Searcharxiv⌕ Search

arXiv subjects

Stephen Sawin

Publications and source records attributed to Stephen Sawin.

16 recordsLinked to original sources
arXiv2024

Gabriel's Theorem for Locally Finite-Dimensional Representations of Infinite Quivers

We prove a version of Gabriel's theorem for locally finite-dimensional representations of infinite quivers. Specifically, we show that if $Ω$ is any connected quiver, the category of locally finite-dimensional representations of $Ω$ has unique representation type (meaning no two indecomposable representations have the same dimension vector) if and only if the underlying graph of $Ω$ is a generalized ADE Dynkin diagram (i.e. one of $A_n, D_n, E_6, E_7, E_8, A_{\infty}, A_{\infty , \infty}$ or $D_\infty$). This result is companion to earlier work of the authors generalizing Gabriel's theorem to infinite quivers with different conditions.

math.RT↗
arXiv2023

Gabriel's Theorem for Infinite Quivers

We prove a version of Gabriel's theorem for (possibly infinite dimensional) representations of infinite quivers. More precisely, we show that the representation theory of quiver $\Omega$ is of unique type (each dimension vector has at most one associated indecomposable) and infinite Krull-Schmidt (every, possibly infinite dimensional, representation is a direct sum of indecomposables) if and only if $\Omega$ is eventually outward and of generalized ADE Dynkin type ($A_n$, $D_n$, $E_6$, $E_7$, $E_8$, $A_\infty$, $A_{\infty, \infty}$, or $D_\infty$). Furthermore we define an analog of the Euler-Tits form on the space of eventually constant infinite roots and show that a quiver is of generalized ADE Dynkin type if and only if this form is positive definite. In this case the indecomposables are all locally finite-dimensional and eventually constant and correspond bijectively to the positive roots (i.e. those of length $1$).

math.RT↗
arXiv2021

Centrality and the KRH Invariant

The purpose of this paper is to discuss the categorical structure for a method of defining quantum invariants of knots, links and three-manifolds. These invariants can be defined in terms of right integrals on certain Hopf algebras. We call such an invariant of 3-manifolds a Hennings invariant. The work reported in this paper has its background in previous work of the authors. The present paper gives an abstract description of these structures and shows how the Hopf algebraic image of a knot lies in the center of the corresponding Hopf algebra. The paper also shows how all the axiomatic properties of a quasi-triangular Hopf algebra are involved in the topology via a functor from the Tangle Category to the Diagrammatic Category of a Hopf Algebra. The invariants described in this paper generalize to invariants of rotational virtual knots. The contents of this paper are an update of the original 1998 version published in JKTR.

math.GT↗
arXiv2017

Path integrals, SUSY QM and the Atiyah-Singer index theorem for twisted Dirac

Feynman's time-slicing construction approximates the path integral by a product, determined by a partition of a finite time interval, of approximate propagators. This paper formulates general conditions to impose on a short-time approximation to the propagator in a general class of imaginary-time quantum mechanics on a Riemannian manifold which ensure these products converge. The limit defines a path integral which agrees pointwise with the heat kernel for a generalized Laplacian. The result is a rigorous construction of the propagator for supersymmetric quantum mechanics, with potential, as a path integral. Further, the class of Laplacians includes the square of the twisted Dirac operator, which corresponds to an extension of N=1/2 supersymmetric quantum mechanics. General results on the rate of convergence of the approximate path integrals suffice in this case to derive the local version of the Atiyah-Singer index theorem.

math-ph↗
arXiv2015

South Pointing Chariot: An Invitation to Differential Geometry

We introduce the south-pointing chariot, an intriguing mechanical device from ancient China. We use its ability to keep track of a global direction as it travels on an arbitrary path as a tool to explore the geometry of curved surfaces. This takes us as far as a famous result of Gauss on the impossibility of a faithful map of the globe, which started off the field of differential geometry. The reader should get a view into how geometers think and an introduction to important early results in the field, but should need no more than a solid background in calculus (ideally through multivariable calculus). This is achieved by relying on the reader's visual intuition.

math.HO↗
arXiv2013

A Rigorous Path Integral for N=1 Supersymmetic Quantum Mechanics on a Riemannian Manifold

Following Feynman's prescription for constructing a path integral representation of the propagator of a quantum theory, a short-time approximation to the propagator for imaginary time, N=1 supersymmetric quantum mechanics on a compact, even-dimensional Riemannian manifold is constructed. The path integral is interpreted as the limit of products, determined by a partition of a finite time interval, of this approximate propagator. The limit under refinements of the partition is shown to converge uniformly to the heat kernel for the Laplace-Beltrami operator on forms. A version of the steepest descent approximation to the path integral is obtained, and shown to give the expected short-time behavior of the supertrace of the heat kernel.

