Quantization on Curves

Deformation quantization on varieties with singularities offers perspectives that are not found on manifolds. The Harrison component of Hochschild cohomology, vanishing on smooth manifolds, reflects information about singularities. The Harrison 2-cochains are symmetric and are interpreted in terms of abelian *-products. This paper begins a study of abelian quantization on plane curves over , being algebraic varieties of the form , where R is a polynomial in two variables; that is, abelian deformations of the coordinate algebra ). To understand the connection between the singularities of a variety and cohomology we determine the algebraic Hochschild (co)homology and its Barr–Gerstenhaber–Schack decomposition. Homology is the same for all plane curves , but the cohomology depends on the local algebra of the singularity of R at the origin. The Appendix, by Maxim Kontsevich, explains in modern mathematical language a way to calculate Hochschild and Harrison cohomology groups for algebras of functions on singular planar curves etc. based on Koszul resolutions.      

Closed star products and cyclic cohomology

We define the notion of a closed star product. A (generalized) star product (deformation of the associative product of functions on a symplectic manifold W) is closed iff integration over W is a trace on the deformed algebra. We show that for these products the cyclic cohomology replaces the Hochschild cohomology in usual star products. We then define the character of a closed star product as the cohomology class (in the cyclic bicomplex) of a well-defined cocycle, and show that, in the case of pseudodifferential operators (standard ordering on the cotangent bundle to a compact Riemannian manifold), the character is defined and given by the Todd class, while in general it fails to satisfy the integrality condition.     

Quantum Invariants

In earlier work, we derived an expression for a partition function ?^(λ), and gave a set of analytic hypotheses under which ?^(λ) does not depend on a parameter λ. The proof that ?^(λ) is invariant involved entire cyclic cohomology and K-theory. Here we give a direct proof that . The considerations apply to non-commutative geometry, to super-symmetric quantum theory, to string theory, and to generalizations of these theories to underlying quantum spaces. Received: 12 January 1998 / Accepted: 1 May 1999     

A Finite Analog of the AGT Relation I: Finite <Emphasis Type="Italic">W</Emphasis>-Algebras and Quasimaps’ Spaces

Recently Alday, Gaiotto and Tachikawa [2] proposed a conjecture relating 4-dimensional super-symmetric gauge theory for a gauge group G with certain 2-dimensional conformal field theory. This conjecture implies the existence of certain structures on the (equivariant) intersection cohomology of the Uhlenbeck partial compactification of the moduli space of framed G-bundles on \mathbbP²{\mathbb{P}^2} . More precisely, it predicts the existence of an action of the corresponding W-algebra on the above cohomology, satisfying certain properties.     

Cohomology and associated deformations for not necessarily co-associative bialgebras

In this Letter, a cohomology and an associated theory of deformations for (not necessarily co-associative) bialgebras are studied. The cohomology was introduced in a previous paper (Lett. Math. Phys. 25, 75–84 (1992)). This theory has several advantages, especially in calculating cohomology spaces and in its adaptability to deformations of quasi-co-associative (qca) bialgebras and even quasi-triangular qca bialgebras.     

A BRST-like operator for space with zero curvature but non-zero torsion tensor

A new type of BRST-like operatorQ has been constructed for spaces with zero Riemann curvature tensor but with non-zero torsion. It is invariant under local coordinate transformation, and its proof depends upon the validity of the first Bianchi identity. Also, there exists a supersymmetry associated withQ. Finally, the cohomology induced byQ has been investigated.     

Kontsevich formality and cohomologies for graphs

A formality on a manifold M is a quasi isomorphism between the space of polyvector fields (T _poly(M)) and the space of multidifferential operators (D _poly(M)). In the case M=R ^d , such a mapping was explicitly built by Kontsevich, using graphs drawn in configuration spaces. Looking for such a construction step by step, we have to consider several cohomologies (Hochschild, Chevalley, and Harrison and Chevalley) for mappings defined on T _poly. Restricting ourselves to the case of mappings defined with graphs, we determine the corresponding coboundary operators directly on the spaces of graphs. The last cohomology vanishes.     

Quantum double and differential calculi

We show that bicovariant bimodules as defined by Woronowicz are in one-to-one correspondence with the Drinfeld quantum double representations. We then prove that a differential calculus associated to a bicovariant bimodule of dimension n is connected to the existence of a particular (n+1)-dimensional representation of the double. An example of bicovariant differential calculus on the nonquasitriangular quantum group E _q (2) is developed. The construction is studied in terms of Hochschild cohomology and a correspondence between differential calculi and 1-cocycles is proved. Some differences of calculi on quantum and finite groups with respect to Lie groups are stressed.     

