An extended square matrix of (x,q)-series indexed by boundary parabolic SL2(C) flat connections completely describes the resurgent structure, Stokes constants, and Borel transform of Chern-Simons perturbation theory at the trivial flat connection for hyperbolic knot complements.
Operators from mirror curves and the quantum dilogarithm
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abstract
Mirror manifolds to toric Calabi-Yau threefolds are encoded in algebraic curves. The quantization of these curves leads naturally to quantum-mechanical operators on the real line. We show that, for a large number of local del Pezzo Calabi-Yau threefolds, these operators are of trace class. In some simple geometries, like local P2, we calculate the integral kernel of the corresponding operators in terms of Faddeev's quantum dilogarithm. Their spectral traces are expressed in terms of multi-dimensional integrals, similar to the state-integrals appearing in three-manifold topology, and we show that they can be evaluated explicitly in some cases. Our results provide further verifications of a recent conjecture which gives an explicit expression for the Fredholm determinant of these operators, in terms of enumerative invariants of the underlying Calabi-Yau threefolds.
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UNVERDICTED 2representative citing papers
Derives weakest quantization conditions in terms of monodromy data for higher-order DEs tied to quantum Toda chain and proves duality predictions for deformed Schrödinger operators.
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Resurgence of Chern-Simons theory at the trivial flat connection
An extended square matrix of (x,q)-series indexed by boundary parabolic SL2(C) flat connections completely describes the resurgent structure, Stokes constants, and Borel transform of Chern-Simons perturbation theory at the trivial flat connection for hyperbolic knot complements.
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Higher-Rank Connections and Deformed Schr\"odinger Operators
Derives weakest quantization conditions in terms of monodromy data for higher-order DEs tied to quantum Toda chain and proves duality predictions for deformed Schrödinger operators.