On the Spectrum of the Local $${\mathbb {P}}^2$$ P 2 Mirror Curve
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nnales Henri Poincar´ e
On the Spectrum of the Local P2 Mirror Curve Rinat Kashaev
and Sergey Sergeev
Abstract. We address the spectral problem of the formally normal quantum mechanical operator associated with the quantised mirror curve of the toric (almost) del Pezzo Calabi–Yau threefold called local P2 in the case of complex values of Planck’s constant. We show that the problem can be approached in terms of the Bethe ansatz-type highly transcendental equations. Mathematics Subject Classification. Primary 39A13, Secondary 33E30.
1. Introduction The recent progress in topological string theory reveals connections between spectral theory, integrable systems and local mirror symmetry. The results on linkings of some quantum mechanical spectral problems with integrable systems and conformal field theory [4,6], together with the relation of topological strings in toric Calabi–Yau manifolds to integrable systems [1,2,10– 13,17,19,20], have lead to the conjecture on the topological string/spectral theory (TS/ST) correspondence [5,8]. In many cases, quantisation of mirror curves produces trace class quantum mechanical operators, and according to the TS/ST correspondence, their spectra seem to contain a great deal of information of the enumerative geometry of the underlying Calabi–Yau manifold; see [16] for a review and references therein. In the case of toric (almost) del Pezzo Calabi–Yau threefold known as local P2 , the corresponding operator is of the form
The work is partially supported by Australian Research Council and Swiss National Science Foundation.
3480
R. Kashaev and S. Sergeev
Ann. Henri Poincar´e
O P2 = u + v + ei/2 v −1 u−1
(1)
with invertible positive self-adjoint operators u and v such that uv = ei vu,
(2)
where is a real strictly positive parameter. With various levels of generality, the spectral problem for similar operators has been addressed in [15] from the perspective of exact WKB approximation, in [9,18] using a matrix integral representation of the eigenfunctions and in [3,14,22] from the standpoint of quantum integrable systems. In this paper, following the approach of [14,22], we consider the so-called strongly coupled regime for the operator (1) with invertible normal operators u and v satisfying the Heisenberg–Weyl relation (2) where = 2πei2θ ,
θ ∈]0, π/2[.
(3)
In this case, we show that a realisation of the operator (1) in the Hilbert space L2 (R) is formally normal so that its spectral problem is still well defined provided the operator admits a normal extension. Our main result is Theorem 2 which describes a set of solutions of our spectral problem in terms of generalised the Bethe ansatz-type transcendental equations. Outline In Sect. 2, we introduce the setup, fix notation and conventions and introduce some objects and constructions that will be used subsequently in the rest of the paper. In Sect. 3, we specify our Hamiltonian as an unbounded operator in L2 (R), show that it is formally normal, rewrite the corresponding spectral problem as a pair of functional difference
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