expansions of conformal blocks, arXiv:0907.3946 [SPIRES].
A. Mironov, S. Mironov, A. Morozov and A. Morozov, CFT exercises for the needs of AGT, arXiv:0908.2064 [SPIRES].
A.Mironov and A.Morozov, On AGT relation in the case of U(3), Nucl. Phys. B 825 (2009) 1 [arXiv:0908.2569] [SPIRES].
S.Iguri and C.Nunez, Coulomb integrals and conformal blocks in the AdS 3 - WZNW model, JHEP 11 (2009) 090 [arXiv:0908.3460] [SPIRES].
A. Mironov, S. Mironov, A. Morozov and A. Morozov, CFT exercises for the needs of AGT, arXiv:0908.2064 [SPIRES].
D. Nanopoulos and D. Xie, Hitchin equation, singularity and N = 2 superconformal field theories, arXiv:0911.1990 [SPIRES].
L.Alday, D.Gaiotto, S.Gukov, Y.Tachikawa and H.Verlinde, Loop and surface operators in N=2 gauge theory and Liouville modular geometry, arXiv:0909.0945 [SPIRES].
N.Drukker, J.Gomis, T.Okuda and J.Teschner, Gauge theory loop operators and Liouville theory, arXiv:0909.1105 [SPIRES].
A.Gadde, E.Pomoni, L.Rastelli and S.Razamat, S-duality and 2d Topological QFT, arXiv:0910.2225 [SPIRES].
G.Bonelli and A.Tanzini, Hitchin systems, N=2 gauge theories and W-gravity, arXiv:0909.4031 [SPIRES].
L.Alday, F.Benini and Y.Tachikawa, Liouville/Toda central charges from M5-branes, arXiv:0909.4776 [SPIRES].
Jian-Feng Wu and Yang Zhou, From Liouville to Chern-Simons, alternative realization of Wilson loop operators in AGT duality., arXiv:0911.1922 [SPIRES].
A. Mironov and A. Morozov, The power of Nekrasov functions, Phys. Lett. B 680 (2009) 188 [arXiv:0908.2190] [SPIRES].
R. Dijkgraaf and C. Vafa, Toda theories, matrix models, topological strings and N = 2 gauge systems, arXiv:0909.2453 [SPIRES].
D. Gaiotto, Asymptotically free N = 2 theories and irregular conformal blocks, arXiv:0908.0307 [SPIRES].
A. Marshakov, A. Mironov and A. Morozov, On non-conformal limit of the AGT relations, Phys. Lett. B 682 (2009) 125 [arXiv:0909.2052] [SPIRES].
H. Awata and Y. Yamada, Five-dimensional AGT conjecture and the deformed Virasoro algebra, arXiv:0910.4431 [SPIRES].
V. Alba and A. Morozov, Non-conformal limit of AGT relation from the 1-point torus conformal block, arXiv:0911.0363 [SPIRES].
A. Marshakov, A. Mironov and A. Morozov, Zamolodchikov asymptotic formula and instanton expansion in N = 2 SUSY N f = 2N c QCD, JHEP 11 (2009) 048 [arXiv:0909.3338] [SPIRES].
R. Poghossian, Recursion relations in CFT and N = 2 SYM theory, JHEP 12 (2009) 038 arXiv:0909.3412 [SPIRES].
L. Hadasz, Z. Jaskolski and P. Suchanek, Recursive representation of the torus 1-point conformal block, arXiv:0911.2353 [SPIRES].
A. Mironov and A. Morozov, Proving AGT relations in the large-c limit, Phys. Lett. B 682 (2009) 118 [arXiv:0909.3531] [SPIRES].
H. Itoyama, K. Maruyoshi and T. Oota, Notes on the quiver matrix model and 2d- 4D conformal connection, arXiv:0911.4244 [SPIRES].
T. Eguchi and K. Maruyoshi, Penner type matrix model and Seiberg-Witten theory, arXiv:0911.4797 [SPIRES].
S. Kanno, Y. Matsuo, S. Shiba and Y. Tachikawa, N = 2 gauge theories and degenerate fields of Toda theory, arXiv:0911.4787 [SPIRES].
A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B 241 (1984) 333 [SPIRES].
