Topological Yang-Mills Theory Research Articles

Mathematical structures of cohomological field theories

A mathematical framework of cohomological field theories (CohFTs) is formulated in the language of bigraded manifolds. Algebraic properties of operators in CohFTs are studied. Methods of constructing CohFTs, with or without gauge symmetries, are discussed. In particular, a generalization of the Mathai-Quillen formalism is given. Examples such as topological quantum mechanics, topological sigma model, topological M-theory, and topological Yang-Mills theory can be obtained uniformly using this new formalism.

Evaluation of Euler number of complex Grassmann manifold G(k,N) via Mathai-Quillen formalism

In this paper, we provide a recipe for computing Euler number of Grassmann manifold G(k,N) by using Mathai-Quillen formalism (MQ formalism) [9] and Atiyah-Jeffrey construction [1]. Especially, we construct path-integral representation of Euler number of G(k,N). Our model corresponds to a finite dimensional toy-model of topological Yang-Mills theory which motivated Atiyah-Jeffrey construction. As a by-product, we construct free fermion realization of cohomology ring of G(k,N).

Semiclassical limit of topological R\xe9nyi entropy in 3d Chern-Simons theory

We study the multi-boundary entanglement structure of the state associated with the torus link complement S3\\Tp,q in the set-up of three-dimensional SU(2)k Chern-Simons theory. The focal point of this work is the asymptotic behavior of the Rényi entropies, including the entanglement entropy, in the semiclassical limit of k → ∞. We present a detailed analysis of several torus links and observe that the entropies converge to a finite value in the semiclassical limit. We further propose that the large k limiting value of the Rényi entropy of torus links of type Tp,pn is the sum of two parts: (i) the universal part which is independent of n, and (ii) the non-universal or the linking part which explicitly depends on the linking number n. Using the analytic techniques, we show that the universal part comprises of Riemann zeta functions and can be written in terms of the partition functions of two-dimensional topological Yang-Mills theory. More precisely, it is equal to the Rényi entropy of certain states prepared in topological 2d Yang-Mills theory with SU(2) gauge group. Further, the universal parts appearing in the large k limits of the entanglement entropy and the minimum Rényi entropy for torus links Tp,pn can be interpreted in terms of the volume of the moduli space of flat connections on certain Riemann surfaces. We also analyze the Rényi entropies of Tp,pn link in the double scaling limit of k → ∞ and n → ∞ and propose that the entropies converge in the double limit as well.

Infinitesimal Gribov copies in gauge-fixed topological Yang-Mills theories

We study the Gribov problem in four-dimensional topological Yang-Mills theories following the Baulieu-Singer approach in the (anti-)self-dual Landau gauges. This is a gauge-fixed approach that allows to recover the topological spectrum, as first constructed by Witten, by means of an equivariant (or constrained) BRST cohomology. As standard gauge-fixed Yang-Mills theories suffer from the gauge copy (Gribov) ambiguity, one might wonder if and how this has repercussions for this analysis. The resolution of the small (infinitesimal) gauge copies, in general, affects the dynamics of the underlying theory. In particular, treating the Gribov problem for the standard Landau gauge condition in non-topological Yang-Mills theories strongly affects the dynamics of the theory in the infrared. In the current paper, although the theory is investigated with the same gauge condition, the effects of the copies turn out to be completely different. In other words: in both cases, the copies are there, but the effects are very different. As suggested by the tree-level exactness of the topological model in this gauge choice, the Gribov copies are shown to be inoffensive at the quantum level. To be more precise, following Gribov, we discuss the path integral restriction to the Gribov horizon. The associated gap equation, which fixes the so-called Gribov parameter, is however shown to only possess a trivial solution, making the restriction obsolete. We relate this to the absence of radiative corrections in both gauge and ghost sectors. We give further evidence by employing the renormalization group which shows that, for this kind of topological model, the gap equation indeed forbids the introduction of a massive Gribov parameter.

