CiteULike: Group: NoncommutativeGeometry - library [81 articles]

Monstrous Moonshine: The first twenty-five years

T Gannon — 2006-03-02T00:49:30-00:00

(14 Apr 2004)

Twenty-five years ago, Conway and Norton published their remarkable paper `Monstrous Moonshine', proposing a completely unexpected relationship between finite simple groups and modular functions. This paper reviews the progress made in broadening and understanding that relationship.

Lectures on conformal field theory and Kac-Moody algebras

J Fuchs — 2006-03-02T00:21:10-00:00

(27 Feb 1997)

This is an introduction to the basic ideas and to a few further selected topics in conformal quantum field theory and in the theory of Kac-Moody algebras.

Deformations, Contractions and Classification of Lie Algebras of Order 3

M Goze — 2006-03-04T02:28:19-00:00

(2 Mar 2006)

Lie algebras of order $F$ (or $F-$Lie algebras) are possible generalisations of Lie algebras (F=1) and Lie superalgebras (F=2). These structures have been used to implement new non-trivial extensions of the Poincaré algebra. In this paper we set the basis of the theory of the deformations (in the Gerstenhaber sense) and contractions for Lie algebras of order 3. We then initiated a general classification for Lie algebras of order 3 and we give all Lie algebras of order 3 based on $\mathfraksl(2,\mathbb C)$ and $\mathfrakiso(1,3)$ the four-dimensional Poincaré algebra.

Fractional supersymmetry and $F$th-roots of representations

Rausch — 2006-03-04T02:25:58-00:00

J. Math. Phys., Vol. 41, No. 7. (2000), pp. 4556-4571.

Fractional supersymmetry of order $F$ in three and more dimensions is discussed. The authors introduce $F$-Lie algebras as a special kind of graded Lie algebras, describe fractional supersymmetry by using $F$-Lie algebras and give a number of examples.

Finite-dimensional Lie algebras of order $F$

Rausch — 2006-03-04T02:25:06-00:00

J. Math. Phys., Vol. 43, No. 10. (2002), pp. 5145-5160.

The authors consider $F$-Lie algebras which formalize the notion of fractional supersymmetry (FSUSY). These algebras are natural generalizations of the Lie algebras ($F=1$) and Lie superalgebras ($F=2$). The authors show how to construct finite-dimensional $F$-Lie algebras with $F>2$ by an inductive process starting from Lie algebras and Lie superalgebras. Many examples of such $F$-Lie algebras and their matrix realizations are constructed. In particular, the first finite-dimensional FSUSY extensions of the Poincaré algebra are presented.

Harmonic cohomology classes of symplectic manifolds

Olivier Mathieu — 2006-02-22T00:04:37-00:00

Comment. Math. Helv., Vol. 70, No. 1. (1995), pp. 1-9.

Perspectives on geometric analysis

Shing-Tung Yau — 2006-02-17T07:00:38-00:00

(16 Feb 2006)

This is a survey paper on several aspects of differential geometry for the last 30 years, especially in those areas related to non-linear analysis. It grew from a talk I gave on the occasion of seventieth anniversary of Chinese Mathematical Society. I dedicate the lecture to the memory of my teacher S.S. Chern who had passed away in December 2004.

The monoid of families of quiver representations

Markus Reineke — 2006-02-03T07:15:24-00:00

(15 May 2001)

A monoid structure on families of representations of a quiver is introduced by taking extensions of representations in families, i.e. subvarieties of the varieties of representations. The study of this monoid leads to interesting interactions between representation theory, algebraic geometry and quantum group theory. For example, it produces a wealth of interesting examples of families of quiver representations, which can be analyzed by representation theoretic and geometric methods. Conversely, results from representation theory, in particular A. Schofield's work on general properties of quiver representations, allow us to relate the monoid to certain degenerate forms of quantized enveloping algebras.

BRST model for equivariant cohomology and representatives for the equivariant Thom class

Jaap Kalkman — 2006-02-01T01:52:29-00:00

Comm. Math. Phys., Vol. 153, No. 3. (1993), pp. 447-463.

One of the fundamental properties of (cohomological) topological field theories is the presence of a nilpotent operator, the BRST operator, whose cohomology models that of some moduli space (of connections in the case of gauge theories). The original space of fields being infinite-dimensional, this is achieved by choosing the BRST operator to be a coboundary operator for equivariant cohomology on the space of fields with respect to some infinite-dimensional symmetry group of automorphisms (gauge transformations). While this interpretation of the BRST operator in topological field theories has, in one way or another, been known for some time, this article, which can also be read as a clear and concise introduction to the algebraic aspects of equivariant cohomology per se, makes this connection much more precise. In particular, in a finite-dimensional context, the "BRST model" is shown to be one member of a one-parameter family connecting the classical Cartan and Weil models for equivariant cohomology. Furthermore, using the concept of "Fourier transform of differential forms", the author shows how to obtain the equivariant Mathai-Quillen representative of the Thom class \ref[V. Mathai and D. G. Quillen, Topology 25 (1986), no. 1, 85--110; <A HREF="/msnmain?fn=105&fmt=doc&r=1&pg1=CNO&s1=836726&loc=fromrevtext">MR0836726 (87k:58006)</A>] as the Fourier transform of a simple BRST closed element, thus connecting BRST cohomology with the interpretation of topological field theory given by M. F. Atiyah and L. C. Jeffrey \ref[J. Geom. Phys. 7 (1990), no. 1, 119--136; <A HREF="/msnmain?fn=105&fmt=doc&r=1&pg1=CNO&s1=1094734&loc=fromrevtext">MR1094734 (92f:58025)</A>].

A Localization Principle for Orbifold Theories

Tommaso de Fernex — 2006-02-01T01:50:46-00:00

(2 Nov 2004)

In this article, written primarily for physicists and geometers, we survey several manifestations of a general localization principle for orbifold theories such as $K$-theory, index theory, motivic integration and elliptic genera.

An Introduction to Gerbes on Orbifolds

Ernesto Lupercio — 2006-02-01T01:50:27-00:00

(19 Feb 2004)

This paper is a gentle introduction to some recent results involving the theory of gerbes over orbifolds for topologists, geometers and physicists. We introduce gerbes on manifolds, orbifolds, the Dixmier-Douady class, Beilinson-Deligne orbifold cohomology, Cheeger-Simons orbifold cohomology and string connections.

Introduction to foliations and Lie groupoids

I Moerdijk — 2006-02-01T01:42:22-00:00

Vol. 91 (2003)

Orbifold String Topology

Ernesto Lupercio — 2006-02-01T01:40:54-00:00

(30 Dec 2005)

In this paper we study the string topology (á la Chas-Sullivan) of an orbifold. We define the string homology ring product at the level of the free loop space of the classifying space of an orbifold. We study its properties (introducing an operad to do so) and do some explicit calculations.

