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Baxter’s relations and spectra of quantum integrable models, Preprint arXiv
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MayerCluster Expansion of Instanton Partition Functions and Thermodynamic Bethe Ansatz
 HAMBURGER BEITRÄGE ZUR MATHEMATIK 497
, 2014
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N = 2 QUIVER GAUGE THEORIES ON ATYPE ALE SPACES
, 2014
"... ABSTRACT. We survey and compare recent approaches to the computation of the partition functions and correlators of chiral BPS observables in N = 2 gauge theories on ALE spaces based on quiver varieties and the minimal resolution Xk of the Ak−1 toric singularity C2/Zk, in light of their recently con ..."
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ABSTRACT. We survey and compare recent approaches to the computation of the partition functions and correlators of chiral BPS observables in N = 2 gauge theories on ALE spaces based on quiver varieties and the minimal resolution Xk of the Ak−1 toric singularity C2/Zk, in light of their recently conjectured duality with twodimensional coset conformal field theories. We review and elucidate the rigorous constructions of gauge theories for a particular family of ALE spaces, using their relation to the cohomology of moduli spaces of framed torsion free sheaves on a suitable orbifold compactification of Xk. We extend these computations to generic N = 2 superconformal quiver gauge theories, obtaining in these instances new constraints on fractional instanton charges, a rigorous proof of the Nekrasov master formula, and new quantizations of Hitchin systems based on the underlying SeibergWitten geometry.
Topological Strings from Quantum Mechanics
, 2014
"... We propose a general correspondence which associates a nonperturbative quantummechanical operator to a toric CalabiYau manifold, and we conjecture an explicit formula for its spectral determinant in terms of an Mtheoretic version of the topological string free energy. As a consequence, we derive ..."
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We propose a general correspondence which associates a nonperturbative quantummechanical operator to a toric CalabiYau manifold, and we conjecture an explicit formula for its spectral determinant in terms of an Mtheoretic version of the topological string free energy. As a consequence, we derive an exact quantization condition for the operator spectrum, in terms of the vanishing of a generalized theta function. The perturbative part of this quantization condition is given by the NekrasovShatashvili limit of the refined topological string, but there are nonperturbative corrections determined by the conventional topological string. We analyze in detail the cases of local P2, local P1xP1 and local F1. In all these cases, the predictions for the spectrum agree with the existing numerical results. We also show explicitly that our conjectured spectral determinant leads to the correct spectral traces of the corresponding operators, which are closely related to topological string theory at orbifold points. Physically, our results provide a Fermi gas picture of topological strings on toric CalabiYau manifolds, which is [...]
Physical mathematics and the future
, 2014
"... These are some thoughts meant to accompany one of the summary talks at Strings2014, Princeton, June 27, 2014. This is a snapshot of a personal and perhaps heterodox view of the relation of Physics and Mathematics, together with some guesses about some of the directions forward in the field of Physic ..."
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These are some thoughts meant to accompany one of the summary talks at Strings2014, Princeton, June 27, 2014. This is a snapshot of a personal and perhaps heterodox view of the relation of Physics and Mathematics, together with some guesses about some of the directions forward in the field of Physical Mathematics. At least, this is my view as of July 21, 2014.
APCTP Pre2014010 Spherical Hecke algebra in the NekrasovShatashvili limit JeanEmile BOURGINE∗
"... The Spherical Hecke central (SHc) algebra has been shown to act on the Nekrasov instanton partition functions of N = 2 gauge theories. Its presence accounts for both integrability and AGT correspondence. On the other hand, a specific limit of the Omega background, introduced by Nekrasov and Shatashv ..."
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The Spherical Hecke central (SHc) algebra has been shown to act on the Nekrasov instanton partition functions of N = 2 gauge theories. Its presence accounts for both integrability and AGT correspondence. On the other hand, a specific limit of the Omega background, introduced by Nekrasov and Shatashvili (NS), leads to the appearance of TBA and Bethe like equations. To unify these two points of view, we study the NS limit of the SHc algebra. We provide an expression of the instanton partition function in terms of Bethe roots, and define a set of operators that generates infinitesimal variations of the roots. These operators obey the commutation relations defining the SHc algebra at first order in the equivariant parameter 2. Furthermore, their action on the bifundamental contributions reproduces the KannoMatsuoZhang transformation. We also discuss the connections with the Mayer cluster expansion approach that leads to TBAlike equations.
Classical Liouville Threepoint Functions from RiemannHilbert Analysis
, 2014
"... We study semiclassical correlation functions in Liouville field theory on a twosphere when all operators have large conformal dimensions. In the usual approach, such computation involves solving the classical Liouville equation, which is known to be extremely difficult for higherpoint functions. T ..."
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We study semiclassical correlation functions in Liouville field theory on a twosphere when all operators have large conformal dimensions. In the usual approach, such computation involves solving the classical Liouville equation, which is known to be extremely difficult for higherpoint functions. To overcome this difficulty, we propose a new method based on the RiemannHilbert analysis, which is applied recently to the holographic calculation of correlation functions in AdS/CFT. The method allows us to directly compute the correlation functions without solving the Liouville equation explicitly. To demonstrate its utility, we apply it to threepoint functions, which are known to be solvable, and confirm that it correctly reproduces the classical limit of the DOZZ formula for quantum threepoint functions. This provides good evidence for the validity of this method.ar X iv
P o
, 2014
"... A pedagogical introduction to quantum integrability with a view towards theoretical highenergy physics ..."
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A pedagogical introduction to quantum integrability with a view towards theoretical highenergy physics