Spin-cobordisms, surgeries and fermionic modular bootstrap

Abstract

We consider general fermionic quantum field theories with a global finite group symmetry $G$, focusing on the case of 2-dimensions and torus spacetime. The modular transformation properties of the family of partition functions with different backgrounds is determined by the 't Hooft anomaly of $G$ and fermion parity. For a general possibly non-abelian $G$ we provide a method to determine the modular transformations directly from the bulk 3d invertible topological quantum field theory (iTQFT) corresponding to the anomaly by inflow. We also describe a method of evaluating the character map from the real representation ring of $G$ to the group which classifies anomalies. Physically the value of the map is given by the anomaly of free fermions in a given representation. We assume classification of the anomalies/iTQFTs by spin-cobordisms. As a byproduct, for all abelian symmetry groups $G$, we provide explicit combinatorial expressions for corresponding spin-bordism invariants in terms of surgery representation of arbitrary closed spin 3-manifolds. We work out the case of $G=\mathbb{Z}_2$ in detail, and, as an application, we consider the constraints that 't Hooft anomaly puts on the spectrum of the infrared conformal field theory.

Figures (21)

• Figure 1: The structure of the action groupoid ${\widetilde{\mathrm{SL}}(2,\mathbb{Z})}\ltimes {\mathrm{Spin}}_{G}(\mathbb{T}^2)$ for $G=\mathbb{Z}^2$. The $(-1)^\mathcal{F}$ and $(-1)^\mathcal{Q}$ generators always act from a groupoid object to itself and are not shown explicitly.
• Figure 2: The structure of the closed manifolds \ref{['defmappingtori']} (mapping tori for non-trivial classes in $\Omega_2^{\mathrm{Spin}}(B\mathbb{Z}_2)$), from which is clear one has to remember the identification $\overline{\overline{X}}\cong X$ via ${(-1)^{\mathcal{F}}}$.
• Figure 3: The mapping tori describing the action of ${(-1)^{\mathcal{F}}}$ on the bordism classes $(0/1,1)$ with the pin$^-$ surface $\Sigma^{(-1)^{\mathcal{F}}}_{(0/1,1)}$ highlighted in green. The horizontal slices correspond to the tori $\{{\mathrm{R}}0,{\mathrm{NS}}/{\mathrm{R}}1\}$ while the vertical direction corresponds to $S^1_{\mathrm{R}}$, with its direction being from bottom to top.
• Figure 4: The closed 3-manifolds (lens spaces) associated to non-trivial entries $(T_{(0,0)})^9_6$ and $(T_{(0,0)})^4_8$. The vertical direction represents the bordism from $\{s_0g_0,s_1 g_1\}$ on the bottom to $\{(s_0+s_1)(g_0+g_1),s_1 g_1\}$ on the top. The red dashed lines represent instead the directions which get contracted to single points.
• Figure 7: The choice of basis configurations of $\mathbb{Z}_2$ charge defects on $\mathbb{T}^2$ with periodic-periodic spin structure.