Determinant bundles and Virasoro algebras

We consider the interplay of infinite-dimensional Lie algebras of Virasoro type and moduli spaces of curves, suggested by string theory. We will see that the infinitesimal geometry of determinant bundles is governed by Virasoro symmetries. The Mumford forms are just invariants of these symmetries. The representations of Virasoro algebra define (twisted)D-modules on moduli spaces; theseD-modules are equations on correlators in conformal field theory.To the memory of Vadik Knizhnik (20. 2. 1962–25. 12. 1987)     

Metrics on the moduli spaces of instantons over Euclidean 4-space

We prove that the natural hyper-Kähler metrics on the moduli space of chargek instantons over Euclidean four-space and on the space of ADHM matrices coincide. We use this to deduce formulae relating expressions in the curvature of a connection to invariant polynomials in the ADHM matrices corresponding to this connection. These arise from consideration of the group of symmetries acting on the moduli spaces.     

Instanton strings and hyper-Kähler geometry

We discuss two-dimensional sigma models on moduli spaces of instantons on K3 surfaces. These N = (4, 4) superconformal field theories describe the near-horizon dynamics of the D1-D5-brane system and are dual to string theory on AdS³. We derive a precise map relating the moduli of the K3 type 1113 string compactification to the moduli of these conformal field theories and the corresponding classical hyper-Kahler geometry. We conclude that in the absence of background gauge fields, the metric on the instanton moduli spaces degenerates exactly to the orbifold symmetric product of K3. Turning on a self-dual NS B-field deforms this symmetric product to a manifold that is diffeomorphic to the Hilbert scheme. We also comment on the mathematical applications of string duality to the global issues of deformations of hyper-Kähler manifolds.     

Observations on the moduli space of superconformal field theories

Some aspects of the moduli space of superconformal field theories are discussed. It is helpful to consider the conformal field theory as a background for propagation of strings and to exploit the space-time interpretation. Using this point of view we show that the metric on the moduli space of N = 4 superconformal field theory with c = 6 is locally that of O(20,4)/O(20) × O(4). We also discover some properties of the moduli space of N = 2 superconformal field theories with c = 9. Particular examples of these conformal field theories are sigma models on four- and six-dimensional Calabi-Yau spaces. Therefore, we can use this technique to learn about the moduli space of these spaces. For c = 6 we recover the known moduli space of K₃. Our analysis of the c = 9 system leads to a new coupling in four dimensional supergravity. As a by-product, we prove that gauge couplings cannot depend on the moduli of N = 1 space-time supersymmetric compactifications.     

Moduli Spaces of Instantons on the Taub-NUT Space

We present ADHM-Nahm data for instantons on the Taub-NUT space and encode these data in terms of Bow Diagrams. We study the moduli spaces of the instantons and present these spaces as finite hyperkähler quotients. As an example, we find an explicit expression for the metric on the moduli space of one SU(2) instanton.We motivate our construction by identifying a corresponding string theory brane configuration. By following string theory dualities we are led to supersymmetric gauge theories with impurities.     

Lattice Wess-Zumino-Witten model and quantum groups

Quantum groups play the role of symmetries of integrable theories in two dimensions. They may be detected on the classical level as Poisson-Lie symmetries of the corresponding phase spaces. We discuss specifically the Wess-Zumino-Witten conformally invariant quantum field model combining two chiral parts which describe the left- and right-moving degrees of freedom. On one hand, the quantum group plays the role of the symmetry of the chiral components of the theory. On the other hand, the model admits a lattice regularization (in Minkowski space) in which the current algebra symmetry of the theory also becomes quantum, providing the simplest example of a quantum group symmetry coupling space-time and internal degrees of freedom. We develop a free field approach to the representation theory of the lattice sl (2)-based current algebra and show how to use it to rigorously construct an exact solution of the quantum SL (2) WZW model on lattice.     

