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Cutting one nonseparating circle on each handle of a closed orientable surface gives a sphere with $g$ pairs of oppositely oriented boundary circles. The topological quantum field theory gluing rule sums over the paired anyon labels $a,\bar{a}$. The resulting anyon fusion matrix formula becomes $\dim V({\Sigma }_g)={\mathcal{D}}^{2g-2}{\sum }_a{d}_a^{2-2g}$ by the Verlinde formula.

A finite semisimple ribbon braided monoidal category with nondegenerate modular S-matrix. For a unitary anyon theory, this means that the only transparent anyon is the vacuum anyon. Such data define a three-dimensional topological quantum field theory.

Cut one nonseparating circle on each handle of the closed orientable surface ${\Sigma }_g$ of genus $g$. The cut surface is a sphere with $2g$ boundary circles. Gluing a pair back together requires an anyon label $a_i$ on one boundary and the antianyon label ${\bar{a}}_i$ on the oppositely oriented boundary. Sum over these labels independently for the $g$ handles.

For a fixed labeling, the dimension of the resulting anyon fusion space on the sphere, with total vacuum anyon charge, isThe gluing rule of the topological quantum field theory therefore gives the handle-cut formula for anyon ground-state degeneracy:Here $1$ denotes the vacuum anyon, and the anyon fusion matrices have entries $(N_b{)}_{ac}=N_{ab}^c$, as in the source. At $g=0$, the empty product is the identity matrix, giving one ground state. At $g=1$, $(N_a{N}_{\bar{a}}{)}_{11}=1$ for each simple charge, so the ground-state degeneracy of the torus is the number of anyon types.

Use the Witten-Reshetikhin-Turaev invariant with the source's normalized Kirby color on all $2g+1$ surgery components. Since the signature of the linking matrix is zero, the framing-anomaly factor is one, and the TQFT partition function isAll evaluations in this part use the isotopy normalization of anyon diagrams and preserve the indicated blackboard framing.

Expand the long component as ${\mathcal{D}}^{-1}{\sum }_a{d}_{a}a$ and fix its charge $a$. Each pair of auxiliary components is in the Borromean rings arrangement with this component. Eliminate such a pair using the two-strand Kirby projector from c: bend the long component to expose its two oppositely oriented portions, viewed as equally oriented strands labeled $a,\bar{a}$. The first encircling normalized Kirby color loop replaces these by a vacuum cup-cap with coefficient $\mathcal{D}/d_a$. This vacuum connection opens the second encircling loop so that the same two-strand Kirby projector applies once more. Removing the resulting vacuum edges and straightening the bends restores the original charge-$a$ strand. Thus the Borromean handle-pair evaluation multiplies the bare strand byThere is no residual twist. This is a relative strand evaluation: the final closure of the long component is counted separately. As a normalization check, when $a=1$ the two auxiliary components become two unlinked normalized Kirby color loops, each of value $\mathcal{D}$, giving $H_1={\mathcal{D}}^2$.

The $g$ pairs occupy disjoint portions of the long component, so their factors multiply. Closing that component gives its quantum dimension $d_a$. Combining this with its normalized Kirby color coefficient $d_a/\mathcal{D}$ givesApplying the overall surgery prefactor yieldsFinally, the topological quantum field theory assigns the identity map to a cylinder over ${\Sigma }_g$. Closing its time direction takes the trace of that identity, soThis independently reproduces b. For $g=0$ the surgery link is one zero-framed normalized Kirby color unknot: its value $\mathcal{D}$ cancels the surgery prefactor and gives $Z(S^2\times S^1)=1$. For $g=1$ the result is the number of simple anyons, as required for the torus.

The scalar assigned to a closed spacetime manifold by a topological quantum field theory. Gluing the two ends of a cylinder over a spatial manifold $\Sigma$ gives $Z(\Sigma \times S^1)=\mathrm{tr}(1_{V(\Sigma )})=\dim V(\Sigma )$. This is the ground-state degeneracy when the state space describes the topological ground sector of a gapped system.

A topological quantum field theory TQFT partition function for a closed oriented three-manifold presented by Dehn surgery on a framed link. With normalized Kirby color on each surgery component and the convention $Z(S^3)=1/\mathcal{D}$, it is ${\mathcal{D}}^{-1}{e}^{-2\pi ic\sigma /8}$ times the colored link evaluation. Here $\sigma$ is the signature of the linking matrix and $c$ is the chiral central charge.