The Interface between Quantum Mechanics and General Relativity

Raymond Chiao — 2006-01-31T22:33:52-00:00

(29 Jan 2006)

The generation, as well as the detection, of gravitational radiation by means of charged superfluids is considered. One example of such a "charged superfluid" consists of a pair of Planck-mass-scale, ultracold "Millikan oil drops," each with a single electron on its surface, in which the oil of the drop is replaced by superfluid helium. When levitated in a magnetic trap, and subjected to microwave-frequency electromagnetic radiation, such a pair of "Millikan oil drops" separated by a microwave wavelength can become an efficient quantum transducer between quadrupolar electromagnetic and gravitational radiation. This leads to the possibility of a Hertz-like experiment, in which the source of microwave-frequency gravitational radiation consists of one pair of "Millikan oil drops" driven by microwaves, and the receiver of such radiation consists of another pair of "Millikan oil drops" in the far field driven by the gravitational radiation generated by the first pair. The second pair then back-converts the gravitional radiation into detectable microwaves. The enormous enhancement of the conversion efficiency for these quantum transducers over that for electrons arises from the fact that there exists macroscopic quantum phase coherence in these charged superfluid systems.

A Bayesian truth serum for subjective data.

D Prelec — 2006-01-31T23:56:29-00:00

Science, Vol. 306, No. 5695. (15 October 2004), pp. 462-466.

Subjective judgments, an essential information source for science and policy, are problematic because there are no public criteria for assessing judgmental truthfulness. I present a scoring method for eliciting truthful subjective data in situations where objective truth is unknowable. The method assigns high scores not to the most common answers but to the answers that are more common than collectively predicted, with predictions drawn from the same population. This simple adjustment in the scoring criterion removes all bias in favor of consensus: Truthful answers maximize expected score even for respondents who believe that their answer represents a minority view.

Automorphisms and Ideals of the Weyl Algebra

Yuri Berest — 2006-01-30T19:59:20-00:00

(25 Feb 2001)

Let $A_1$ be the (first) Weyl algebra, and let $G$ be its automorphism group. We study the natural action of $G$ on the space of isomorphism classes of right ideals of $A_1$ (equivalently, of finitely generated rank 1 torsion-free right $A_1$-modules). We show that this space breaks up into a countable number of orbits each of which is a finite dimensional algebraic variety. Our results are strikingly similar to those for the commutative algebra of polynomials in two variables; however, we do not know of any general principle that would allow us to predict this in advance. As a key step in the proof, we obtain a new description of the bispectral involution of \citeW1. We also make some comments on the group $G$ from the viewpoint of Shafaravich's theory of infinite dimensional algebraic groups.

Ideal Classes of the Weyl Algebra and Noncommutative Projective Geometry (with an Appendix by M. Van den Bergh)

Yuri Berest — 2006-01-30T19:57:48-00:00

(8 Aug 2001)

Let R be the set of isomorphism classes of ideals in the Weyl algebra $A=A_1$, and let C be the set of isomorphism classes of triples (V; X, Y), where V is a finite-dimensional (complex) vector space, and X, Y are endomorphisms of V such that [X,Y]+I has rank 1. Following a suggestion of L. Le Bruyn, we define a map $θ: R \to C$ by appropriately extending an ideal of A to a sheaf over a quantum projective plane, and then using standard methods of homological algebra. We prove that $θ$ is inverse to a bijection $ω: C \to R$ constructed in \citeBW by a completely different method. The main step in the proof is to show that $θ$ is equivariant with respect to natural actions of the group G=Aut(A) on R and C: for that we have to study also the extensions of an ideal to certain weighted quantum projective planes. Along the way, we find an elementary description of θ.

Non-commutative Simplicial Complexes and the Baum-Connes Conjecture

Joachim Cuntz — 2006-01-30T19:56:14-00:00

(30 Mar 2001)

We associate a non-commutative $C^*$-algebra with any locally finite simplicial complex. We determine the $K$-theory of these algebras and show that they can be used to obtain a conceptual explanation for the Baum-Connes conjecture.

Lie groupoids and Lie algebroids in differential geometry

K Mackenzie — 2006-01-20T02:36:31-00:00

Vol. 124 (1987)

Topological and differentiable (or smooth) groupoids (and more generally categories) were introduced by C. Ehresmann in 1958 \ref[in Colloque de Géometrie Differentielle Globale (Bruxelles, 1958), 137--150, Centre Belge Rech. Math., Louvain, 1959; <A HREF="/msnmain?fn=105&fmt=doc&r=1&pg1=CNO&s1=116360&loc=fromrevtext">MR0116360 (22 \#7148)</A>]. The main motivations at that period were the structure of jet manifolds (with the composition of jets) and the gauge groupoid of a principal bundle, which is a transitive and locally trivial groupoid. The holonomy groupoid of a foliation (later named "graph of the foliation") gives an important motivation for considering groupoids which are not locally transitive and also groupoids endowed with other structures (such as measure, Riemannian metric, symplectic or Poisson structure), and which may be "structured" in a sense different from Ehresmann's. In the book under review, the term of "Lie groupoids" (following Ngô Van Quê) means transitive locally trivial groupoids (note that other authors, such as Kumpera and Weinstein, use it for general differentiable groupoids). As indicated by the title and clearly claimed in the preface, the book is mainly centered on that special case of smooth groupoids and on Ehresmann's approach of the theory of principal bundles via their gauge groupoid (disregarding certain phenomena which are specific to the nontransitive case). The corresponding infinitesimal object is in that case equivalent to the so-called Atiyah sequence of the bundle; its elements were named "infinitesimal displacements of the fibres" by Ehresmann who used them for a geometric formulation of infinitesimal connections. In the general case the term of Lie algebroid was introduced by the reviewer to describe the precise algebraic-differentiable structure of this object (which is more precise than a general Lie pseudo-algebra) and to emphasize that it is the candidate for generalizing the role of the Lie algebra of a Lie group. In the transitive case, the infinitesimal connections correspond to the splittings of the Atiyah sequence of Lie algebroids, so that this geometric situation may be viewed as a generalization of the purely algebraic one consisting of the splittings of exact sequences of Lie algebras. The book under review is focused on this formal analogy which is exploited with much virtuosity and leads to a very rich and elegant cohomological theory which unifies the equivariant de Rham cohomology for principal bundles and the Hochschild-Serre theory for Lie algebras. This is a very important and illuminating contribution. Note that it relies deeply on the fact that the kernel in the Atiyah sequence is indeed a Lie algebra bundle, which is specific to the transitive case. By a curious irony of fate, that strategy had been developed by the author in order to give a van Est style cohomological proof of Lie's third integrability theorem for Lie algebroids, a statement announced by the reviewer in 1968. As pointed out much later by \n P. Molino\en and \n R. Almeida\en in 1985, it turns out that the global form of this statement is erroneous, counterexamples arising from the transverse theory of foliations. Then the author immediately realized that the precise measure of the obstruction lies in a third order cohomology class resulting from his previous constructions, while Molino and Almeida gave an independent direct and "more elementary" description of this obstruction, pointing out its connection with other geometrical problems considered earlier by A. Weil, Aragnol and Kostant (in quantization theory) \ref[Almeida and Molino, in Séminaire de Géométrie Différentielle (Montpellier, 1984/85), 39--59, Univ. Sci. Tech. Languedoc, Montpellier, 1985; Zbl 596:57017]. To conclude, this book is very important and stimulating and must be read by geometers and physicists. It draws attention to the gauge groupoid approach for bundles (inaugurated by Ehresmann, but too much disregarded by geometers and rediscovered by physicists) as well as to the infinitesimal corresponding notion of Lie algebroid (now extensively used, in the nontransitive case, in symplectic geometry). It is certainly not a definitive textbook which closes a classical subject (the author himself announces improvements and developments to appear in Cahiers Topologie de Géometrie et Différentielle Catégoriques and in J. Pure Appl. Algebra), but a stimulating guide for new researches. {Reviewer's remarks: The reviewer regrets that the Lie functorial correspondence between Lie groupoids and Lie algebroids is developed only with a fixed base: it is certainly illuminating to interpret conditions of Maurer-Cartan type as defining more general morphisms of Lie algebroids and to state Lie's second theorem in this wider framework. {On the other hand, the reader is warned that the assumption of paracompactness for manifolds is not sufficient for certain statements (notably Corollary 1.9, p. 89) and should be replaced by the second countability condition.}

