Abstract
We demonstrate an equivalence between two integrable flows defined in a polynomial ring quotiented by an ideal generated by a polynomial. This duality of integrable systems allows us to systematically exploit the Korteweg-de Vries hierarchy and its tau-function to propose amplitudes for non-compact topological gravity on Riemann surfaces of arbitrary genus. We thus quantize topological gravity coupled to non-compact topologica matter and demonstrate that this phase of topological gravity at N = 2 matter central charge larger than three is equivalent to the phase with matter of central charge smaller than three.
Highlights
Field theories with N = 2 supersymmetry in two dimensions give rise to topological quantum field theories after twisting [1]
When the starting point is a non-compact conformal field theory, the correlation functions of the resulting topological quantum field theories were recently computed [2]. These theories were coupled to topological gravity [3, 4] and the gravitational theory was solved on the sphere
One motivation for studying topological gravity coupled to non-compact matter is to test the gravitational consequences of going beyond the central charge bound c = 3 for N = 2 minimal matter in two dimensions
Summary
We analyze the extent to which the change of variables proves an equivalence between the compact and non-compact topological quantum field theories, and their coupling to gravity. We observe that the inversion of variables comes down to an analytic continuation of the exponent of the superpotential from kc to −k, where both kc and k are positive integers.3 This analytic continuation was proposed as a method for obtaining results about models of topological gravity coupled to a topological non-compact coset conformal field theory [18], or matrix models with a negative power monomial potential [19]. In appendices A and B we provide illustrations that may help to reveal aspects of our paper as either subtle, or simple
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