Matrix product states and equivariant topological field theories for bosonic symmetry-protected topological phases in (1+1) dimensions

Abstract Fusion category symmetries are finite symmetries in 1+1 dimensions described by unitary fusion categories. We classify 1+1d time-reversal invariant bosonic symmetry protected topological (SPT) phases with fusion category symmetry by using topological field theories. We first formulate two-dimensional unoriented topological field theories whose symmetry splits into time-reversal symmetry and fusion category symmetry. We then solve them to show that SPT phases are classified by equivalence classes of quintuples (Z, M, i, s, ϕ) where (Z, M, i) is a fiber functor, s is a sign, and ϕ is the action of orientation- reversing symmetry that is compatible with the fiber functor (Z, M, i). We apply this classification to SPT phases with Kramers-Wannier-like self-duality.

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M-theoretic genesis of topological phases

Journal of High Energy Physics ◽

10.1007/jhep11(2020)115 ◽

2020 ◽

Vol 2020 (11) ◽

Author(s):

Gil Young Cho ◽

Dongmin Gang ◽

Hee-Cheol Kim

Keyword(s):

Topological Field Theories ◽

Field Theories ◽

Theoretic Approach ◽

Topological Phases ◽

World Volume ◽

Topological Field ◽

Topological Phases Of Matter ◽

Topological Data

Abstract We present a novel M-theoretic approach of constructing and classifying anyonic topological phases of matter, by establishing a correspondence between (2+1)d topological field theories and non-hyperbolic 3-manifolds. In this construction, the topological phases emerge as macroscopic world-volume theories of M5-branes wrapped around certain types of non-hyperbolic 3-manifolds. We devise a systematic algorithm for identifying the emergent topological phases from topological data of the internal wrapped 3-manifolds. As a benchmark of our approach, we reproduce all the known unitary bosonic topological orders up to rank 4. Remarkably, our construction is not restricted to an unitary bosonic theory but it can also generate fermionic and/or non-unitary anyon models in an equivalent fashion. Hence, we pave a new route toward the classification of topological phases of matter.

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A COMMENT ON TOPOLOGICAL FIELD THEORY QUANTIZATION AND STOCHASTIC PROCESSES

International Journal of Modern Physics A ◽

10.1142/s0217751x90001604 ◽

1990 ◽

Vol 05 (19) ◽

pp. 3777-3786 ◽

Cited By ~ 2

Author(s):

L.F. CUGLIANDOLO ◽

G. LOZANO ◽

H. MONTANI ◽

F.A. SCHAPOSNIK

Keyword(s):

Field Theory ◽

Equilibrium State ◽

Stochastic Processes ◽

Topological Field Theories ◽

Topological Charge ◽

Field Theories ◽

Langevin Equations ◽

Topological Field Theory ◽

Topological Field

We discuss the relation between different quantization approaches to topological field theories by deriving a connection between Bogomol’nyi and Langevin equations for stochastic processes which evolve towards an equilibrium state governed by the topological charge.

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Topological field theories on manifolds with Wu structures

Reviews in Mathematical Physics ◽

10.1142/s0129055x17500155 ◽

2017 ◽

Vol 29 (05) ◽

pp. 1750015 ◽

Cited By ~ 6

Author(s):

Samuel Monnier

Keyword(s):

Topological Field Theories ◽

Geometric Quantization ◽

Discrete Symmetry ◽

Local System ◽

Field Theories ◽

Simons Theory ◽

General Point ◽

Topological Field ◽

Generalized Cohomology ◽

Chern Simons

We construct invertible field theories generalizing abelian prequantum spin Chern–Simons theory to manifolds of dimension [Formula: see text] endowed with a Wu structure of degree [Formula: see text]. After analyzing the anomalies of a certain discrete symmetry, we gauge it, producing topological field theories whose path integral reduces to a finite sum, akin to Dijkgraaf–Witten theories. We take a general point of view where the Chern–Simons gauge group and its couplings are encoded in a local system of integral lattices. The Lagrangian of these theories has to be interpreted as a class in a generalized cohomology theory in order to obtain a gauge invariant action. We develop a computationally friendly cochain model for this generalized cohomology and use it in a detailed study of the properties of the Wu Chern–Simons action. In the 3-dimensional spin case, the latter provides a definition of the “fermionic correction” introduced recently in the literature on fermionic symmetry protected topological phases. In order to construct the state space of the gauged theories, we develop an analogue of geometric quantization for finite abelian groups endowed with a skew-symmetric pairing. The physical motivation for this work comes from the fact that in the [Formula: see text] case, the gauged 7-dimensional topological field theories constructed here are essentially the anomaly field theories of the 6-dimensional conformal field theories with [Formula: see text] supersymmetry, as will be discussed elsewhere.

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