A Quantum Goldman Bracket for Loops on Surfaces

Abstract

In the context of (2+1)-dimensional gravity, we use holonomies of constant connections which generate a q-deformed representation of the fundamental group to derive signed area phases which relate the quantum matrices assigned to homotopic loops. We use these features to determine a quantum Goldman bracket (commutator) for intersecting loops on surfaces, and discuss the resulting quantum geometry.

Lambda models from Chern-Simons theories

“…One of the main results of this approach and summarized in the diagram (3.140) is that the precursor of the would-be Poisson algebra of the lambda model monodromy matrix turns out to be closely related to some sort of spectral parameter extension of the Goldman bracket [29], which is used to study the intersection properties of homotopy classes of loops on Riemann surfaces (this certainly deserves further study as our theory is by hypothesis defined on the disc having a trivial fundamental group, but see the comments about this issue in the concluding remarks below). Contrary to the exchange algebra of monodromy matrices in non-ultralocal integrable field theories, which is unknown and ambiguous due to their non-ultralocality [6], the Goldman bracket has been studied for quite a long time and even quantized [30][31][32][33], mostly within the context of 2+1 dimensional quantum gravity. This opens the possibility for using the vast amount of results available on CS theories to developed a first principle quantization setup (at least) for the AdS 5 × S 5 lambda model.…”

Tullio Regge: An Eclectic Genius

“…It is linear, but subject to non-linear constraints (equation (53)). It has led us to a new quantum group structure, some interesting properties in quantum geometry, a quantisation of the Goldman bracket [22], and a theory of intersecting loops on surfaces [23].…”

“…Roger Picken and I have studied this second version, and it has led us to a new quantum group structure, some interesting properties in quantum geometry, and a quantisation of the Goldman bracket[23].…”

2+1 dimensional gravity

“…It is linear, but subject to non-linear constraints (equation (5.3)). It has led us to a new quantum group structure, some interesting properties in quantum geometry, a quantisation of the Goldman bracket [22], and a theory of intersecting loops on surfaces [23].…”