### Constant connections, quantum holonomies and the Goldman bracket

J. E. Nelson and R. F. Picken
Source: Adv. Theor. Math. Phys. Volume 9, Number 3 (2005), 407-433.

#### Abstract

In the context of $2+1$-dimensional quantum gravity with negative cosmological constant and topology $\IR \times T^2$, constant matrix-valued connections generate a $q$-deformed representation of the fundamental group, and signed area phases relate the quantum matrices assigned to homotopic loops. Some features of the resulting quantum geometry are explored, and as a consequence a quantum version of the Goldman bracket is obtained.

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