Semi-classical properties of the quantum representations of mapping class groups

By Søren Fuglede Jørgensen
PhD Dissertations
August 2013

The theory discussed in this dissertation may be traced back to Edward Witten's 1989 study of Chern--Simons theory as a $(2+1)$-dimensional quantum field theory. Let $G = \mathrm{SU}(N)$, let $M$ be a compact oriented $3$-manifold containing an oriented link $L$, and let $P \to M$ be a principal $G$-bundle. Let $\mathcal{A}_P$ denote the space of connections in $P \to M$, and let $\mathcal{G}_P$ denote the space of gauge transformations. The action functional of Chern--Simons theory then defines a map $\mathrm{CS} : \mathcal{A}_P/\mathcal{G}_P \to \mathbb{R}/\mathbb{Z}$.

Let moreover $k \in \mathbb{Z}_{ > 0}$ and choose for every component $L_i$ of $L$ a finite-dimensional representation $R_i$ of $G$. Let $\mathrm{hol}_A(L_i) \in G$ denote the holonomy of a connection $A \in \mathcal{A}_P$ along $L_i$. Witten argued that the path integral \[ Z_{k,G}^{\mathrm{phys}}(M,L,R) = \int_{\mathcal{A}_P/\mathcal{G}_P} \prod_i \mathrm{tr}(R_i(\mathrm{hol}_{[A]}(L_i))) \exp(2\pi i k \mathrm{CS}([A])) \, \mathcal{D} A, \] known as the partition function of quantum Chern-Simons theory, defines a topological invariant of the pair $(M,L)$ and that, moreover, this invariant extends to a topological quantum field theory (TQFT), which roughly means that if $M = M_1 \cup_\Sigma M_2$ is a union of two $3$-manifolds along a surface $\Sigma$, then $Z_k(M,L)$ is described in terms of $Z_k(M,L\cap M_1)$, $Z_k(M_2,L \cap M_2)$ and boundary data $V_k(\Sigma,L \cap \Sigma)$ associated to the surface $\Sigma$.

The path integral however is not defined: at the time of writing there is no method of associating in a natural way a measure $\mathcal{D} A$ to the infinite-dimensional space $\mathcal{A}_P/\mathcal{G}_P$. Nonetheless, after a few years, Reshetikhin and Turaev defined, using the representation theory of quantum groups, topological quantum field theories $(Z_k^G,V_k^G)$ with the properties prescribed by Witten's quantum theory. This fact contrasts what is the case for most other gauge theories and makes Chern-Simons theory particularly interesting from the point of view of mathematical physics.

The dissertation at hand consists of a study of the two-dimensional part of this mathematical theory; the axioms of a $(2+1)$-dimensional TQFT ensures for each $k$ the existence of a projective representation - known as the quantum representation - of the mapping class group of any compact surface with suitable boundary conditions. The main part of the dissertation will revolve around two particular problems. On one hand we will investigate the Asymptotic Expansion Conjecture (AEC) which may be viewed as a reality check for quantum topology: even though the path integral $Z_{k,G}^{\mathrm{phys}}$ is not defined, it is possible to deduce through heuristic means a mathematical conjecture for its behaviour for large values of $k$; one of the reasons for this is that the space of classical solutions of Chern-Simons theory forms a finite dimensional space: the moduli space of flat $G$-connections on $M$. This fact furthermore means that geometric quantization plays a key role in the study of TQFT. The asymptotic behaviour resulting from this study may then be compared with that of the mathematically well-defined invariants; the AEC makes this precise.

On the other hand, we discuss the AMU conjecture which states that the quantum representations contain information about the dynamics of the mapping class group action on the underlying surface. As the mathematical definition of TQFT is based on the Jones polynomial of knot theory, this can be viewed as a counterpart to the problem of describing the geometrical meaning of the Jones polynomial (compare with the Volume Conjecture).

Thesis advisor: Jørgen Ellegaard Andersen
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