Tensor Fields of Mixed Young Symmetry Type¶and N-Complexes

We construct N-complexes of non-completely antisymmetric irreducible tensor fields on ℝ ^D which generalize the usual complex (N=2) of differential forms. Although, for N≥ 3, the generalized cohomology of these N-complexes is nontrivial, we prove a generalization of the Poincaré lemma. To that end we use a technique reminiscent of the Green ansatz for parastatistics. Several results which appeared in various contexts are shown to be particular cases of this generalized Poincaré lemma. We furthermore identify the nontrivial part of the generalized cohomology. Many of the results presented here were announced in [10]. Received: 25 October 2001 / Accepted: 13 November 2001     

String center of mass operator and its effect on BRST cohomology

We consider the theory of bosonic closed strings on the flat background ℝ^25,1. We show how the BRST complex can be extended to a complex where the string center of mass operator,x ₀ ^μ is well defined. We investigate the cohomology of the extended complex. We demonstrate that this cohomology has a number of interesting features. Unlike in the standard BRST cohomology, there is no doubling of physical states in the extended complex. The cohomology of the extended complex is more physical in a number of aspects related to the zero-momentum states. In particular, we show that the ghost number one zero-momentum cohomology states are in one to one correspondence with the generators of the global symmetries of the backgroundi.e., the Poincaré algebra. Supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under cooperative agreement #DF-FC02-94ER40818     

PV Cohomology of the Pinwheel Tilings, their Integer Group of Coinvariants and Gap-Labeling

In this paper, we first remind how we can see the “hull” of the pinwheel tiling as an inverse limit of simplicial complexes (Anderson and Putnam in Ergod Th Dynam Sys 18:509–537, 1998) and we then adapt the PV cohomology introduced in Savinien and Bellissard (Ergod Th Dynam Sys 29:997–1031, 2009) to define it for pinwheel tilings. We then prove that this cohomology is isomorphic to the integer Čech cohomology of the quotient of the hull by S ¹ which let us prove that the top integer Čech cohomology of the hull is in fact the integer group of coinvariants of the canonical transversal Ξ of the hull. The gap-labeling for pinwheel tilings is then proved and we end this article by an explicit computation of this gap-labeling, showing that m^t ( C(X,\mathbb Z) ) = \frac1264\mathbb Z [ \frac15]{\mu^t \left( C(\Xi,\mathbb {Z}) \right) = \frac{1}{264}\mathbb {Z} \left [ \frac{1}{5}\right ]}.     

The Cohomology Algebra of the Semi-Infinite Weil Complex

In 1993, Lian-Zuckerman constructed two cohomology operations on the BRST complex of a conformal vertex algebra with central charge 26. They gave explicit generators and relations for the cohomology algebra equipped with these operations in the case of the c = 1 model. In this paper, we describe another such example, namely, the semi-infinite Weil complex of the Virasoro algebra. The semi-infinite Weil complex of a tame -graded Lie algebra was defined in 1991 by Feigin-Frenkel, and they computed the linear structure of its cohomology in the case of the Virasoro algebra. We build on this result by giving an explicit generator for each non-zero cohomology class, and describing all algebraic relations in the sense of Lian-Zuckerman, among these generators.     

Stokes Matrices and Monodromy of the Quantum Cohomology of Projective Spaces

In this paper we compute Stokes matrices and monodromy of the quantum cohomology of projective spaces. This problem can be formulated in a “classical” framework, as the problem of computation of Stokes matrices and monodromy of differential equations with regular and irregular singularities. We prove that the Stokes' matrix of the quantum cohomology coincides with the Gram matrix in the theory of derived categories of coherent sheaves. We also study the monodromy group of the quantum cohomology and we show that it is related to hyperbolic triangular groups. Received: 24 October 1998 / Accepted: 27 April 1999     

BRST cohomology for certain reducible topological symmetries

In this paper, two different definitions of the BRST complex are connected. We obtain the BRST complex of topological quantum field theories (leading to equivariant cohomology) from the standard definition of the classical BRST complex (leading to Lie algebra cohomology) provided that we include ghosts for ghosts. Hereby, we use a finite dimensional model with a semi-direct product action ofH DiffM on a configuration spaceM, whereH is a compact Lie group representing the gauge symmetry in this model.     