A. Zamolodchikov and Al. Zamolodchikov, Conformal field theory and critical phenomena in 2d systems (in Russian) (2009).
V.S. Dotsenko and V.A. Fateev, Conformal algebra and multipoint correlation functions in 2D statistical models, Nucl. Phys. B 240 (1984) 312 [SPIRES].
A. Gerasimov, A. Morozov, M. Olshanetsky, A. Marshakov and S.L. Shatashvili, Wess-Zumino-Witten model as a theory of free fields, Int. J. Mod. Phys. A 5 (1990) 2495 [SPIRES].
A. Gerasimov, A. Marshakov and A. Morozov, Free field representation of parafermions and related coset models, Nucl. Phys. B 328 (1989) 664 [Theor. Math. Phys. 83 (1990) 466] [Teor. Mat. Fiz. 83 (1990) 186] [SPIRES].
Al.B. Zamolodchikov, Conformal symmetry in two-dimensions: an explicit recurrence formula for the conformal partial wave amplitude, Commun. Math. Phys. 96 (1984) 419 [SPIRES].
Al.B. Zamolodchikov, Two-dimensional conformal symmetry and critical four-spin correlation functions in the Ashkin-Teller model, JETP 63 (1986) 1061.
Al.B. Zamolodchikov, Conformal symmetry in two-dimensional space: Recursion representation of conformal block, Theor. Math. Phys. 73 (1987) 1088.
Al.B. Zamolodchikov, Conformal scalar field on the hyperelliptic curve and critical Ashkin-Teller multipoint correlation functions, Nucl. Phys. B 285 (1987) 481 [SPIRES].
V.A. Fateev, A.V. Litvinov, A. Neveu and E. Onofri, Differential equation for four-point correlation function in Liouville field theory and elliptic four-point conformal blocks, J. Phys. A 42 (2009) 304011 [arXiv:0902.1331] [SPIRES].
A. Gerasimov, A. Marshakov, A. Mironov, A. Morozov and A. Orlov, Matrix models of 2-D gravity and Toda theory, Nucl. Phys. B 357 (1991) 565 [SPIRES].
A. Mironov and A. Morozov, On the origin of Virasoro constraints in matrix models: Lagrangian approach, Phys. Lett. B 252 (1990) 47 [SPIRES].
F. David, Loop equations and nonperturbative effects in two-dimensional quantum gravity, Mod. Phys. Lett. A 5 (1990) 1019 [SPIRES].
J. Ambjorn and Yu. Makeenko, Properties of loop equations for the Hermitean matrix model and for two-dimensional quantum gravity, Mod. Phys. Lett. A 5 (1990) 1753 [SPIRES].
H. Itoyama and Y. Matsuo, Noncritical Virasoro algebra of d < 1 matrix model and quantized string field, Phys. Lett. B 255 (1991) 202 [SPIRES].
A. Marshakov, A. Mironov and A. Morozov, From Virasoro constraints in Kontsevich’s model to W constraints in two matrix model, Mod. Phys. Lett. A 7 (1992) 1345 [hep-th/9201010] [SPIRES].
Y. Makeenko, A. Marshakov, A. Mironov and A. Morozov, Continuum versus discrete Virasoro in one matrix models, Nucl. Phys. B 356 (1991) 574 [SPIRES].
M. Kontsevich, Theory of intersections on the moduli space of curves, Funk. Anal. Prilozh. 25(2) (1991) 50.
M. Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function, Commun. Math. Phys. 147 (1992) 1 [SPIRES].
S. Kharchev, A. Marshakov, A. Mironov, A. Morozov and A. Zabrodin, Unification of all string models with C < 1, Phys. Lett. B 275 (1992) 311 [hep-th/9111037] [SPIRES].
S. Kharchev, A. Marshakov, A. Mironov, A. Morozov and A. Zabrodin, Towards unified theory of 2 − D gravity, Nucl. Phys. B 380 (1992) 181 [hep-th/9201013] [SPIRES].
S. Kharchev, A. Marshakov, A. Mironov and A. Morozov, Generalized Kontsevich model versus Toda hierarchy and discrete matrix models, Nucl. Phys. B 397 (1993) 339 [hep-th/9203043] [SPIRES].
