On moduli space of symmetric orthogonal matrices and exclusive Racah matrix $\bar S$ for representation $R=[3,1]$ with multiplicities

TL;DR

This paper derives the exclusive Racah matrix for a complex representation with multiplicities, revealing its operator-valued nature and the structure of the moduli space of symmetric orthogonal matrices, advancing understanding in knot theory and representation theory.

Contribution

It provides the first explicit construction of the operator-valued Racah matrix for representation R=[3,1] with multiplicities, and analyzes the moduli space of symmetric orthogonal matrices involved.

Findings

The Racah matrix depends on basis choices in intertwiner spaces.

The moduli space of symmetric orthogonal matrices has about half the dimension of ordinary orthogonal matrices.

The method combines orthogonality conditions with educated guesses to reconstruct the matrix.

Abstract

Racah matrices and higher $j$-symbols are used in description of braiding properties of conformal blocks and in construction of knot polynomials. However, in complicated cases the logic is actually inverted: they are much better deduced from these applications than from the basic representation theory. Following the recent proposal of arXiv:1612.00422, we obtain the exclusive Racah matrix $\bar S$ for the currently-front-line case of representation $R=[3,1]$ with non-trivial multiplicities, where it is actually operator valued, i.e. depends on the choice of basises in the intertwiner spaces. Effective field theory for arborescent knots in this case possesses gauge invariance, which is not yet properly described and understood. Because of this lack of knowledge a big part (about a half) of $\bar S$ needs to be reconstructed from orthogonality conditions. Therefore we discuss the abundance of symmetric orthogonal matrices, to which $\bar S$ belong, and explain that dimension of their moduli space is also about a half of that for the ordinary orthogonal matrices. Thus the knowledge approximately matches the freedom and this explains why the method can work -- with some limited addition of educated guesses. A similar calculation for $R=[r,1]$ for $r>3$ should also be doable.