Orbifold conformal blocks and the stack of pointed G-covers

Starting with a vertex algebra V, a finite group G of automorphisms of V, and a suitable collection of twisted V-modules, we construct (twisted) D-modules on the stack of pointed G-covers, introduced by Jarvis, Kaufmann, and Kimura. The fibers of these sheaves are spaces of orbifold conformal blocks defined in joint work with Edward Frenkel. The key ingredient is a G-equivariant version of the Virasoro uniformization theorem.     

Momentum lattice for CHL string

We propose some analogue of the Narain lattice for CHL string. The symmetries of this lattice are the symmetries of the perturbative spectrum. We explain in this language the known results about the possible gauge groups in compactified theory. For the four-dimensional theory, we explicitly describe the action of S-duality on the background fields. We show that the moduli spaces of the six-, seven- and eight-dimensional compactifications coincide with the moduli spaces of the conjectured type IIA, M-theory and F-theory duals. We classify the rational components of the boundary of the moduli space in seven, eight and nine dimensions.     

Non-trivial extension of the (1+2)-Poincaré algebra and conformal invariance on the boundary of {\mathrm{AdS}}_3

Using recent results on strings on AdS$_3\times N^d$, where N is a d dimensional compact manifold, we re-examine the derivation of the non-trivial extension of the (1+2)-dimensional-Poincaré algebra obtained by Rausch de Traubenberg and Slupinsky. We show by explicit computation that this new extension is a special kind of fractional supersymmetric algebra which may be derived from the deformation of the conformal structure living on the boundary of AdS. The two so(1,2) Lorentz modules of spin used in building of the generalization of the (1+2) Poincaré algebra are re-interpreted in our analysis as highest weight representations of the left and right Virasoro symmetries on the boundary of AdS. We also complete known results on 2d-fractional supersymmetry by using spectral flow of affine Kac–Moody and superconformal symmetries. Finally we make preliminary comments on the trick of introducing Fth roots of g-modules to generalize the so(1,2) result to higher rank Lie algebras g. Received: 20 July 2000 / Published online: 19 September 2001     

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.     

Quaternionic monopoles

We present the simplest non-abelian version of Seiberg-Witten theory: Quaternionic monopoles. These monopoles are associated withSpin ^h (4)-structures on 4-manifolds and form finite-dimensional moduli spaces. On a Kähler surface the quaternionic monopole equations decouple and lead to the projective vortex equation for holomorphic pairs. This vortex equation comes from a moment map and gives rise to a new complex-geometric stability concept. The moduli spaces of quaternionic monopoles on Kähler surfaces have two closed subspaces, both naturally isomorphic with moduli spaces of canonically stable holomorphic pairs. These components intersect along a Donaldson instanton space and can be compactified with Seiberg-Witten moduli spaces. This should provide a link between the two corresponding theories.Partially supported by: AGE-Algebraic Geometry in Europe, contract No ERBCHRXCT940557 (BBW 93.0187), and by SNF, nr. 21-36111.92     

Automorphism Group of k((t)):¶Applications to the Bosonic String

This paper is concerned with the formulation of a non-pertubative theory of the bosonic string. We introduce a formal group G which we propose as the “universal moduli space” for such a formulation. This is motivated because G establishes a natural link between representations of the Virasoro algebra and the moduli space of curves. Received: 26 March 1999 / Accepted: 10 September 2000     

Quantum Group as Semi-infinite Cohomology

We obtain the quantum group SL _q (2) as semi-infinite cohomology of the Virasoro algebra with values in a tensor product of two braided vertex operator algebras with complementary central charges c+[`(c)]=26{c+\bar{c}=26}. Each braided VOA is constructed from the free Fock space realization of the Virasoro algebra with an additional q-deformed harmonic oscillator degree of freedom. The braided VOA structure arises from the theory of local systems over configuration spaces and it yields an associative algebra structure on the cohomology. We explicitly provide the four cohomology classes that serve as the generators of SL _q (2) and verify their relations. We also discuss the possible extensions of our construction and its connection to the Liouville model and minimal string theory.     

Chiral de Rham Complex

We define natural sheaves of vertex algebras over smooth manifolds which may be regarded as semi-infinite de Rham complexes of certain D-modules over the loop spaces. For Calabi–Yau manifolds they admit N=2 supersymmetry. Connection with Wakimoto modules is discussed. Received: 29 April 1998 / Accepted: 7 February 1999     

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.     

Letter: Models of Relativistic Particle with Curvature and Torsion Revisited

Models, describing relativistic particles, where Lagrangian densities depend linearly on both the curvature and the torsion of the trajectories, are revisited in D=3 space forms. The moduli spaces of trajectories are completely and explicitly determined using the Lancret program. The moduli subspaces of closed solitons in the three sphere are also determined.     

