Workshop: Dualities in Field Theories and the Role of K-Theory

Lectures by Jonathan Rosenberg (University of Maryland)

Short biography

Professor Jonathan Rosenberg obtained his Ph.D. from University of California (Berkeley) in 1976 under the supervision of Professor Marc A. Rieffel. He subsequently spent some time at University of Pennsylvania at the rank of an Assistant Professor between 1977-81. He joined University of Maryland in 1981 where he currently holds the Ruth M. Davis Professorship. He has won several honours and awards in his career and he is widely regarded as a world leading researcher and a first-rate expositor on topics related to operator algebras, geometry, topology and T-duality in string theory. He is renowned for his work on the Gromov-Lawson-Rosenberg conjecture.

He was the CBMS principal speaker at TCU, Fortworth, Texas, entitled Topology, C*- algebras, and String Duality, June 2-6, 2008, where he gave a series of 10 highly successful lectures, which have been written up into a book published by the AMS. He was also one the three principal speakers at the prestigious Oberwolfach Seminar on Topological K- theory for Noncommutative Algebras and Applications, May 2005, which eventually became an authoritative book on the subject. He is a managing editor of the Journal of K-theory since 2007 and is on the editorial boards of several other prestigious international journals.

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Titles of individual lectures

• The basic idea of dualities in field theories and T-duality

It is now known (or in some cases just believed) that many quantum field theories exhibit dualities, equivalences with the same or a different theory in which things appear very different, but the overall physical implications are the same. We will discuss some of these dualities from the point of view of a mathematician, and then give a (simplified) outline of the case of string theory and T-duality ("target space" duality). This duality is closely related to other dualities in string theory, such as mirror symmetry.

• K-Theory and its relevance to physics

K-theory is the extraordinary cohomology theory that arises out of classifying vector bundles. It thus has a very simple geometric interpretation. We will discuss some of the ways vector bundles arise in physics and how the study of "charge conservation" forces one to consider the role played by K-theory.

• Basics of C*-algebras and crossed products

This lecture will be an elementary introduction to some aspects of C*-algebras, which are needed for formulating the idea of a "noncommutative space." I will also explain what crossed products are, give some examples, and illustrate what they are good for.

• Continuous-trace algebras and twisted K-theory

We will review the definition of continuous-trace algebras, C*-algebras locally Morita equivalent to commutative algebras, and the basic classification theory for them as developed by Dixmier and Douady. The K-theory of continuous-trace algebras is known as twisted K-theory. We will explain why, in the presence of background fluxes, the K-groups appearing in string theory must be replaced by twisted K-groups.

• The topology of T-duality and the Bunke-Schick construction

This lecture will introduce the idea, originally due to Bouwknegt, Evslin, and Mathai, of topological T-duality. While T-duality for physicists is a metric notion, this concentrates on just the topological aspect of T-duality, the part that is independent of the metric. This can be viewed as the "leading term" in the physicists' (metric) T-duality. Bunke and Schick gave a very nice way, which we will also discuss, of axiomatizing topological T-duality.

• T-duality via crossed products

An alternative approach to topological T-duality, due to Mathai and the author, depends on crossed products and continuous-trace algebras, which were introduced in Lectures 3 and 4. We will quickly explain the basic idea and how it sometimes leads to non-classical T-duals in the higher-dimensional case.

• Problems presented by S-duality

S-duality is in many ways a much more mysterious duality than T-duality, but it seems to involve very deep mathematics. (For example, Kapustin and Witten have related it to the Langlands program.) In this lecture we will discuss some of the puzzles it raises in terms of K-theory constraints, and a possible approach to handling some of them, due to Mendez-Diez and the author.

• The AdS/CFT correspondence and problems it raises

Still another mysterious duality in string theory is the AdS/CFT correspondence, which was the main topic of the parallel lecture series by Gopakumar. The main question we will try to address is how to relate the K-theoretic classification of branes in string theory to invariants of a similar nature in gauge theory.

• KR-theory and some KR calculations

KR-theory, introduced by Atiyah, the K-theory of complex vector bundles with a conjugate-linear involution compatible with a fixed involution on the base. This is the appropriate K-theory for studying, for example, algebraic varieties defined over R. We will discuss a few facts about this variant of K-theory and discuss some calculations of it.

• T-duality for orientifolds and applications of KR-theory

In the final talk, we will discuss string theory on orientifolds, spacetime manifolds equipped with an involution. In such a theory, charges take their values in KR-theory. We will concentrate on the case where the spacetime is an elliptic curve defined over R (crossed with R^8 with trivial involution) and discuss how T-duality matches up with the KR calculations. This is joint work with Doran and Mendez-Diez.

Downloads

• Event poster
• Abstracts
• List of participants

Schedule and slides

Click here for the slides from Prof Rosenberg's lectures.

Participant information

Venue

Conference Room 7.15, Level 7, Ingkarni Wardli (see on the map)

Registration

There will be no registration fees: all are welcome. However, if you are interested in attending, kindly send an e-mail to Snigdhayan Mahanta by 14 February 2012, with the following information:

• Name
• Position and Affiliation
• E-mail

Please also indicate in your e-mail whether you would like to attend the workshop dinner (Thursday evening, 22 March), which will be $5 per person.

Funding

This event is co-sponsored by the Australian Mathematical Sciences Institute (AMSI). AMSI allocates a travel allowance to each of its member universities (for a list of members, click here). Students or Early Career Researchers from AMSI member universities without access to a suitable research grant or other source of funding may apply to their Head of Mathematical Sciences for subsidy of travel and accommodation out of the departmental travel allowance. No other funding is available.

Accommodation

Participants, other than the invited speakers, are asked to make their own travel and accommodation arrangements.

Internet access

The University of Adelaide provides wireless access for visitors whose home institutions have eduroam. For more information on how to set up eduroam, please click here.

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