Workshop: Noncommutative Geometry, K-theory, and String Theory

Workshop on Noncommutative Geometry, K-theory, and String Theory at the Institute for Geometry and its Applications on 7 - 8 January, 2002.

Titles and abstracts

Speaker: Ross Street (Macquarie University)
Title: Basic structures for higher-dimensional algebra
Lecture 1: Substitudes and convolution.
Lecture 2: Higher dimensional categories.
Abstract: Emmy Noether emphasized that Euler characteristic, genus, and other numerical invariants of spaces should be extracted after passing to invariant algebraic structures such as homology and homotopy groups. Categories provided a language for such functorial passages from geometry to algebra. Sometimes, as when we pass from a compact Hausdorff space to its commutative C*-algebra, the passage is perfect and we have a duality, thereby diminishing the division between geometry and algebra. The distinction between the language and the mathematics must also be abandoned as we see an increase in the use of categorical structures themselves as invariants. Examples are the fundamental groupoid of a space and the monoidal category of appropriate representations of a topological group. When we work in a traditional algebra we may view it as 0-dimensional: the elements are points that may or may not be equal. When we work in a category we consider paths of morphisms that may be equal or not: this is 1-dimensional algebra. In a 2-category or bicategory, we work with "pasting diagrams" of 2-dimensional regions containing 2-cells. In this way we are led to higher-dimensional algebra. In the first lecture I will describe some low dimensional categorical structures and show how we work in them to understand processes lying behind such areas as Tannaka duality and vertex algebras. In the second lecture I will define the basic structures for doing n-dimensional algebra for all n.

Speaker: Alexei Davydov (Macquarrie University)
Title: K-theory of braided monoidal categories
Abstract: The commutativity constraint of braided category gives rise to an algebraic structure on its K-theory known as Gerstenhaber algebra. If, in addition, the braiding has a compatible balanced structure the Gerstenhaber bracket on K-theory is generated by a Batalin-Vilkovisky differential. We use these algebraic structures to prove a generalization of Anderson-Moore-Vafa theorem which says that the order of the twist in a semi-simple ribbon (tortile) category with finitely many simple objects is finite.

Speaker: Valerio Toledano (University of Paris, Jussieu, Paris)
Title: Lectures on Operator Algebras and Conformal Field Theory
Abstract: I will describe an operator algebraic solution to the problem of fusion for the Wess-Zumino-Witten model of Conformal Field Theory. This approach, initiated by A. Wassermann, relies on the use of Connes' tensor product of bimodules over a von Neumann algebra to define a multiplicative operation (Connes fusion) on the positive energy representations of a loop group LG = C(S1,G) at a given level. The notion of bimodules arises by restricting these representations to loops with support contained in an interval I of S1 or of its complement.
This method has also been used by T. Loke to define and compute the fusion ring corresponding to the discrete series representations of Diff(S1) and is also applicable to the case where the target group G isn't simply connected. One of the interesting problems it poses is to relate it to Kazhdan and Lustzig's definition of fusion.
I will show that the multiplicities of tensor product decompositions (fusion rules) are given by the Verlinde rules. The computation is done in the spirit of invariant theory à l H. Weyl and rests ultimately on the detailed study of the primary fields of the theory, which allows to prove that they define operator-valued distributions and on the explicit computation of the monodromy of a family of Knizhnik-Zamolodchikov equations corresponding to a suitable 'vector' representation of the loop group. An interesting, and perhaps novel feature here is that the exceptional groups E6,E7,F4,G2 turn out to possess such vector representations.

Speaker: Paolo Piazza (La Sapienza, Rome)
Title: Dirac index classes and the non commutative spectral flow
Abstract: I shall present some joint work with Eric Leichtnam. Let M--->B be a fibration of compact manifolds, for example a fibration of spin manifolds. Let t be in [0,1] and assume that for each t we are given a vertical family of generalized Dirac operators D(t) = (D(t)b)b in B. D(t) can be given, for example, by a smooth variation of the vertical metrics defining the Dirac operators on the fibres. We assume that D(t) depends continuously on t and that Ind(D(0)) = 0 in Ki (B), i = 0,1 depending of the dimension of the fibres. Of course we then have Ind(D(t)) = 0 for each t. Under these assumptions Dai and Zhang have defined the notion of higher spectral flow of {D(t)}t in [0,1]; this is an element in Ki+1(B). I shall present a generalization of this notion to a noncommutative context, with D(t) a 1-parameter family of operators acting on the sections of a A-bundle, with A a C*-algebra. We shall be mainly interested in the reduced C*-algebra of a discrete group and to the 1-parameter family of operators induced by a family of G-invariant Dirac operators on a Galois G-covering. In this case the noncommutative spectral flow is, when defined , an element in K*(C*r (G)).
I shall give geometric applications of this concept to index theoretic and geometric situations, including the problem of establishing when the Novikov's higher signatures of a closed manifolds are cut-and-paste invariants.