Poincaré Polynomial of Moduli Spaces of Framed Sheaves on (Stacky) Hirzebruch Surfaces

We perform a study of the moduli space of framed torsion-free sheaves on Hirzebruch surfaces by using localization techniques. We discuss some general properties of this moduli space by studying it in the framework of Huybrechts-Lehn theory of framed modules. We classify the fixed points under a toric action on the moduli space, and use this to compute the Poincaré polynomial of the latter. This will imply that the moduli spaces we are considering are irreducible. We also consider fractional first Chern classes, which means that we are extending our computation to a stacky deformation of a Hirzebruch surface. From the physical viewpoint, our results provide the mathematical framework for the counting of D4-D2-D0 branes bound states on total spaces of the bundles O_{\mathbb P¹}(-p){\mathcal {O}_{\mathbb {P}^1}(-p)} .     

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     

Open-Closed Moduli Spaces and Related Algebraic Structures

We set up a Batalin–Vilkovisky Quantum Master Equation (QME) for open-closed string theory and show that the corresponding moduli spaces give rise to a solution, a generating function for their fundamental chains. The equation encodes the topological structure of the compactification of the moduli space of bordered Riemann surfaces. The moduli spaces of bordered J-holomorphic curves are expected to satisfy the same equation, and from this viewpoint, our paper treats the case of the target space equal to a point. We also introduce the notion of a symmetric Open-Closed Topological Conformal Field Theory (OC TCFT) and study the L _∞ and A _∞ algebraic structures associated to it.     

Topological Recursions of Eynard–Orantin Type for Intersection Numbers on Moduli Spaces of Curves

We prove that the Virasoro constraints satisfied by the higher Weil–Petersson volumes of moduli spaces of curves are equivalent to Eynard–Orantin topological recursions for some spectral curve. This provides a geometric proof of a result originally derived using a matrix model by Eynard.     

The topology of homophase misorientation spaces

Homophase misorientation spaces are investigated with a focus on the effect of symmetry operations on their topology and their minimum embedding dimensions in Euclidean space. Whereas the topology of rotation space is well established and requires a minimum of five variables for a one-to-one and continuous mapping, the spaces of orientations and misorientations are quotient spaces of the rotation space and are obtained by applying various equivalence relations. The equivalence relations for orientation spaces only involve the rotational symmetries of the underlying crystals. These spaces are classified under the three-dimensional manifolds called the spherical 3-manifolds, which have a non-trivial fundamental group, are not simply connected spaces, and do not embed in three-dimensional Euclidean space. In the case of homophase misorientation spaces, however, in addition to rotational symmetry operations there is a further ‘grain exchange symmetry’, which is shown to simplify the topology considerably. In some important cases this symmetry also reduces the number of Euclidean dimensions required to embed these misorientation spaces. The homophase misorientation spaces for the dihedral point groups D ₂(222), D ₄(422) and D ₆(622), the tetrahedral point group T(23), and the octahedral group O(432) are all found to be embeddable in only three dimensions, two dimensions less than required for rotations. Hence, these misorientation systems can be represented using three variables in a one-to-one and continuous manner.     

Geometry of Virasoro constraints in nonperturbative 2-d quantum gravity

The string equation and the Virasoro constraints for arbitrary hermitian multimatrix models are derived using the Lie-Bäcklund symmetries of the generalised KdV equations. From this point of view the origin of the string equation and the meaning of the Virasoro constraints are explained. Some speculation about the appearance of extra constraints, which we suspect to be theW-constraints, is also given.     

On the Uniqueness of Diffeomorphism Symmetry in Conformal Field Theory

A Möbius covariant net of von Neumann algebras on S¹ is diffeomorphism covariant if its Möbius symmetry extends to diffeomorphism symmetry. We prove that in case the net is either a Virasoro net or any at least 4-regular net such an extension is unique: the local algebras together with the Möbius symmetry (equivalently: the local algebras together with the vacuum vector) completely determine it. We draw the two following conclusions for such theories. (1) The value of the central charge c is an invariant and hence the Virasoro nets for different values of c are not isomorphic as Möbius covariant nets. (2) A vacuum preserving internal symmetry always commutes with the diffeomorphism symmetries. We further use our result to give a large class of new examples of nets (even strongly additive ones), which are not diffeomorphism covariant; i.e. which do not admit an extension of the symmetry to Diff⁺(S¹).Supported in part by the Italian MIUR and GNAMPA-INDAM.