math-ph↗
arXiv2013

Efficient Computation of the Kauffman Bracket

This paper bounds the computational cost of computing the Kauffman bracket of a link in terms of the crossing number of that link. Specifically, it is shown that the image of a tangle with $g$ boundary points and $n$ crossings in the Kauffman bracket skein module is a linear combination of $O(2^g)$ basis elements, with each coefficient a polynomial with at most $n$ nonzero terms, each with integer coefficients, and that the link can be built one crossing at a time as a sequence of tangles with maximum number of boundary points bounded by $C\sqrt{n}$ for some $C.$ From this it follows that the computation of the Kauffman bracket of the link takes time and memory a polynomial in $n$ times $2^{C\sqrt{n}}.$

math.GT↗
arXiv2007

A Rigorous Path Integral for Supersymmetric Quantum Mechanics and the Heat Kernel

In a rigorous construction of the path integral for supersymmetric quantum mechanics on a Riemann manifold, based on Bär and Pfäffle's use of piecewise geodesic paths, the kernel of the time evolution operator is the heat kernel for the Laplacian on forms. The path integral is approximated by the integral of a form on the space of piecewise geodesic paths which is the pullback by a natural section of Mathai and Quillen's Thom form of a bundle over this space. In the case of closed paths, the bundle is the tangent space to the space of geodesic paths, and the integral of this form passes in the limit to the supertrace of the heat kernel.

math-ph↗
arXiv1999

Three-Dimensional 2-Framed TQFTs and Surgery

The notion of 2-framed three-manifolds is defined. The category of 2-framed cobordisms is described, and used to define a 2-framed three-dimensional TQFT. Using skeletonization and special features of this category, a small set of data and relations is given that suffice to construct a 2-framed three-dimensional TQFT. These data and relations are expressed in the language of surgery.

math.QA↗
arXiv1997

Diffeomorphism-Invariant Spin Network States

We extend the theory of diffeomorphism-invariant spin network states from the real-analytic category to the smooth category. Suppose that G is a compact connected semisimple Lie group and P -> M is a smooth principal G-bundle. A `cylinder function' on the space of smooth connections on P is a continuous complex function of the holonomies along finitely many piecewise smoothly immersed curves in M. We construct diffeomorphism-invariant functionals on the space of cylinder functions from `spin networks': graphs in M with edges labeled by representations of G and vertices labeled by intertwining operators. Using the `group averaging' technique of Ashtekar, Marolf, Mourao and Thiemann, we equip the space spanned by these `diffeomorphism-invariant spin network states' with a natural inner product.

q-alg↗
arXiv1995

Links, Quantum Groups, and TQFT's

The Jones polynomial and the Kauffman bracket are constructed, and their relation with knot and link theory is described. The quantum groups and tangle functor formalisms for understanding these invariants and their descendents are given. The quantum group $U_q(sl_2)$, which gives rise to the Jones polynomial, is constructed explicitly. The $3$-manifold invariants and the axiomatic topological quantum field theories which arise from these link invariants at certain values of the parameter are constructed.

q-alg↗
arXiv1995

On Finiteness of Certain Vassiliev Invariants

The best known examples of Vassiliev invariants are the coefficients of a Jones-type polynomial expanded after exponential substitution. We show that for a given knot, the first $N$ Vassiliev invariants in this family determine the rest for some integer $N$.

q-alg↗
arXiv1995

Functional Integration on Spaces of Connections

Let $G$ be a compact connected Lie group and $P \to M$ a smooth principal $G$-bundle. Let a `cylinder function' on the space $\A$ of smooth connections on $P$ be a continuous function of the holonomies of $A$ along finitely many piecewise smoothly immersed curves in $M$, and let a generalized measure on $\A$ be a bounded linear functional on cylinder functions. We construct a generalized measure on the space of connections that extends the uniform measure of Ashtekar, Lewandowski and Baez to the smooth case, and prove it is invariant under all automorphisms of $P$, not necessarily the identity on the base space $M$. Using `spin networks' we construct explicit functions spanning the corresponding Hilbert space $L^2(\A/\G)$, where $\G$ is the group of gauge transformations.

q-alg↗
arXiv1995

Direct Sum Decompositions and Indecomposable TQFT's

The decomposition of an arbitrary axiomatic topological quantum field theory or TQFT into indecomposable theories is given. In particular, unitary TQFT's in arbitrary dimensions are shown to decompose into a sum of theories in which the Hilbert space of the sphere is one-dimensional, and indecomposable two-dimensional theories are classified.

q-alg↗
arXiv1995

Path Integration in Two-Dimensional Toplogical Quantum Field Theory

A positive, diffeomorphism-invariant generalized measure on the space of metrics of a two-dimensional smooth manifold is constructed. We use the term generalized measure analogously with the generalized measures of Ashtekar and Lewandowski and of Baez. A family of actions is presented which, when integrated against this measure, give the two-dimensional axiomatic topological quantum field theories, or TQFT's, in terms of which Durhuus and Jonsson decompose every two-dimensional unitary TQFT as a direct sum.

gr-qc↗