Aspects of NT ⩾ 2 topological gauge theories and D-branes (original) (raw)
Two-dimensional topological Yang-Mills theory
Physics Letters B, 1990
Two-dimensional euclidean (topological) quantum Yang-Mills theory on the compact manifold in the Lorentz gauge is analysed in the framework of the covariant path-integral approach. The Nicolai map for the partition function and for the Wilson loop observables is explicitly given. Topological quantum field theory (TQFT) is a fascinating and fashionable subject nowadays. Each "theory of nothing", i.e. possessing zero degrees of freedom from the "non-topological point of view" (particularly, a theory with a local symmetry), is a potential candidate for TQFT. Apparently, there are three categories of TQFTs: ( 1 ) TQFT with (very large) topological symmetry, e.g. topological Chern-Witten theory in four dimensions
Instanton calculus, topological field theories and N = 2 super Yang-Mills theories
Journal of High Energy Physics, 2000
The results obtained by Seiberg and Witten for the low-energy Wilsonian effective actions of N = 2 supersymmetric theories with gauge group SU(2) are in agreement with instanton computations carried out for winding numbers one and two. This suggests that the instanton saddle point saturates the non-perturbative contribution to the functional integral. A natural framework in which corrections to this approximation are absent is given by the topological field theory built out of the N = 2 Super Yang-Mills theory. After extending the standard construction of the Topological Yang-Mills theory to encompass the case of a non-vanishing vacuum expectation value for the scalar field, a BRST transformation is defined (as a supersymmetry plus a gauge variation), which on the instanton moduli space is the exterior derivative. The topological field theory approach makes the so-called "constrained instanton" configurations and the instanton measure arise in a natural way. As a consequence, instanton-dominated Green's functions in N = 2 Super Yang-Mills can be equivalently computed either using the constrained instanton method or making reference to the topological twisted version of the theory. We explicitly compute the instanton measure and the contribution to u = Trφ 2 for winding numbers one and two. We then show that each non-perturbative contribution to the N = 2 low-energy effective action can be written as the integral of a total derivative of a function of the instanton moduli. Only instanton configurations of zero conformal size contribute to this result. Finally, the 8k-dimensional instanton moduli space is built using the hyperkähler quotient procedure, which clarifies the geometrical meaning of our approach.
Twisted N = 2 supergravity as topological gravity in four dimensions
Nuclear Physics B, 1993
We show that the BRST quantum version of pure D=4 N=2 supergravity can be topologically twisted, to yield a formulation of topological gravity in four dimensions. The topological BRST complex is just a rearrangement of the old BRST complex, that partly modifies the role of physical and ghost fields: indeed, the new ghost number turns out to be the sum of the old ghost number plus the internal U(1) charge. Furthermore, the action of N=2 supergravity is retrieved from topological gravity by choosing a gauge fixing that reduces the space of physical states to the space of gravitational instanton configurations, namely to self-dual spin connections. The descent equations relating the topological observables are explicitly exhibited and discussed. Ours is a first step in a programme that aims at finding the topological sector of matter coupled N=2 supergravity, viewed as the effective Lagrangian of type II superstrings and, as such, already related to 2D topological field-theories. As it stands the theory we discuss may prove useful in describing gravitational instantons moduli-spaces. supergravity, whose general form has been obtained in , further generalizing the results of conformal tensor calculus , should be liable to a topological twist and have a topological sector. Although , the systematic programme of topologically twisting D=4, N=2 theories has not yet been carried through. In this paper we try to fill the gap beginning with pure N=2 supergravity. Before addressing some of the technical and conceptual details of our derivation, let us spend few words on motivations. They are essentially three: i) The construction and the analysis of a well founded four-dimensional topologically gravity may furnish a gravitational analogue of Donandson theory. In other words, it may provide a new tool to study intersection theory on the moduli space of gravitational instantons. ii) The topological interpretation should provide new calculational tools in N=2 supergravity. Finally, to our taste the most exciting, although still vague motivation is the third iii) The special Kaähler geometry of Calabi-Yau moduli-space is related, as we already recalled, to D=2 topological field-theories. On the other hand, it also follows from the requirement of N=2 supersymmetry in D=4. From the superstring point of view, this is understood in terms of the h-map , stating that on the same Calabi-Yau manifold we can compactify both the heterotic and the type II string. The latter has N=2 matter coupled supergravity as an effective lagrangian. Hence the topological interpretation of this theory should shed new light on the relation between topological field-theories in two and in four dimensions. Let us now outline the conceptual set up and the contents of our paper. Our purpose is to show that the topological twist of N=2 pure supergravity defines a gauge-fixed version of pure topological gravity where the gauge-fixing condition is ω -ab = 0, ω -ab denoting the antiselfdual part of the spin connection. To this effect we utilize the BRST-approach, having, as final goal, the comparison of the abstract gauge-theory a la
Lectures on topological quantum field theory
1998
In these lectures we present a general introduction to topological quantum field theories. These theories are discussed in the framework of the Mathai-Quillen formalism and in the context of twisted N = 2 supersymmetric theories. We discuss in detail the recent developments in Donaldson-Witten theory obtained from the application of results based on duality for N = 2 supersymmetric Yang-Mills theories. This involves a description of the computation of Donaldson invariants in terms of Seiberg-Witten invariants. Generalizations of Donaldson-Witten theory are reviewed, and the structure of the vacuum expectation values of their observables is analyzed in the context of duality for the simplest case.