General theory of Lie groupoids and Lie algebroids

Kirill Mackenzie — 2006-01-20T02:34:55-00:00

Vol. 213 (2005)

The graded ring of quantum theta functions for noncommutative torus with real multiplication

Mariya Vlasenko — 2006-01-20T02:33:06-00:00

(17 Jan 2006)

For quantum torus generated by unitaries $UV = e(θ)VU$ there exist nontrivial strong Morita autoequivalences in case when $θ$ is real quadratic irrationality. A.Polishchuk introduced and studied the graded ring of holomorphic sections of powers of the respective bimodule (depending on the choice of a complex structure). We define and study another graded ring whose components are linear spaces of quantum theta functions in sense of Yu.Manin. The respective spaces of quantum thetas are spanned by Rieffel scalar products of Polishchuk's holomorphic vectors.

Equivariant cohomology and localization of path integrals

Richard Szabo — 2006-01-20T02:23:05-00:00

Vol. 63 (2000)

A topological space $X$ with the action of a Lie group $G$ has equivariant cohomology groups $H_G^*(X)$, which admit various de Rham type models if $X$ is a smooth finite-dimensional manifold. This cohomology theory mixes the ordinary cohomology of $X$ with the group action in nontrivial ways. Most importantly, equivariant cohomology has a localization formula, due to Atiyah and Bott, which reduces equivariant integrals of equivariantly closed forms to integrals over the fixed point set of the action. This has many applications, the most famous being the reinterpretation and reproof of the Duistermaat-Heckman formula for the exactness of the stationary phase expansion of classical partition functions on symplectic manifolds with circle actions. In the 1980s, various formal proofs of the Atiyah-Singer index theorem based on path integral techniques appeared in the physics literature. These proofs are intriguing but difficult for mathematicians to evaluate, since path integrals have resisted rigorous interpretation for over fifty years. In any case, Atiyah and Witten \ref[M. F. Atiyah, Astérisque No. 131 (1985), 43--59; <A HREF="/msnmain?fn=105&fmt=doc&r=1&pg1=CNO&s1=816738&loc=fromrevtext">MR0816738 (87h:58206)</A>] realized that these formal proofs were most cleanly presented as a formal application of Atiyah-Bott/Duistermaat-Heckman localization to path integrals on the infinite-dimensional manifold of loops on $X$. Here "formal" means: take results valid in finite dimensions and use them without further justification in infinite dimensions. Physicists view this approach as another instance of the predictive power of path integrals, so formal applications of equivariant cohomology to infinite-dimensional settings of physical interest have continued over the past fifteen years. The driving force here is the search for exactly solvable field theories (called integrable theories in the book), so that perturbative expansions of path integrals can be avoided. Such field theories are closely related to topological quantum field theories, although precise statements are hard to find. The book under review is a thorough exposition of the current state of applying equivariant cohomology to quantum field theory. A large quantity of intriguing material for mathematicians, including index theory, coadjoint orbit and Bott-Borel-Weil theory, modular functors, and Mathai-Quillen formalism, is covered. However, the methods are formal, as this is a text written by a physicist in a mathematical physics series. The style may be difficult for mathematicians: calculations, often in local coordinates, take precedence over statements of theorems; there is often no distinction between a rigorous proof and a formal demonstration; and mathematical and physical concepts are introduced so rapidly that at least half the audience will probably lack the motivational background at any one time. Some details will make mathematicians nervous, such as the implicit but nontrivial switch from Lorentzian to Riemannian signature in §4.2, or the statement in §2.4 that "a nontrivial vector bundle can always be considered as a trivial one endowed with a nontrivial curvature". Nevertheless, if one takes the attitude that this material may make mathematical sense within the next fifty years, the book can be appreciated as a well-organized exposition of the topological content of quantum field theory from a physics viewpoint. In more detail, after a brief introduction in Chapter 1, the book reviews rigorous equivariant cohomology and Atiyah-Bott localization in Chapter 2. An alternative formulation of localization due to Berline-Vergne is also discussed. In Chapter 3, applications of localization to symplectic geometry are given, including the Duistermaat-Heckman formula and Witten's proposed generalization to nonabelian group actions. In Chapter 4, the book enters the formal realm of path integrals. A nice treatment of the formal proofs of the Atiyah-Singer index theorem is given, although mathematicians will prefer Atiyah's exposition. A formal version, due to Niemi-Tirkkonen, of the Duistermaat-Heckman formula for degenerate functions is used to formally derive Lefschetz-type generalizations of the index theorem. In Chapter 5, Hamiltonian systems whose path integrals can be equivariantly localized are treated. In particular, Niemi-Tirkkonen localization is used to rederive the Kirillov character formula and the Weyl character formula. The phase spaces treated here are simply connected, and Chapter 6 deals with non-simply connected phase spaces, primarily Riemann surfaces. Modular functors appear in this formal setting. Chapter 7 contains a more technical approach to not quite integrable field theories, where the perturbative expansion stops after the one-loop contributions. Chapter 8 discusses the relationship between formal equivariant localization and cohomological/topological field theories. There is a brief discussion of Mathai-Quillen formalism, where the Lagrangians of certain toplogical field theories are seen to be infinite-dimensional versions of Mathai-Quillen forms. Finally, there are brief appendices on BRST quantization and the various models of equivariant cohomology.