P. Di Francesco, C. Itzykson and J.-B. Zuber, Polynomial averages in the Kontsevich model, Comm. Math. Phys. 151 (1993) 193 [hep-th/9206090] [SPIRES].
A. Marshakov, A. Mironov and A. Morozov, Generalized matrix models as conformal field theories: discrete case, Phys. Lett. B 265 (1991) 99 [SPIRES].
S. Kharchev, A. Marshakov, A. Mironov, A. Morozov and S. Pakuliak, Conformal matrix models as an alternative to conventional multimatrix models, Nucl. Phys. B 404 (1993) 717 [hep-th/9208044] [SPIRES].
S. Kharchev, A. Marshakov, A. Mironov and A. Morozov, Generalized Kazakov-Migdal-Kontsevich model: group theory aspects, Int. J. Mod. Phys. A 10 (1995) 2015 [hep-th/9312210] [SPIRES].
A. Marshakov and N. Nekrasov, Extended Seiberg-Witten theory and integrable hierarchy, JHEP 01 (2007) 104 [hep-th/0612019] [SPIRES].
A. Marshakov, On microscopic origin of integrability in Seiberg-Witten theory, Theor. Math. Phys. 154 (2008) 362 [arXiv:0706.2857] [SPIRES].
A. Marshakov, Non Abelian gauge theories, prepotentials and Abelian differentials, arXiv:0810.1536 [SPIRES].
B. Eynard, All order asymptotic expansion of large partitions, J. Stat. Mech. (2008) P07023 [arXiv:0804.0381] [SPIRES].
I.K. Kostov, Gauge invariant matrix model for the A-D-E closed strings, Phys. Lett. B 297 (1992) 74 [hep-th/9208053] [SPIRES].
I.K. Kostov, Solvable statistical models on a random lattice, Nucl. Phys. Proc. Suppl. 45A (1996) 13 [hep-th/9509124] [SPIRES].
A. Morozov, String theory: what is it?, Phys. Usp. 35 (1992) 671.
A. Morozov, Integrability and matrix models, Phys. Usp. 37 (1994) 1 [hep-th/9303139] [SPIRES].
A. Morozov, Matrix models as integrable systems, hep-th/9502091 [SPIRES].
A. Morozov, Challenges of matrix models, hep-th/0502010 [SPIRES].
A. Mironov, 2-d gravity and matrix models. 1. 2-d gravity., Int. J. Mod. Phys. A 9 (1994) 4355 [hep-th/9312212] [SPIRES].
A. Mironov, Quantum deformations of tau functions, bilinear identities and representation theory, hep-th/9409190 [SPIRES].
A. Mironov, τ-function within group theory approach and its quantization, Theor. Math. Phys. 114 (1998) 127 [q-alg/9711006].
A.S. Alexandrov, A. Mironov and A. Morozov, Partition functions of matrix models as the first special functions of string theory. I: Finite size Hermitean 1-matrix model, Int. J. Mod. Phys. A 19 (2004) 4127 [Teor. Mat. Fiz. 142 (2005) 419] [hep-th/0310113] [SPIRES].
A.S. Alexandrov, A.D. Mironov and A.Yu. Morozov, Partition functions of matrix models as the first special functions of string theory: Finite Hermitian one-matrix model., Theor. Math. Phys. 142 (2005) 349 [SPIRES].
A. Morozov and Sh. Shakirov, Exact 2-point function in Hermitian matrix model, JHEP 12 (2009) 003 [arXiv:0906.0036] [SPIRES].
A.S. Alexandrov, A. Mironov, A. Morozov and P. Putrov, Partition functions of matrix models as the first special functions of string theory. II. Kontsevich Model, Int. J. Mod. Phys. A 24 (2009) 4939 [arXiv:0811.2825] [SPIRES].
A.S. Alexandrov, A. Mironov and A. Morozov, Solving Virasoro constraints in matrix models, Fortsch. Phys. 53 (2005) 512 [hep-th/0412205] [SPIRES].
A. Givental, Semisimple Frobenius structures at higher genus, math.AG/0008067.
I.K. Kostov, Conformal field theory techniques in random matrix models, hep-th/9907060 [SPIRES].
