Groups and Related Topics pp 189-196 | Cite as

# State sum invariants of compact 3-manifolds with boundary and 6j-symbols

## Abstract

In a recent article Turaev and Viro [TV] have constructed nontrivial “quantum”
invariants of closed compact 3-manifolds *M* ^{3} in the form of state sums (called partition
functions in statistical physics and vacuum functionals in quantum field theory) associated
with the quantized universal enveloping algebra *Uq*(sl(2, C)) when *q* is a complex
root of unity of a certain degree 2*r* > 4. The state sum is first defined for a given
triangulation *X* of *M* and then shown to be independent of the triangulation *X* of *M*
thus giving rise to a well defined invariant of *M*, which thus depends on *q*. This result
may be viewed as a rigorous mathematical construction of what is called a topological
quantum field theory. In fact, in the language of physicists, a triangulation corresponds
to the introduction of a high-energy cut-off. Now topological quantum field theories
have trivial dynamics, in other words they only deal with the vacuum sector, are scale
invariant and more generally independent of any metrics. Invariance under subdivision
is just the statement that the associated renormalization group transformation is trivial.
This result suggests that the familiar techniques from algebraic topology should become
useful to construct and discuss other topological quantum field theories. Moreover, similar
combinatorial techniques could become helpful in discussing particle structures of
low dimensional quantum field theories, whose vacuum sector is given in terms of a
topological quantum field theory.

## Keywords

Conformal Field Theory Edge Colouring Vacuum Sector Topological Quantum Field Theory Trivial Dynamic## Preview

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