scholarly journals A GNS construction of three-dimensional abelian Dijkgraaf–Witten theories

2018 ◽  
Vol 30 (02) ◽  
pp. 1850005
Author(s):  
Lukas Müller ◽  
Christoph Schweigert
Keyword(s):  
Topological Field Theories ◽  
Three Dimensional ◽  
Detailed Account ◽  
Field Theories ◽  
Vector Spaces ◽  
Additional Structure ◽  
Principal Bundles ◽  
Topological Field ◽  
Universal Construction ◽  
Gns Construction

We give a detailed account of the so-called “universal construction” that aims to extend invariants of closed manifolds, possibly with additional structure, to topological field theories and show that it amounts to a generalization of the GNS construction. We apply this construction to an invariant defined in terms of the groupoid cardinality of groupoids of bundles to recover Dijkgraaf–Witten theories, including the vector spaces obtained as a linearization of spaces of principal bundles.

scholarly journals Exceptional Chern-Simons-Matter dualities

2019 ◽  
Vol 7 (4) ◽  
Author(s):  
Clay Cordova ◽  
Po-Shen Hsin ◽  
Kantaro Ohmori
Keyword(s):  
Gauge Theory ◽  
Topological Field Theories ◽  
Time Reversal ◽  
Three Dimensional ◽  
Field Theories ◽  
Chiral Algebra ◽  
Gauge Groups ◽  
Topological Field ◽  
Conformal Embeddings ◽  
Chern Simons

We use conformal embeddings involving exceptional affine Kac-Moody algebras to derive new dualities of three-dimensional topological field theories. These generalize the familiar level-rank duality of Chern-Simons theories based on classical gauge groups to the setting of exceptional gauge groups. For instance, one duality sequence we discuss is (E_{N})_{1}\leftrightarrow SU(9-N)_{-1}(EN)1↔SU(9−N)−1. Others such as SO(3)_{8}\leftrightarrow PSU(3)_{-6},SO(3)8↔PSU(3)−6, are dualities among theories with classical gauge groups that arise due to their embedding into an exceptional chiral algebra. We apply these equivalences between topological field theories to conjecture new boson-boson Chern-Simons-matter dualities. We also use them to determine candidate phase diagrams of time-reversal invariant G_{2}G2 gauge theory coupled to either an adjoint fermion, or two fundamental fermions.

scholarly journals The Chromatic Brauer Category and Its Linear Representations

2020 ◽  
Author(s):  
L. Felipe Müller ◽  
Dominik J. Wrazidlo
Keyword(s):  
Representation Theory ◽  
Tensor Product ◽  
Topological Field Theories ◽  
Monoidal Category ◽  
Field Theories ◽  
Vector Spaces ◽  
Real Vector ◽  
Linear Representations ◽  
Linear Maps ◽  
Topological Field

AbstractThe Brauer category is a symmetric strict monoidal category that arises as a (horizontal) categorification of the Brauer algebras in the context of Banagl’s framework of positive topological field theories (TFTs). We introduce the chromatic Brauer category as an enrichment of the Brauer category in which the morphisms are component-wise labeled. Linear representations of the (chromatic) Brauer category are symmetric strict monoidal functors into the category of real vector spaces and linear maps equipped with the Schauenburg tensor product. We study representation theory of the (chromatic) Brauer category, and classify all its faithful linear representations. As an application, we use indices of fold lines to construct a refinement of Banagl’s concrete positive TFT based on fold maps into the plane.

scholarly journals A CONNECTION BETWEEN LATTICE AND SURGERY CONSTRUCTIONS OF THREE-DIMENSIONAL TOPOLOGICAL FIELD THEORIES

1995 ◽  
Vol 10 (37) ◽  
pp. 2831-2842
Author(s):  
MASAKO ASANO ◽  
SABURO HIGUCHI
Keyword(s):  
Topological Field Theories ◽  
Three Dimensional ◽  
Field Theories ◽  
Similar Relation ◽  
Topological Field ◽  
Lattice Construction

We study the relation between lattice construction and surgery construction of three-dimensional topological field theories. We show that a class of the Chung-Fukuma-Shapere theory on the lattice has representation theoretic reformulation which is closely related to the Altschuler-Coste theory constructed by surgery. There is a similar relation between the Turaev-Viro theory and the Reshetikhin-Turaev theory.

scholarly journals THREE-DIMENSIONAL TOPOLOGICAL FIELD THEORIES AND STRINGS

1991 ◽  
Vol 06 (03) ◽  
pp. 171-181 ◽  
Author(s):  
S. CARLIP ◽  
I. KOGAN
Keyword(s):  
String Theory ◽  
Topological Field Theories ◽  
Three Dimensional ◽  
Field Theories ◽  
The Status ◽  
Topological Field ◽  
Topological Fields

We discuss the status of the program to reformulate string theory as a theory of topological fields and gravity in 3 dimensions.

scholarly journals ON THREE-DIMENSIONAL TOPOLOGICAL FIELD THEORIES CONSTRUCTED FROM Dω(G) FOR FINITE GROUPS

1994 ◽  
Vol 09 (25) ◽  
pp. 2359-2369 ◽  
Author(s):  
MASAKO ASANO ◽  
SABURO HIGUCHI
Keyword(s):  
Partition Function ◽  
Hopf Algebra ◽  
Topological Field Theories ◽  
Finite Groups ◽  
Three Dimensional ◽  
Commutative Group ◽  
Field Theories ◽  
Quantum Double ◽  
Topological Field

We investigate the 3-D lattice topological field theories defined by Chung, Fukuma and Shapere. We concentrate on the model defined by taking a deformation Dω(G) of the quantum double of a finite commutative group G as the underlying Hopf algebra. It is suggested that Chung-Fukuma-Shapere partition function is related to that of Dijkgraaf-Witten by Z CFS =|Z DW |2 when G=ℤ2N+1. For G=ℤ2N, such a relation does not hold.

scholarly journals On dualizability of braided tensor categories

2021 ◽  
Vol 157 (3) ◽  
pp. 435-483
Author(s):  
Adrien Brochier ◽  
David Jordan ◽  
Noah Snyder
Keyword(s):  
Field Theory ◽  
Quantum Group ◽  
Topological Field Theories ◽  
Characteristic Zero ◽  
Three Dimensional ◽  
Field Theories ◽  
Topological Field Theory ◽  
Tensor Categories ◽  
Fusion Categories ◽  
Topological Field

We study the question of dualizability in higher Morita categories of locally presentable tensor categories and braided tensor categories. Our main results are that the 3-category of rigid tensor categories with enough compact projectives is 2-dualizable, that the 4-category of rigid braided tensor categories with enough compact projectives is 3-dualizable, and that (in characteristic zero) the 4-category of braided multi-fusion categories is 4-dualizable. Via the cobordism hypothesis, this produces respectively two-, three- and four-dimensional framed local topological field theories. In particular, we produce a framed three-dimensional local topological field theory attached to the category of representations of a quantum group at any value of $q$ .

A COMMENT ON TOPOLOGICAL FIELD THEORY QUANTIZATION AND STOCHASTIC PROCESSES

1990 ◽  
Vol 05 (19) ◽  
pp. 3777-3786 ◽  
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.

scholarly journals Topological field theories on manifolds with Wu structures

2017 ◽  
Vol 29 (05) ◽  
pp. 1750015 ◽  
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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