Modules associated to disconnected surfaces by quantization functors (Q1292723)

Topological quantum field theory (TQFT) is a functor, from a cobordism theory to a category of modules, satisfying certain axioms. A cobordism category in dimension 2+1 has objects consisting of surfaces with certain structure. The morphisms from \(\Sigma_1\) to \(\Sigma_2\) are 3-dimensional cobordisms between \(\Sigma_1\) and \(\Sigma_2\). TQFT assigns a module \(V(\Sigma)\) over a fixed ring to each object \(\Sigma\). To each morphism \(M\) from \(\Sigma_1\) to \(\Sigma_2\) it assignes \(Z_M\), a linear map from \(V(\Sigma_1)\) to \(V(\Sigma_2)\). The letter \(Z\) denotes the partition function often used in physics. \textit{C. Blanchet, N. Habegger, G. Masbaum} and \textit{P. Vogel} [Topology 31, No. 4, 685-699 (1992; Zbl 0771.57004)] defined a quantization functor \((V_p,Z_p)\) on a category whose objects are oriented closed surfaces together with a collection of colored banded points and \(p_1\) structure. The modules are over a ring \(k_p\): \[ k_p=Z[1/d,A,\kappa]/\phi_{2p},(A),\kappa^6-\mu), \] where \(\phi_{2p}\) is the \(2p\)-cyclotomic polynomial in the indeterminant \(A\) and \[ \mu=A^{-6-p(p+1)/2}, \quad \text{and}\quad d=\begin{cases} p &\text{for \(p\neq 3\)}, \\ 1, &\text{for \(p=3\)}. \end{cases} \] For \(p\) even, it satisfies the tensor product axiom, which defines the modules associated to a disconnected surface as the tensor product of the modules asssociated to its components. The same authors introduced also the notion of a generalized tensor product. They defined certain spin TQFT's for surfaces and cobordisms with spin structures and showed that their spin theories satisfied this formula. In the paper under review the author shows that for \(p\) odd a quantization functor on a category whose objects are oriented closed surfaces together with a collection of colored banded points and \(p_1\) structure satisfies a generalized tensor product formula. The proof is based on the fusion technique. Let \(\widehat V(\Sigma)\) denote \(V(\Sigma\cup \widehat S^2)\) where \(\widehat S^2\) is a sphere with one banded point colored \(p-2\). The determination of the rank of \(\widehat V(\Sigma)\) is reduced to a combinatorial problem of counting the admissible colorings of trivalent graphs. The generalized tensor product formula expresses \(V_p(\Sigma_1 \cup \Sigma_2)\) in terms of \(V_p(\Sigma_1)\), \(V_p(\Sigma_2)\), \(\widehat V_p(\Sigma_1)\) and \(\widehat V_p(\Sigma_1)\). The author reduces the calculation of \(\widehat V_p(\Sigma)\) to known results and calculates it in many cases.