Supersymmetry and equivariant de Rham theory

Victor Guillemin — 2006-01-20T02:22:22-00:00

(1999)

With an appendix containing two reprints by Henri Cartan [ MR0042426 (13,107e); MR0042427 (13,107f)]

Manifolds, tensor analysis, and applications

R Abraham — 2006-01-20T02:21:22-00:00

Vol. 75 (1988)

Foundations of mechanics

Ralph Abraham — 2006-01-20T02:20:08-00:00

(1978)

Second edition, revised and enlarged, With the assistance of Tudor Ra\c tiu and Richard Cushman

Algebraic topology

Allen Hatcher — 2006-01-20T02:18:41-00:00

(2002)

This book is one of the most interesting and accessible texts to come out in recent years. The subject is presented in three broad areas: elementary notions including fundamental group, homology and cohomology, and homotopy theory. In most cases, some of the more sophisticated topics, such as $H$-spaces, Bockstein homomorphisms and Hopf invariants, are placed in sections called "Additional Topics" at the end of the appropriate chapter. The style is refreshingly informal, although precise definitions are always given. The author always prepares the reader for the next idea, and there are basic, simple black and white pictures which really help. He is not afraid to repeat ideas with increasing generality. For example, there are treatments of Postnikov towers which begin with a cellular version, go on to the usual tower of principal fibrations and end with Moore-Postnikov decompositions. The reviewer finds this a very "student-friendly" approach (very different from the texts which always look for the greatest generality). There are many excellent examples. The author talks about two further volumes involving spectral sequences and $K$-theory. When complete, this will be a major achievement which will help a great many serious students.

Lecture notes in algebraic topology

James Davis — 2006-01-20T02:17:40-00:00

Vol. 35 (2001)

This is a textbook for a second-year graduate course in topology, highlighting the basic tools of homotopy theory. The authors remark that, at their home institution, most topology research students are in geometric topology, and the book does emphasize the geometric origins of homotopy theory, as opposed to a more abstract, categorical approach. Prerequisites are the fundamental group, covering spaces (especially covering transformations), CW-complexes, singular homology and cohomology, and some elementary notions from differential topology, such as transversality. The first third of the book is occupied with background material, including universal coefficient theorems, products in singular (co)homology, fiber bundles, and homology with local coefficients. The core material then begins with a chapter on fibrations, cofibrations, homotopy groups, and the basic exact sequences in which these objects are involved. This is followed by a chapter on obstruction theory, leading to Eilenberg-Mac Lane spaces and their role in representing ordinary cohomology. Next come generalized (co)homology theories, beginning with framed bordism, then stable homotopy, then spectra in general. Perhaps the most important material is contained in the two chapters on spectral sequences, or, really, on one particular spectral sequence, namely, a combination of the Leray-Serre and Atiyah-Hirzebruch sequences, which, given a fibration $F \to E \to B$, converges to the generalized (co)homology $h(E)$. The focus here is on understanding how to use spectral sequences rather than on setting up the theory behind how they arise. The point is that the consequences of spectral sequences are abundant and wide-ranging. The topics presented to support this include the Hurewicz theorem, the Freudenthal suspension theorem, some calculations of homotopy groups of spheres, the Steenrod squares, the Thom isomorphism theorem, Stiefel-Whitney classes, and the relation between intersection numbers and cup products. The book concludes on a somewhat different note with a chapter on simple-homotopy theory, reaching Reidemeister torsion and the classification of lens spaces, first up to homotopy, then up to homeomorphism. The book is written in a down-to-earth, non-formal style, and the material has been kept to a manageable length. In a handful of places, full proofs of complicated results are omitted, and the reader is referred to the literature, for instance, G. W. Whitehead's book \ref[ Elements of homotopy theory, Springer, New York, 1978; <A HREF="/msnmain?fn=105&fmt=doc&r=1&pg1=CNO&s1=516508&loc=fromrevtext">MR0516508 (80b:55001)</A>]. The most conspicuous example of this is the construction of the Leray-Serre-Atiyah-Hirzebruch spectral sequence mentioned above. Roughly 200 exercises are included; they are well integrated into the exposition. In addition, each chapter concludes with one or two "projects," in which the reader is asked to look up and digest a substantial result and which could serve as sources for student presentations. Some of these involve material which is used later in the text (limits and colimits, classifying spaces for fiber bundles), and some are included more for breadth (Postnikov systems, the $EHP$ sequence). The authors have made a special effort to keep the presentation flexible, holding down the amount of technical terminology and notation "to minimize the need to shuffle pages back and forth when reading the book". This feature is likely to be appealing not only to students, but also to practicing mathematicians in need of a refresher on a particular topic.

Cyclic homology

Jean-Louis Loday — 2006-01-20T02:16:23-00:00

Vol. 301 (1992)

Appendix E by Mar\i a O. Ronco

The wild world of 4-manifolds

Alexandru Scorpan — 2006-01-18T02:33:21-00:00

(2005)

Deformation quantization of gerbes, I

P Bressler — 2006-01-18T02:30:57-00:00

(6 Dec 2005)

This is the first in a series of articles devoted to deformation quantization of gerbes. Here we give basic definitions and interpret deformations of a given gerbe as Maurer-Cartan elements of a differential graded Lie algebra (DGLA). We classify all deformations of a given gerbe on a symplectic manifold, as well as provide a deformation-theoretic interpretation of the first Rozansky-Witten class.

Fukaya Type Categories for Associative Algebras

Ryszard Nest — 2006-01-18T02:29:06-00:00

(27 Mar 1998)

We define for an associative algebra an $A_∞$ category whose objects are automorphisms of this algebra. This construction has some resemblance with Fukaya'a categories related to Floer cohomology.

The cohomology structure of an associative ring

Murray Gerstenhaber — 2006-01-18T02:27:25-00:00

Ann. of Math. (2), Vol. 78 (1963), pp. 267-288.

