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167
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 393 (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.
Curve counting via stable pairs in the derived category
, 2009
"... For a nonsingular projective 3fold X, we define integer invariants virtually enumerating pairs (C,D) where C ⊂ X is an embedded curve and D ⊂ C is a divisor. A virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of X. The resulting in ..."
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Cited by 114 (21 self)
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For a nonsingular projective 3fold X, we define integer invariants virtually enumerating pairs (C,D) where C ⊂ X is an embedded curve and D ⊂ C is a divisor. A virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of X. The resulting invariants are conjecturally equivalent, after universal transformations, to both the GromovWitten and DT theories of X. For CalabiYau 3folds, the latter equivalence should be viewed as a wallcrossing formula in the derived category. Several calculations of the new invariants are carried out. In the Fano case, the local contributions of nonsingular embedded curves are found. In the local toric CalabiYau case, a completely new form of the topological vertex is described. The virtual enumeration of pairs is closely related to the geometry underlying the BPS state counts of Gopakumar and Vafa. We
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 93 (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.
GromovWitten/DonaldsonThomas correspondence for toric 3folds
, 2008
"... We prove the equivariant GromovWitten theory of a nonsingular toric 3fold X with primary insertions is equivalent to the equivariant DonaldsonThomas theory of X. As a corollary, the topological vertex calculations by Agangic, Klemm, Mariño, and Vafa of the GromovWitten theory of local CalabiYau ..."
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Cited by 61 (17 self)
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We prove the equivariant GromovWitten theory of a nonsingular toric 3fold X with primary insertions is equivalent to the equivariant DonaldsonThomas theory of X. As a corollary, the topological vertex calculations by Agangic, Klemm, Mariño, and Vafa of the GromovWitten theory of local CalabiYau toric 3folds are proven to be correct in the full 3leg setting.
Topological string amplitudes, complete intersection Calabi–Yau spaces and threshold corrections
, 2005
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Wall crossing in local Calabi Yau manifolds
, 2008
"... We study the BPS states of a D6brane wrapping the conifold and bound to collections of D2 and D0 branes. We find that in addition to the complexified Kähler parameter of the rigid P 1 it is necessary to introduce an extra real parameter to describe BPS partition functions and marginal stability wa ..."
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Cited by 46 (3 self)
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We study the BPS states of a D6brane wrapping the conifold and bound to collections of D2 and D0 branes. We find that in addition to the complexified Kähler parameter of the rigid P 1 it is necessary to introduce an extra real parameter to describe BPS partition functions and marginal stability walls. The supergravity approach to BPS statecounting gives a simple derivation of results of Szendrői concerning DonaldsonThomas theory on the noncommutative conifold. This example also illustrates some interesting limitations on the supergravity approach to BPS statecounting and wallcrossing.
Instanton counting, Macdonald function and the moduli space
 of Dbranes,” JHEP 0505 (2005) 039, arXiv:hepth/0502061
"... We argue the connection of Nekrasov’s partition function in the Ω background and the moduli space of Dbranes, suggested by the idea of geometric engineering and GopakumarVafa invariants. In the instanton expansion of N = 2 SU(2) YangMills theory the Nakrasov’s partition function with equivariant ..."
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Cited by 40 (5 self)
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We argue the connection of Nekrasov’s partition function in the Ω background and the moduli space of Dbranes, suggested by the idea of geometric engineering and GopakumarVafa invariants. In the instanton expansion of N = 2 SU(2) YangMills theory the Nakrasov’s partition function with equivariant parameters ǫ1,ǫ2 of toric action on C 2 factorizes correctly as the character of SU(2)L ×SU(2)R spin representation. We show that up to two instantons the spin contents are consistent with the Lefschetz action on the moduli space of D2branes on (local) F0. We also present an attempt at constructing a refined topological vertex in terms of the Macdonald function. The refined topological vertex with two parameters of T 2 action allows us to obtain the generating functions of equivariant χy and elliptic Gravity/gauge theory correspondence is one of important subjects in the recent developments of nonperturbative string theory and Dbranes are the crucial dynamical objects for understanding the correspondence. In topological theory the correspondence
The D5brane effective action and superpotential in N = 1 Compactifications
, 2008
"... The fourdimensional effective action for D5branes in generic compact CalabiYau orientifolds is computed by performing a KaluzaKlein reduction. The N = 1 Kähler potential, the superpotential, the gaugekinetic coupling function and the Dterms are derived in terms of the geometric data of the int ..."
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Cited by 37 (4 self)
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The fourdimensional effective action for D5branes in generic compact CalabiYau orientifolds is computed by performing a KaluzaKlein reduction. The N = 1 Kähler potential, the superpotential, the gaugekinetic coupling function and the Dterms are derived in terms of the geometric data of the internal space and of the twocycle wrapped by the D5brane. In particular, we obtain the D5brane and flux superpotential by integrating out fourdimensional threeforms which couple via the ChernSimons action. Also the infinitesimal complex structure deformations of the twocycle induced by the deformations of the ambient space contribute to the Fterms. The superpotential can be expressed in terms of relative periods depending on both the open and closed moduli. To analyze this dependence we blow up along the twocycle and obtain a rigid divisor in an auxiliary compact threefold with negative first Chern class. The variation of the mixed Hodge structure on this blownup geometry is equivalent to the original deformation problem and can be analyzed by PicardFuchs equations. We exemplify the blowup procedure for a noncompact CalabiYau threefold given by the canonical bundle over del Pezzo surfaces.