On the integral representation of states on a C*-algebra

The purpose of this paper is to give some complements to the various extremal decompositions of states on aC*-dynamical system i.e. a pair (A, G) whereA is aC*-algebra andG is a group acting on aA by *-automorphisms. We shall see for instance that the method of decomposition associated with a maximal abelianW*-algebra does not give all the extremal measures in the general case. We also give the explicit form of the greatest lower bound of all the extremal measures and a certain form of continuity of the decomposition. Finally we characterize various systems in the literature (G-abelian algebras, large systems and quasi-large systems) in terms of the equivalence of different notions of ergodicity.     

On the Relation Between Open and Closed Topological Strings

We discuss the relation between open and closed string correlators using topological string theories as a toy model. We propose that one can reconstruct closed string correlators from the open ones by considering the Hochschild cohomology of the category of D-branes. We compute the Hochschild cohomology of the category of D-branes in topological Landau-Ginzburg models and partially verify the conjecture in this case.Acknowledgement A.K. would like to thank Volodya Baranovsky, Ezra Getzler, Kentaro Hori, Dima Orlov, and Sasha Voronov for help at various stages. A.K. is also grateful to the Department of Mathematics of Northwestern University and the Erwin Schrödinger Institute for hospitality while this work was being completed. L. R. is very grateful to Mikhail Khovanov for numerous discussions of the category of matrix factorizations. This work was supported in part by the DOE grant DE-FG03-92-ER40701 and by the NSF grant DMS-0196131.     

Compound Basis Arising from the Basic {A^{(1)}_{1}}-Module

Given a conditionally completely positive map on a unital *-algebra , we find an interesting connection between the second Hochschild cohomology of with coefficients in the bimodule of adjointable maps, where M is the GNS bimodule of , and the possibility of constructing a quantum random walk [in the sense of (Attal et al. in Ann Henri Poincar 7(1):59–104, 2006; Lindsay and Parthasarathy in Sankhya Ser A 50(2):151–170, 1988; Sahu in Quantum stochastic Dilation of a class of Quantum dynamical Semigroups and Quantum random walks. Indian Statistical Institute, 2005; Sinha in Banach Center Publ 73:377–390, 2006)] corresponding to . D. Goswami was supported by a project funded by the Indian National Academy of Sciences. L. Sahu had research support from the National Board of Higher Mathematics, DAE (India) is gratefully acknowledged.     

A Pairing Between Super Lie-Rinehart and Periodic Cyclic Homology

We consider a pairing producing various cyclic Hochschild cocycles, which led Alain Connes to cyclic cohomology. We are interested in geometrical meaning and homological properties of this pairing. We define a non-trivial pairing between the homology of a Lie-Rinehart (super-)algebra with coefficients in some partial traces and relative periodic cyclic homology. This pairing generalizes the index formula for summable Fredholm modules, the Connes-Kubo formula for the Hall conductivity and the formula computing the K⁰-group of a smooth noncommutative torus. It also produces new homological invariants of proper maps contracting each orbit contained in a closed invariant subset in a manifold acted on smoothly by a connected Lie group. Finally we compare it with the characteristic map for the Hopf-cyclic cohomology. The author was partially supported by the KBN grant 1P03A 036 26.     

Chiral Equivariant Cohomology of a Point: A First Look

The chiral equivariant cohomology contains and generalizes the classical equivariant cohomology of a manifold M with an action of a compact Lie group G. For any simple G, there exist compact manifolds with the same classical equivariant cohomology, which can be distinguished by this invariant. When M is a point, this cohomology is an interesting conformal vertex algebra whose structure is still mysterious. In this paper, we scratch the surface of this object in the case G = SU(2).     