Let $A$ be an associative ring (not necessarily with unit) and $P$ a two-sided $S$-module. In addition, assume that $A$ and $P$ are modules over a commutative ring $S$ and that all operations of $A$ on $A$ and $P$ are $S$-homomorphisms. Defining the cohomology groups of the $S$-algebra $A$ with coefficients in $P$ by the usual cochain formulae, the author is primarily interested in the ring structure of $H^*(A,A)$ and the module structure over $H^*(A,A)$ of $H^*(A,P)$ given by the usual cup product multiplication. Among other things, it is shown that $H^*(A,A)$ is a commutative ring in the sense of graded rings. A bracket product $[ , ]$ is introduced in $H^*(A,A)$ under which $H^*(A,A)$ becomes a graded Lie ring with $[H^m(A,A),H^n(A,A)]⊂ H^m+n-1(A,A)$ and such that the bracket operation of $H^1(A,A)$ into itself is the ordinary Poisson bracket of derivations of $A$ into itself. Various other properties of this operation are derived and its role in the author's theory of deformations of algebras indicated.

Methods of homological algebra

Sergei Gelfand — 2006-01-17T02:22:28-00:00

(2003)

Continued fractions, modular symbols, and noncommutative geometry

Yuri Manin — 2006-01-17T02:20:17-00:00

Selecta Math. (N.S.), Vol. 8, No. 3. (2002), pp. 475-521.