Speaker: Danny Stevenson (Adelaide University)
Title: The K-theory of Bundle Gerbes
Abstract: In the presence of a non-trivial B-field, D-brane charges in certain string theories are classified by twisted K-theory. The appearance of twisted K-theory is due to the fact that D-branes have `twisted' bundles on their world volumes. This twisting arises from a class in H3(M; Z ) coming from the field strength of the B-field. We shall describe some recent joint work with Bouwknegt, Carey, Mathai and Murray which allows for a description of these twisted bundles as honest bundles on the total space of a projective unitary bundle together with an action of a certain gerbe. We shall give an example of this in the case where the twisting class in H3(M; Z ) is torsion. We shall also discuss the twisted Chern character in this setting.

Speaker: Peter Bouwknegt (Adelaide University)
Title: Branes on Group Manifolds and Fusion Rules
Abstract: We discuss the classification of D-brane charges for branes on group manifolds in the context of boundary conformal field theory. We compare the results to the K-theoretic classification of D-brane charges.

Speaker: Krzysztof Wysocki (Melbourne University)
Title: Pseudoholmorphic curves and Hamiltonian dynamics
Abstract: We shall describe new useful tools in the study of three-dimensional flows. The tools are based on a first order partial differential equations of Cauchy-Riemann type. Families of solutions of the partial differential equations can be used to construct a natural foliation of a star-shaped energy surface. This foliation gives rise to a geometric decomposition of the star-shaped energy surface into a few very special periodic orbits P of the Hamiltonian vector field and a 2-dimensional foliation in the complement consisting of embedded surfaces transversal to the flow and bounded by periodic orbits from P. As a consequence for dynamical systems one concludes, for example, that every non-degenerate Hamiltonian vector field on the star-shaped energy surface possesses either 2 or infinitely many periodic orbits. Joint work with H. Hofer and E.Zehnder.

Schedule

Monday 7th January 2002

9:30-9:45   Refreshments
9:45-10:00    Opening
10:00-11:00    Ross Street (Macquarie University)
Basic structures for higher-dimensional algebra
Lecture 1: Substitudes and convolution.
11:00-12:00    Daniel Stevenson (Adelaide University)
The K-theory of Bundle Gerbes
12:00-14:00    Lunch Break
14:00-15:00    Valerio Toledano (University of Paris, Jussieu, Paris)
Lectures on Operator Algebras and Conformal Field Theory
15:00-15:30    Refreshments
15:30-16:30    Alexei Davydov (Macquarrie University)
K-theory of braided monoidal categories

Tuesday 8th January 2002

10:00-11:00    Ross Street (Macquarie University)
Basic structures for higher-dimensional algebra
Lecture 2: Higher dimensional categories.
11:00-12:00    Paolo Piazza (La Sapienza, Rome)
Dirac index classes and the non commutative spectral flow
12:00-14:00    Lunch Break
14:00-15:00   Krzysztof Wysocki (Melbourne University)
Pseudoholmorphic curves and Hamiltonian dynamics
15:00-15:15    Refreshments
15:15-16:00   Peter Bouwknegt (Adelaide University)
Branes on Group Manifolds and Fusion Rules
16:00-16:10    Concluding remarks

Participant information

Venue

Coffee Breaks will be held in Engineering and Mathematics Building, Room 07 (see campus map).

Registration

There will be no registration fees. Please send an email to mathai.varghese@adelaide.edu.au to ascertain numbers for coffee/refreshments.

Funding

Some support towards traveling and local expenses of postgraduate students might be available. Please contact M Varghese.

Accommodation

Some accommodation is available at Kathleen Lumley College, 51 Finniss Str, North Adelaide. Please contact M Varghese.

Organiser