[Paper Review] Trace functionals on non-commutative deformations of moduli spaces of flat connections

Philippe Roché, András Szenes|ArXiv.org|Aug 18, 2000

Algebraic structures and combinatorial models22 references4 citations

TL;DR

This paper constructs a non-commutative deformation of the algebra of functions on moduli spaces of flat connections on punctured Riemann surfaces, using a geometric quantization approach based on ribbon graphs and quantum group invariants. It proves an asymptotic correspondence between a formal trace on the deformation algebra and a holomorphic trace on a $ q $-deformed algebra for $ G = SU(2) $, providing a conjectural lifting of the Fedosov-Nest-Tsygan index theorem to the $ q $-setting.

ABSTRACT

We describe an efficient construction of a canonical non-commutative deformation of the algebraic functions on the moduli spaces of flat connections on a Riemann surface. We show that this algebra, which is a variant of the quantum moduli algebra introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche, has a trace functional which is related to the canonical trace in the formal index theory of Fedosov and Nest-Tsygan via the Verlinde formula.

Motivation & Objective

• To construct a non-commutative deformation algebra $ A_q $ of the algebra of functions on moduli spaces of flat $ G $-connections on punctured Riemann surfaces.
• To lift the canonical trace from the formal deformation $ A_\hbar $ (over $ \mathbb{C}[[\hbar]] $) to a holomorphic trace on $ A_q $, valued in functions holomorphic in the unit disc.
• To establish an asymptotic expansion relating the $ q $-trace to the formal trace at $ q = 1 $, generalizing the Verlinde formula via index theory.
• To provide a geometric and computationally efficient quantization procedure using ribbon graphs and Reshetikhin-Turaev invariants.
• To lay the foundation for extending the index-theoretic approach to higher-rank groups and to study the characteristic class of $ A_q $.

Proposed method

• Constructs the algebra $ A_q $ via a geometric variant of the Reshetikhin-Turaev quantization, using colored ribbon graphs to represent invariants of the moduli space.
• Applies the graphical calculus of Fock and Rosly and the theory of quantum groups to define a Poisson structure and Poisson trace on the classical moduli space.
• Uses the universal $ R $-matrix and $ J $-matrix from quantum group theory to define the fusion matrix $ J(\lambda) $, which encodes the quantum structure.
• Derives the universal element $ Q(\lambda) $ as $ \sum_i S(a_i)b_i $, where $ J(\lambda) = \sum_i a_i \otimes b_i $, and shows it acts as a trace operator on representations.
• Employs the dynamical quantum Weyl group operators $ A_w(\lambda) $ to relate different weight spaces and construct intertwiners between modules.
• Performs explicit computations in the $ \mathfrak{sl}_2 $ case using basic hypergeometric functions and $ q $-special functions to verify the trace asymptotics.

Experimental results

Research questions

• RQ1 Can the canonical trace on the formal deformation $ A_\hbar $ be lifted to a holomorphic trace on a $ q $-deformed algebra $ A_q $, with values in functions holomorphic in the unit disc?
• RQ2 What is the asymptotic relationship between the $ q $-trace and the formal trace at $ q = 1 $, and how does it relate to Verlinde's formula?
• RQ3 How can the Reshetikhin-Turaev construction be adapted to yield a transparent, geometric, and computationally efficient quantization of the moduli space algebra?
• RQ4 What is the role of the dynamical quantum Weyl group in realizing the trace and intertwiners in the $ q $-deformed setting?
• RQ5 To what extent can the index-theoretic framework of Fedosov and Nest-Tsygan be extended to non-formal, holomorphic $ q $-deformations?

Key findings

• For $ G = SU(2) $, the paper proves that the $ q $-trace on $ A_q $, valued in holomorphic functions on the unit disc, asymptotically expands to the canonical trace on $ A_\hbar $ as $ q \to 1 $, confirming the conjectured lifting.
• An explicit formula for the trace operator $ Q(\lambda) $ is derived in the $ \mathfrak{sl}_2 $ case using basic hypergeometric functions: $ Q(\lambda)v^m_k = {}_2\varphi_1(q^{-2(m-k+1)}, q^{2k}; q^{2(\lambda-m+2k)}; q^{-2}) v^m_k $.
• The relation $ Q(-\lambda-2)Q(\lambda+h) = q^{h^2/2 + h} v $ is verified, showing consistency with quantum group duality and trace identities.
• The action of the dynamical quantum Weyl group operator $ A_{V_m}(\lambda) $ is computed explicitly as $ A_{V_m}(\lambda)(v^m_k) = (-1)^k q^{-k(m-k+1)} \frac{(q^{2(\lambda+k+1)}; q^{-2})_k}{(q^{2(\lambda-m+2k)}; q^{-2})_k} v^m_{m-k} $, using the $ q $-Saalschütz formula.
• The trace on $ A_q $ is shown to be compatible with the Verlinde formula via the asymptotic expansion, suggesting a deeper link between quantum invariants and index theory.
• The construction provides a geometric and computationally efficient alternative to earlier algebraic quantizations, with explicit formulas for the Poisson structure and trace using ribbon graph techniques.