Cohomological Yang–Mills theories on Kähler 3-folds

We study topological gauge theories with N_c=(2,0) supersymmetry based on stable bundles on general Kähler 3-folds. In order to have a theory that is well defined and well behaved, we consider a model based on an extension of the usual holomorphic bundle by including a holomorphic 3-form. The correlation functions of the model describe complex 3-dimensional generalizations of Donaldson–Witten type invariants. We show that the path integral can be written as a sum of contributions from stable bundles and a complex 3-dimensional version of Seiberg–Witten monopoles. We study certain deformations of the theory, which allow us to consider the situation of reducible connections. We shortly discuss situations of reduced holonomy. After dimensional reduction to a Kähler 2-fold, the theory reduces to Vafa–Witten theory. On a Calabi–Yau 3-fold, the supersymmetry is enhanced to N_c=(2,2). This model may be used to describe classical limits of certain compactifications of (matrix) string theory.     

Curve counting,instantons and McKay correspondences

We survey some features of equivariant instanton partition functions of topological gauge theories on four and six dimensional toric Kähler varieties, and their geometric and algebraic counterparts in the enumerative problem of counting holomorphic curves. We discuss the relations of instanton counting to representations of affine Lie algebras in the four-dimensional case, and to Donaldson–Thomas theory for ideal sheaves on Calabi–Yau threefolds. For resolutions of toric singularities, an algebraic structure induced by a quiver determines the instanton moduli space through the McKay correspondence and its generalizations. The correspondence elucidates the realization of gauge theory partition functions as quasi-modular forms, and reformulates the computation of noncommutative Donaldson–Thomas invariants in terms of the enumeration of generalized instantons. New results include a general presentation of the partition functions on ALE spaces as affine characters, a rigorous treatment of equivariant partition functions on Hirzebruch surfaces, and a putative connection between the special McKay correspondence and instanton counting on Hirzebruch–Jung spaces.     

Master Spaces and the Coupling Principle:¶From Geometric Invariant Theory to Gauge Theory

We introduce a general mathematical principle, with roots in Geometric Invariant Theory, which provides a unified way for understanding several celebrated results and conjectures like e. g. the Verlinde formula, the Vafa-Intriligator formula, and Witten's conjecture about the relation between Donaldson theory and Seiberg–Witten theory. This principle also suggests new results about Gromov invariants of moduli spaces of stable bundles over curves, and shows that gauge theoretical invariants associated with moduli spaces of PU(2)-monopoles are determined by Seiberg–Witten and Donaldson invariants. Received: 17 November 1998 / Accepted: 7 March 1999     

The background field method and the linearization problem for Poisson manifolds

The background field method (BFM) for the Poisson sigma model (PSM) is studied as an example of the application of the BFM technique to open gauge algebras. The relationship with Seiberg–Witten maps arising in non-commutative gauge theories is clarified. It is shown that the implementation of the BFM for the PSM in the Batalin–Vilkovisky formalism is equivalent to the solution of a generalized linearization problem (in the formal sense) for Poisson structures in the presence of gauge fields. Sufficient conditions for the existence of a solution and a constructive method to derive it are presented.     

On Twisted N = 2 5D Super Yang–Mills Theory

On a five-dimensional simply connected Sasaki–Einstein manifold, one can construct Yang–Mills theories coupled to matter with at least two supersymmetries. The partition function of these theories localises on the contact instantons, however, the contact instanton equations are not elliptic. It turns out that these equations can be embedded into the Haydys–Witten equations (which are elliptic) in the same way the 4D anti-self-dual instanton equations are embedded in the Vafa–Witten equations. We show that under some favourable circumstances, the latter equations will reduce to the former by proving some vanishing theorems. It was also known that the Haydys–Witten equations on product manifolds \({M_5 = M_4 \times \mathbb{R}}\) arise in the context of twisting the 5D maximally supersymmetric Yang–Mills theory. In this paper, we present the construction of twisted N = 2 Yang–Mills theory on Sasaki–Einstein manifolds, and more generally on K-contact manifolds. The localisation locus of this new theory thus provides a covariant version of the Haydys–Witten equation.     

Geometric singularities and spectra of Landau-Ginzburg models

Some mathematical and physical aspects of superconformal string compactification in weighted projective space are discussed. In particular, we recast the path integral argument establishing the connection between Landau-Ginzburg conformal theories and Calabi-Yau string compactification in a geometric framework. We then prove that the naive expression for the vanishing of the first Chern class for a complete intersection (adopted from the smooth case) is sufficient to ensure that the resulting variety, which is generically singular, can be resolved to a smooth Calabi-Yau space. This justifies much analysis which has recently been expended on the study of Landau-Ginzburg models. Furthermore, we derive some simple formulae for the determination of the Witten index in these theories which are complimentary to those derived using semiclassical reasoning by Vafa. Finally, we also comment on the possible geometrical significance ofunorbifolded Landau-Ginzburg theories.     

Monopole and dyon bound states in N = 2 supersymmetric Yang-Mills theories

We study the existence of monopole bound states saturating the BPS bound in N = 2 supersymmetric Yang-Mills theories. We describe how the existence of such bound states relates to the topology of index bundles over the moduli space of BPS solutions. Using an L² index theorem, we prove the existence of certain BPS states predicted by Seiberg and Witten based on their study of the vacuum structure of N = 2 Yang-Mills theories.     

