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Pin TQFT and Grassmann integral · Around

2019

Pin TQFT and Grassmann integral

Kobayashi, Ryohei

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We discuss a recipe to produce a lattice construction of fermionic phases of matter on unoriented manifolds.

This is performed by extending the construction of spin TQFT via the Grassmann integral proposed by Gaiotto and Kapustin, to the unoriented pin$_\pm$ case.
As an application, we construct gapped boundaries for time-reversal-invariant Gu-Wen fermionic SPT phases.
In addition, we provide a lattice definition of (1+1)d pin$_-$ invertible theory whose partition function is the Arf-Brown-Kervaire invariant, which generates the $\mathbb{Z}_8$ classification of (1+1)d topological superconductors.

Built on

R. Kobayashi, K. Ohmori, and Y. Tachikawa, On gapped boundaries for SPT phases beyond group cohomology,
Original
1905
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E. H. Brown, Generalizations of the Kervaire invariant,
1972
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R. Kirby and L. Taylor, Pin Structure on Low-dimensional Manifolds,
1989
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L. Fidkowski and A. Kitaev, Topological phases of fermions in one dimension,
Original
2011
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X. Chen, Z. C. Gu, Z. X. Liu, and X. G. Wen, Symmetry protected topological orders and the group cohomology of their symmetry group,
Original
2013
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Z.-C. Gu and X.-G. Wen, Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinear σ \sigma models and a special group supercohomology theory,
Original
2014
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Z. C. Gu, Z. Wang, and X. G. Wen, Lattice model for fermionic toric code,
Original
2014
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L. Fidkowski, X. Chen, and A. Vishwanath, Non-Abelian topological order on the surface of a 3d topological superconductor from an exactly solved model,
Original
2014
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Then

L. Bhardwaj, Unoriented 3d TFTs,
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2016

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C. Wang, C. H. Lin, and Z. C. Gu, Interacting fermionic symmetry-protected topological phases in two dimensions,

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M. A. Metlitski, L. Fidkowski, X. Chen, and A. Vishwanath, Interaction effects on 3D topological superconductors: surface topological order from vortex condensation, the 16 fold way and fermionic Kramers doublets,

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M. Guo, K. Ohmori, P. Putrov, Z. Wan, and J. Wang, Fermionic Finite-Group Gauge Theories and Interacting Symmetric/Crystalline Orders via Cobordisms,

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A. Turzillo, Diagrammatic State Sums for 2D Pin-Minus TQFTs,

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D. S. Freed and M. J. Hopkins, Reflection Positivity and Invertible Topological Phases,

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2018

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J. Wang, X.-G. Wen, and E. Witten, Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions,

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T. D. Ellison and L. Fidkowski, Disentangling Interacting Symmetry-Protected Phases of Fermions in Two Dimensions,

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