Grassmannian and String Theory

The infinite-dimensional Grassmannian manifold contains moduli spaces of Riemann surfaces of all genera. This well known fact leads to a conjecture that non-perturbative string theory can be formulated in terms of the Grassmannian. We present new facts supporting this hypothesis. In particular, it is shown that Grassmannians can be considered as generalized moduli spaces; this statement permits us to define corresponding “string amplitudes” (at least formally). One can conjecture that it is possible to explain the relation between non-perturbative and perturbative string theory by means of localization theorems for equivariant cohomology; this conjecture is based on the characterization of moduli spaces, relevant to string theory, as sets consisting of points with large stabilizers in certain groups acting on the Grassmannian. We describe an involution on the Grassmannian that could be related to S-duality in string theory. Received: 19 December 1996 / Accepted: 27 March 1998     

Bundle gerbes and moduli spaces

In this paper, we construct the index bundle gerbe of a family of self-adjoint Dirac-type operators, refining a construction of Segal. In a special case, we construct a geometric bundle gerbe called the caloron bundle gerbe, which comes with a natural connection and curving, and show that it is isomorphic to the analytically constructed index bundle gerbe. We apply these constructions to certain moduli spaces associated to compact Riemann surfaces, constructing on these moduli spaces, natural bundle gerbes with connection and curving, whose 3-curvature represent Dixmier-Douady classes that are generators of the third de Rham cohomology groups of these moduli spaces.     

Symmetries and couplings in heterotic superconformal field theories

We discuss the symmetries of the superpotential in comfactified heterotic superstring theories formed from the product of minimal N = 2 superconformal field theories. It is shown how these symmetries can ensure flatness of the potential involving the moduli, and we derive new results (going beyond those obtained by superconformal techniques alone) for the flatness of the potential involving other massless fields.     

Dynamical stabilization of runaway potentials at finite density

We study four-dimensional non-Abelian gauge theories with classical moduli. Introducing a chemical potential for a flavor charge causes moduli to become unstable and start condensing. We show that the moduli condensation in the presence of a chemical potential generates non-Abelian field strength condensates. These condensates are homogeneous but non-isotropic. The end point of the condensation process is a stable homogeneous, but non-isotropic, vacuum in which both gauge and flavor symmetries and the rotational invariance are spontaneously broken. Possible applications of this phenomenon for the gauge theory/string theory correspondence and in cosmology are briefly discussed.     

A note on Wilson-?t Hooft operators

We find explicitly the geometry of certain moduli spaces of BPS configurations in the presence of ?t Hooft operators. This is required to compute the Operator Product Expansion of Wilson-?t Hooft operators in N=4 super-Yang-Mills theory with gauge group G=PSU(3). We verify our results by reproducing the OPE of ?t Hooft operators predicted by S-duality.     

New moduli spaces from string background independence consistency conditions

In string field theory an infinitesimal background deformation is implemented as a canonical transformation whose hamiltonian function is defined by moduli spaces of punctured Riemann surfaces having one special puncture. We show that the consistency conditions associated to the commutator of two deformations are implemented by virtue of the existence of moduli spaces of punctured surfaces with two special punctures. The spaces are antisymmetric under the exchange of the special punctures, and satisfy recursion relations relating them to moduli spaces with one special puncture and to string vertices. We develop the theory of moduli spaces of surfaces with arbitrary number of special punctures and indicate their relevance to the construction of a string field theory that makes no reference to a conformal background. Our results also imply a partial antibracket cohomology theorem for the string action.     

Gauge Theory on Infinite Connected Sum and Mean Dimension

We study the geometry of infinite dimensional moduli spaces in the Yang-Mills gauge theory over infinite connected sum spaces. We develop the technique of gluing infinitely many instantons, and apply it to the evaluation of the mean dimension of the moduli spaces.     

