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Three-dimensional TQFTs via string-nets and two-dimensional surgery
[article]

2022
*
arXiv
*
pre-print

We show how

arXiv:2206.13262v1
fatcat:qnejlt7skbeorgov53oclrqxv4
*to*extend this assignment*to*a 3-dimensional topological quantum field theory (TQFT), by defining how*the*surgery generators in Juhász' presentation*of**the*oriented 3-dimensional*bordism*category ... If C is a spherical fusion category,*the*string-*net*construction associates*to*each closed oriented surface Σ*the*vector space Z_SN(Σ)*of*linear combinations*of*C-labelled graphs on Σ modulo local relations ... String-*nets*In this section, we review*the*notion*of**the*string-*net*space Z SN (Σ)*of*an oriented surface Σ, given*the*initial data*of*a spherical fusion category C. ...##
###
Assembly maps in bordism-type theories
[chapter]

*
Novikov Conjectures, Index Theorems and Rigidity
*

*The*basic point is that a simple property

*of*manifolds gives rise

*to*an elaborate and rich structure including

*bordism*, homology, and "assembly maps." ...

*The*use

*of*obstruction spaces instead

*of*groups was

*the*major ingredient

*of*

*the*extension

*to*more general base spaces. ... Here "isomorphism"

*of*cycles

*means*

*the*following: cycles are functions

*from*

*the*nerve

*of*

*the*star cover

*to*PL manifolds. ...

##
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The equivariant Euler characteristic

1991
*
Journal of Pure and Applied Algebra
*

We describe an equivariant version

doi:10.1016/0022-4049(91)90071-9
fatcat:xj2ebjdenrcgbi2vnjyzvmw2pe
*of**the*Euler characteristic in order*to*extend*to**the*equivariant case classical results relating*the*Euler characteristic*to*vector field (Reinhart)*bordism**of*smooth ... Wu,*The*equib riant Euler characteristic, Journal*of*Pure and Applied Algebra 70 (1991) 227-249. ... We are also indebted*to*A. Assadi for suggesting*the*method we use in*the**proof**of*Proposition 6.2. ...##
###
Field theories with defects and the centre functor
[article]

2011
*
arXiv
*
pre-print

This note is intended as an introduction

arXiv:1107.0495v1
fatcat:fqf3wxn3onf43cxmf7ub7ru7mi
*to**the*functorial formulation*of*quantum field theories with defects. ... Finally, we give an application in algebra, where*the*defect TFT provides us with a functorial definition*of**the*centre*of*an algebra. ... In [BDH, Def. 4] , 2-morphisms are sectors between*the*defect*nets*-this*means*a Hilbert space with compatible actions*of*all four conformal*nets*involved:*the*two defect*nets*and*the*two conformal*nets*...##
###
Remarks on Chern-Simons theory

2009
*
Bulletin of the American Mathematical Society
*

We include, in

doi:10.1090/s0273-0979-09-01243-9
fatcat:ge6rjrpv2bhv5frdgmtgmqxmzi
*the*introduction and*the*last section, some general discussion about*the*current interaction between geometry and quantum theories*of*fields and gravity. ... We describe some*of**the*mathematical structure which has been built around this and other topological field theories. ... In topology we say X is a*bordism**from*Y 0*to*Y 1 and write X : Y 0 → Y 1 . These*bordisms*compose by gluing, which corresponds*to**the*evolution*of*time. (See Figure 1 .) ...##
###
Dualizable tensor categories
[article]

2018
*
arXiv
*
pre-print

We show that

arXiv:1312.7188v2
fatcat:65cp7wctk5fylj6stjykl7pu7e
*the*1-dimensional loop*bordism*, which exhibits a single full rotation, acts as*the*double dual autofunctor*of*a tensor category. ... We prove that*the*2-dimensional belt-trick*bordism*, which unravels a double rotation, operates on any finite tensor category, and therefore supplies a trivialization*of**the*quadruple dual. ...*Proof*. Such a bimodule equivalence α is in particular a right D-module functor*from*D*to*D and is therefore equivalent*to**the*left*multiplication*func- tor d → α(1) ⊗ d. ...##
###
Remarks on Chern-Simons Theory
[article]

2008
*
arXiv
*
pre-print

In

arXiv:0808.2507v2
fatcat:gzzdfddcbrdl3gbvjulspxjv5q
*the*late 1980s Witten used*the*Chern-Simons form*of*a*connection**to*construct new invariants*of*3-manifolds and knots, recovering in particular*the*Jones invariants. ... An appendix gives a lightening treatment*of**the*Chern-Simons-Weil theory*of**connections*.*The*paper concludes with general remarks about*the*Geometry-QFT-Strings interaction. ... In topology we say X is a*bordism**from*Y 0*to*Y 1 and write X : Y 0 → Y 1 . These*bordisms*compose by gluing, which corresponds*to**the*evolution*of*time. (See Figure 1.) ...##
###
Short-range entanglement and invertible field theories
[article]

