ESI Higher Structures in String Theory and Quantum Field Theory

Mini Workshop on KK-theory, twists and applications Dec. 14 - Dec. 16, 2015


Paolo Aschieri (Piemonte)
Alan Carey (ANU)
Jens Kaad (Nijmegen)
Bram Mesland (Leibniz University Hannover)
Jonathan Rosenberg (Maryland)
Hisham Sati (Pittsburgh)
Mahmoud Zeinalian (Long Island University)


All talks will be in the Boltzmann Lecture Theatre at the ESI

Dec. 14 - 16

10:00 - 11:00 am
Morning Tea 11:00 - 11:30 am
11:30 am - 12:30 pm
Lunch 12:30 - 2:00 pm
2:00 - 3:00 pm

Titles and Abstracts

Paolo Aschieri : `Deformation quantization of Noncommutative Principal Bundles'

Abstract: Drinfeld twist deformation theory of modules and algebras that carry a representation of a Hopf Algebra H can be extended to deform also morphisms and connections that are not H-equivariant. In this talk I present how similar techniques allow to canonically deform principal G-bundles, and in general how Hopf-Galois extensions are canonically deformed to new Hopf-Galois extensions. Twisting the structure group we obtain principal bundles with noncommutative fiber and where the structure group is a quantum group. Twisting the automorphism group of the principal bundle we further obtain a noncommutative base space. Examples include homogeneous spaces and the instanton bundle on the four sphere.

Alan Carey: `Spectral flow in the skew adjoint Fredholm operators on a real Hilbert space'

Abstract: Topological arguments show that in the real case spectral flow is Z2 valued. I will explain a way to give an analytic definition. This is of interest in condensed matter theory and there should be connections to Kasparov modules.

Jens Kaad: `Morita equivalences of spectral triples'

Abstract: Classically, two algebras are Morita equivalent when they are linked by a pair of finitely generated projective bimodules via the module tensor product. In particular, two such algebras will have the same representation theory. At the C*-algebraic level this concept was extended by Marc Rieffel using the interior tensor product of C*-correspondences and the extra analytic flexibility then allows for the use of modules with infinitely many generators. The idea that Morita equivalent algebras should also have the same non-commutative geometry (meaning that they admit the same spectral triples) has been successfully applied to the algebraic case where the correspondence takes place via a finitely generated projective module. In this talk I will show how the unbounded Kasparov product can be applied to obtain Morita equivalence results for spectral triples via infinitely generated modules. As an application we shall see that the following pairs of objects admits the same (twisted) spectral triples:

  1. Two Riemannian manifolds with conformally equivalent metrics.
  2. A hereditary subalgebra and its full algebra.
  3. The crossed product of a discrete group acting on a manifold and the quotient manifold.

Bram Mesland: `The noncommutative geometry of Bianchi groups',

Abstract: A Bianchi group is a group of the form PSL(2,OK), where OK is the ring of integers in an imaginary quadratic field. Consider a torsion free finite index subgroup Γ⊂ PSL(2,OK) and its action on hyperbolic 3-space H. The quotient M:=H/Γ is a noncompact hyperbolic 3-manifold with cusps. The action of Γ on H extends to an ergodic action on the sphere at infinity and we let A be the corresponding crossed product algebra, which is simple, nuclear and purely infinite. On the level of K-homology, a Gysin sequence relates the K-homology groups of M, C*r(Γ) and A. The maps in this sequence can be made explicit on the level of unbounded Fredholm modules, yielding a complete description of the K-homology group K1(A). The construction features the use of harmonic measures, the Poisson kernel and geometric K-cycles on M on one side, and group cocycles and the unbounded γ-element on the other side. The unbounded Kasparov product features as a patching tool. Using a natural notion of Hecke operator on K-homology, the result is a Hecke equivariant isomorphism K1(A)→ H1(Γ,Z2).

Jonathan Rosenberg: `T-duality and the Baum-Connes conjecture'

Abstract: The phenomenon of T-duality in string theory is actually closely associated to KK-theory via the following line of logic. D-brane charges in string theory take their values in twisted K-theory of one sort or another (depending on precisely what string theory one is looking at), and a T-duality sets up an isomorphism between the (twisted) K-theory groups of the two different spacetimes, usually with a degree shift depending on the number of dualized directions. In all cases where this has been worked out, the isomorphism turns out to be given by a special case of the Baum-Connes conjecture for some group (possibly with coefficients). We will discuss a number of concrete examples. An open problem is to try to determine exactly how and why this happens in general.

Hisham Sati: `Higher twisted spectra and applications'

Abstract: In this talk I will consider higher twists from the point of view of generalized cohomology theories of higher chromatic and categorical levels with applications. I will first describe joint work with Craig Westerland on twists of Morava K-theory and E-theory at all chromatic levels and applications to M-theory and string theory. Then I will describe recent work with John Lind and Craig Westerland on a new periodic form of the iterated algebraic K-theory of ku, generalizing the K-theory of 2-vector bundles K(ku) to all categorical degrees, as well as a natural twisting of this cohomology theory by higher gerbes. As an application, this leads to topological T-duality for sphere bundles oriented with respect to this theory as an isomorphism of the twisted forms, vastly generalizing existing results.

Mahmoud Zeinalian: `Poisson geometry of moduli stack of twists'

Abstract: Trace of holonomy along a closed loop on an oriented surface defines a function, known as the Wilson line on the moduli stack of flat connections on the surface. Bill Goldman showed the Poisson bracket of two Wilson lines is a linear combination of Wilson lines. This gave rise to a Poisson structure, known as the Goldman bracket, on the algebra generated by the free homotopy classes of closed curves on the surface. Chas and Sullivan generalized this structure to the equivariant homology of the free loop space of any oriented manifold. Abbaspour and I gave a Poisson geometry interpretation of the Chas-Sullivan string bracket. This was based on a physical interpretation by Cattaneo, Froehlich, Padroni and Rossi. Our description at the time was unnaturally limited to the even dimensional part of the equivariant homology of the free loop space of even dimensional manifolds. I will cover the background and describe how these restrictions can be removed. This is the beginning of current work with Gregory Ginot on Wilson line observable on the derived moduli stack of perfect complexes. There is a relationship between the above objects and twists of (differential) cohomology theories and field theories which is the subject of a current joint work with Peter Teichner.