Geometry and Quantum Theory (GQT)

Conference

The GQT colloquium takes place on the final two days, June 7 and 8, and consists of nine research talks. These are attended by Ph.D. students, members of the GQT staff, and visitors from abroad. We aim at organizing research talks which feature topics covered during the school, though some of the talks are unrelated.

The program is as follows:

Thursday 7 June

9:00-10:00: arrival
10:00-11:00: Lennart Meier (Utrecht), Elliptic cohomology of level n
11:30 -12:30: Claire Debord (Université Clermont-Auvergne), Index theory through Lie groupoids
lunch
14:00-15:00: Arne Smeets (Nijmegen), Tame degenerations of genus 1 curves
15:30-16:30: Mario Garcia Fernandez (ICMAT Madrid), Canonical metrics in complex non-Kähler geometry
18:30: dinner

Friday 8 June
9:00-10:00: Ralph Klaasse (ULB), Constructing stable generalized complex four-manifolds
10:15 – 11:15: Thomas Grimm (ITF, Utrecht), Evidence for a Quantum Gravity Conjecture using limiting mixed Hodge structures 
11:30 – 12:30: Jens Kaad (Syddansk Universitet Odense), Group cocycles on loop groups
lunch
14:00-15:00: Vito Zenobi (Göttingen), Lie groupoids, Dirac operators and metrics with positive scalar curvature
15:30 – 16:30: Niek de Kleijn (Copenhagen), Deformation Quantization and the Algebraic Index Theorem

 

Abstracts

Lennart Meier (Utrecht)

Elliptic cohomology of level n

Elliptic genera have played an important role in algebraic topology and algebraic geometry since the 1980s. They associate modular forms for the congruence subgroups $\Gamma_1(n)$ to (almost-complex) manifolds. More recently, elliptic cohomology theories have been built that are natural targets of elliptic genera for families. I will give an overview of these theories and report in particular on certain $C_2$-equivariant refinements that allow, for example, to treat families of real algebraic varieties.

 

Claire Debord (Université Clermont-Auvergne)

Index theory through Lie groupoids

We will explain why Lie groupoids are very naturally linked to Atiyah-Singer index theory. After introducing Lie groupoids, and giving various examples, I will explain how these geometrical objects can be involved in index theory :

  • to generalize index problems,
  • to construct the index of pseudodifferential operators without using the pseudodifferential calculus,
  • to prove index theorems,
  • and if time remains to construct the pseudodifferential calculus.

This talk, inspired by ideas of A. Connes, will be based on joint works with JM. Lescure and with G. Skandalis.

 

Arne Smeets (Nijmegen)

Tame degenerations of genus 1 curves

Let K be a local field of mixed or positive characteristic. I will explain some results on proper degenerations of (torsors under) abelian varieties from the point of view of logarithmic algebraic geometry; in particular, I will give a complete characterisation of curves of genus 1 over K with ‘logarithmic good reduction’. This is based on joint work with Alberto Bellardini, and with Kentaro Mitsui.

 

Mario Garcia Fernandez (ICMAT Madrid)

Canonical metrics in complex non-Kähler geometry

In the 1950s Calabi asked the question of whether a compact  complex manifold admits a preferred Kähler metric, adjusted to the holomorphic structure and distinguished by natural conditions on the volume or the Ricci tensor. Following recent advances in Kähler geometry, such as the solution of the Kähler-Einstein problem, there is a renewed interest in extending Calabi’s Programme to the case of compact complex manifolds which do not admit a Kähler metric. After a brief account of the history of this question, in this talk I will overview different proposals for a theory of canonical metrics in complex non-Kähler geometry, based on generalized Kähler geometry and holomorphic Courant algebroids.

 

Ralph Klaasse (ULB)
Constructing stable generalized complex four-manifolds
A generalized complex structure is said to be stable if its defining anticanonical section vanishes transversally, giving rise to a codimension-two type-change locus. These structures are those generalized complex structures that are closest to being symplectic, while exhibiting further interesting behavior. An alternate and fruitful viewpoint is to consider them as zero-residue symplectic structures in the elliptic tangent bundle, a Lie algebroid naturally associated to their type-change locus. We develop Gompf-Thurston methods for Lie algebroids to relate their existence to that of fibration-like maps. We define boundary Lefschetz fibrations and use them to construct stable structures out of log-symplectic structures. Moreover we discuss how to obtain such boundary Lefschetz fibrations on concrete four-manifolds, and in fact classify all such fibrations over the disk. This is based on joint work with Stefan Behrens and Gil Cavalcanti.
Thomas Grimm (ITF, Utrecht)

Evidence for a Quantum Gravity Conjecture using limiting mixed Hodge structures 

It has been conjectured that in theories consistent with quantum gravity infinite distances in field space coincide with an infinite tower of physical states becoming massless exponentially fast. In this talk I present non-trivial evidence for this conjecture from the study of String Theory on complex three-dimensional manifolds. Our main results arise from the analysis of limits in complex structure deformation space by using limiting mixed Hodge structures and nilpotent orbits.

 

Jens Kaad (Syddansk Universitet Odense)

Group cocycles on loop groups

The Connes-Karoubi multiplicative character is an invariant of higher algebraic K-theory associated to any finitely summable Fredholm module. In the one-dimensional case this invariant can be computed using central extensions and group 2-cocycles. Due to the work of Carey-Pincus this links the multiplicative character to the tame symbol of a pair of meromorphic functions on a Riemann surface. In this talk I will discuss the two-dimensional setting and explain how explicit group 3-cocycles on theta deformed tori can be obtained from the setting of groups acting on categories. Our results thereby extend results of Frenkel and Zhu from the algebraic setting of the formal bidisc to the analytic setting of smooth functions on the noncommutative 2-torus. Fundamental ingredients in the analytic approach are determinant lines of Fredholm operators and canonical isomorphisms coming from their trace class perturbations. It is then conjectured that our group 3-cocycles compute the Connes-Karoubi multiplicative character and that they are linked to the tame symbol of triples of meromorphic functions on a 2-dimensional complex manifold. The talk is based on joint work with Ryszard Nest and Jesse Wolfson.

 

Vito Zenobi (Göttingen)

Lie groupoids, Dirac operators and metrics with positive scalar curvature

We will see how to construct a hierarchy of K-theoretic invariants (fundamental classes, index classes and secondary classes) associated to Dirac operators on Lie groupoids. We will see how these invariants apply to the study of metric with positive scalar curvature and how, in the framework of the Lie groupoids approach, one can deal with many geometrical different situations at once.

Niek de Kleijn (Copenhagen)

Deformation Quantization and the Algebraic Index Theorem

There are many roads that lead to the index theorem. In this talk I will walk the algebraic road. This roads starts with the work of Fedosov and was fully realized by Nest–Tsygan in the 90’s. The main idea comes form the fact that pseudo-differential operators, or at least their symbol algebra, may be obtained through deformation quantization. Since the index of a pseudo-differential operator depends only on its (principal) symbol one may try to formulate the index theorem (and its proof) using merely the symbol algebra. This leads to the algebraic version of the index theorem phrased in terms of cyclic homology and obtained for deformation quantizations of more general manifolds than cotangent bundles. In this talk I will give a history of the theorem. This will lead us through the topics of deformation quantization and formal geometry and will have us ending up in recent mathematical research.