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484
Liouville Correlation Functions from Fourdimensional Gauge Theories
 SIMONS CENTER FOR GEOMETRY AND PHYSICS, STONY BROOK UNIVERSITY
, 2009
"... We conjecture an expression for the Liouville theory conformal blocks and correlation functions on a Riemann surface of genus g and n punctures as the Nekrasov partition function of a certain class of N = 2 SCFTs recently defined by one of the authors. We conduct extensive tests of the conjecture ..."
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Cited by 394 (22 self)
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We conjecture an expression for the Liouville theory conformal blocks and correlation functions on a Riemann surface of genus g and n punctures as the Nekrasov partition function of a certain class of N = 2 SCFTs recently defined by one of the authors. We conduct extensive tests of the conjecture at genus 0, 1.
Localization of gauge theory on a foursphere and supersymmetric Wilson loops
, 2007
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The topological vertex
, 2003
"... We construct a cubic field theory which provides all genus amplitudes of the topological Amodel for all noncompact toric CalabiYau threefolds. The topology of a given Feynman diagram encodes the topology of a fixed CalabiYau, with Schwinger parameters playing the role of Kähler classes of the th ..."
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Cited by 169 (24 self)
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We construct a cubic field theory which provides all genus amplitudes of the topological Amodel for all noncompact toric CalabiYau threefolds. The topology of a given Feynman diagram encodes the topology of a fixed CalabiYau, with Schwinger parameters playing the role of Kähler classes of the threefold. We interpret this result as an operatorial computation of the amplitudes in the Bmodel mirror which is the quantum KodairaSpencer theory. The only degree of freedom of this theory is an unconventional chiral scalar on a Riemann surface. In this setup we identify the Bbranes on the mirror Riemann
AN−1 conformal Toda field theory correlation functions from conformal N=2 SU(N) quiver gauge theories
, 2009
"... We propose a relation between correlation functions in the 2d AN−1 conformal Toda theories and the Nekrasov instanton partition functions in certain conformal N = 2 SU(N) 4d quiver gauge theories. Our proposal generalises the recently uncovered relation between the Liouville theory and SU(2) quivers ..."
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Cited by 159 (4 self)
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We propose a relation between correlation functions in the 2d AN−1 conformal Toda theories and the Nekrasov instanton partition functions in certain conformal N = 2 SU(N) 4d quiver gauge theories. Our proposal generalises the recently uncovered relation between the Liouville theory and SU(2) quivers [1]. New features appear in the analysis that have no counterparts in the Liouville case.
Quantization of Integrable Systems and Four Dimensional Gauge Theories
, 2009
"... We study four dimensional N = 2 supersymmetric gauge theory in the Ωbackground with the two dimensional N = 2 superPoincare invariance. We explain how this gauge theory provides the quantization of the classical integrable system underlying the moduli space of vacua of the ordinary four dimension ..."
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Cited by 115 (3 self)
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We study four dimensional N = 2 supersymmetric gauge theory in the Ωbackground with the two dimensional N = 2 superPoincare invariance. We explain how this gauge theory provides the quantization of the classical integrable system underlying the moduli space of vacua of the ordinary four dimensional N = 2 theory. The εparameter of the Ωbackground is identified with the Planck constant, the twisted chiral ring maps to quantum Hamiltonians, the supersymmetric vacua are identified with Bethe states of quantum integrable systems. This four dimensional gauge theory in its low energy description has two dimensional twisted superpotential which becomes the YangYang function of the integrable system. We present the thermodynamicBetheansatz like formulae for these functions and for the spectra of commuting Hamiltonians following the direct computation in gauge theory. The general construction is illustrated at the examples of the manybody systems, such as the periodic Toda chain, the elliptic CalogeroMoser system, and their relativistic versions, for which we present a complete characterization of the L²spectrum. We very briefly discuss the quantization of Hitchin system.
Black holes, qdeformed 2d YangMills, and nonperturbative topological strings,” Nucl. Phys. B715
, 2005
"... We count the number of bound states of BPS black holes on local CalabiYau threefolds involving a Riemann surface of genus g. We show that the corresponding gauge theory on the brane reduces to a qdeformed YangMills theory on the Riemann surface. Following the recent connection between the black h ..."