Towards the Proof of AGT Relations with the Help of the Generalized Jack Polynomials

Original proofs of the AGT relations with the help of the Hubbard–Stratanovich duality of the modified Dotsenko–Fateev matrix model did not work for β ≠ 1, because Nekrasov functions were not properly reproduced by Selberg–Kadell integrals of Jack polynomials. We demonstrate that if the generalized Jack polynomials, depending on the N-ples of Young diagrams from the very beginning, are used instead of the N-linear combinations of ordinary Jacks, this resolves the problem. Such polynomials naturally arise as special elements in the equivariant cohomologies of the GL(N)-instanton moduli spaces, and this also establishes connection to alternative ABBFLT approach to the AGT relations, studying the action of chiral algebras on the instanton moduli spaces. In this paper, we describe a complete proof of AGT in the simple case of GL(2) (N = 2) Yang–Mills theory, i.e., the 4-point spherical conformal block of the Virasoro algebra.     

Moduli of ADHM sheaves and the local Donaldson-Thomas theory

The ADHM construction establishes a one-to-one correspondence between framed torsion free sheaves on the projective plane and stable framed representations of a quiver with relations in the category of complex vector spaces. This paper studies the geometry of moduli spaces of representations of the same quiver with relations in the abelian category of coherent sheaves on a smooth complex projective curve X. In particular it is proven that this moduli space is virtually smooth and related by relative Beilinson spectral sequence to the curve counting construction via stable pairs of Pandharipande and Thomas. This yields a new conjectural construction for the local Donaldson-Thomas theory of curves as well as a natural higher rank generalization.     

From the Dirac operator to Wess-Zumino models on spatial lattices

We investigate two-dimensional Wess-Zumino models in the continuum and on spatial lattices in detail. We show that a non-antisymmetric lattice derivative not only excludes chiral fermions but in addition introduces supersymmetry breaking lattice artifacts. We study the non-local and antisymmetric SLAC derivative which allows for chiral fermions without doublers and minimizes those artifacts. The supercharges of the lattice Wess-Zumino models are obtained by dimensional reduction of Dirac operators in high-dimensional spaces. The normalizable zero modes of the models with N=1 and N=2 supersymmetry are counted and constructed in the weak- and strong-coupling limits. Together with known methods from operator theory this gives us complete control of the zero mode sector of these theories for arbitrary coupling.     

Nonperturbative model of Liouville gravity

We formulate nonperturbative 2D gravity in the framework of Liouville theory. In particular, we express the specific heat of pure gravity in terms of an expansion of integrals on moduli spaces of punctured Riemann spheres. We recognize the relevant divisors on moduli spaces and write the integrands in terms of the Liouville action. We evaluate the integrals (rational intersections) and show that satisfies the Painlevé I.     

Notes on generalized global symmetries in QFT

It was recently argued that quantum field theories possess one‐form and higher‐form symmetries, labelled ‘generalized global symmetries.’ In this paper, we describe how those higher‐form symmetries can be understood mathematically as special cases of more general 2‐groups and higher groups, and discuss examples of quantum field theories admitting actions of more general higher groups than merely one‐form and higher‐form symmetries. We discuss analogues of topological defects for some of these higher symmetry groups, relating some of them to ordinary topological defects. We also discuss topological defects in cases in which the moduli ‘space’ (technically, a stack) admits an action of a higher symmetry group. Finally, we outline a proposal for how certain anomalies might potentially be understood as describing a transmutation of an ordinary group symmetry of the classical theory into a 2‐group or higher group symmetry of the quantum theory, which we link to WZW models and bosonization.     

Conformal correlation functions,Frobenius algebras and triangulations

We formulate two-dimensional rational conformal field theory as a natural generalization of two-dimensional lattice topological field theory. To this end we lift various structures from complex vector spaces to modular tensor categories. The central ingredient is a special Frobenius algebra object A in the modular category that encodes the Moore–Seiberg data of the underlying chiral CFT. Just like for lattice TFTs, this algebra is itself not an observable quantity. Rather, Morita equivalent algebras give rise to equivalent theories. Morita equivalence also allows for a simple understanding of T-duality.We present a construction of correlators, based on a triangulation of the world sheet, that generalizes the one in lattice TFTs. These correlators are modular invariant and satisfy factorization rules. The construction works for arbitrary orientable world sheets, in particular, for surfaces with boundary. Boundary conditions correspond to representations of the algebra A. The partition functions on the torus and on the annulus provide modular invariants and NIM-reps of the fusion rules, respectively.