Differentials on graph complexes III: hairy graphs and deleting a vertex

We continue studying the cohomology of the hairy graph complexes which compute the rational homotopy of embedding spaces, generalizing the Vassiliev invariants of knot theory, after the second part in this series. In that part we have proven that the hairy graph complex \(\mathrm {HGC}_{m,n}\) with the extra differential is almost acyclic for even m. In this paper, we give the expected same result for odd m. As in the previous part, our results yield a way to construct many hairy graph cohomology classes by the waterfall mechanism also for odd m. However, the techniques are quite different. The main tool used in this paper is a new differential, deleting a vertex in non-hairy Kontsevich’s graphs, and a similar map for hairy vertices. We hope that the new differential can have further applications in the study of Kontsevich’s graph cohomology. Namely it is conjectured that the Kontsevich’s graph complex with deleting a vertex as an extra differential is acyclic.

     

The Extended Moduli Space of¶Special Lagrangian Submanifolds

It is well known that the moduli space of all deformations of a compact special Lagrangian submanifold X in a Calabi–Yau manifold Y within the class of special Lagrangian submanifolds is isomorphic to the first de Rham cohomology group of X. Reinterpreting the embedding data X⊂Y within the mathematical framework of the Batalin–Vilkovisky quantization, we find a natural deformation problem which extends the above moduli space to the full de Rham cohomology group of X. Received: 29 June 1998 / Accepted: 7 June 1999     

Inverse backscattering in two dimensions

We interpretN=2 superconformal field theories (SCFTs) formulated by Kazama and Suzuki via Goddard-Kent-Olive (GKO) construction from a viewpoint of the Lie algebra cohomology theory for the affine Lie algebra. We determine the cohomology group completely in terms of a certain subset of the affine Weyl group. We find that this subset describing the cohomology group can be obtained from its classical counterpart by the action of the Dynkin diagram automorphisms. Some algebra automorphisms of theN=2 superconformal algebra are also formulated. Utilizing the algebra automorphisms, we study the field identification problem for the branching coefficient modules in the GKO-construction. Also the structure of the Poincaré polynomial defined for eachN=2 theory is revealed.Dedicated to Professor Noboru Tanaka on his sixtieth birthday     

The Maslov Gerbe

Let Lag(E) be the Grassmannian of Lagrangian subspaces of a complex symplectic vector space E. We construct a Maslov class which generates the second integral cohomology of Lag(E), and we show that its mod 2 reduction is the characteristic class of a flat gerbe with structure group Z ₂. We explain the relation of this gerbe to the well-known flat Maslov line bundle with structure group Z ₄ over the real Lagrangian Grassmannian, whose characteristic class is the mod 4 reduction of the real Maslov class.     

A Riemann–Roch Theorem¶for One-Dimensional Complex Groupoids

We consider a smooth groupoid of the form Σ⋊Γ, where Σ is a Riemann surface and Γ a discrete pseudogroup acting on Σ by local conformal diffeomorphisms. After defining a K-cycle on the crossed product C ₀(Σ)⋊Γ generalising the classical Dolbeault complex, we compute its Chern character in cyclic cohomology, using the index theorem of Connes and Moscovici. This involves in particular a generalisation of the Euler class constructed from the modular automorphism group of the von Neumann algebra L ^∞(Σ)⋊Γ. Received: 1 February 2000 / Accepted: 3 December 2000     

Lie Conformal Algebra Cohomology and the Variational Complex

We find an interpretation of the complex of variational calculus in terms of the Lie conformal algebra cohomology theory. This leads to a better understanding of both theories. In particular, we give an explicit construction of the Lie conformal algebra cohomology complex, and endow it with a structure of a \mathfrakg{\mathfrak{g}}-complex. On the other hand, we give an explicit construction of the complex of variational calculus in terms of skew-symmetric poly-differential operators.     

Mirror Symmetry in Two Steps: A–I–B

We suggest an interpretation of mirror symmetry for toric varieties via an equivalence of two conformal field theories. The first theory is the twisted sigma model of a toric variety in the infinite volume limit (the A–model). The second theory is an intermediate model, which we call the I–model. The equivalence between the A–model and the I–model is achieved by realizing the former as a deformation of a linear sigma model with a complex torus as the target and then applying to it a version of the T–duality. On the other hand, the I–model is closely related to the twisted Landau-Ginzburg model (the B–model) that is mirror dual to the A–model. Thus, the mirror symmetry is realized in two steps, via the I–model. In particular, we obtain a natural interpretation of the superpotential of the Landau-Ginzburg model as the sum of terms corresponding to the components of a divisor in the toric variety. We also relate the cohomology of the supercharges of the I–model to the chiral de Rham complex and the quantum cohomology of the underlying toric variety.Partially supported by the DARPA grant HR0011-04-1-0031 and the NSF grant DMS-0303529.Partially supported by the Federal Program 40.052.1.1.1112, by the Grants INTAS 03-51-6346, NSh-1999/2003.2 and RFFI-04-01- 00637.     

