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# Advances in Theoretical and Mathematical Physics

## Volume 9 (2005)

### Number 3

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

Pages: 407 – 433

DOI: http://dx.doi.org/10.4310/ATMP.2005.v9.n3.a2

#### Authors

#### 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.