Quantum Theta Functions Research Articles

Spectral determinants and quantum theta functions

It has been recently conjectured that the spectral determinants of operators associated to mirror curves can be expressed in terms of a generalization of theta functions, called quantum theta functions. In this paper we study the symplectic properties of these spectral determinants by expanding them around the point , where the quantum theta functions become conventional theta functions. We find that they are modular invariant, order by order, and we give explicit expressions for the very first terms of the expansion. Our derivation requires a detailed understanding of the modular properties of topological string free energies in the Nekrasov–Shatashvili limit. We derive these properties in a diagrammatic form. Finally, we use our results to provide a new test of the duality between topological strings and spectral theory.

Quantum Super Theta Vectors and Theta Functions

There are many concepts around classical theta functions, theta vectors and quantum theta functions. Manin clarifled the relation among these concepts with the sym- metry of functional equations. We extend his results to the super torus.

Quantum Theta Functions and Gabor Frames for Modulation Spaces

Quantum Theta Functions and Gabor Frames for Modulation Spaces

Quantum Theta Functions and Gabor Frames for Modulation Spaces

Representations of the celebrated Heisenberg commutation relations in quantum mechanics and their exponentiated versions form the starting point for a number of basic constructions, both in mathematics and mathematical physics (geometric quantization, quantum tori, classical and quantum theta functions) and signal analysis (Gabor analysis). In this paper we try to bridge the two communities, represented by the two co--authors: that of noncommutative geometry and that of signal analysis. After providing a brief comparative dictionary of the two languages, we will show e.g. that the Janssen representation of Gabor frames with generalized Gaussians as Gabor atoms yields in a natural way quantum theta functions, and that the Rieffel scalar product and associativity relations underlie both the functional equations for quantum thetas and the Fundamental Identity of Gabor analysis.

Quantum thetas on noncommutative with general embeddings

In this paper, we construct quantum theta functions over noncommutative with general embeddings. Manin has constructed quantum theta functions from the lattice embedding into vector space × finite group. We extend Manin's construction of quantum thetas to the case of general embedding of vector space × lattice × torus. It turns out that only for the vector space part of the embedding there exists the holomorphic theta vector, while for the lattice part there does not. Furthermore, the so-called quantum translations from embedding into the lattice part become non-additive, while those from the vector space part are additive.

Quantum thetas on noncommutative from embeddings into lattice

In this paper, we investigate the theta vector and quantum theta function over noncommutative from the embedding of . Manin has constructed the quantum theta functions from the lattice embedding into vector space (× finite group). We extend Manin's construction of the quantum theta function to the embedding of vector space × lattice case. We find that the holomorphic theta vector exists only over the vector space part of the embedding, and over the lattice part we can only impose the condition for the Schwartz function. The quantum theta function built on this partial theta vector satisfies the requirement of the quantum theta function. However, two subsequent quantum translations from the embedding into the lattice part are nonadditive, contrary to the additivity of those from the vector space part.

Symmetry of quantum torus with crossed product algebra

In this paper, we study the symmetry of quantum torus with the concept of crossed product algebra. As a classical counterpart, we consider the orbifold of classical torus with complex structure and investigate the transformation property of classical theta function. An invariant function under the group action is constructed as a variant of the classical theta function. Then our main issue, the crossed product algebra representation of a quantum torus with complex structure under the symplectic group, is analyzed as a quantum version of orbifolding. We perform this analysis with Manin’s so-called model II quantum theta function approach. The symplectic group Sp(2n,Z) satisfies the consistency condition of a crossed product algebra representation of quantum torus times the algebra of functions on the Siegel space. However, only a subgroup of Sp(2n,Z) satisfies the consistency condition for orbifolding of the quantum torus.

The graded ring of quantum theta functions for noncommutative torus with real multiplication

For the quantum torus generated by unitaries UV = e(θ)V U there exist nontrivial strong Morita autoequivalences in the case when θ is a real quadratic irrationality. A.Polishchuk introduced and studied the graded ring of holomorphic sections of powers of the respective bimodule (depending on the choice of a complex structure). We consider a Segre square of this ring whose graded components are spanned by Rieffel scalar products of Polishchuk’s holomorphic vectors as in [5] and [8]. These graded components are linear spaces of quantum theta functions in the sense of Yu. Manin.

Theta vectors and quantum theta functions

In this paper, we clarify the relation between Manin's quantum theta function and Schwarz's theta vector in comparison with the kq representation, which is equivalent to the classical theta function, and the corresponding coordinate space wavefunction. We first explain the equivalence relation between the classical theta function and the kq representation in which the translation operators of the phase space are commuting. When the translation operators of the phase space are not commuting, then the kq representation is no more meaningful. We explain why Manin's quantum theta function obtained via algebra (quantum tori) valued inner product of the theta vector is a natural choice for quantum version of the classical theta function (kq representation). We then show that this approach holds for a more general theta vector with constant obtained from a holomorphic connection of constant curvature than the simple Gaussian one used in the Manin's construction. We further discuss the properties of the theta vector and of the quantum theta function, both of which have similar symmetry properties under translation.