FEATURED REVIEW. The rotation algebra is a well-studied $C^*$-algebra, providing a ready example on which to test notions related to foliations and crossed products. It is defined as the $C^*$-algebra $A_θ$, where $θ∈\Bbb R$, generated by two unitaries $U$ and $V$ satisfying $$VU=\exp(2π i θ)UV.$$ By an important result of M. A. Rieffel\ \ref[Pacific J. Math. 93 (1981), no. 2, 415--429; <A HREF="/msnmain?fn=105&fmt=doc&r=1&pg1=CNO&s1=623572&loc=fromrevtext">MR0623572 (83b:46087)</A>; J. Pure Appl. Algebra 5 (1974), 51--96; <A HREF="/msnmain?fn=105&fmt=doc&r=1&pg1=CNO&s1=367670&loc=fromrevtext">MR0367670 (51 \#3912)</A>], the algebras $A_θ$ and $A_θ'$, $θ, θ'∈\Bbb R$, are strongly Morita equivalent if and only if $θ$ and $θ'$ are in the same orbit of the fractional linear action of $\text PSL(2,\Bbb Z)$ on the projective real line $ P_1(\Bbb R)$. Strongly Morita equivalent $C^*$-algebras have the same space of classes of irreducible representations, canonically isomorphic $K$-theory groups, and share many other properties \ref[see A. Connes, Noncommutative geometry, Academic Press, San Diego, CA, 1994; <A HREF="/msnmain?fn=105&fmt=doc&r=1&pg1=CNO&s1=1303779&loc=fromrevtext">MR1303779 (95j:46063)</A> (Chapter II, Appendix A)]. On putting $U=\exp(2π i x)$, $V=\exp(2π iy)$, $x,y∈ \Bbb R$ and $θ=0$, we see that $A_0$ is the algebra of continuous functions on the 2-torus $\Bbb R^2/\Bbb Z^2$. In Connes's theory of noncommutative geometry, a dense subalgebra $\scr A_θ$ of "smooth elements" of the rotation algebra $A_θ$ is known as the "noncommutative 2-torus $\Bbb T_θ$" and is an important case study in this theory. A generic element of $a∈\scr A_θ$ is a formal sum $$a=∑_(m,n)∈\Bbb Z^2a(m,n)U^mV^n,$$ where the sequence $(a(m,n))_(m,n)∈\Bbb Z^2$ is of rapid decay. The algebra $A_θ$ has a canonical trace function $τ(·)$ determined by $$τ≤ft(a\right)=a(0,0).$$ In \ref[Inst. Hautes Études Sci. Publ. Math. No. 62 (1985), 257--360; <A HREF="/msnmain?fn=105&fmt=doc&r=1&pg1=CNO&s1=823176&loc=fromrevtext">MR0823176 (87i:58162)</A>] Connes computed the Hochschild and periodic cyclic cohomology of $\scr A_θ$. He showed that the dimension of the Hochschild cohomology spaces depends on the Diophantine properties of $θ$, whereas the periodic cyclic cohomology is complex 2-dimensional in both odd and even degrees. The bases of the periodic cyclic cohomology can be described in terms of the trace $τ$ and the natural derivations on $\scr A_θ$ determined by $$δ_1(U^mV^n)=2π imU^mV^n,\quad δ_2(U^mV^n)=2π i nU^mV^n.$$ The noncommutative torus $\Bbb T_θ$ was used by J. Bellissard \ref[in Statistical mechanics and field theory\thinspace: mathematical aspects (Groningen, 1985), 99--156, Lecture Notes in Phys., 257, Springer, Berlin, 1986; <A HREF="/msnmain?fn=105&fmt=doc&r=1&pg1=CNO&s1=862832&loc=fromrevtext">MR0862832 (88e:46053)</A>; in Localization in disordered systems (Bad Schandau, 1986), 61--74, Teubner, Leipzig, 1988; <A HREF="/msnmain?fn=105&fmt=doc&r=1&pg1=CNO&s1=965981&loc=fromrevtext">MR0965981 (91c:82005)</A>] in an application of noncommutative geometry to the Quantum Hall Effect (QHE) in physics. Bellissard gave a complete explanation of certain stable numerical quantities described by the QHE in terms of noncommutative topological invariants of $\Bbb T_θ$, thereby completing earlier work of Novikov and Thouless. The 2-torus $\Bbb R^2/\Bbb Z^2$ can be given a complex structure. The different possibilities for this structure are parametrized by the Poincaré upper half plane $\scr H$ of complex numbers with positive imaginary part. To $z∈\scr H$, we may associate the complex 1-dimensional torus $$T_z=\Bbb C^2/(\Bbb Z+z\Bbb Z),$$ given by the complex numbers modulo translation by the lattice $\Bbb Z+z\Bbb Z$. The complex isomorphism classes of these tori, also known as elliptic curves from their geometric description, are in bijective correspondence with the orbits of the fractional linear action of $\text PSL(2,\Bbb Z)$ on $\scr H$. The quotient space $\text PSL(2,\Bbb Z)\backslash\scr H$ can be compactified by adding a point at infinity corresponding to the 1-point quotient or "cusp" $\text PSL(2,\Bbb Z)\backslash P_1(\Bbb Q)$. Adding extra structure to the isomorphism classes of elliptic curves leads to replacing $\text PSL(2,\Bbb Z)$ by certain of its finite index subgroups $G_0$. The corresponding modular curves $G_0\backslash\scr H$ can be compactified by adding the finite set of cusps $G_0\backslash P_1(\Bbb Q)$. The central point of the paper under review is that this traditional picture bypasses the Morita classes of noncommutative tori that would appear if the boundary of the modular curve $G_0\backslash\scr H$ were considered instead to be the "noncommutative modular curve" $\text PSL(2,\Bbb Z)\backslash P_1(\Bbb R)$. This is in the spirit of J. Bost and Connes \ref[Selecta Math. (N.S.) 1 (1995), no. 3, 411--457; <A HREF="/msnmain?fn=105&fmt=doc&r=1&pg1=CNO&s1=1366621&loc=fromrevtext">MR1366621 (96m:46112)</A>], Connes \ref[Selecta Math. (N.S.) 5 (1999), no. 1, 29--106; <A HREF="/msnmain?fn=105&fmt=doc&r=1&pg1=CNO&s1=1694895&loc=fromrevtext">MR1694895 (2000i:11133)</A>], Y. Soibelman \ref[Lett. Math. Phys. 56 (2001), no. 2, 99--125 <A HREF="/msnmain?fn=105&fmt=doc&pg1=MR&s1=1854130&v1=MR%3DMR1854130%20%282004c%3A58019%29&loc=fromrevtext">MR1854130 (2004c:58019)</A> ] and others, as discussed in the paper under review and its references. The paper contributes completely new concrete examples of how mathematics usually applied to the commutative case relates to that traditionally applied to the noncommutative case. It opens up a new field in "noncommutative number theory", aimed at combining the mathematics of classical spaces of automorphic functions with that of noncommutative algebras. For further work in this direction by the authors see \ref[Yu. I. Manin, "Real multiplication and noncommutative geometry", preprint, arXiv.org/abs/math/0202109; "Von Zahlen und Figuren", preprint, arXiv.org/abs/math/0201005; M. Marcolli, J. Number Theory 98 (2003), no. 2, 348--376 <A HREF="/msnmain?fn=105&fmt=doc&pg1=MR&s1=1955422&v1=MR%3DMR1955422%20%282004b%3A11062%29&loc=fromrevtext">MR1955422 (2004b:11062)</A> ]. We now outline the main results of the paper. A matrix $A∈\text PSL(2,\Bbb R)$ is hyperbolic if its trace has absolute value greater than 2. In this case, it has two hyperbolic fixed points $θ$, with $A'(θ)<1$, and $θ'$, with $A'(θ')>1$. The oriented geodesic in $\scr H$ from $θ'$ to $θ$ is invariant under the action of $A$ and is called the axis of $A$. If $A∈ G_0$, then the axis of $A$ becomes a closed geodesic in $G_0\backslash\scr H$. Moreover $θ$ and $θ'$ are then irrational conjugates in a real quadratic field. Conversely, every closed geodesic in $G_0\backslash\scr H$ represents the conjugacy class of a primitive hyperbolic transformation in $G_0$. Furthermore, closed geodesics for the modular group are known to be coded by "minus" continued fractions \ref[D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, Berlin, 1981; <A HREF="/msnmain?fn=105&fmt=doc&r=1&pg1=CNO&s1=631688&loc=fromrevtext">MR0631688 (82m:10002)</A>]. For $θ∈ \Bbb R$, we may try to understand in what sense $\Bbb T_θ$ is a limit of $T_z$ as $z$ tends to $θ$ along a geodesic in $\scr H$. In this vein, the authors extend the classical definition of modular symbols to "limiting modular symbols", with limits along geodesics in the upper half plane ending at points on the "noncommutative boundary". They show that quadratic irrationalities give rise to limiting cycles whereas generic irrational points give rise to cycles vanishing in a suitable averaged sense. Let $X_G_0=X_G_0(\Bbb C)$ denote the smooth compactification of $G_0\backslash\scr H$ by the finite number of cusps in bijection with $G_0\backslash P_1(\Bbb Q)$ and let $φ$ be the corresponding covering map. As in \ref[Yu. I. Manin, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 19--66; <A HREF="/msnmain?fn=105&fmt=doc&r=1&pg1=CNO&s1=314846&loc=fromrevtext">MR0314846 (47 \#3396)</A>; also in Selected papers of Yu. I. Manin, World Sci. Publishing, River Edge, NJ, 1996; <A HREF="/msnmain?fn=105&fmt=doc&r=1&pg1=CNO&s1=1408904&loc=fromrevtext">MR1408904 (97m:01108)</A> (pp. 202--247); L. Merel, Manuscripta Math. 80 (1993), no. 3, 283--289; <A HREF="/msnmain?fn=105&fmt=doc&r=1&pg1=CNO&s1=1240651&loc=fromrevtext">MR1240651 (94k:14014)</A>], for any two points $α,β$ in $\scr H∪ P_1(\Bbb Q)$, we can define a real homology class or "modular symbol" ${α,β}∈ H^1(X_G_0,\Bbb R)$ by integrating lifts $φ^*(ω)$ of differentials $ω$ of the first kind on $X_G_0$ along the geodesic path connecting $α$ to $β$: $$∫_α,βω\coloneq ∫_α^βφ(ω).$$ When $α$, $β$ are cusps, the modular symbol represents a rational homology class. To extend the definition to "limiting modular symbols", when either endpoint is real irrational, the authors define $${{*, β}}_G_0\coloneq \lim\frac1T(x,y){x,y}_G_0∈ H^1(X_G_0,\Bbb R),$$ where $x,y∈ \scr H$ are two points on the geodesic joining $α$ to $β$, $x$ is arbitrary but fixed, $T(x,y)$ is the geodesic distance between them, and the limit is taken as $y$ tends to $β$. If the limit exists, the authors show that it depends neither on $x$ nor on $α$, which justifies the notation. These integrals can be related to finite (when $α$, $β$ are cusps), stably periodic (when $α$, $β$ are two fixed points of a hyperbolic element of $G_0$ as described above), or general infinite continued fractions. The different cases are treated using results from \ref[Yu. I. Manin, op. cit., 1972] and \ref[J. B. Lewis and D. B. Zagier, in The mathematical beauty of physics (Saclay, 1996), 83--97, World Sci. Publishing, River Edge, NJ, 1997; <A HREF="/msnmain?fn=105&fmt=doc&r=1&pg1=CNO&s1=1490850&loc=fromrevtext">MR1490850 (99c:11108)</A>]. Continued fractions that eventually agree up to a shift of index can be identified by $\textPGL(2,\Bbb Z)\backslash P_1(\Bbb R)$. In this paper all of this noncommutative boundary is considered. A striking result of the paper (Theorem 0.2.2), which is derived from certain averaging techniques over successive convergents in infinite continued fractions, uses modular symbols to relate Mellin transforms of weight-two cusp forms for $G_0=Γ_0(N)$ to quantities defined entirely on the noncommutative boundary of the corresponding modular curve. This gives a concrete example of how the two types of tori $\Bbb T_θ$, $θ∈\Bbb R$, and $T_z$, $z∈\scr H$, give information about each other. These results on the limiting modular symbols rely on certain properties, involving spectral analysis, of the Ruelle transfer operator or Gauss-Kuzmin operator for the shift of the continued fraction expansion, generalized to subgroups $G$ of finite index in $\text GL(2,\Bbb Z)$. In particular the authors generalize the Gauss-Kuzmin-Lévy formula (Theorem 0.1.2). This result gives a formula for the limit of the pullback of the Lebesgue measure on $(0,1)× \text GL(2,\Bbb Z)/G$ with respect to $g_n(α)$ acting on $α$ and $t$ simultaneously, where $$g_n(α)=±atrix p_n-1(α)& p_n(α)\cr q_n-1(α)& q_n(α)\endpmatrix,$$ and $p_n(α)/q_n(α)$ are the successive convergents to $α$. A different direction, with a related philosophy, is the study of the $K$-theory of the noncommutative modular curves in the spirit of Connes noncommutative geometry [A. Connes, op. cit., 1985]. In this theory, the quotient space $G\backslash P_1(\Bbb R)$, where $G$ is of finite index in $\text PSL(2,\Bbb Z)$, can be studied "topologically" via its associated crossed product algebra $C( P_1(\Bbb R))\rtimes G$ or the strongly Morita equivalent $C(\widehatX)\rtimes\text PSL(2,\Bbb Z)$, where $\widehatX= P_1(\Bbb R)× \text PSL(2,\Bbb Z)/G$. M. V. Pimsner \ref[Invent. Math. 86 (1986), no. 3, 603--634; <A HREF="/msnmain?fn=105&fmt=doc&r=1&pg1=CNO&s1=860685&loc=fromrevtext">MR0860685 (88f:22022)</A>] (see also \ref[M. Laca and J. S. Spielberg, J. Reine Angew. Math. 480 (1996), 125--139; <A HREF="/msnmain?fn=105&fmt=doc&r=1&pg1=CNO&s1=1420560&loc=fromrevtext">MR1420560 (98a:46085)</A>]) has studied the $K$-theory of $C(\widehatX)$ and its crossed product with $Γ=\text PSL(2,\Bbb Z)=\Bbb Z/2*\Bbb Z/3$, $Γ_0=\Bbb Z/2$ and $Γ_1=\Bbb Z/3$. The $K$-theory in degrees 0 and 1 is related by a six-term exact sequence. On the other hand, in [Yu. I. Manin, op. cit., 1972] and [L. Merel, op. cit.] the homology groups $H_1(X_G;\Bbb Z)$ and relative homology groups $H^\textcusps\coloneq H_1(X_G,\textcusps; \Bbb Z)$ were studied via the "modular complex" and "relative modular complex" (with respect to the elliptic and parabolic (cuspidal) fixed points of $\text PSL(2,\Bbb Z)$). This homology is based on the $n$-cells, $n=0,1,2,$ of the $\text PSL(2,\Bbb Z)/G$-orbit of the fundamental region of $\text PSL(2,\Bbb Z)$ built from geodesics joining those fixed points. The authors show (Theorem 4.4.1) that there is a natural isomorphism between a four-term exact sequence derived from Pimsner's exact sequence and an exact sequence derived from the modular complexes. Essentially, this relates $H^\textcusps$ to the noncommutative topology of $G\backslash P_1(\Bbb R)$. The authors also relate the modular complex to homological constructions of noncommutative geometry via the periodic cyclic cohomology of the "smooth" crossed product algebras associated to $G\backslash P_1(\Bbb R)$. These innovative and ground-breaking results reveal the mutual influence of the mathematics of a commutative geometric object and that of its natural noncommutative boundary.

