Half De Rham complexes and line fields on odd-dimensional manifolds

In this paper we introduce a new elliptic complex on an odd-dimensional manifold with a self-dual line field. The notion of a self-dual line field is a generalization of the notion of a conformal line field. Ellipticity, Fredholm properties and Hodge decompositions of these new complexes are proved both in the case of a closed manifold and in the case of a manifold with boundary. The cohomology groups of these elliptic complexes are computed in some cases. In addition, in this paper, we generalize the notion of an anti-self-dual connection on a smooth 4-manifold to a 3-manifold with a line field and a smooth 5-manifold with a line field. The above new elliptic complexes can be twisted by anti-self-dual connections in dimensions 3 and 5, but only by flat connections in dimensions above 5. This reveals a special feature of dimensions 3 and 5.

     

Deformations and generalized derivations of Hom-Lie conformal algebras

The purpose of this paper is to extend the cohomology and conformal derivation theories of the classical Lie conformal algebras to Hom-Lie conformal algebras. In this paper, we develop cohomology theory of Hom-Lie conformal algebras and discuss some applications to the study of deformations of regular Hom-Lie conformal algebras. Also, we introduce α~k-derivations of multiplicative Hom-Lie conformal algebras and study their properties.     

Convergence in Conformal Field Theory

Convergence and analytic extension are of fundamental importance in the mathematical construction and study of conformal field theory. The author reviews some main convergence results, conjectures and problems in the construction and study of conformal field theories using the representation theory of vertex operator algebras. He also reviews the related analytic extension results, conjectures and problems. He discusses the convergence and analytic extensions of products of intertwining operators (chiral conformal fields) and of q-traces and pseudo-q-traces of products of intertwining operators. He also discusses the convergence results related to the sewing operation and the determinant line bundle and a higher-genus convergence result. He then explains conjectures and problems on the convergence and analytic extensions in orbifold conformal field theory and in the cohomology theory of vertex operator algebras.     

Four-manifold systoles and surjectivity of period map

P. Buser and P. Sarnak showed in 1994 that the maximum, over the moduli space of Riemann surfaces of genus $s$, of the least conformal length of a nonseparating loop, is logarithmic in $s$. We present an application of (polynomially) dense Euclidean packings, to estimates for an analogous 2-dimensional conformal systolic invariant of a 4-manifold $X$ with indefinite intersection form. The estimate turns out to be polynomial, rather than logarithmic, in $\chi(X)$, if the conjectured surjectivity of the period map is correct. Such surjectivity is targeted by the current work in gauge theory. The surjectivity allows one to insert suitable lattices with metric properties prescribed in advance, into the second de Rham cohomology group of $X$, as its integer lattice. The idea is to adapt the well-known Lorentzian construction of the Leech lattice, by replacing the Leech lattice by the Conway-Thompson unimodular lattices which define asymptotically dense packings. The final step can be described, in terms of the successive minima $\lambda_i$ of a lattice, as deforming a $\lambda_2$-bound into a $\lambda_1$-bound, illustrated by Figure 1.     

Self-duality operators on odd dimensional manifolds

In this paper we construct a new elliptic operator associated to any nowhere zero vector field on an odd-dimensional manifold and study its index theory. It turns out this operator has several geometric applications to conformal vector fields, self-dual vector fields, locally free -actions and transversal hypersurfaces of these vector fields in an odd-dimensional manifold. In particular, we reveal a non-stable phenomena about the existence of conformal vector fields and self-dual vector fields in odd dimensions above 3. This is in sharp contrast to the stable phenomena about the existence of nowhere zero vector fields in odd dimensions. Besides these applications, the index formula of this new operator also gives the formulas for the dimensions of self-duality cohomology groups and for the virtual dimensions of the moduli spaces of anti-self-dual connections on 5-cobordisms, which are introduced in author's previous papers.

     

Laplaza sets, or how to select coherence diagrams for pseudo algebras

We define a general concept of pseudo algebras over theories and 2-theories. A more restrictive such a notion was introduced in [Po Hu, Igor Kriz, Conformal field theory and elliptic cohomology, Adv. Math. 189 (2) (2004) 325-412, http://www.math.lsa.umich.edu/~ikriz/], but as noticed by M. Gould, did not capture the desired examples. The approach taken in this paper corrects the mistake by introducing a more general concept, allowing more flexibility in selecting coherence diagrams for pseudo algebras.     

The variational Poisson cohomology

It is well known that the validity of the so called Lenard–Magri scheme of integrability of a bi-Hamiltonian PDE can be established if one has some precise information on the corresponding 1st variational Poisson cohomology for one of the two Hamiltonian operators. In the first part of the paper we explain how to introduce various cohomology complexes, including Lie superalgebra and Poisson cohomology complexes, and basic and reduced Lie conformal algebra and Poisson vertex algebra cohomology complexes, by making use of the corresponding universal Lie superalgebra or Lie conformal superalgebra. The most relevant are certain subcomplexes of the basic and reduced Poisson vertex algebra cohomology complexes, which we identify (non-canonically) with the generalized de Rham complex and the generalized variational complex. In the second part of the paper we compute the cohomology of the generalized de Rham complex, and, via a detailed study of the long exact sequence, we compute the cohomology of the generalized variational complex for any quasiconstant coefficient Hamiltonian operator with invertible leading coefficient. For the latter we use some differential linear algebra developed in the Appendix.     

