arXiv subjects

Stephen F. Sawin

Publications and source records attributed to Stephen F. Sawin.

7 recordsLinked to original sources

arXiv2018

Relationship of the Hennings and Chern-Simons Invariants For Higher Rank Quantum Groups

The Hennings invariant for the small quantum group associated to an arbitrary simple Lie algebra at a root of unity is shown to agree with Jones- Witten-Reshetikhin-Turaev invariant arising from Chern-Simons filed theory for the same Lie algebra and the same root of unity on all integer homol- ogy three-spheres, at roots of unity where both are defined. This partially generalizes the work of Chen, et al. ([CYZ12, CKS09]) which relates the Hennings and Chern-Simons invariants for SL(2) and SO(3) for arbitrary rational homology three-spheres.

math.GT↗

arXiv2018

Rigorous path integrals for supersymmetric quantum mechanics: completing the path integral proof of the index theorem

Many introductory courses in quantum mechanics include Feynman's time-slicing definition of the path integral, with a complete derivation of the propagator in the simplest of cases. However, attempts to generalize this, for instance to non-quadratic potentials, encounter formidable analytic issues in showing the successive approximations in fact converge to a definite expression for the path integral. The present work describes how to carry out the analysis for a class of Lagrangians broad enough to include the evolution, in imaginary time, of spinors constrained to live on a Riemannian manifold. For these Lagrangians, the successive time-slicing approximations converge. The limit provides a definition of the path integral which agrees with the imaginary-time Feynman propagator. With this as the definition, the steepest-descent approximation to the path integral for twisted $N=1/2$ supersymmetric quantum mechanics is provably correct. These results complete a new proof of the Atiyah-Singer index theorem for the twisted Dirac operator.

hep-th↗

arXiv2010

Quantum Groups at Roots of Unity and Modularity

For each compact, simple, simply-connected Lie group and each integer level we construct a modular tensor category from a quotient of a certain subcategory of the category of representations of the corresponding quantum group. We determine when at fractional levels the corresponding category is modular. We also give a quantum version of the Racah formula for the decomposition of the tensor product. This work relies on developing the basic representation theory of quantum groups at roots of unity, including Harish-Chandra's Theorem. It generalizes previous work which applied only to fractional levels or only to the projective form of the group.

math.QA↗

arXiv2005

Closed subsets of the Weyl alcove and TQFTs

For an arbitrary simple Lie algebra $\g$ and an arbitrary root of unity $q,$ the closed subsets of the Weyl alcove of the quantum group $U_q(\g)$ are classified. Here a closed subset is a set such that if any two weights in the Weyl alcove are in the set, so is any weight in the Weyl alcove which corresponds to an irreducible summand of the tensor product of a pair of representations with highest weights the two original weights. The ribbon category associated to each closed subset admits a ``quotient'' by a trivial subcategory as described by Bruguières and Müger, to give a modular category and a framed three-manifold invariant or a spin modular category and a spin three-manifold invariant. Most of these theories are equivalent to theories defined in previous work of the author, but several exceptional cases represent the first nontrivial examples to the author's knowledge of theories which contain noninvertible trivial objects, making the theory much richer and more complex.

math.QA↗

arXiv2005

Witten's Nonabelian Localization for Noncompact Hamiltonian Spaces

For a finite-dimensional (but possibly noncompact) symplectic manifold with a compact group acting with a proper moment map, we show that the square of the moment map is an equivariantly perfect Morse function in the sense of Kirwan, and that the set of critical points of the square of the moment map is a countable discrete union of compact sets. We show that certain integrals of equivariant cohomology classes localizes as a sum of contributions from these compact critical sets, and we bound the contribution from each critical set. In the case (1) that the contribution from higher critical sets grows slowly enough that the overall integral converges rapidly and (2) that 0 is a regular value of the moment map, we recover Witten's result \cite{Witten92} identifying the polynomial part of these integrals as the ordinary integral of the image of the class under the Kirwan map to the symplectic quotient.

math.SG↗

arXiv2000

Invariants of Spin Three-Manifolds From Chern-Simons Theory and Finite-Dimensional Hopf Algebras

A version of Kirby calculus for spin and framed three-manifolds is given and is used to construct invariants of spin and framed three-manifolds in two situations. The first is ribbon *-categories which possess odd degenerate objects. This case includes the quantum group situations corresponding to the half-integer level Chern-Simons theories conjectured to give spin TQFTs by Dijkgraaf and Witten \cite{DW90}. In particular, the spin invariants constructed by Kirby and Melvin \cite{KM91} are shown to be identical to the invariants associated to $\mathrm{SO}(3).$ Second, an invariant of spin manifolds analogous to the Hennings invariant is constructed beginning with an arbitrary factorizable, unimodular quasitriangular Hopf algebra. In particular a framed manifold invariant is associated to every finite-dimensional Hopf algebra via its quantum double, and is conjectured to be identical to Kuperberg's noninvolutory invariant of framed manifolds associated to that Hopf algebra.

math.QA↗

arXiv1999

Jones-Witten Invariants for Nonsimply-Connected Lie Groups and The Geometry of the Weyl Alcove

The quotient process of Müger and Bruguières is used to construct modular categories and TQFTs out of closed subsets of the Weyl alcove of a simple Lie algebra. In particular it is determined at which levels closed subsets associated to nonsimply-connected groups lead to TQFTs. Many of these TQFTs are shown to decompose into a tensor product of TQFTs coming from smaller subsets. The "prime" subsets among these are classified, and apart from some giving TQFTs depending on homology as described by Murakami, Ohtsuki and Okada, they are shown to be in one-to-one correspondence with the TQFTs predicted by Dijkgraaf and Witten to be associated to Chern-Simons theory with a nonsimply-connected Lie group. Thus in particular a rigorous construction of the Dijkgraaf-Witten TQFTs is given. As a byproduct, a purely quantum groups proof of the modularity of the full Weyl alcove for arbitrary quantum groups at arbitrary levels is given.

math.QA↗