Arithmetic noncommutative geometry

Matilde Marcolli — 2006-01-17T02:19:17-00:00

Vol. 36 (2005)

With a foreword by Yuri Manin

Operads and PROPs

Martin Markl — 2006-01-17T02:16:22-00:00

(6 Jan 2006)

We review definitions and basic properties of operads, PROPs and algebras over these structures.

Twisted $K$-theory and $K$-theory of bundle gerbes

Peter Bouwknegt — 2006-01-17T02:13:06-00:00

Comm. Math. Phys., Vol. 228, No. 1. (2002), pp. 17-45.

Chern character in twisted $K$-theory: equivariant and holomorphic cases

Varghese Mathai — 2006-01-17T02:12:27-00:00

Comm. Math. Phys., Vol. 236, No. 1. (2003), pp. 161-186.

It was shown by E. Witten \ref[J. High Energy Phys. 1998, no. 12, Paper 19, 41 pp. (electronic); <A HREF="/msnmain?fn=105&fmt=doc&r=1&pg1=CNO&s1=1674715&loc=fromrevtext">MR1674715 (2000e:81151)</A>] that the D-brane charge in type IIB string theory over a space-time $M$ should be regarded as an element of a twisted $K$-theory group $K_[H](M)$, where $H$ is a global $3$-form associated to the $B$-field. Furthermore, in \ref[P. Bouwknegt et al., Comm. Math. Phys. 228 (2002), no. 1, 17--45; <A HREF="/msnmain?fn=105&fmt=doc&r=1&pg1=CNO&s1=1911247&loc=fromrevtext">MR1911247 (2003g:58049)</A>], the authors and their collaborators provided a way to interpret the twisted $K$-theory within the category of bundle gerbes and stable isomorphisms. The paper under review discusses in detail the Chern-Weil term associated to the $K$-theoretic bundle gerbe, extending the construction of \ref[P. Bouwknegt et al., op. cit.] to the equivariant and holomorphic cases.

Entire cyclic homology of continuous trace algebras

Varghese Mathai — 2006-01-17T02:11:16-00:00

(15 Feb 2005)

A central result here is the computation of the entire cyclic homology of canonical smooth subalgebras of stable continuous trace C*-algebras having smooth manifolds M as their spectrum. More precisely, the entire cyclic homology is shown to be canonically isomorphic to the continuous periodic cyclic homology for these algebras. By an earlier result of the authors, one concludes that the entire cyclic homology of the algebra is canonically isomorphic to the twisted de Rham cohomology of M.

