Abstract
Abstract We study the correspondence between four-dimensional supersymmetric gauge theories and two-dimensional conformal field theories in the case of $ \mathcal{N}={2^{*}} $ gauge theory. We emphasize the genus expansion on the gauge theory side, as obtained via geometric engineering from the topological string. This point of view uncovers modular properties of the one-point conformal block on a torus with complexified intermediate momenta: in the large intermediate weight limit, it is a power series whose coefficients are quasimodular forms. The all-genus viewpoint that the conformal field theory approach lends to the topological string yields insight into the analytic structure of the topological string partition function in the field theory limit.
Highlights
Gauge theory side to an expansion in a complex structure parameter of the corresponding Riemann surface with punctures on the conformal field theory side [1, 9,10,11]
Combining recursion relations satisfied by the toroidal conformal blocks with modular results for N = 2∗ and N = 4 gauge theory [20], we have obtained new insights on both sides of the correspondence
We have demonstrated that the gauge theory results imply an infinite sequence of constraints, non-perturbative in q = e2πiτ, on the residua of the conformal blocks
Summary
We exhibit the role of the one-point toroidal conformal block in conformal field theory, discuss the corresponding quantity in topological string theory, and review how they are expected to match [1]. We will recall the engineering of gauge theory within topological string theory and freely use the language of the latter setup in the following. The only parameter which enters is the central charge c. It will be useful to introduce the following parameterizations that have their origins in Liouville theory. The semi-classical Liouville limit c → ∞ has the incarnations b → 0 or b → ∞. In the semi-classical limit, we connect to a classical Liouville theory with action principle.
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