KYUNGPOOK Math. J. 2019; 59(3): 403-414
Published online September 23, 2019
Copyright © Kyungpook Mathematical Journal.
Super Theta Vectors and Super Quantum Theta Operators
Department of Mathematics and Institute for Mathematical Convergence, Kyungpook National University, Daegu 41566, Korea
e-mail : firstname.lastname@example.org
Received: March 20, 2019; Accepted: June 5, 2019
Theta functions are the sections of line bundles on a complex torus. Noncommutative versions of theta functions have appeared as theta vectors and quantum theta operators. In this paper we describe a super version of theta vectors and quantum theta operators. This is the natural unification of Manin’s result on bosonic operators, and the author’s previous result on fermionic operators.
Keywords: super theta vectors, quantum theta operators, super Heisenberg group.
In physics, one considers two related notions: observables and states. In classical theory, the observables are real valued functions on a phase space (position with momentum), and the states are probability measures on the phase space. In quantum theory, the observables are self-adjoint (Hermitian) operators on a Hilbert space, and the pure states are vectors in the Hilbert space with length one, mixed states are a mixture of pure states. Note that pure states correspond to Dirac measures on the phase space and mixed states are correspond to probability measures. Roughly speaking, in both classical and quantum theories, observables contain states.
Theta functions are functions on complex spaces, but more precisely, are sections of line bundles on the complex torus. The Noncommutative torus was introduced in ; however, the concept of the noncommutative torus had already been developed in terms of the Heisenberg group and Schrödinger representation, in . Noncommutative tori are used in physics in the toroidal compactification by Connes, Douglas and Schwarz in . Later the concept of theta vectors was introduced by Schwarz in . Finally, Manin studied an operator version of theta functions, called quantum theta functions in . Classically, theta functions play the role of observables (and states) or a Hilbert space in the geometric quantization. Theta vectors are the vacuum states which have the minimum energy and from which the other states are constructed, in the Hilbert space and quantum theta functions are observables. For more explanation and interpretation, see .
A classical theta function of
Two questions were raised by Schwarz in . The first one was of the connection between quantum theta functions and theta vectors, and the second one was of the existence of a quantum analogue of the classical functional equation for thetas. Manin answered both of these questions in .
Manin showed the follwing in .
Also he showed that
What the author obtained in  on ℝ0|2
As in the bosonic case we define
Then we get our main theorem.
There are several different approaches to define super theta functions [1, 9]. However their approaches and the concepts are different from ours in the sense that they deal with classical super theta functions while our quantum super theta functions are operators (observables) coming from super theta vectors which are vacuum states.
The contents of this paper are as follows. In section 2, we describe the necessary materials on superspaces for this paper. In section 3, we discuss the super Heisenberg groups and the super theta vector which is a vacuum state generalizing the classical theta vector. In section 4, we construct the super quantum theta functions on a superspace ℝ
In this section, we explain necessary materials for later sections including superspace, superlinear algebra, integration on superspace.
For any supermatrix
Here we introduce how to integrate on odd space called Berezin integral. Let Λ
Super Theta Vectors
In this section, we want to define a super theta vector which is a vacuum state. First we want to define the super Heisenberg group
We choose such that and . We define as the space of super holomorphic functions
Now we determine the representation of the Heisenberg algebra associated to the above Heisenberg group representation. Let
In the Heisenberg representation
(Super Theta Vector) We define this as the super theta vector.
Super theta operator
In this section, we generalize Manin’s result on quantum theta function for ℝ
(Super Quantum Theta Operator) Let , the super theta vector. As in the bosonic case we define C∞(
The following is the proof of our main theorem (Theorem 1.2).
The proof of (1) is obtained by the computation of
The proof of (2) comes from the following observation.
The proof of (3) is a generalization of Manin, by using the Fourier transform of
- M. Bergvelt, and J. Rabin.
Supercurves, their Jacobians, and super KP equations. Duke Math. J., 98(1)(1999), 1-57.
- A. Connes, M. Douglas, and A. Schwarz.
Noncommutative geometry and matrix theory. J. High Energy Phys., JHEP9802(1998) 003, 35.
- H. Kim.
Quantum super theta vectors and theta functions. Kyungpook Math. J., 56(1)(2016), 249-256.
- Y. Manin.
Theta functions, quantum tori and Heisenberg Groups. Lett. Math. Phys., 56(3)(2001), 295-320.
- Y. Manin.
Functional equations for quantum theta functions. Publ. Res. Inst. Math. Sci., 40(3)(2004), 605-624.
- D. Mumford.
Tata lectures on theta III. Progress in Math, 97, Birkhäuser, Boston, 1991. (with M Nori and P Norman),.
- M. Rieffel.
Projective modules over higher dimensional noncommutative tori. Canad. J. Math., 40(2)(1988), 257-338.
- A. Schwarz.
Theta functions on noncommutative tori. Lett Math Phys., 58(2001), 81-90.
- Y. Tsuchimoto.
On super theta functions. J. Math. Kyoto Univ., 34(3)(1994), 641-694.
- Tyurin. A. Quantization and “theta functions”.