Conformally invariant gauge fixed actions for 2-D topological gravity

We show that invariants of Mumford for moduli spaces of curves are obtainable from a gauge fixed action of a topological quantum field theory in two dimensions. The method is completely analogous to the relation of Donaldson invariants with the topological quantum field theory for gauge theories in four dimensions.Supported by D.O.E. Grant DE-FG02-88ER 25066     

TFT construction of RCFT correlators I: partition functions

We formulate rational conformal field theory in terms of a symmetric special Frobenius algebra A and its representations. A is an algebra in the modular tensor category of Moore–Seiberg data of the underlying chiral CFT. The multiplication on A corresponds to the OPE of boundary fields for a single boundary condition. General boundary conditions are A-modules, and (generalised) defect lines are A–A-bimodules.The relation with three-dimensional TFT is used to express CFT data, like structure constants or torus and annulus coefficients, as invariants of links in three-manifolds. We compute explicitly the ordinary and twisted partition functions on the torus and the annulus partition functions. We prove that they satisfy consistency conditions, like modular invariance and NIM-rep properties.We suggest that our results can be interpreted in terms of non-commutative geometry over the modular tensor category of Moore–Seiberg data.     

The volume conjecture and topological strings

In this paper, we discuss a relation between Jones‐Witten theory of knot invariants and topological open string theory on the basis of the volume conjecture. We find a similar Hamiltonian structure for both theories, and interpret the AJ conjecture as the 𝒟‐module structure for a D‐brane partition function. In order to verify our claim, we compute the free energy for the annulus contributions in the topological string using the Chern‐Simons matrix model, and find that it coincides with the Reidemeister torsion in the case of the figure‐eight knot complement and the SnapPea census manifold m009.     

Non-Commutative Periods and Mirror Symmetry¶in Higher Dimensions

We study an analog for higher-dimensional Calabi–Yau manifolds of the standard predictions of Mirror Symmetry. We introduce periods associated with “non-commutative” deformations of Calabi–Yau manifolds. These periods define a map on the moduli space of such deformations which is a local isomorphism. Using these non-commutative periods we introduce invariants of variations of semi-infinite generalized Hodge structures living over the moduli space ℳ. It is shown that the generating function of such invariants satisfies the system of WDVV-equations exactly as in the case of Gromov–Witten invariants. We prove that the total collection of rational Gromov–Witten invariants of complete intersection Calabi–Yau manifold can be identified with the collection of invariants of variations of generalized (semi-infinite) Hodge structures attached to the mirror variety. The basic technical tool utilized is the deformation theory. Received: 6 April 2000 / Accepted: 15 January 2002     

A Geometric Approach to Boundaries and Surface Defects in Dijkgraaf–Witten Theories

Dijkgraaf–Witten theories are extended three-dimensional topological field theories of Turaev–Viro type. They can be constructed geometrically from categories of bundles via linearization. Boundaries and surface defects or interfaces in quantum field theories are of interest in various applications and provide structural insight. We perform a geometric study of boundary conditions and surface defects in Dijkgraaf–Witten theories. A crucial tool is the linearization of categories of relative bundles. We present the categories of generalized Wilson lines produced by such a linearization procedure. We establish that they agree with the Wilson line categories that are predicted by the general formalism for boundary conditions and surface defects in three-dimensional topological field theories that has been developed in Fuchs et al. (Commun Math Phys 321:543–575, 2013)     

From Zwiebach Invariants to Getzler Relation

We introduce the notion of Zwiebach invariants that generalize Gromov- Witten invariants and homotopical algebra structures. We outline the induction procedure that induces the structure of Zwiebach invariants on the sub-bicomplex, that gives the structure of Gromov-Witten invariants on sub-bicomplex with zero differentials. We propose to treat Hodge dGBV with 1/12 axiom as the simplest set of Zwiebach invariants, and explicitly prove that it induces WDVV and Getzler equations in genera 0 and 1 respectively.     

Holonomy groups,complex structures andD=4 topological Yang-Mills theory

We study conditions for the existence of extended supersymmetry in topological Yang-Mills theory. These conditions are most conveniently formulated in terms of the holonomy group of the underlying manifold, on which the topological Yang-Mills theory is defined. For irreducible manifolds we find that extended supersymmetries are in 1–1 correspondence with covariantly constant complex structures. Therefore, the topological Yang-Mills theory on any Kähler manifold possesses one additional supersymmetry and on any hyper Kähler manifold there are three additional supersymmetries. The Donaldson map, which plays a crucial role in the construction of the topological invariants, is generalized for Kähler manifolds, thus providing candidates for new invariants of complex manifolds.     

A bigraded version of the Weil algebra and of the Weil homomorphism for Donaldson invariants. Elementary algebra and cohomology behind the Baulieu-Singer approach to Witten's topological Yang-Mills quantum field theory

We describe a bigraded generalization of the Weil algebra, of its basis and of the characteristic homomorphism which besides ordinary characteristic classes also maps on cohomology classes leading to Donaldson invariants in the appropriate context. Furthermore these cohomology classes exhaust the image of the generalized characteristic homomorphisms.     

The quantum orbifold cohomology of toric stack bundles

We study Givental’s Lagrangian cone for the quantum orbifold cohomology of toric stack bundles. Using Gromov–Witten invariants of the base and combinatorics of the toric stack fibers, we construct an explicit slice of the Lagrangian cone defined by the genus 0 Gromov–Witten theory of a toric stack bundle.