T-duality for torus bundles with H-fluxes via noncommutative topology, II: the high-dimensional case and the T-duality group

Varghese Mathai — 2006-01-17T02:10:45-00:00

(11 Aug 2005)

We use noncommutative topology to study T-duality for principal torus bundles with H-flux. We characterize precisely when there is a "classical" T-dual, i.e., a dual bundle with dual H-flux, and when the T-dual must be "non-classical," that is, a continuous field of noncommutative tori. The duality comes with an isomorphism of twisted $K$-theories, required for matching of D-brane charges, just as in the classical case. The isomorphism of twisted cohomology which one gets in the classical case is replaced in the non-classical case by an isomorphism of twisted cyclic homology. An important part of the paper contains a detailed analysis of the classifying space for topological T-duality, as well as the T-duality group and its action. The issue of possible non-uniqueness of T-duals can be studied via the action of the T-duality group.

Tropical Geometry and its applications

Grigory Mikhalkin — 2006-01-04T13:13:40-00:00

(3 Jan 2006)

These notes outline some basic notions of Tropical Geometry and survey some of its applications for problems in classical (real and complex) geometry. To appear in the Proceedings of the Madrid ICM.

The Meaning of Einstein's Equation

John Baez — 2006-01-06T18:41:40-00:00

(5 Jan 2006)

This is a brief introduction to general relativity, designed for both students and teachers of the subject. While there are many excellent expositions of general relativity, few adequately explain the geometrical meaning of the basic equation of the theory: Einstein's equation. Here we give a simple formulation of this equation in terms of the motion of freely falling test particles. We also sketch some of its consequences, and explain how the formulation given here is equivalent to the usual one in terms of tensors. Finally, we include an annotated bibliography of books, articles and websites suitable for the student of relativity.

Differential forms on loop spaces and the cyclic bar complex

Ezra Getzler — 2006-01-17T02:05:27-00:00

Topology, Vol. 30, No. 3. (1991), pp. 339-371.

Poisson geometry and Morita equivalence

Henrique Bursztyn — 2006-01-17T02:02:12-00:00

(30 Mar 2004)

These notes discuss various aspect of the “representation theory” of Poisson manifolds, with focus on Morita equivalence and Picard groups. We give a brief introduction to Poisson geometry (including Dirac and twisted Poisson structures) and algebraic Morita theory before presenting the geometric Morita theory of Poisson manifolds. We also point out the connections with the theory of symplectic groupoids and hamiltonian actions.

The Lefschetz property, formality and blowing up in symplectic geometry

Gil Cavalcanti — 2006-01-17T02:00:42-00:00

(6 Apr 2005)

In this paper we study the behaviour of the Lefschetz property under the blow-up construction. We show that it is possible to reduce the dimension of the kernel of the Lefschetz map if we blow up along a suitable submanifold satisfying the Lefschetz property. We use that, together with results about Massey products, to construct nonformal (simply connected) symplectic manifolds satisfying the Lefschetz property.

New aspects of the ddc-lemma

Gil Cavalcanti — 2006-01-17T01:59:42-00:00

(24 Jan 2005)

We produce examples of generalized complex structures on manifolds by generalizing results from symplectic and complex geometry. We produce generalized complex structures on symplectic fibrations over a generalized complex base. We study in some detail different invariant generalized complex structures on compact Lie groups and provide a thorough description of invariant structures on nilmanifolds, achieving a classification on 6-nilmanifolds. We study implications of the `dd^c-lemma' in the generalized complex setting. Similarly to the standard dd^c-lemma, its generalized version induces a decomposition of the cohomology of a manifold and causes the degeneracy of the spectral sequence associated to the splitting d = \del + \delbar at E_1. But, in contrast with the dd^c-lemma, its generalized version is not preserved by symplectic blow-up or blow-down (in the case of a generalized complex structure induced by a symplectic structure) and does not imply formality.

Generalized complex geometry

Marco Gualtieri — 2006-01-17T01:59:00-00:00

(18 Jan 2004)

Generalized complex geometry, as developed by Hitchin, contains complex and symplectic geometry as its extremal special cases. In this thesis, we explore novel phenomena exhibited by this geometry, such as the natural action of a B-field. We provide new examples, including some on manifolds admitting no known complex or symplectic structure. We prove a generalized Darboux theorem which yields a local normal form for the geometry. We show that there is an elliptic deformation theory and establish the existence of a Kuranishi moduli space. We then define the concept of a generalized Kahler manifold. We prove that generalized Kahler geometry is equivalent to a bi-Hermitian geometry with torsion first discovered by physicists. We then use this result to solve an outstanding problem in 4-dimensional bi-Hermitian geometry: we prove that there exists a Riemannian metric on the complex projective plane which admits exactly two distinct Hermitian complex structures with equal orientation. Finally, we introduce the concept of generalized complex submanifold, and show that such sub-objects correspond to D-branes in the topological A- and B-models of string theory.

Generalized complex structures on nilmanifolds

Gil Cavalcanti — 2006-01-17T01:58:11-00:00

(25 Apr 2004)

We show that all 6-dimensional nilmanifolds admit generalized complex structures. This includes the five classes of nilmanifold which admit no known complex or symplectic structure. Furthermore, we classify all 6-dimensional nilmanifolds according to which of the four types of left-invariant generalized complex structure they admit. We also show that the two components of the left-invariant complex moduli space for the Iwasawa manifold are no longer disjoint when they are viewed in the generalized complex moduli space. Finally, we provide an 8-dimensional nilmanifold which admits no left-invariant generalized complex structure.

Generalized geometry and the Hodge decomposition

Marco Gualtieri — 2006-01-17T01:57:38-00:00

(7 Sep 2004)

In this lecture, we review some of the concepts of generalized geometry, as introduced by Hitchin and developed in the speaker's thesis. We also prove a Hodge decomposition for the twisted cohomology of a compact generalized Kähler manifold, as well as a generalization of the $dd^c$-lemma of Kähler geometry.

Reduction of Courant algebroids and generalized complex structures

Henrique Bursztyn — 2006-01-17T01:57:04-00:00

(27 Sep 2005)

We present a theory of reduction for Courant algebroids as well as Dirac structures, generalized complex, and generalized Kähler structures which interpolates between holomorphic reduction of complex manifolds and symplectic reduction. The enhanced symmetry group of a Courant algebroid leads us to define extended actions and a generalized notion of moment map. Key examples of generalized Kähler reduced spaces include new explicit bi-Hermitian metrics on $\CC P^2$.