arXiv subjects

George Thompson

Publications and source records attributed to George Thompson.

At least 19 recordsLinked to original sources

arXiv2024

On the Evaluation of the Ray-Singer Torsion Path Integral

There are very few explicit evaluations of path integrals for topological gauge theories in more than 3 dimensions. Here we provide such a calculation for the path integral representation of the Ray-Singer Torsion of a flat connection on a vector bundle on base manifolds that are themselves $S^{1}$ bundles of any dimension. The calculation relies on a suitable algebraic choice of gauge which leads to a convenient factorisation of the path integral into horizontal and vertical parts.

hep-th↗

arXiv2022

3 Definitions of BF Theory on Homology 3-Spheres

3-dimensional BF theory with gauge group $G$ (= Chern-Simons theory with non-compact gauge group $TG$) is a deceptively simple yet subtle topological gauge theory. Formally, its partition function is a sum/integral over the moduli space $\mathcal{M}$ of flat connections, weighted by the Ray-Singer torsion. In practice, however, this formal expression is almost invariably singular and ill-defined. In order to improve upon this, we perform a direct evaluation of the path integral for certain classes of 3-manifolds (namely integral and rational Seifert homology spheres). By a suitable choice of gauge, we sidestep the issue of having to integrate over $\mathcal{M}$ and reduce the partition function to a finite-dimensional Abelian matrix integral which, however, itself requires a definition. We offer 3 definitions of this integral, firstly via residues, and then via a large $k$ limit of the corresponding $G\times G$ or $G_C$ Chern-Simons matrix integrals (obtained previously). We then check and discuss to which extent the results capture the expected sum/integral over all flat connections.

hep-th↗

arXiv2022

Massive Ray-Singer Torsion and Path Integrals

Zero modes are an essential part of topological field theories, but they are frequently also an obstacle to the explicit evaluation of the associated path integrals. In order to address this issue in the case of Ray-Singer Torsion, which appears in various topological gauge theories, we introduce a massive variant of the Ray-Singer Torsion which involves determinants of the twisted Laplacian with mass but without zero modes. This has the advantage of allowing one to explicitly keep track of the zero mode dependence of the theory. We establish a number of general properties of this massive Ray-Singer Torsion. For product manifolds $M=N \times S^1$ and mapping tori one is able to interpret the mass term as a flat $\mathbb{R}_{+}$ connection and one can represent the massive Ray-Singer Torsion as the path integral of a Schwarz type topological gauge theory. Using path integral techniques, with a judicious choice of an algebraic gauge fixing condition and a change of variables which leaves one with a free action, we can evaluate the torsion in closed form. We discuss a number of applications, including an explicit calculation of the Ray-Singer Torsion on $S^1$ for $G=PSL(2,R)$ and a path integral derivation of a generalisation of a formula of Fried for the torsion of finite order mapping tori.

hep-th↗

arXiv2018

Chern-Simons Theory on a General Seifert 3-Manifold

The path integral for the partition function of Chern-Simons gauge theory with a compact gauge group is evaluated on a general Seifert 3-manifold. This extends previous results and relies on abelianisation, a background field method and local application of the Kawasaki Index theorem.

hep-th↗

arXiv2016

Chern-Simons Theory with Complex Gauge Group on Seifert Fibred 3-Manifolds

We consider Chern-Simons theory with complex gauge group and present a complete non-perturbative evaluation of the path integral (the partition function and certain expectation values of Wilson loops) on Seifert fibred 3-Manifolds. We use the method of Abelianisation. In certain cases the path integral can be seen to factorize neatly into holomorphic and anti-holomorphic parts. We obtain closed formulae of this factorization for the expectation values of torus knots.

hep-th↗

arXiv2013

Chern-Simons Theory on Seifert 3-Manifolds

We study Chern-Simons theory on 3-manifolds M that are circle-bundles over 2-dimensional orbifolds S by the method of Abelianisation. This method, which completely sidesteps the issue of having to integrate over the moduli space of non-Abelian flat connections, reduces the complete partition function of the non-Abelian theory on M to a 2-dimensional Abelian theory on the orbifold S which is easily evaluated.

hep-th↗

arXiv2010

Intersection Pairings on Spaces of Connections and Chern-Simons Theory on Seifert Manifolds