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Cited by 101 (10 self)
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We count the number of bound states of BPS black holes on local CalabiYau threefolds involving a Riemann surface of genus g. We show that the corresponding gauge theory on the brane reduces to a qdeformed YangMills theory on the Riemann surface. Following the recent connection between the black hole entropy and the topological string partition function, we find that for a large black hole charge N, up to corrections of O(e−N), ZBH is given as a sum of a square of chiral blocks, each of which corresponds to a specific Dbrane amplitude. The leading chiral block, the vacuum block, corresponds to the closed topological string amplitudes. The subleading chiral blocks involve topological string amplitudes with Dbrane insertions at (2g − 2) points on the Riemann surface analogous to the Ω points in the large N 2d YangMills theory. The finite N amplitude provides a nonperturbative definition of topological strings in these backgrounds. This also leads to a novel nonperturbative formulation of c = 1 noncritical string at the selfdual radius. November
Topological strings and (almost) modular forms
, 2007
"... The Bmodel topological string theory on a CalabiYau threefold X has a symmetry group Γ, generated by monodromies of the periods of X. This acts on the topological string wave function in a natural way, governed by the quantum mechanics of the phase space H 3 (X). We show that, depending on the cho ..."
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Cited by 94 (10 self)
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The Bmodel topological string theory on a CalabiYau threefold X has a symmetry group Γ, generated by monodromies of the periods of X. This acts on the topological string wave function in a natural way, governed by the quantum mechanics of the phase space H 3 (X). We show that, depending on the choice of polarization, the genus g topological string amplitude is either a holomorphic quasimodular form or an almost holomorphic modular form of weight 0 under Γ. Moreover, at each genus, certain combinations of genus g amplitudes are both modular and holomorphic. We illustrate this for the local CalabiYau manifolds giving rise to SeibergWitten gauge theories in four dimensions and local IP2 and IP1×IP1. As a byproduct, we also obtain a simple way of relating the topological string amplitudes near different points in the moduli space, which we use to give predictions for GromovWitten invariants of the orbifold C 3 / Z3.
Instanton counting on blowup, I. 4Dimensional pure gauge theory
 INVENT. MATH
, 2005
"... We give a mathematically rigorous proof of Nekrasov’s conjecture: the integration in the equivariant cohomology over the moduli spaces of instantons on R 4 gives a deformation of the SeibergWitten prepotential for N = 2 SUSY YangMills theory. Through a study of moduli spaces on the blowup of R 4, ..."
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Cited by 85 (5 self)
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We give a mathematically rigorous proof of Nekrasov’s conjecture: the integration in the equivariant cohomology over the moduli spaces of instantons on R 4 gives a deformation of the SeibergWitten prepotential for N = 2 SUSY YangMills theory. Through a study of moduli spaces on the blowup of R 4, we derive a differential equation for the Nekrasov’s partition function. It is a deformation of the equation for the SeibergWitten prepotential, found by Losev et al., and further studied by Gorsky et al.
Instanton Counting and ChernSimons Theory
 Adv. Theor. Math. Phys
"... The instanton partition function of N = 2, D = 4 SU(2) gauge theory is obtained by taking the field theory limit of the topological open string partition function, given by a ChernSimons theory, of a CY3fold. The CY3fold on the open string side is obtained by geometric transition from local IP 1 ..."
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Cited by 69 (3 self)
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The instanton partition function of N = 2, D = 4 SU(2) gauge theory is obtained by taking the field theory limit of the topological open string partition function, given by a ChernSimons theory, of a CY3fold. The CY3fold on the open string side is obtained by geometric transition from local IP 1 × IP 1 which is used in the geometric engineering of the SU(2) theory. The partition function obtained from the ChernSimons theory agrees with the closed topological string partition function of local IP 1 × IP 1 proposed recently by Nekrasov. We also obtain the partition functions for local F1 and F2 CY3folds and show that the topological string amplitudes of all three local Hirzebruch surfaces give rise to the same field theory limit. It is shown that a generalization of the topological closed string partition function whose field theory limit is the generalization of the instanton partition function, proposed by Nekrasov, Large N dualities in the context of closed and open topological strings on different CY3fold backgrounds have been the source of much excitement recently [1, 2]. These dualities