Exact eigenfunctions and the open topological string

Abstract

Mirror curves to toric Calabi-Yau threefolds can be quantized and lead to\ntrace class operators on the real line. The eigenvalues of these operators are\nencoded in the BPS invariants of the underlying threefold, but much less is\nknown about their eigenfunctions. In this paper we first develop methods in\nspectral theory to compute these eigenfunctions. We also provide a matrix\nintegral representation which allows to study them in a 't Hooft limit, where\nthey are described by standard topological open string amplitudes. Based on\nthese results, we propose a conjecture for the exact eigenfunctions which\ninvolves both the WKB wavefunction and the standard topological string\nwavefunction. This conjecture can be made completely explicit in the maximally\nsupersymmetric, or self-dual case, which we work out in detail for local P1xP1.\nIn this case, our conjectural eigenfunctions turn out to be closely related to\nBaker-Akhiezer functions on the mirror curve, and they are in full agreement\nwith first-principle calculations in spectral theory.\n