[Paper Review] On Regularised Quantum Dimensions of the Singlet Vertex Operator Algebra and False Theta Functions

Thomas Creutzig, Antun Milas+1 more|arXiv (Cornell University)|Nov 12, 2014

Algebraic structures and combinatorial modelsMathematics48 references26 citations

TL;DR

This paper introduces a regularisation procedure for quantum dimensions in non-C2-cofinite singlet vertex operator algebras using a complex parameter 𝜖, linking them to false theta functions and quantum modular forms. By deforming characters via 𝜖 and analyzing their modular transformation properties, the authors show that regularised quantum dimensions yield a ring isomorphism for Re(𝜖) > 0 and recover the fusion ring of a rational VOAs for Re(𝜖) < 0, with vector-valued quantum modular forms encoding modular tensor category structure constants.

ABSTRACT

We study a family of non-C2-cofinite vertex operator algebras, called the singlet vertex operator algebras, and connect several important concepts in the theory of vertex operator algebras, quantum modular forms, and modular tensor categories. More precisely, starting from explicit formulae for characters of modules over the singlet vertex operator algebra, which can be expressed in terms of false theta functions and their derivatives, we first deform these characters by using a complex parameter {\epsilon}. We then apply modular trans- formation properties of regularised partial theta functions to study asymptotic behaviour of regularised characters of irreducible modules and compute their regularised quantum dimensions. We also give a purely geometric description of the regularisation parameter as a uniformisation parameter of the fusion variety coming from atypical blocks. It turns out that the quantum dimensions behave very differently depending on the sign of the real part of {\epsilon}. The map from the space of characters equipped with the Verlinde product to the space of regularised quantum dimensions turns out to be a genuine ring isomorphism for positive real part of {\epsilon} while for sufficiently negative real part of {\epsilon} its surjective image gives the fusion ring of a rational vertex operator algebra. The category of modules of this rational vertex operator algebra should be viewed as obtained through the process of a semi-simplification procedure widely used in the theory of quantum groups. Interestingly, the modular tensor category structure constants of this vertex operator algebra, can be also detected from vector valued quantum modular forms formed by distinguished atypical characters.

Motivation & Objective

• To understand the asymptotic behaviour of characters in non-C2-cofinite vertex operator algebras, particularly the singlet VOAs.
• To define and compute regularised quantum dimensions for irreducible modules in the singlet VOAs using a complex deformation parameter 𝜖.
• To establish a geometric interpretation of the regularisation parameter as a uniformisation parameter of the fusion variety from atypical blocks.
• To determine how the Verlinde product structure maps to regularised quantum dimensions under different signs of Re(𝜖).
• To connect modular tensor category structure constants of a rational VOA to vector-valued quantum modular forms formed by atypical characters.

Proposed method

• Deform the characters of singlet VOAs using a complex parameter 𝜖 to define regularised characters.
• Apply modular transformation properties of regularised partial theta functions to study the asymptotic behaviour of these regularised characters as 𝑞 → 0.
• Use Eichler integral constructions and non-holomorphic Eichler integrals to relate holomorphic characters to their analytic continuations in the upper and lower half-planes.
• Construct vector-valued quantum modular forms from atypical characters and derive their transformation laws under SL(2, ℤ).
• Relate the S-matrix of the (p+, p−)-minimal model to the transformation of these quantum modular forms via integral kernels.
• Prove that the map from the Verlinde product space to regularised quantum dimensions is a ring isomorphism for Re(𝜖) > 0 and surjective onto the fusion ring for Re(𝜖) < 0.

Experimental results

Research questions

• RQ1 How do regularised quantum dimensions behave asymptotically for irreducible modules of the singlet VOAs under complex deformation by 𝜖?
• RQ2 What is the geometric meaning of the regularisation parameter 𝜖 in terms of the fusion variety of atypical blocks?
• RQ3 Does the map from the Verlinde product to regularised quantum dimensions preserve ring structure, and if so, under what conditions on 𝜖?
• RQ4 Can modular tensor category structure constants of a rational VOA be recovered from vector-valued quantum modular forms formed by atypical characters?
• RQ5 How do the modular transformation properties of false theta functions and their derivatives relate to the S-matrix of minimal models?

Key findings

• For Re(𝜖) > 0, the map from the Verlinde product space to regularised quantum dimensions is a genuine ring isomorphism.
• For sufficiently negative Re(𝜖), the image of the Verlinde product map is surjective onto the fusion ring of a rational vertex operator algebra.
• The regularisation parameter 𝜖 geometrically corresponds to a uniformisation parameter of the fusion variety arising from atypical blocks.
• The S-matrix of the (p+, p−)-minimal model is encoded in the transformation law of vector-valued quantum modular forms formed by atypical characters.
• The components of the vector-valued quantum modular form G(w) satisfy a transformation law involving the S-matrix and a correction term g(w), with G(w) analytic on H ∪ H and smooth on R.
• The quantum modular forms F(τ) and G(τ) are shown to have radial limits agreeing with their Eichler integral counterparts at rational points, confirming their quantum modular nature.