More about the renormalization properties of topological Yang-Mills theories

Quantum properties of topological Yang-Mills theory in (anti-)self-dual Landau gauge were recently investigated by the authors. We extend the analysis of renormalizability for two generalized classes of gauges; each of them depending on one gauge parameter. The (anti-)self-dual Landau gauge is recovered at the vanishing of each gauge parameter. The theory shows itself to be renormalizable in these classes of gauges. Moreover, we discuss the ambiguity on the choice of the renormalization factors (which is not present in usual Yang-Mills theories) and argue a possible origin of such ambiguity.

Absence of radiative corrections in four-dimensional topological Yang-Mills theories

We prove that all connected n-point Green functions of four-dimensional topological Yang-Mills theories quantized in the (anti-)self-dual Landau gauges are tree-level exact, i.e. there are no radiative corrections in this gauge choice.

Topological Yang-Mills theories in self-dual and anti-self-dual Landau gauges revisited

We reconsider the renormalizability of topological Yang-Mills field theories in (anti-)self-dual Landau gauges. By employing algebraic renormalization techniques, we show that there is only one independent renormalization. Moreover, due to the rich set of Ward identities, we are able to obtain some important exact features of the (connected and one-particle irreducible) two-point functions. Specifically, we show that all two-point functions are tree-level exact.

Chains of topological oscillators with instantons and calculable topological observables in topological quantum mechanics

We extend to a possibly infinite chain the conformally invariant mechanical system that was introduced earlier as a toy model for understanding the topological Yang–Mills theory. It gives a topological quantum model that has interesting and computable zero modes and topological invariants. It confirms the recent conjecture by several authors that supersymmetric quantum mechanics may provide useful tools for understanding robotic mechanical systems (Vitelli et al.) and condensed matter properties (Kane et al.), where trajectories are allowed or not by the conservation of topological indices. The absences of ground state and mass gaps are special features of such systems.

Generalized zeta function representation of groups and 2-dimensional topological Yang-Mills theory: The example of GL(2, 𝔽q) and PGL(2, 𝔽q)

We recall the relation between zeta function representation of groups and two-dimensional topological Yang-Mills theory through Mednikh formula. We prove various generalisations of Mednikh formulas and define generalization of zeta function representations of groups. We compute some of these functions in the case of the finite group GL(2, 𝔽q) and PGL(2, 𝔽q). We recall the table characters of these groups for any q, compute the Frobenius-Schur indicator of their irreducible representations, and give the explicit structure of their fusion rings.

DIFFERENTIAL GEOMETRY FROM QUANTUM FIELD THEORY

We review the historical development and physical ideas of topological Yang–Mills theory and explain how quantum field theory, a physical framework describing subatomic physics, can be used as a tool to study differential geometry. We further emphasize that this phenomenon demonstrates that the inter-relation between theoretical physics and mathematics have come into a new stage.

On topological field theory representation of higher analogs of classical special functions

Looking for a quantum field theory model of Archimedean algebraic geometry a class of infinite-dimensional integral representations of classical special functions was introduced. Precisely the special functions such as Whittaker functions and Gamma-function were identified with correlation functions in topological field theories on a two-dimensional disk. Mirror symmetry of the underlying topological field theory leads to a dual finite-dimensional integral representations reproducing classical integral representations for the corresponding special functions. The mirror symmetry interchanging infinite- and finite-dimensional integral representations provides an incarnation of the local Archimedean Langlands duality on the level of classical special functions. In this note we provide some directions to higher-dimensional generalizations of our previous results. In the first part we consider topological field theory representations of multiple local L-factors introduced by Kurokawa and expressed through multiple Barnes's Gamma-functions. In the second part we are dealing with generalizations based on consideration of topological Yang-Mills theories on non-compact four-dimensional manifolds. Presumably, in analogy with the mirror duality in two-dimensions, S-dual description should be instrumental for deriving integral representations for a particular class of quantum field theory correlation functions and thus providing a new interesting class of special functions supplied with canonical integral representations.