Let M be a U(1) bundle over a smooth Riemann surface. I show that for Chern-Simons theory on M, with structure group G, the path integral is an integral over the space of G-connections on the Riemann surface involving characteristic classes as well as a certain 4-dimensional class that comes from a universal bundle. When M is the product of a Riemann surface with a circle the 4-dimensional class does not enter and the path integral takes the form of a Riemann-Roch formula albeit in infinite dimensions. The discussion is generalised to include Wilson lines along the fibre direction in M.

math.DG↗

arXiv2008

Skew Invariant Theory of Symplectic Groups, Pluri-Hodge Groups and 3-Manifold Invariants

This article deals with a number of topics which are, somewhat surprisingly, related. Firstly, the fundamental theorem of skew invariant theory for the symplectic group giving the generators and relations of symplectic invariants is established. The relations are the so called P_n relations which appear in the study of certain 3-manifold invariants. Next a class of cohomology groups are introduced, called Pluri-Hodge groups (somewhat in keeping with the notion of pluri-canonical groups). These are Dolbeault groups on a complex manifold X with values in tensor powers of sheaves of holomorphic forms of various degrees. By Riemann-Roch one shows that knowledge of the Pluri-Hodge groups gives precise formulae for all Chern numbers of the manifold. When X is holomorphic symplectic the Pluri-Hodge groups form representations of Sp(g) where g counts the number of tensor products. Still with X holomorphic symplectic, the Pluri-Hodge groups are vectors in the vector space H_g(X) of states in the topological field theory of Rozansky and Witten for a 3-manifold with boundary a Riemann surface of genus g. A formulation of the Murakami-Ohtsuki invariants, due to Sawon, allows us to show that quotients of the spaces of trivalent graphs B_g carry representations of the symplectic group. There is a weight system W_X: B_g -> H_g(X) and we show that W_X preserves the various symplectic group actions.

math.DG↗

arXiv2006

Chern-Simons Theory on S^1-Bundles: Abelianisation and q-deformed Yang-Mills Theory

We study Chern-Simons theory on 3-manifolds $M$ that are circle-bundles over 2-dimensional surfaces $Σ$ and show that the method of Abelianisation, previously employed for trivial bundles $Σ\times S^1$, can be adapted to this case. This reduces the non-Abelian theory on $M$ to a 2-dimensional Abelian theory on $Σ$ which we identify with q-deformed Yang-Mills theory, as anticipated by Vafa et al. We compare and contrast our results with those obtained by Beasley and Witten using the method of non-Abelian localisation, and determine the surgery and framing presecription implicit in this path integral evaluation. We also comment on the extension of these methods to BF theory and other generalisations.

hep-th↗

arXiv2003

The Universal Connection and Metrics on Moduli Spaces

We introduce a class of metrics on gauge theoretic moduli spaces. These metrics are made out of the universal matrix that appears in the universal connection construction of M. S. Narasimhan and S. Ramanan. As an example we construct metrics on the c_{2}=1 SU(2) moduli space of instantons on R^4 for various universal matrices.

math.DG↗

arXiv2001

Holomorphic Vector Bundles, Knots and the Rozansky-Witten Invariants

Link invariants, for 3-manifolds, are defined in the context of the Rozansky-Witten theory. To each knot in the link one associates a holomorphic bundle over a holomorphic symplectic manifold X. The invariants are evaluated for b_{1}(M) \geq 1 and X Hyper-Kaehler. To obtain invariants of Hyper-Kaehler X one finds that the holomorphic vector bundles must be hyper-holomorphic. This condition is derived and explained. Some results for X not Hyper-Kaehler are presented.

hep-th↗

arXiv2001

Instantons, the Information Metric, and the AdS/CFT Correspondence

We describe some remarkable properties of the so-called Information Metric on instanton moduli space. This Metric is manifestly gauge and conformally invariant and coincides with the Euclidean AdS_5 metric on the one-instanton SU(2) moduli space for the standard metric on R^4. We propose that for an arbitrary boundary metric the AdS/CFT bulk space-time is the instanton moduli space equipped with the Information Metric. To test this proposal, we examine the variation of the instanton moduli and the Information Metric for first-order perturbations of the boundary metric and obtain three non-trivial and somewhat surprising results: (1) The perturbed Information Metric is Einstein. (2) The perturbed instanton density is the corresponding massless boundary-to-bulk scalar propagator. (3) The regularized boundary-to-bulk geodesic distance is proportional to the logarithm of the perturbed instanton density. The Hamilton-Jacobi equation implied by (3) equips the moduli space with a rich geometrical structure which we explore. These results tentatively suggest a picture in which the one-instanton sector of SU(2) Yang-Mills theory (rather than some large-N limit) is in some sense holographically dual to bulk gravity.

hep-th↗

arXiv2000

On the Relationship between the Rozansky-Witten and the 3-Dimensional Seiberg-Witten Invariants

The Seiberg-Witten analysis of the low-energy effective action of d=4 N=2 SYM theories reveals the relation between the Donaldson and Seiberg-Witten (SW) monopole invariants. Here we apply analogous reasoning to d=3 N=4 theories and propose a general relationship between Rozansky-Witten (RW) and 3-dimensional Abelian monopole invariants. In particular, we deduce the equality of the SU(2) Casson invariant and the 3-dimensional SW invariant (this includes a special case of the Meng-Taubes theorem relating the SW invariant to Milnor torsion). Since there are only a finite number of basic RW invariants of a given degree, many different topological field theories can be used to represent essentially the same topological invariant. This leads us to advocate using higher rank Abelian gauge theories to shed light on the higher (non-Abelian) RW invariants and we write down candidate higher rank SW equations.

hep-th↗

arXiv1999

The Universal Perturbative Quantum 3-manifold Invariant, Rozansky-Witten Invariants, and the Generalized Casson Invariant

Let Z^{LMO} be the 3-manifold invariant of [LMO]. It is shown that Z^{LMO}(M)=1, if the first Betti number of M, b_{1}(M), is greater than 3. If b_{1}(M)=3, then Z^{LMO}(M) is completely determined by the cohomology ring of M. A relation of Z^{LMO} with the Rozansky-Witten invariants Z_{X}^{RW}[M] is established at a physical level of rigour. We show that Z_{X}^{RW}[M] satisfies appropriate connected sum properties suggesting that the generalized Casson invariant ought to be computable from the LMO invariant.

math.GT↗

arXiv1999

A Geometric Interpretation of the χ-{y} Genus on Hyper-Kahler Manifolds

The group SL(2) acts on the space of cohomology groups of any hyper-Kahler manifold X. The χ_{y} genus of a hyper-Kahler X is shown to have a geometric interpretation as the super trace of an element of SL(2). As a by product one learns that the generalized Casson invariant for a mapping torus is essentially the χ_{y} genus.

math.AG↗

arXiv1998

On the Generalized Casson Invariant

The path integral generalization of the Casson invariant as developed by Rozansky and Witten is investigated. The path integral for various three manifolds is explicitly evaluated. A new class of topological observables is introduced that may allow for more effective invariants. Finally it is shown how the dimensional reduction of these theories corresponds to a generalization of the topological B sigma model.

hep-th↗

arXiv1998

One Loop Effects in Various Dimensions and D-Branes

We calculate some one loop corrections to the effective action of theories in $d$ dimensions that arise on the dimensional reduction of a Weyl fermion in $D$ dimensions. The terms that we are interested in are of a topological nature. Special attention is given to the effective actions of the super Yang Mills theories that arise on dimensional reduction of the N=1 theory in six dimensions or on the dimensional reduction of the N=1 theory in ten dimensions. In the latter case we suggest an interpretation of the quantum effect as a coupling of the gauge field on the brane to a relative background gauge field.

hep-th↗

arXiv1997

Euclidean SYM Theories by Time Reduction and Special Holonomy Manifolds

Euclidean supersymmetric theories are obtained from Minkowskian theories by performing a reduction in the time direction. This procedure elucidates certain mysterious features of Zumino's N=2 model in four dimensions, provides manifestly hermitian Euclidean counterparts of all non-mimimal SYM theories, and is also applicable to supergravity theories. We reanalyse the twists of the 4d N=2 and N=4 models from this point of view. Other applications include SYM theories on special holonomy manifolds. In particular, we construct a twisted SYM theory on Kaehler 3-folds and clarify the structure of SYM theory on hyper-Kaehler 4-folds.

hep-th↗