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

Bruce Bartlett
2022 arXiv   pre-print
We show how 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.  ... 
arXiv:2206.13262v1 fatcat:qnejlt7skbeorgov53oclrqxv4

Assembly maps in bordism-type theories [chapter]

Frank Quinn, Steven C. Ferry, Andrew Ranicki, Jonathan M. Rosenberg
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.  ... 
doi:10.1017/cbo9780511662676.011 fatcat:s22ne5kzjnb5xc2hxqc44vmsfe

The equivariant Euler characteristic

Steven R. Costenoble, Stefan Waner, Yihren Wu
1991 Journal of Pure and Applied Algebra  
We describe an equivariant version 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.  ... 
doi:10.1016/0022-4049(91)90071-9 fatcat:xj2ebjdenrcgbi2vnjyzvmw2pe

Field theories with defects and the centre functor [article]

Alexei Davydov, Liang Kong, Ingo Runkel
2011 arXiv   pre-print
This note is intended as an introduction 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  ... 
arXiv:1107.0495v1 fatcat:fqf3wxn3onf43cxmf7ub7ru7mi

Remarks on Chern-Simons theory

Daniel S. Freed
2009 Bulletin of the American Mathematical Society  
We include, in 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 .)  ... 
doi:10.1090/s0273-0979-09-01243-9 fatcat:ge6rjrpv2bhv5frdgmtgmqxmzi

Dualizable tensor categories [article]

Christopher L. Douglas, Christopher Schommer-Pries, Noah Snyder
2018 arXiv   pre-print
We show that 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.  ... 
arXiv:1312.7188v2 fatcat:65cp7wctk5fylj6stjykl7pu7e

Remarks on Chern-Simons Theory [article]

Daniel S. Freed
2008 arXiv   pre-print
In 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.)  ... 
arXiv:0808.2507v2 fatcat:gzzdfddcbrdl3gbvjulspxjv5q

Short-range entanglement and invertible field theories [article]

Daniel S. Freed
2014 arXiv   pre-print
This leads 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.  ... 
arXiv:1406.7278v2 fatcat:vehwbxgqkjccpklj6uhlk4yiw4

Three-tier CFTs from Frobenius algebras [article]

Andre Henriques
2013 arXiv   pre-print
These are lecture notes 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.  ... 
arXiv:1304.7328v2 fatcat:pfthcwliurh2vbdojc7ppsx7pi

Analytic Pontryagin duality

Johnny Lim
2019 Journal of Geometry and Physics  
We propose a 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).  ... 
doi:10.1016/j.geomphys.2019.103483 fatcat:qoywd6lqdba57cvic5zhbwnyeq

Internal bicategories [article]

Christopher L. Douglas, André G. Henriques
2016 arXiv   pre-print
This framework is well suited 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.  ... 
arXiv:1206.4284v2 fatcat:uyd6wh54kbcpngl4feedftpoja

A hardness of approximation result in metric geometry [article]

Zarathustra Brady and Larry Guth and Fedor Manin
2020 arXiv   pre-print
We show that it is NP-hard 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.  ... 
arXiv:1908.02824v2 fatcat:gz2kuiytwzg2zjzfmxiwnqxexa

Controlled Mather-Thurston theorems [article]

Michael Freedman
2021 arXiv   pre-print
Small scale, UV, "distortions" 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).  ... 
arXiv:2006.00374v5 fatcat:bkkwzg3vdnd6ngya4ezvdten5m

Knot concordance and higher-order Blanchfield duality

Tim D Cochran, Shelly Harvey, Constance Leidy
2009 Geometry and Topology  
Teichner defined a filtration F_n 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.  ... 
doi:10.2140/gt.2009.13.1419 fatcat:h6kdflihbjhujhxvst3fop43qi

Generalized Gauss maps and integrals for three-component links: Toward higher helicities for magnetic fields and fluid flows

Dennis DeTurck, Herman Gluck, Rafal Komendarczyk, Paul Melvin, Clayton Shonkwiler, David Shea Vela-Vick
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 .  ... 
doi:10.1063/1.4774172 fatcat:yfmplsdtvnaprdo6g5o7wxvcmi
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