Singular Dimensions¶of the N= 2 Superconformal Algebras II:¶The Twisted N= 2 Algebra

We introduce a suitable adapted ordering for the twisted N= 2 superconformal algebra (i.e. with mixed boundary conditions for the fermionic fields). We show that the ordering kernels for complete Verma modules have two elements and the ordering kernels for G-closed Verma modules just one. Therefore, spaces of singular vectors may be two-dimensional for complete Verma modules whilst for G-closed Verma modules they can only be one-dimensional. We give all singular vectors for the levels , 1, and for both complete Verma modules and G-closed Verma modules. We also give explicit examples of degenerate cases with two-dimensional singular vector spaces in complete Verma modules. General expressions are conjectured for the relevant terms of all (primitive) singular vectors, i.e. for the coefficients with respect to the ordering kernel. These expressions allow to identify all degenerate cases as well as all G-closed singular vectors. They also lead to the discovery of subsingular vectors for the twisted N= 2 superconformal algebra. Explicit examples of these subsingular vectors are given for the levels , 1, and . Finally, the multiplication rules for singular vector operators are derived using the ordering kernel coefficients. This sets the basis for the analysis of the twisted N= 2 embedding diagrams. Received: Received: 15 March 1999 / Accepted: 12 November 2000     

Spectral flow invariants and twisted cyclic theory for the Haar state on

In [A.L. Carey, J. Phillips, A. Rennie, Twisted cyclic theory and an index theory for the gauge invariant KMS state on Cuntz algebras. arXiv:0801.4605], we presented a K-theoretic approach to finding invariants of algebras with no non-trivial traces. This paper presents a new example that is more typical of the generic situation. This is the case of an algebra that admits only non-faithful traces, namely SU_q(2) and also KMS states. Our main results are index theorems (which calculate spectral flow), one using ordinary cyclic cohomology and the other using twisted cyclic cohomology, where the twisting comes from the generator of the modular group of the Haar state. In contrast to the Cuntz algebras studied in [A.L. Carey, J. Phillips, A. Rennie, Twisted cyclic theory and an index theory for the gauge invariant KMS state on Cuntz algebras. arXiv:0801.4605], the computations are considerably more complex and interesting, because there are non-trivial ‘eta’ contributions to this index.     

A Variant of the Mukai Pairing via Deformation Quantization

Let X be a smooth projective complex variety. The Hochschild homology HH_?(X) of X is an important invariant of X, which is isomorphic to the Hodge cohomology of X via the Hochschild?CKostant?CRosenberg isomorphism. On HH_?(X), one has the Mukai pairing constructed by Caldararu. An explicit formula for the Mukai pairing at the level of Hodge cohomology was proven by the author in an earlier work (following ideas of Markarian). This formula implies a similar explicit formula for a closely related variant of the Mukai pairing on HH_?(X). The latter pairing on HH_?(X) is intimately linked to the study of Fourier?CMukai transforms of complex projective varieties. We give a new method to prove a formula computing the aforementioned variant of Caldararu??s Mukai pairing. Our method is based on some important results in the area of deformation quantization. In particular, we use part of the work of Kashiwara and Schapira on Deformation Quantization modules together with an algebraic index theorem of Bressler, Nest and Tsygan. Our new method explicitly shows that the ??Noncommutative Riemann?CRoch?? implies the classical Riemann?CRoch. Further, it is hoped that our method would be useful for generalization to settings involving certain singular varieties.     

Integrality theorems in the theory of topological strings

We give a simplified derivation of the expression of instanton numbers and of mirror map in terms of Frobenius map on p-adic cohomology and use this expression to prove integrality theorems. Modifying this proof we verify that the Aganagic–Vafa formulas for the number of holomorphic disks can be expressed in terms of Frobenius map on p-adic relative cohomology; this expression permits us to prove integrality of this number.     

Cohomology and deformation of Leibniz pairs

Cohomology and deformation theories are developed for Poisson algebras starting with the more general concept of a Leibniz pair, namely of an associative algebraA together with a Lie algebraL mapped into the derivations ofA. A bicomplex (with both Hochschild and Chevalley-Eilenberg cohomologies) is essential.     

Bulk Universality and Related Properties of Hermitian Matrix Models

We give a new proof of universality properties in the bulk of spectrum of the hermitian matrix models, assuming that the potential that determines the model is globally C ² and locally C ³ function (see Theorem 3.1). The proof as our previous proof in (Pastur and Shcherbina in J. Stat. Phys. 86:109–147, 1997) is based on the orthogonal polynomial techniques but does not use asymptotics of orthogonal polynomials. Rather, we obtain the sin -kernel as a unique solution of a certain non-linear integro-differential equation that follows from the determinant formulas for the correlation functions of the model. We also give a simplified and strengthened version of paper (Boutet de Monvel, et al. in J. Stat. Phys. 79:585–611, 1995) on the existence and properties of the limiting Normalized Counting Measure of eigenvalues. We use these results in the proof of universality and we believe that they are of independent interest.     

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.