Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

SKK groups of manifolds and non-unitary invertible TQFTs

Renee S. Hoekzema, Luuk Stehouwer, Simona Veselá

TL;DR

The paper develops a general framework for computing SKK groups \(\mathrm{SKK}^\xi_n\) of manifolds with tangential structure \(\xi\), via a short exact sequence relating them to bordism groups \(\Omega^\xi_n\) and a disc-bounding sphere kernel. It resolves when odd Euler-characteristic \(\xi\)-manifolds exist and identifies precise splitting criteria in odd and even dimensions, employing Wu classes and Kervaire semi-characteristics as key invariants. These results yield a complete classification of not-necessarily-unitary invertible TQFTs in spacetime dimensions 1–5 for the tenfold-way tangential structures, extending Freed–Hopkins-type pictures to non-unitary settings. The work also connects discrete SKK data to continuous invertible TQFTs via vector-field data on bordisms and the MTξ spectrum, offering a unified perspective on anomalies and topological phases across physics and geometry.

Abstract

This work considers the computation of controllable cut-and-paste groups $\mathrm{SKK}^ξ_n$ of manifolds with tangential structure $ξ:B_n\to BO_n$. To this end, we apply the work of Galatius-Madsen-Tillman-Weiss, Genauer and Schommer-Pries, who showed that for a wide range of structures $ξ$ these groups fit into a short exact sequence that relates them to bordism groups of $ξ$-manifolds with kernel generated by the disc-bounding $ξ$-sphere. The order of this sphere can be computed by knowing the possible values of the Euler characteristic of $ξ$-manifolds. We are thus led to address two key questions: the existence of $ξ$-manifolds with odd Euler characteristic of a given dimension and conditions for the exact sequence to admit a splitting. We resolve these questions in a wide range of cases. $\mathrm{SKK}$ groups are of interest in physics as they play a role in the classification of non-unitary invertible topological quantum field theories, which classify anomalies and symmetry protected topological (SPT) phases of matter. Applying our topological results, we give a complete classification of non-unitary invertible topological quantum field theories in the tenfold way in dimensions 1-5.

SKK groups of manifolds and non-unitary invertible TQFTs

TL;DR

The paper develops a general framework for computing SKK groups of manifolds with tangential structure , via a short exact sequence relating them to bordism groups and a disc-bounding sphere kernel. It resolves when odd Euler-characteristic -manifolds exist and identifies precise splitting criteria in odd and even dimensions, employing Wu classes and Kervaire semi-characteristics as key invariants. These results yield a complete classification of not-necessarily-unitary invertible TQFTs in spacetime dimensions 1–5 for the tenfold-way tangential structures, extending Freed–Hopkins-type pictures to non-unitary settings. The work also connects discrete SKK data to continuous invertible TQFTs via vector-field data on bordisms and the MTξ spectrum, offering a unified perspective on anomalies and topological phases across physics and geometry.

Abstract

This work considers the computation of controllable cut-and-paste groups of manifolds with tangential structure . To this end, we apply the work of Galatius-Madsen-Tillman-Weiss, Genauer and Schommer-Pries, who showed that for a wide range of structures these groups fit into a short exact sequence that relates them to bordism groups of -manifolds with kernel generated by the disc-bounding -sphere. The order of this sphere can be computed by knowing the possible values of the Euler characteristic of -manifolds. We are thus led to address two key questions: the existence of -manifolds with odd Euler characteristic of a given dimension and conditions for the exact sequence to admit a splitting. We resolve these questions in a wide range of cases. groups are of interest in physics as they play a role in the classification of non-unitary invertible topological quantum field theories, which classify anomalies and symmetry protected topological (SPT) phases of matter. Applying our topological results, we give a complete classification of non-unitary invertible topological quantum field theories in the tenfold way in dimensions 1-5.

Paper Structure

This paper contains 40 sections, 62 theorems, 119 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Splitting results for odd-dimensional SKK groups
  3. Splitting results for even-dimensional SKK groups
  4. Implications for physics
  5. Tangential structures and the parity of the Euler characteristic
  6. Background on tangential structures on manifolds and cobordisms
  7. Twice stabilised tangential structures
  8. The parity of the Euler characteristic of a manifold
  9. Relative Wu and Stiefel-Whitney classes
  10. k-orientability
  11. Relationship between k-orientability and b-parallelisability
  12. Other tangential structures
  13. Unstable and stable framings
  14. Pin-manifolds
  15. Tangential structures relevant for physics
  16. ...and 25 more sections

Key Result

Theorem 1

[thm:iffForMfldInvariant] Let be $\kappa$ a homomorphism $\mathcal{M}^\xi_n \to \mathbb{Z}/2,$ for $\mathcal{M}^\xi_n$ the monoid of closed $n$-dimensional $\xi$-manifolds under disjoint union. Then $\kappa$ induces a splitting of seq:introZ2 if and only if for all $(n+1)$-dimensional $\xi$-manifold

Figures (3)

  • Figure 1.1: A pair of manifolds glued along a diffeomorphism $\phi$ differs in $\mathrm{SKK}$ from the pair glued along a different diffeomorphism $\psi$ by a manifold $f(\varphi,\psi)$ which depends only on $\phi$ and $\psi$.
  • Figure 3.1: Sequence of chimaera moves proving the following relations: (a) $2[D^2]=[S_b^1\times I]+2[D^2]$ or $2[D^2]=[M\ddot{o}b]+2[D^2]$ (b) $[\Sigma]+[S_b^1\times I]=[\Sigma\setminus D^2]+[D^2]$ in particular $[S^2]+[S_b^1\times I]=2[D^2]$ or $[\mathbb{RP}^2]+[S_b^1\times I]=[M\ddot{o}b]+[D^2]$.; (c) $[\Sigma_{g+1}]+2[D^2]=[\Sigma_g]+[S_b^1\times I]$
  • Figure 4.1: (a) The manifold $V_{1,2}$. (b) A manifold with boundary $W_{1,2,3,4}=V_{1,2}\cup_{T_{fg^{-1}}}V_{3,4}$.

Theorems & Definitions (180)

  • Theorem 1
  • Theorem 2: \ref{['thm:OddSplittingTopWuclass']}
  • Theorem 3: Results in odd dimensions
  • Conjecture 4
  • Theorem 5: \ref{['thm:slittingSKKevenEvenChi']} and \ref{['thm:torsionWithOddEuler']}
  • Theorem 6: Results in even dimensions
  • Definition 2.1
  • Definition 2.2
  • Remark 2.1
  • Definition 2.3
  • ...and 170 more