2014
*
arXiv
*
pre-print

This leads

arXiv:1406.7278v2
fatcat:vehwbxgqkjccpklj6uhlk4yiw4
*to*concrete topological invariants*of*gapped SRE phases which are finer than existing invariants. Computations in examples demonstrate their effectiveness. ... For those with short range entanglement (SRE)*the*effective topological theory is invertible, and so amenable*to*study via stable homotopy theory. ... Recall that this*means*that we augment*the*set*of*fields*to*include a G-*connection*on a principal G-bundle. ...##
###
Three-tier CFTs from Frobenius algebras
[article]

2013
*
arXiv
*
pre-print

These are lecture notes

arXiv:1304.7328v2
fatcat:pfthcwliurh2vbdojc7ppsx7pi
*of*a course given at*the*Summer School on Topology and Field Theories held at*the*Centre for Mathematics*of**the*University*of*Notre Dame, Indiana,*from*May 29*to*June 2, 2012. ... In*the*case*of*conformal field theory (CFT), we are talking*of*an extension*of**the*Atiyah-Segal axioms, where one replaces*the**bordism*category*of*Riemann surfaces by a suitable*bordism*bicategory, whose ... I am also very grateful*to*my student Jules Lamers for compiling a first draft*of*these notes, and for drawing all*the*pictures. ...##
###
Analytic Pontryagin duality

2019
*
Journal of Geometry and Physics
*

We propose a

doi:10.1016/j.geomphys.2019.103483
fatcat:qoywd6lqdba57cvic5zhbwnyeq
*geometric*model for*the*group K^0(X,R/Z). ... We also study two special cases*of**the*analytic pairing*of*this form in*the*cohomology group H^1(X,R/Z) and H^2(X,R/Z). ... [4] ), measuring*the**net*change between*the*positive crossing (*from*negative*to*positive eigenvalues across 0) and*the*negative crossing (*from*positive*to*negative eigenvalues across 0). ...##
###
Internal bicategories
[article]

2016
*
arXiv
*
pre-print

This framework is well suited

arXiv:1206.4284v2
fatcat:uyd6wh54kbcpngl4feedftpoja
*to*examples arising in geometry and algebra, such as*the*3-category*of**bordisms*or*the*3-category*of*conformal*nets*. ... When*the*ambient 2-category is symmetric monoidal categories, this provides a convenient framework for encoding*the*structures*of*a symmetric monoidal 3-category. ...*The*first author was partially supported by a Miller Research Fellowship, and*the*second author was partially supported by ERC Horizon 2020 grant 674978. ...##
###
A hardness of approximation result in metric geometry
[article]

2020
*
arXiv
*
pre-print

We show that it is NP-hard

arXiv:1908.02824v2
fatcat:gz2kuiytwzg2zjzfmxiwnqxexa
*to*approximate*the*hyperspherical radius*of*a triangulated manifold up*to*an almost-polynomial factor. ... We would like*to*thank an anonymous referee for correcting a number*of*typos and offering other useful suggestions. F. Manin was partially supported by NSF individual grant DMS-2001042. ... So in each slab D, choose an ε-*net*V D and a forest T D*connecting*all*the*points*of*V D*to**the*boundary*of*D. ...##
###
Controlled Mather-Thurston theorems
[article]

2021
*
arXiv
*
pre-print

Small scale, UV, "distortions"

arXiv:2006.00374v5
fatcat:bkkwzg3vdnd6ngya4ezvdten5m
*of**the*base topology and structure group allow flat*connections**to*simulate curvature at larger scales. ... Classical results*of*Milnor, Wood, Mather, and Thurston produce flat*connections*in surprising places. ...*Geometrically*these 4-fold products translate into longitudinal sums*of*four copies*of*C 1 (and*of*C, at*the*level*of**bordisms*). ...##
###
Knot concordance and higher-order Blanchfield duality

2009
*
Geometry and Topology
*

Teichner defined a filtration F_n

doi:10.2140/gt.2009.13.1419
fatcat:h6kdflihbjhujhxvst3fop43qi
*of**the*classical knot concordance group C.*The*filtration is important because*of*its strong*connection**to**the*classification*of*topological 4-manifolds. ... We establish*the*same result for*the*corresponding filtration*of**the*smooth concordance group. ... Acknowledgments*The*first author was partially supported by*the*NSF DMS-0406573 and DMS-0706929.*The*second author was partially supported by DMS-0539044 and*The*Alfred P Sloan Foundation. ...##
###
Generalized Gauss maps and integrals for three-component links: Toward higher helicities for magnetic fields and fluid flows

2013
*
Journal of Mathematical Physics
*

*To*each three-component link in Euclidean 3-space, we associate a

*geometrically*natural generalized Gauss map

*from*

*the*3-torus

*to*

*the*2-sphere, and show that

*the*pairwise linking numbers and Milnor triple ... This can be viewed as a natural extension

*of*

*the*familiar fact that

*the*linking number

*of*a two-component link in 3-space is

*the*degree

*of*its associated Gauss map

*from*

*the*2-torus

*to*

*the*2-sphere. ... This

*means*that

*the*vector

*from*x z

*to*y z points straight up, and so lies in some page P

*of*

*the*standard open book in R 3 . ...

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