We study matrix models in the β-ensemble by building on the refined recursion relation proposed by Chekhov and Eynard. We present explicit results for the first β-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 ${\cal N}=1$ gauge theories, and the calculation of vevs of surface operators in superconformal ${\cal N}=2$ theories and their Liouville duals. Finally, we study the β-deformation of the Chern–Simons matrix model. Our results indicate that this model does not provide an appropriate description of the Ω-deformed topological string on the resolved conifold, and therefore that the β-deformation might provide a different generalization of topological string theory in toric Calabi–Yau backgrounds.
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In particular, it continues to hold true when we vary the choice of A and B cycles, thereby changing the very definition of dS(p, q) and k; for example, in the context of Seiberg-Witten curves, this would allow us to find expansions in any S-duality frame, also at strong coupling.
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This form of the amplitude is dictated by an implicit choice of gluing along one of the unpreferred legs of the refined topological vertex. Other choices of gluing only result in minor differences in this particular case, which for the
$\mathcal {O}(1)$
term in (4.37) amount to an overall rescaling by a factor of et; this obviously leaves unchanged the discussion about the comparison with the Chern-Simons matrix model computation.© 2011 American Institute of Physics.
2011
American Institute of Physics
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