Generalized Geometry and Applications
26 of March 2020
Hamburg
This fiveday workshop is addressed to researchers and advanced graduate students in differential geometry, global analysis, algebra and mathematical physics. Its main focus is on generalized geometry, understood as the study of Courant algebroids and associated geometric structures, as well as on its applications to the study of the systems of partial differential equations and dualities occurring in supergravity and string theory.
The workshop is funded by the Deutsche Forschungsgemeinschaft under Germany's Excellence Strategy  Excellence Cluster 2121 "Quantum Universe" and the Research Training Group 1670 "Mathematics inspired by String Theory and Quantum Field Theory".
The workshop poster is available here.
List of speakers, titles and abstracts:
Luigi Alfonsi (London)
Title: Global Double Field Theory as gerbe geometry
Abstract:
I will review the "Global Double Field Theory is Higher KaluzaKlein Theory" proposal from
[arXiv:1912.07089]. In this framework the doubled space is a higher principal bundle: the bundle
gerbe of KalbRamond field. The invariance under its higher principal action is exactly the (global)
strong constraint and the patching problem of DFT is naturally solved by gluing the doubled
space with a higher group of symmetries. Moreover, the infinitesimal symmetries of the doubled
space are encoded by a familiar Courant algebroid twisted by a gerbe (also known as Extended
Riemannian Geometry). I will also show that this picture gives automatically rise to Tduality and
that the NS5brane is exactly a higher version of the GrossPerry monopole.
Severin Bunk (Hamburg)
Title: From Nonassociative Translations to Symmetries of Gerbes
Abstract:
Gerbes are geometric objects describing the third integer cohomology of a manifold and the Bfield in string theory. They are related to line bundles in many ways, but their structure is significantly richer. Infinitesimal symmetries of gerbes on a manifold M are associated with algebroid structures on M. In this talk, we investigate finite symmetries of gerbes; we demonstrate how, in the presence of a Lie group action on M, gerbes on M lead to higher group extensions which have several applications in geometry and physics.
Gil Cavalcanti (Utrecht)
Title: Stable Generalized Complex Structures with SelfCrossing
Abstract:
Stable generalized complex structures are those for which the anticanonical section is transverse
to the zero section. Many four dimensional examples exist, but in higher dimension the
transversality condition proves to be too strong and rules out interesting manifolds, such as Fano
varieties. To overcome this problem, we weaken the stable condition and allow the zero locus of
the anticanonical section to have self crossings. I will present the basic theory of these structures,
give examples in dimension 4 and higher and review what stable structures with self crossing
can tell us about genuine stable structures.
Martin Cederwall (Gothenburg)
Title: Extended geometry
Abstract:
I plan to, as calmly as possible, walk through the ingredients of extended geometry (structure algebras, modules, physical content, generalised diffeomorphisms, strong constraint, ...) and the fundamental role played by certain superalgebras. Maybe I also have some time to outline some thoughts and plans for the future.
Mario GarcíaFernández (Madrid)
Title: Morita equivalence for string algebroids.
Abstract:
I will present a gaugetheoretic picture for string algebroids,
a special class of holomorphic Courant algebroids introduced in arXiv:1807.10329.
Our approach is based on a new notion of Morita equivalence for these objects.
If time allows, I will comment on some applications to the HullStrominger system
problem which follow from our theory. This is joint work with Roberto Rubio and
Carl Tipler in arXiv2003.
Gueo V. Grantcharov (Florida)
Title: Killing vector fields and parahyperhermitian surfaces
Abstract:
In a 4dimensional vector space with metric of signature (2,2), two null vectors spanning a nullplane determine a canonical action of the split quaternions. We noticed that on an oriented
manifold with two null Killing vector fields spanning an isotropic plane everywhere, the induced
almost parahypercomplex structure is integrable. Based on the classification of compact
complex surfaces this allows to describe the topology of the compact 4manifolds with such
vector fields. In the talk I'll discuss the result and present examples of parahyperhermitian
structures admitting 2 null Killing vector fields on most of these manifolds. If time permits, I'll
explain a reduction procedure for parahyperhermitian structures and how the instanton moduli
space on compact parahyperhermitian surfaces carries a parahyperhermitian structure.