Issues in topological gauge theory
Nuclear Physics B, 1998
We discuss topological theories, arising from the general N = 2 twisted gauge theories. We initiate a program of their study in the Gromov-Witten paradigm. We re-examine the low-energy effective abelian theory in the presence of sources and study the mixing between the various p-observables. We present the twisted superfield formalism which makes duality transformations transparent. We propose a scheme
Four-Dimensional Yang-Mills Theory as a Deformation of Topological BF Theory
Communications in Mathematical Physics, 1998
The classical action for pure Yang-Mills gauge theory can be formulated as a deformation of the topological BF theory where, beside the two-form field B, one has to add one extra-field η given by a one-form which transforms as the difference of two connections. The ensuing action functional gives a theory that is both classically and quantistically equivalent to the original Yang-Mills theory. In order to prove such an equivalence, it is shown that the dependency on the field η can be gauged away completely. This gives rise to a field theory that, for this reason, can be considered as semi-topological or topological in some but not all the fields of the theory. The symmetry group involved in this theory is an affine extension of the tangent gauge group acting on the tangent bundle of the space of connections. A mathematical
Topological Yang-Mills Theories
2003
Abstract: Using topological Yang-Mills theory as example, we discuss the definition and determination of observables in topological field theories (of Witten-type) within the superspace formulation proposed by Horne. This approach to the equivariant cohomology leads to a set of bi-descent equations involving the BRST and supersymmetry operators as well as the exterior derivative. This allows us to determine superspace expressions for all observables, and thereby to recover the Donaldson-Witten polynomials when choosing
Lectures in Topological Quantum Field Theory
1997
In these lectures we present a general introduction to topological quantum field theories. These theories are discussed in the framework of the Mathai-Quillen formalism and in the context of twisted N = 2 supersymmetric theories. We discuss in detail the recent developments in Donaldson-Witten theory obtained from the application of results based on duality for N = 2 supersymmetric Yang-Mills theories. This involves a description of the computation of Donaldson invariants in terms of Seiberg-Witten invariants. Generalizations of Donaldson-Witten theory are reviewed, and the structure of the vacuum expectation values of their observables is analyzed in the context of duality for the simplest case.
Duality In Topological Quantum Field Theories
Arxiv preprint hep-th/9907123, 1999
This thesis presents a thorough analysis of the links between the N = 4 supersymmetric gauge theory in four dimensions and its three topological twisted counterparts. Special emphasis is put in deriving explicit results in terms of the vacuum structure and low-energy effective ...
2000
The results obtained by Seiberg and Witten for the low–energy Wilsonian effective actions of N = 2 supersymmetric theories with gauge group SU(2) are in agreement with instanton computations carried out for winding numbers one and two. This suggests that the instanton saddle point saturates the non–perturbative contribution to the functional integral. A natural framework in which corrections to this approximation are absent is given by the topological field theory built out of the N = 2 Super Yang–Mills theory. After extending the standard construction of the Topological Yang–Mills theory to encompass the case of a non–vanishing vacuum expectation value for the scalar field, a BRST transformation is defined (as a supersymmetry plus a gauge variation), which on the instanton moduli space is the exterior derivative. The topological field theory approach makes the so–called “constrained instanton ” configurations and the instanton measure arise in a natural way. As a consequence, insta...