George Thompson

Topological Gauge Theories of Antisymmetric Tensor Fields

Abstract We introduce a new class of topological gauge field theories in any dimension, based on anti-symmetric tensor fields, and discuss the BRST-quantization of these reducible systems as well as the equivalence of BRST-quantization and Schwarzs method of resolvents in detail. As a consequence we can use path-integral techniques and BRST-symmetry to prove metric independence and other properties of the Ray-Singer torsion. We pay particular attention to the presence of zero modes and discuss various methods of treating them in these models and other topological field theories. Non-Abelian models in two dimensions provide us with a complete Nicolai map for Yang-Mills theory on an arbitrary two-surface, as well as with a theory of topological gravity which is closely related to Hitchins self-duality equation on a Riemann surface. Candidate observables for Abelian models in any dimension are linking and intersection numbers of manifolds for which we give explicit path-integral representations.

Derivation of the Verlinde Formula from Chern-Simons Theory and the G/G Model

Abstract We give a derivation of the Verlinde formula for the G k WZW model from Chern-Simons theory, without taking recourse to CFT, by calculating explicitly the partition function Z Σ ×S 1 of Σ × S 1 with an arbitrary number of labelled punctures. By what is essentially a suitable gauge choice, Z Σ ×S 1 is reduced to the partition function of an abelian topological field theory on Σ (a deformation of non-abelian BF and Yang-Mills theory) whose evaluation is straightforward. This relates the Verlinde formula to the Ray-Singer torsion of Σ × S 1 . We derive the G k /G k model from Chern-Simons theory, proving their equivalence, and give an alternative derivation of the Verlinde formula by calculating the G k /G k path integral via a functional version of the Weyl integral formula. From this point of view the Verlinde formula arises from the corresponding jacobian, the Weyl determinant. Also, a novel derivation of the shift k → k + h is given, based on the index of the twisted Dolbeault complex.

Aspects of NT ⩾ 2 topological gauge theories and D-branes

Abstract We comment on various aspects of topological gauge theories possessing N T ⩾ 2 topological symmetry: (1) We show that the construction of Vafa-Witten and Dijkgraaf-Moore of ‘balanced’ topological field theories is equivalent to an earlier construction in terms of N T = 2 superfields inspired by supersymmetric quantum mechanics. (2) We explain the relation between topological field theories calculating signed and unsigned sums of Euler numbers of moduli spaces. (3) We show that the topological twist of N = 4 d = 4 Yang-Mills theory recently constructed by Marcus is formally a deformation of four-dimensional super-BF theory. (4) We construct a novel N T = 2 topological twist of N = 4 d = 3 Yang-Mills theory, a ‘mirror’ of the Casson invariant model, with certain unusual features (e.g. no bosonic scalar field and hence no underlying equivariant cohomology) (5) We give a complete classification of the topological twists of N = 8 d = 3 Yang-Mills theory and show that they are realised as world-volume theories of Dirichlet two brane instantons wrapping supersymmetric three-cycles of Calabi-Yau three-folds and G 2 -holonomy Joyce manifolds. (6) We describe the topological gauge theories associated to D-string instantons on holomorphic curves in K3s and Calabi-Yau 3-folds.

Euclidean SYM theories by time reduction and special holonomy manifolds

Abstract Euclidean supersymmetric theories are obtained from Minkowskian theories by performing a reduction in the time direction. This procedure elucidates certain mysterious features of Zuminos 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 Kahler 3-folds and clarify the structure of SYM theory on hyper-Kahler 4-folds.

Localization and Diagonalization: A Review of Functional Integral Techniques for Low-Dimensional Gauge Theories and Topological Field Theories

Localization techniques for functional integrals which have recently been used to perform calculations in and gain insight into the structure of certain topological field theories and low‐dimensional gauge theories are reviewed. These are the functional integral counterparts of the Mathai–Quillen formalism, the Duistermaat–Heckman theorem, and the Weyl integral formula, respectively. In each case, the necessary mathematical background (Euler classes of vector bundles, equivariant cohomology, topology of Lie groups) is introduced, and the finite dimensional integration formulae described. Then some applications to path integrals are discussed and an overview of the relevant literature is given. The applications include supersymmetric quantum mechanics, cohomological field theories, phase space path integrals, and two‐dimensional Yang–Mills theory.

Quantum Yang-Mills theory on arbitrary surfaces

We study quantum Maxwell and Yang-Mills theory on orientable two-dimensional surfaces with an arbitrary number of handles and boundaries. Using path integral methods we derive general and explicit expressions for the partition function and expectation values of contractible and noncontractible Wilson loops on closed surfaces of any genus, as well as for the kernels on manifolds with handles and boundaries. In the Abelian case we also compute correlation functions of intersecting and self-intersecting loops on closed surfaces, and discuss the role of large gauge transformations and topologically nontrivial bundles.

N=2 topological gauge theory, the Euler characteristic of moduli spaces, and the Casson invariant

We discuss gauge theory with a topologicalN=2 symmetry. This theory captures the de Rham complex and Riemannian geometry of some underlying moduli space ℳ and the partition function equals the Euler number χ(ℳ) of ℳ. We explicitly deal with moduli spaces of instantons and of flat connections in two and three dimensions. To motivate our constructions we explain the relation between the Mathai-Quillen formalism and supersymmetric quantum mechanics and introduce a new kind of supersymmetric quantum mechanics based on the Gauss-Codazzi equations. We interpret the gauge theory actions from the Atiyah-Jeffrey point of view and relate them to supersymmetric quantum mechanics on spaces of connections. As a consequence of these considerations we propose the Euler number χ(ℳ) of the moduli space of flat connections as a generalization to arbitrary three-manifolds of the Casson invariant. We also comment on the possibility of constructing a topological version of the Penner matrix model.

The heavy mass limit in field theory and the heavy quark effective theory

Abstract We present a derivation of a large mass effective action from the complete theory of elementary interactions, via a sequence of field redefinitions. In this way one derives straightforwardly the “matching conditions” of the static effective field theory. Gauge dependent properties of the effective theory are analysed, and relationships between the complete and effective theory are established. We give a general proof that there are no 1/ m corrections to the absolute normalization condition of current-induced flavour changing transitions.

BRST quantization of topological field theories

Abstract We consider in detail the construction of a variety of topological quantum field theories through BRST quantization. In particular, we show that supersymmetric quantum mechanics on an arbitrary riemannian manifold can be obtained as the BRST quantization of a purely bosonic theory. The introduction of a new local symmetry allows for the possibility of different gauge choices, and we show how this freedom can simplify the evaluation of the Witten index in certain cases. Topological sigma models are also constructed via the same mechanism. In three dimensions, we consider a Yang-Mills-Higgs model related to the four-dimensional TQFT of Witten.

Chern-Simons theory on S1-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 × S1, 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.