Marco Gualtieri (Toronto)
Title: Quantization of generalized KÃ¤hler structures
Abstract:
I will explain an approach to generalized Kahler metrics in which a holomorphic Poisson
structure may be viewed as a polarization giving rise to a geometric quantization. Applying this
idea to multiples of a symplectic form, we obtain a graded algebra, which unlike the usual case
of a homogeneous coordinate ring, may be noncommutative.
Madeleine Jotz Lean (Göttingen)
Title: Dorfman connections and Courant algebroids
Abstract:
This talk surveys a few results on the notion of connection in the context of Courant algebroids.
Splittings of the standard Courant algebroid over a vector bundle are described by `Dorfman
connections'  in the same manner as linear connections split the tangent Lie algebroid of a
vector bundle. This is used, for instance, in the study of linear generalised complex structures.
Then the case of principal bundles is considered. As principal connections split the Atiyah
sequence of a principal bundle, principal Dorfman connections split the AtiyahDorfman
sequence of a principal bundle. The full picture is here still work in progress, but some aspects of
the theory are presented. This research is partly joint with Malte Heuer and with Jan Nöller.
Magdalena Larfors (Uppsala)
Title: Heterotic perspectives on G2 geometry
Abstract:
Heterotic string compactifications on integrable G2 structure manifolds provide an interesting
class of threedimensional supergravity models. Solutions of this type preserve minimal
supersymmetry, and result in Minkowski or AdS$_3$ geometries. As in the more studied fourdimensional StromingerHull system, the effective field theory describing the threedimensional
supergravity is largely determined by the geometry of the compactification.
In this talk, I will discuss two aspects of these threedimensional supergravity models. Starting
from the mathematical structure of the infinitesimal deformations, captured by a vector bundle
$Q$, with a G2 instanton connection $\check {\cal D}$, I will show how one may determine a
superpotential for the threedimensional supergravity. I will then discuss the relation between
these models and the fourdimensional StromingerHull system, with some remarks on the
connection to generalised geometry.
Ruben Minasian (Saclay)
Title: Beyond generalised geometry
Abstract:
I'll review some recent stringtheoretic computations of quantum corrections to the low energy
actions, and the challenges they present for generalised geometry.
Michela Petrini (Paris)
Title: Systematics of consistent truncations
Abstract:
I will describe how generalised geometry provides a general framework based on a generalized
version of Gstructure to construct consistent truncations with different amount of
supersymmetry. This approach allows to reproduce known truncations and to construct new
ones, where the Gstructure is a truly generalised one.
Pavol Severa (Geneva)
Title: Generalized Laplacian, generalized Ricci flow, and PoissonLie Tduality
Abstract:
To a generalized metric in any Courant algebroid one can associate a natural Laplace operator
acting on halfdensities. Using this operator one can then easily show that PoissonLie Tduality
(a nonabelian generalization of Tduality) is compatible with Ricci flow and, under certain mild
conditions, also with supergravity equations. Based on a joint work with Fridrich Valach.
Thomas Strobl (Lyon)
Title: Sigma models and generalized geometry
Abstract:
Twodimensional sigma models with gauge symmetries are intimately related to generalized
geometry. This applies to the standard notion of gauge theories, where a group is acting on the
target space M of the sigma model, but also to the novel, purely geometric ones where more
general singular foliations can be gauged. The choice of a (possibly small and singular) Dirac
structure give rise to the orbits or leaves that are gauged in the target manifold M of the sigma
model. The generalized metric, which is induced by the metric and the 3form on M, is compatible
with the gauging if it gives rise to what we call a generalized Riemannian foliation (as the name
suggests, it generalizes the classical notion of a Riemannian foliation, as we will explain).
Moreover, there exists a universal gauge theory, which is the (not necessarily topological) Dirac
sigma model. In the purely topological case, it generalizes the Poisson sigma model to arbitrary
Dirac structures, but also any other known nontopological gauged sigma models within the
class described above factor through it.