A. Alexandrov, A. Mironov and A. Morozov, M-theory of matrix models, Teor. Mat. Fiz. 150 (2007) 179 [hep-th/0605171] [SPIRES].
A.S. Alexandrov, A. Mironov and A. Morozov, Instantons and merons in matrix models, Physica D 235 (2007) 126 [hep-th/0608228] [SPIRES].
A. Alexandrov, A. Mironov and A. Morozov, BGWM as second constituent of complex matrix model., JHEP 12 (2009) 053 [arXiv:0906.3305] [SPIRES].
A. Mironov, Matrix models vs. matrix integrals, Theor. Math. Phys. 146 (2006) 63 [Teor. Mat. Fiz. 146 (2006) 77] [hep-th/0506158] [SPIRES].
N. Orantin, Symplectic invariants, Virasoro constraints and Givental decomposition, arXiv:0808.0635 [SPIRES].
A. Klemm and P. Sulkowski, Seiberg-Witten theory and matrix models, Nucl. Phys. B 819 (2009) 400 [arXiv:0810.4944] [SPIRES].
P. Sulkowski, Matrix models for 2* theories., Phys. Rev. D 80 (2009) 086006 arXiv:0904.3064 [SPIRES].
F. Cachazo, K.A. Intriligator and C. Vafa, A large-N duality via a geometric transition, Nucl. Phys. B 603 (2001) 3 [hep-th/0103067] [SPIRES].
F. Cachazo and C. Vafa, N = 1 and N = 2 geometry from fluxes, hep-th/0206017 [SPIRES].
R. Dijkgraaf and C. Vafa, Matrix models, topological strings, and supersymmetric gauge theories, Nucl. Phys. B 644 (2002) 3 [hep-th/0206255] [SPIRES].
R. Dijkgraaf and C. Vafa, On geometry and matrix models, Nucl. Phys. B 644 (2002) 21 [hep-th/0207106] [SPIRES].
R. Dijkgraaf and C. Vafa, A perturbative window into non-perturbative physics, hep-th/0208048 [SPIRES].
L. Chekhov and A. Mironov, Matrix models versus Seiberg-Witten/Whitham theories., Phys. Lett. B 552 (2003) 293 [hep-th/0209085] [SPIRES].
H. Itoyama and A. Morozov, The Dijkgraaf-Vafa prepotential in the context of general Seiberg-Witten theory, Nucl. Phys. B 657 (2003) 53 [hep-th/0211245] [SPIRES].
H. Itoyama and A. Morozov, Experiments with the WDVV equations for the gluino-condensate prepotential: The cubic (two-cut) case, Phys. Lett. B 555 (2003) 287 [hep-th/0211259] [SPIRES].
H. Itoyama and A. Morozov, Calculating gluino condensate prepotential, Prog. Theor. Phys. 109 (2003) 433 [hep-th/0212032] [SPIRES].
L. Chekhov, A. Marshakov, A. Mironov and D. Vasiliev, DV and WDVV, Phys. Lett. B 562 (2003) 323 [hep-th/0301071] [SPIRES].
L. Chekhov, A. Marshakov, A. Mironov and D. Vasiliev, Complex geometry of matrix models, Proc. Steklov Inst. Math. 251 (2005) 254 [hep-th/0506075] [SPIRES].
H. Itoyama and A. Morozov, Gluino-condensate (CIV-DV) prepotential from its Whithamtime derivatives, Int. J. Mod. Phys. A 18 (2003) 5889 [hep-th/0301136] [SPIRES].
A. Klemm, M. Mariño and S. Theisen, Gravitational corrections in supersymmetric gauge theory and matrix models, JHEP 03 (2003) 051 [hep-th/0211216] [SPIRES].
M. Matone and L. Mazzucato, Branched matrix models and the scales of supersymmetric gauge theories, JHEP 07 (2003) 015 [hep-th/0305225] [SPIRES].
R. Argurio, G. Ferretti and R. Heise, An introduction to supersymmetric gauge theories and matrix models, Int. J. Mod. Phys. A 19 (2004) 2015 [hep-th/0311066] [SPIRES].
M. Gomez-Reino, Exact superpotentials, theories with flavor and confining vacua, JHEP 06 (2004) 051 [hep-th/0405242] [SPIRES].
H. Itoyama and H. Kanno, Whitham prepotential and superpotential, Nucl. Phys. B 686 (2004) 155 [hep-th/0312306] [SPIRES].
H. Itoyama, K. Maruyoshi and M. Sakaguchi, N = 2 quiver gauge model and partial supersymmetry breaking, Nucl. Phys. B 794 (2008) 216 [arXiv:0709.3166] [SPIRES].
N.A. Nekrasov and S.L. Shatashvili, Quantization of integrable systems and four dimensional gauge theories, arXiv:0908.4052 [SPIRES].
A. Mironov and A. Morozov, Nekrasov functions and exact Bohr-Zommerfeld integrals, arXiv:0910.5670 [SPIRES].
A. Mironov and A. Morozov, Nekrasov functions from exact BS periods: the case of SU(N), arXiv:0911.2396 [SPIRES].
M. Semenov-Tyan-Shansky, Izv. RAN ser. Phys. 40 (1976) 562.
J.J. Duistermaat and G.J. Heckman, On the variation in the cohomology of the symplectic form of the reduced phase space, Inv. Math. 72 (1983) 153.
M. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984) 1 [SPIRES].
M.F. Atiyah, Circular symmetry and stationary phase approximation, Asterisque 131 (1985) 43.
E. Witten, Topological quantum field theory, Commun. Math. Phys. 117 (1988) 353 [SPIRES].
E. Witten, Introduction to cohomological field theories, Int. J. Mod. Phys. A 6 (1991) 2775 [SPIRES].
A. Alekseev, L. Faddeev and S. Shatashvili, Quantization of symplectic orbits of compact Lie groups by means of the functional integral, J. Geom. Phys. 1 (1989) 3.
M. Blau, E. Keski-Vakkuri and A. Niemi, Path integrals and geometry of trajectories., Phys. Lett. B 246 (1990) 92 [SPIRES].
A. Hietamaki, A. Morozov, A. Niemi and K. Palo, Geometry of N = 1/2 supersymmetry and the Atiyah-Singer index theorem, Phys. Lett. B 263 (1991) 417 [SPIRES].
A.Y. Morozov, A.J. Niemi and K. Palo, Supersymmetry and loop space geometry, Phys. Lett. B 271 (1991) 365 [SPIRES].
A.Y. Morozov, A. Niemi and K. Palo, Supersymplectic geometry of supersymmetric quantum field theories, Nucl. Phys. B 377 (1992) 295 [SPIRES].
A.Y. Morozov, A.J. Niemi and K. Palo, Geometric approach to supersymmetry, Int. J. Mod. Phys. B 6 (1992) 2149 [SPIRES].
M.F. Atiyah, N.J. Hitchin, V.G. Drinfeld and Y.I. Manin, Construction of instantons, Phys. Lett. A 65 (1978) 185 [SPIRES].
A.A. Belavin, A.M. Polyakov, A.S. Schwartz and Y.S. Tyupkin, Pseudoparticle solutions of the Yang-Mills equations, Phys. Lett. B 59 (1975) 85 [SPIRES].
A. Morozov, Unitary integrals and related matrix models, arXiv:0906.3518 [SPIRES].
R. Dijkgraaf, The moduli spaces of curves, Brikhauser, Prog. Math. 129 (1995) 149.
S. Lando and D. Zvonkine, Counting ramified coverings and intersection theory on spaces of rational functions I (Cohomology of Hurwitz spaces), Funk. Anal. Appl. 33 (1999) 178 [math.AG/0303218].
A. Givental, Gromov-Witten invariants and quantization of quadratic hamiltonians, math/0108100.
M. Kazarian, KP hierarchy for Hodge integrals, arXiv:0809.3263.
V. Bouchard and M. Marino, Hurwitz numbers, matrix models and enumerative geometry, arXiv:0709.1458 [SPIRES].
A. Mironov and A. Morozov, Virasoro constraints for Kontsevich-Hurwitz partition function, JHEP 02 (2009) 024 [arXiv:0807.2843] [SPIRES].
A. Okounkov, Toda equations for Hurwitz numbers, Math. Res. Lett. 7 (2000) 447.
A. Morozov and S. Shakirov, Generation of matrix models by W-operators, JHEP 04 (2009) 064 [arXiv:0902.2627] [SPIRES].
A. Morozov and S. Shakirov, On equivalence of two Hurwitz matrix models, Mod. Phys. Lett. A 24 (2009) 2659 [arXiv:0906.2573] [SPIRES].
G. Borot, B. Eynard, M. Mulase and B. Safnuk, A Matrix model for simple Hurwitz numbers, and topological recursion, arXiv:0906.1206 [SPIRES].
A. Mironov, A. Morozov and S. Natanzon, Complete set of cut-and-join operators in Hurwitz-Kontsevich theory, arXiv:0904.4227 [SPIRES].
A. Mironov, A. Morozov and S. Natanzon, Universal algebras of Hurwitz numbers, arXiv:0909.1164 [SPIRES].
E. Witten, Solutions of four-dimensional field theories via M-theory, Nucl. Phys. B 500 (1997) 3 [hep-th/9703166] [SPIRES].
A. Marshakov, M. Martellini and A. Morozov, Insights and puzzles from branes: 4D SUSY Yang-Mills from 6D models, Phys. Lett. B 418 (1998) 294 [hep-th/9706050] [SPIRES].
D. Gaiotto, N=2 dualities, arXiv:0904.2715 [SPIRES].
Y. Tachikawa, Six-dimensional D N theory and four-dimensional SO-USp quivers, JHEP 07 (2009) 067 [arXiv:0905.4074] [SPIRES].
D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin systems and the WKB approximation, arXiv:0907.3987 [SPIRES].
A. Okounkov, Generating functions for intersection numbers on moduli spaces of curves, math.AG/0101201.
P. Di Francesco, M. Gaudin, C. Itzykson and F. Lesage, Laughlin’s wave functions, Coulomb gases and expansions of the discriminant, Int. J. Mod. Phys. A 9 (1994) 4257 [hep-th/9401163] [SPIRES].
A. Zabrodin, Random matrices and Laplacian growth, arXiv:0907.4929.
C. Itoi, Universal wide correlators in non-Gaussian orthogonal, unitary and symplectic random matrix ensembles, Nucl. Phys. B 493 (1997) 651 [cond-mat/9611214] [SPIRES].
A. Marshakov and A. Mironov, 5D and 6D supersymmetric gauge theories: Prepotentials from integrable systems, Nucl. Phys. B 518 (1998) 59 [hep-th/9711156] [SPIRES].
B. De Wit and G. t’Hooft, Nonconvergence of the 1/N expansion for SU(N) gauge fields on a lattice, Phys. Lett. B 69 (1977) 61 [SPIRES].
A. Mironov, A. Morozov and G.W. Semenoff, Unitary matrix integrals in the framework of generalized Kontsevich model. 1. Brezin-Gross-Witten model, Int. J. Mod. Phys. A 11 (1996) 5031 [hep-th/9404005] [SPIRES].
M. Aganagic, R. Dijkgraaf, A. Klemm, M. Mariño and C. Vafa, Topological strings and integrable hierarchies, Commun. Math. Phys. 261 (2006) 451 [hep-th/0312085] [SPIRES].
T.W. Grimm, A. Klemm, M. Mariño and M. Weiss, Direct integration of the topological string, JHEP 08 (2007) 058 [hep-th/0702187] [SPIRES].
E.K. Sklyanin, Separation of variables – new trends, Prog. Theor. Phys. Suppl. 118 (1995) 35 [solv-int/9504001] [SPIRES].
A. Gerasimov, S. Khoroshkin, D. Lebedev, A. Mironov and A. Morozov, Generalized Hirota equations and representation theory. 1. The case of SL(2) and SL q (2), Int. J. Mod. Phys. A 10 (1995) 2589 [hep-th/9405011] [SPIRES].
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Mironov, A., Morozov, A. & Shakirov, S. Matrix model conjecture for exact BS periods and Nekrasov functions. J. High Energ. Phys. 2010, 30 (2010). https://doi.org/10.1007/JHEP02(2010)030
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DOI: https://doi.org/10.1007/JHEP02(2010)030