Bubble Divergences from Cellular Cohomology

We consider a class of lattice topological field theories, among which are the weak-coupling limit of 2d Yang–Mills theory, the Ponzano–Regge model of 3d quantum gravity and discrete BF theory, whose dynamical variables are flat discrete connections with compact structure group on a cell 2-complex. In these models, it is known that the path integral measure is ill-defined in general, because of a phenomenon called ‘bubble divergences’. A common expectation is that the degree of these divergences is given by the number of ‘bubbles’ of the 2-complex. In this note, we show that this expectation, although not realistic in general, is met in some special cases: when the 2-complex is simply connected, or when the structure group is Abelian – in both cases, the divergence degree is given by the second Betti number of the 2-complex.     

The fuzzy supersphere

We introduce the fuzzy supersphere as sequence of finite-dimensional, noncommutative ₂-graded algebras tending in a suitable limit to a dense subalgebra of the ₂-graded algebra of ^∞-functions on the (2|2)-dimensional supersphere. Noncommutative analogues of the body map (to the (fuzzy) sphere) and the super-deRham complex are introduced. In particular we reproduce the equality of the super-deRham cohomology of the supersphere and the ordinary deRham cohomology of its body on the “fuzzy level”.     

Stable Ergodic Properties of Cocycles¶over Hyperbolic Attractors

We consider a hyperbolic flow φ ^t defined on an attracting basic set Λ. A map from the first (Čech) cohomology group of Λ into the dynamic cohomology group is constructed. This map is used to discuss the stable ergodicity and mixing of compact Lie group extensions and velocity changes of φ ^t . Received: 17 June 1998 / Accepted: 24 February 1999     

Symplectic Fibrations and the Abelian Vortex Equations

The n ^th symmetric product of a Riemann surface carries a natural family of K?hler forms, arising from its interpretation as a moduli space of abelian vortices. We give a new proof of a formula of Manton–Nasir [10] for the cohomology classes of these forms. Further, we show how these ideas generalise to families of Riemann surfaces. These results help to clarify a conjecture of D. Salamon [13] on the relationship between Seiberg–Witten theory on 3–manifolds fibred over the circle and symplectic Floer homology.     

Cohomology of Canonical Projection Tilings

We define the cohomology of a tiling as the cocycle cohomology of its associated groupoid and consider this cohomology for the class of tilings which are obtained from a higher dimensional lattice by the canonical projection method in Schlottmann's formulation. We prove the cohomology to be equivalent to a certain cohomology of the lattice. We discuss one of its qualitative features, namely that it provides a topological obstruction for a generic tiling to be substitutional. We develop and demonstrate techniques for the computation of cohomology for tilings of codimension smaller than or equal to 2, presenting explicit formulae. These in turn give computations for the $K$-theory of certain associated non-commutative C ^* algebras. Received: 24 June 1999 / Accepted: 18 October 2001     

Conformal Operators on Forms and Detour Complexes on Einstein Manifolds

For even dimensional conformal manifolds several new conformally invariant objects were found recently: invariant differential complexes related to, but distinct from, the de Rham complex (these are elliptic in the case of Riemannian signature); the cohomology spaces of these; conformally stable form spaces that we may view as spaces of conformal harmonics; operators that generalise Branson’s Q-curvature; global pairings between differential form bundles that descend to cohomology pairings. Here we show that these operators, spaces, and the theory underlying them, simplify significantly on conformally Einstein manifolds. We give explicit formulae for all the operators concerned. The null spaces for these, the conformal harmonics, and the cohomology spaces are expressed explicitly in terms of direct sums of subspaces of eigenspaces of the form Laplacian. For the case of non-Ricci flat spaces this applies in all signatures and without topological restrictions. In the case of Riemannian signature and compact manifolds, this leads to new results on the global invariant pairings, including for the integral of Q-curvature against the null space of the dimensional order conformal Laplacian of Graham et al.