Quantum modular group in (2+1)-dimensional gravity

Abstract

The role of the modular group in the holonomy representation of (2+1)-dimensional quantum gravity is studied. This representation can be viewed as a "Heisenberg picture," and for simple topologies, the transformation to the ADM "Schrödinger picture" may be found. For spacetimes with the spatial topology of a torus, this transformation and an explicit operator representation of the mapping class group are constructed. It is shown that the quantum modular group splits the holonomy representation Hilbert space into physically equivalent orthogonal "fundamental regions" that are interchanged by modular transformations. *

Quantum Gravity in 2 + 1 Dimensions: The Case of a Closed Universe

“…To obtain the corresponding Schröodinger picture, we proceed as in ordinary quantum mechanics: We diagonalize , obtaining a transition matrix that allows us to transform between representations [68, 88]. The resulting “time”-dependent wave functions obey a Schrödinger equation of the form (52, 53), but with the Laplacian in Ĥ replaced by the weight 1/2 Maass Laplacian Δ 1/2 of Equation (54).…”

“…As a useful byproduct, this analysis allows us to solve the problem of the poorly-behaved action of the modular group discussed at the end of Section 3.2 [88, 89]. If we start with a reduced phase space wave function and use the transition matrix K to determine a Chern-Simons wave function , we find, indeed, that is not modular invariant.…”

Combinatorial quantisation of the Euclidean torus universe

“…It would also be interesting to establish a relation with previous work on the (Lorentzian) torus universe [16,17,18,19,20,21,22,23]. This would require relating our holonomy-based formalism to time-dependent formalisms along the lines of [19,20,22].…”

“…It would be interesting to understand how our result is related to the descriptions in [16,17,18,21,22] and to identify a common source of this problem. This would require relating the timeless formulation in terms of holonomies to time dependent quantisation formalisms.…”

Periodic $ \lambda$-rings and exponents of finite groups

“…Instead of a Gaussian we can use the modularinvariant wavefunctions of Carlip and Nelson [13,30,31] (λ, µ) = andψ ν (τ,τ ) are the automorphic modular forms of weight −1/2 on the torus, with eigenvalues λ ν with respect to −1/2 , the Maass Laplacian of weight −1/2 (see [13] for definitions):…”