Topological Chern-Simons sigma model

We consider topological twisting of recently constructed Chern-Simons-matter theories in three dimensions with N=4 or higher supersymmetry. We enumerate physically inequivalent twistings for each N, and find two different twistings for N=4, one for N=5,6, and four for N=8. We construct the two types of N=4 topological theories, which we call A/B-models, in full detail. The A-model has been recently studied by Kapustin and Saulina. The B-model is new and it consists solely of a Chern-Simons term of a complex gauge field up to BRST-exact terms. We also compare the new theories with topological Yang-Mills theories and find some interesting connections. In particular, the A-model seems to offer a new perspective on Casson invariant and its relation to Rozansky-Witten theory.

Cohomological gauge theory, quiver matrix models and Donaldson–Thomas theory

We study the relation between Donaldson–Thomas theory of Calabi–Yau threefolds and a six-dimensional topological Yang–Mills theory. Our main example is the topological U ( N ) gauge theory on flat space in its Coulomb branch. To evaluate its partition function we use equivariant localization techniques on its noncommutative deformation. As a result the gauge theory localizes on noncommutative instantons which can be classified in terms of N-coloured three-dimensional Young diagrams. We give to these noncommutative instantons a geometrical description in terms of certain stable framed coherent sheaves on projective space by using a higher-dimensional generalization of the ADHM formalism. From this formalism we construct a topological matrix quantum mechanics which computes an index of BPS states and provides an alternative approach to the six-dimensional gauge theory.

Instantons and Donaldson–Thomas invariants

AbstractWe review some recent progress in understanding the relation between a six dimensional topological Yang–Mills theory and the enumerative geometry of Calabi–Yau threefolds. The gauge theory localizes on generalized instanton solutions and is conjecturally equivalent to Donaldson–Thomas theory. We evaluate the partition function of the U(N) theory in its Coulomb branch on flat space by employing equivariant localization techniques on its noncommutative deformation. Geometrically this corresponds to a higher dimensional generalization of the ADHM formalism. This formalism can be extended to a generic toric Calabi–Yau.

Observables in topological Yang-Mills theorieswith extended shift supersymmetry

We present a complete classification, at the classical level, of the observables of topological Yang-Mills theories with an extended shift supersymmetry of N generators, in any space-time dimension. The observables are defined as the Yang-Mills BRST cohomology classes of shift supersymmetry invariants. These cohomology classes turn out to be solutions of an N-extension of Witten's equivariant cohomology. This work generalizes results known in the case of shift supersymmetry with a single generator.

The Kodama state for topological quantum field theory beyond instantons

Constructing a symplectic structure that preserves the ordinary symmetries and the topological invariance for topological Yang–Mills theory, it is shown that the Kodama (Chern–Simons) state traditionally associated with a topological phase of unbroken diffeomorphism invariance for instantons, exists actually for the complete topological sector of the theory. The case of gravity is briefly discussed.

OBSERVABLES IN TOPOLOGICAL YANG–MILLS THEORIES

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 find the most general solution to the problem of equivariant cohomology, and to recover the Donaldson–Witten polynomials when choosing a Wess–Zumino-type gauge.

Observables in topological theories: A superspace formulation

Observables of topological Yang-Mills theory were defined by Witten as the classes of an equivariant cohomology. We propose to define them alternatively as the BRST cohomology classes of a superspace version of the theory, where BRST invariance is associated to super Yang-Mills invariance. We provide and discuss the general solution of this cohomology.

Superfield algorithm for higher order gauge field theories

We propose an algorithm for the construction of higher order gauge field theories from a superfield formulation within the Batalin-Vilkovisky formalism. This is a generalization of the superfield algorithm recently considered by Batalin and Marnelius. This generalization seems to allow for non-topological gauge field theories as well as alternative representations of topological ones. A five dimensional non-abelian Chern-Simons theory and a topological Yang-Mills theory are treated as examples.

Generalized BRST models and topological Yang–Mills theories

We present a general framework for models admitting a decomposition of the type d=[δ,b], with b the BRST operator and δ a certain (even) derivation. We focus our attention on models whose fields can be described as components of two ladders W=c+A+⋯ and F=φ+ψ+⋯ and show how they relate to some aspects of topological Yang–Mills theory. We relate our construction to the standard mathematical ideas of Cartan’s G-operation and interpret W and F as pair of algebraic connection and curvature in a certain bigraded differential algebra.