Richard Joseph Szabo (Edinburgh)
Title: The geometry of double field theory
Abstract:
I will give an overview of recent developments aimed at providing a rigorous framework for
"doubled geometry" in dualityinvariant formulations of string theory, focusing on the
approaches based on paraHermitian geometry and gauged worldsheet sigmamodels in Born
manifolds.
Daniel Waldram (London)
Title: The HullStrominger system, moment maps and generalised geometry
Abstract:
We show that the conditions for solutions of the HullStrominger system can be reformulated
as an integrable SU(3)xSpin(6+n) structure on a suitable Courant algebroid. Geometrically they
correspond to the existence of an involutive subbundle and the vanishing of a moment map for
the action of generalised diffeomorphisms. The space of involutive structures admits a natural
Kahler metric implying that solving the moment map is equivalent to a GIT quotient. The dilation
functional of GarciaFernandez et al plays the role of of the norm functional and there is an
analogue of the Futaki invariant. If time permits we will discuss how analogous structures arise
in the description of (generalisations of) G2 manifolds.
Selected contributions:
Miquel Cueca Ten (Göttingen)
Title: The Cohomology of Courant algebroids and their characteristic classes
Abstract:
Courant algebroids originated over 20 years ago motivated by constrained mechanics but now
play an important role in Poisson geometry and related areas. Courant algebroids have an
associated cohomology, which is hard to describe concretely. Building on work of Keller and
Waldmann, I will show an explicit description of the complex of a Courant algebroid where the
differential satisfies a Cartantype formula. This leads to a new viewpoint on connections and
representations of Courant algebroids and allows us to define new invariants as secondary
charcateristic classes, analogous to what Crainic and Fernandes did for Lie algebroids. This is joint
work with R. Mehta.
Charlotte KirchhoffLukat (Leuven)
Title: Generalized complex branes, coisotropic submanifolds and deformations
Abstract:
I will begin by giving an overview of generalized complex branes as natural submanifolds of generalized complex manifolds and summarising known results on their deformations. After a general introduction, the focus of the talk will be coisotropic Abranes in
symplectic manifolds. It is known that such objects should be additional objects in the Fukaya category. Abranes are coisotropic submanifolds with an additional structure.
I will discuss initial results on the deformation theory of such objects in examples, as well as in comparison to that of coisotropic manifolds.
Marco Zambon (Leuven)
Title: Deformations of Lagrangian submanifolds in bsymplectic manifolds.
Abstract:
bsymplectic manifolds are certain kinds of Poisson manifolds that are symplectic outside a
codimension 1 subset. In many respects they behave similarly to symplectic manifolds. We focus
on Lagrangian submanifolds contained in the singular codimension 1 subset, show that their
deformations are governed by a certain DGLA, and that  unlike the symplectic case 
infinitesimal deformations are generally obstructed. Based on joint work with my Ph.D. student
Stephane Geudens.
Programme:
The talks will start on Monday, March 2, at 10:15 am, and end on Friday, March 6, at 4 pm. Registration will be possible on Monday from 9 am to 10:15 am.
On Tuesday evening at 7 pm, we will hold a
dinner. The registration fee includes the workshop dinner.
Here is the workshop programme.
Venue:
The workshop will take place in Lecture Hall 2 (H2) of the
Department
of Mathematics, Bundesstr. 55, of the University of Hamburg.
Here is a
map.
The closest metro station is "Schlump" on the red line U2 and the
yellow line U3. The train station "Sternschanze" (train lines S11,
S21 and S31) is also in walking distance. See also the
website
for public transport in Hamburg.
Registration and fee payment:
Please fill out this form to register. The registration fee is 50 euros, payable in cash on arrival.
The deadline for registration is 17 February 2020.
Accommodation:
Participants of the workshop are kindly asked to arrange
their accommodation on their own. Here is a list of some hotels.
Organizers:
Vicente Cortés (Hamburg), Liana David (Bucharest), Carlos Shahbazi (Hamburg)
Administrative support:
Heike Wessling, Birgit Mehrabadi
If you have any questions or special needs, please feel free to
contact the organizers at gg2020.math [at] lists [dot] unihamburg [dot] de.
Sponsored by:
