11,566 research outputs found
Reflection positivity and invertible topological phases
We implement an extended version of reflection positivity (Wick-rotated
unitarity) for invertible topological quantum field theories and compute the
abelian group of deformation classes using stable homotopy theory. We apply
these field theory considerations to lattice systems, assuming the existence
and validity of low energy effective field theory approximations, and thereby
produce a general formula for the group of Symmetry Protected Topological (SPT)
phases in terms of Thom's bordism spectra; the only input is the dimension and
symmetry group. We provide computations for fermionic systems in physically
relevant dimensions. Other topics include symmetry in quantum field theories, a
relativistic 10-fold way, the homotopy theory of relativistic free fermions,
and a topological spin-statistics theorem.Comment: 136 pages, 16 figures; minor changes/corrections in version 2; v3
major revision; v4 minor revision: corrected proof of Lemma 9.55, many small
changes throughout; v5 version for publication in Geometry & Topolog
An invitation to 2D TQFT and quantization of Hitchin spectral curves
This article consists of two parts. In Part 1, we present a formulation of
two-dimensional topological quantum field theories in terms of a functor from a
category of Ribbon graphs to the endofuntor category of a monoidal category.
The key point is that the category of ribbon graphs produces all Frobenius
objects. Necessary backgrounds from Frobenius algebras, topological quantum
field theories, and cohomological field theories are reviewed. A result on
Frobenius algebra twisted topological recursion is included at the end of Part
1.
In Part 2, we explain a geometric theory of quantum curves. The focus is
placed on the process of quantization as a passage from families of Hitchin
spectral curves to families of opers. To make the presentation simpler, we
unfold the story using SL_2(\mathbb{C})-opers and rank 2 Higgs bundles defined
on a compact Riemann surface of genus greater than . In this case,
quantum curves, opers, and projective structures in all become the same
notion. Background materials on projective coordinate systems, Higgs bundles,
opers, and non-Abelian Hodge correspondence are explained.Comment: 53 pages, 6 figure
A Prehistory of n-Categorical Physics
This paper traces the growing role of categories and n-categories in physics,
starting with groups and their role in relativity, and leading up to more
sophisticated concepts which manifest themselves in Feynman diagrams, spin
networks, string theory, loop quantum gravity, and topological quantum field
theory. Our chronology ends around 2000, with just a taste of later
developments such as open-closed topological string theory, the
categorification of quantum groups, Khovanov homology, and Lurie's work on the
classification of topological quantum field theories.Comment: 129 pages, 8 eps figure
Two-dimensional quantum Yang-Mills theory with corners
The solution of quantum Yang-Mills theory on arbitrary compact two-manifolds
is well known. We bring this solution into a TQFT-like form and extend it to
include corners. Our formulation is based on an axiomatic system that we hope
is flexible enough to capture actual quantum field theories also in higher
dimensions. We motivate this axiomatic system from a formal
Schroedinger-Feynman quantization procedure. We also discuss the physical
meaning of unitarity, the concept of vacuum, (partial) Wilson loops and
non-orientable surfaces.Comment: 31 pages, 6 figures, LaTeX + AMS; minor corrections, reference
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