Twisted smooth Deligne cohomology

Deligne cohomology can be viewed as a differential refinement of integral cohomology, hence captures both topological and geometric information. On the other hand, it can be viewed as the simplest nontrivial version of a differential cohomology theory. While more involved differential cohomology theories have been explicitly twisted, the same has not been done to Deligne cohomology, although existence is known at a general abstract level. We work out what it means to twist Deligne cohomology, by taking degree one twists of both integral cohomology and de Rham cohomology. We present the main properties of the new theory and illustrate its use with examples and applications. Given how versatile Deligne cohomology has proven to be, we believe that this explicit and utilizable treatment of its twisted version will be useful.     

Differential conformal superalgebras and their forms

We introduce the formalism of differential conformal superalgebras, which we show leads to the “correct” automorphism group functor and accompanying descent theory in the conformal setting. As an application, we classify forms of N=2 and N=4 conformal superalgebras by means of Galois cohomology.     

Oriented Cohomology Theories of Algebraic Varieties

This article contains proofs of the results announced by Panin and Smirnov (http://www. math.uiuc.edu/k-theory/0459/2000) in the part concerning general properties of oriented cohomology theories of algebraic varieties. It is constructed one-to-one correspondences between orientations, Chern structures and Thom structures on a given ring cohomology theory. The theory is illustrated by motivic cohomology, algebraic K-theory, algebraic cobordism theory and by other examples.     

Difference Forms

Currently, there is much interest in the development of geometric integrators, which retain analogues of geometric properties of an approximated system. This paper provides a means of ensuring that finite difference schemes accurately mirror global properties of approximated systems. To this end, we introduce a cohomology theory for lattice varieties, on which finite difference schemes and other difference equations are defined. We do not assume that there is any continuous space, or that a scheme or difference equation has a continuum limit. This distinguishes our approach from theories of “discrete differential forms” built on simplicial approximations and Whitney forms, and from cohomology theories built on cubical complexes. Indeed, whereas cochains on cubical complexes can be mapped injectively to our difference forms, a bijection may not exist. Thus our approach generalizes what can be achieved with cubical cohomology. The fundamental property that we use to prove our results is the natural ordering on the integers. We show that our cohomology can be calculated from a good cover, just as de Rham cohomology can. We postulate that the dimension of solution space of a globally defined linear recurrence relation equals the analogue of the Euler characteristic for the lattice variety. Most of our exposition deals with forward differences, but little modification is needed to treat other finite difference schemes, including Gauss-Legendre and Preissmann schemes. Dedicated to Professor Arieh Iserles on the Occasion of his Sixtieth Birthday.     

Nonlinear BVPS with functional parameters

We study the existence, nonexistence and properties of solutions for a certain class of second-order ODEs and their dependence on functional parameters, also in the case when nonlinearities are, in some sense, singular. This approach is based on variational methods and cover both sublinear and superlinear cases. We develop a duality theory and variational principles for this problem. As a consequence of the duality theory we give a numerical version of the variational principle which enables approximation of the solution for our problem. We apply these results to obtain the existence of bounded, radial and positive classical solutions for the BVP of elliptic type. Observe that our method allows us to investigate a certain class of elliptic systems in both bounded annular domain and exterior domain.     

The modular hierarchy of the Toda lattice

The modular vector field plays an important role in the theory of Poisson manifolds and is intimately connected with the Poisson cohomology of the space. In this paper we investigate its significance in the theory of integrable systems. We illustrate in detail the case of the Toda lattice both in Flaschka and natural coordinates.     

Modular Data: The Algebraic Combinatorics of Conformal Field Theory

This paper is primarily intended as an introduction for mathematicians to some of the rich algebraic combinatorics arising in for instance conformal field theory (CFT). It tries to refine, modernise, and bridge the gap between papers [6] and [55]. Our paper is essentially self-contained, apart from some of the background motivation (Section 1) and examples (Section 3) which are included to give the reader a sense of the context. Detailed proofs will appear elsewhere. The theory is still a work-in-progress, and emphasis is given here to several open questions and problems.     

正合范畴中的Gorenstein性和Tate上同调（英文）

Motivated by the classical Gorenstein homological theory and structure of Tate cohomology, we develop a theory of Gorenstein projective objects and Tate cohomology in an exact category A with enough projectives. We study some properties of Gorenstein projective objects and establish Tate cohomology of objects with finite Gorenstein projective dimension.     

A lower bound of cohomologic dimension for an elliptic space

Our goal in this paper is to prove that, under appropriate hypotheses, the sum of the Betti numbers of a 1-connected elliptic space is greater or equal to the dimension of its Q vector space of homotopy. The paper concludes with some examples for which the inequality is strict.     

Power operations in elliptic cohomology and representations of loop groups

Part I of this paper describes power operations in elliptic cohomology in terms of isogenies of the underlying elliptic curve. Part II discusses a relationship between equivariant elliptic cohomology and representations of loop groups. Part III investigates the representation of theoretic considerations which give rise to the power operations discussed in Part I.

     

A local homology theory for linearly compact modules

We introduce a local homology theory for linearly compact modules which is in some sense dual to the local cohomology theory of A. Grothendieck [A. Grothendieck, Local Cohomology, Lecture Notes in Math., vol. 20, Springer-Verlag, Berlin/Tokyo/New York, 1967. [10]]. Some basic properties such as the noetherianness, the vanishing and non-vanishing of local homology modules of linearly compact modules are proved. A duality theory between local homology and local cohomology modules of linearly compact modules is developed by using Matlis duality and Macdonald duality. As consequences of the duality theorem we obtain some generalizations of well-known results in the theory of local cohomology for semi-discrete linearly compact modules.