Perturvative Quantum Field Theory
Asymptotic symmetries and integrable systems in gravity
Cobordism and K-theory in the string swampland
The topology of T-duality
Marco Gualtieri (Toronto)
Groupoids and Stacks in Generalized Geometry
Introductory courses
Sylvie Paycha (Postdam)
Prerequisites in Mathematics
Diego Mauricio Gallego (Tunja)
Prerequisites in Physics
Marcela Cárdenas Lisperguer, Universidad de Santiago de Chile, Chile.
Title: Asymptotic symmetries and integrable systems in gravity.
Abstract: This mini-course will offer basic tools to study asymptotic
symmetries in gravity, and it will conclude with a recent result that
relates these symmetries to integrable systems. In order to do that, we
first construct the result by J.D. Brown and M. Henneaux for 3D gravity
with negative cosmological constant, that leads us to find the two
copies of the Virasoro algebra. We will study their role in the
statistical properties of black holes. From that point, we will present
the canonical tools that allow the generalization of such asymptotic
conditions to include integrable systems. Special attention will be paid
to the integrable system of Korteweg-de Vries (KDV), and the relation
between its generators of symmetry and the ones associated to the
Virasoro algebra. We will show that gravity may as well contain the
integrable system of Ablowitz-Kaup-Newell-Segur (AKNS) as asymptotic
conditions that lead to an infinite set of abelian charges. This
integrable system includes the KDV hierarchy, Sine-Gordon, non-lineal
Schrödinger, among others. I will suggest a geometric interpretation of
those equations in the space-time.
Marco Gualtieri, University of Toronto, Canada.
Title: Groupoids and stacks in generalized geometry.
Abstract: This will be an introductory series of talks on generalized
geometry, focusing on generalized complex and Kähler structures, which
have seen applications in the physics of 2-dimensional sigma models. I
hope to explain how these structures provide simple examples of shifted
symplectic stacks, through their relation to symplectic groupoids. I
will assume that students have a familiarity with complex and symplectic
structures. To prepare for this minicourse, it would be useful to learn
the definition of a Poisson structure and the related concept of Dirac
structure, which I will review at the start of the course.
Oscar Loaiza Brito, University of Guanajuato, Mexico.
Title: Cobordism and K-theory in the string swampland
Abstract: In this course we shall first review the notes of bordism and
bordism groups and their relation to the cobordism conjecture in string
theory, which states that there is a correspondence between effective
theories - extendable to quantum gravity- and trivial cobordism groups.
By the use of the Atiyah-Hirzebruch Spectral Sequence (AHSS), we shall
also relate the cobordism groups to K-theory, which in turn classifies
the charges of objects as D-branes and non-BPS states. By these means,
we shall study a conexion between the cobordism conjecture and the
global symmetry conjecture, which states the absence of any global
symmetry. Finally we shall extend these results to the presence of
orientifold planes inducing an involution map in the AHSS and the
corresponding relation between cobordism and K-theory.
Kasia Rejzner, University of York, England.
Title: Perturbative quantum field theory.
Abstract: Quantum field theory (QFT) is the theory underlying particle
physics, which combines quantum mechanics and special relativity. Since
the mathematical foundations of QFT are not completely settled, there
are many misconceptions about which aspects of the theory can be made
mathematically precise and which not. In this series of lectures, I will
introduce a rigorous approach to perturbative quantum field theory,
called perturbative algebraic quantum field theory (pAQFT) and show how
it can be applied in a range of physically relevant examples, including
gauge theories and effective quantum gravity. The very nature of the
topic calls for the use of a combination of mathematical and physics
prerequisites, to which I will dedicate part of the lectures that should
therefore be accessible to both mathematics and physics students.
Thomas Schick, University of Göttingen, Germany.
Title: The topology of T-duality.
Abstract: Inspired by symmetries expected in certain models of quantum
field theory and string theory, T-duality is the idea that two different
space-time manifolds (with extra data) should be physically equivalent.
Mathematically, this is modelled by the idea that over a given base
manifold M a first circle bundles
E->M (together with extra the extra information of a so-called H-flux B)
and a second circle bundle F->M (with flux C) might be T-dual to each
other. The motivation for this particular structure of the space-time
manifold is from string theory: we expect a high-dimensional space-time
manifold, but certain directions are rolled up and compactified
(modelled by the circle directions of the circle bundle). The T-duality
between (E,B) and (F,C) should induce an isomorphism between the twisted
K-theory of E (twisted by B) and of F (twisted by C). These twisted
K-theory groups model physically relevant charges on the background
space-time manifolds E and F, respectively.
The course will start with the introduction of concepts like circle
bundles and K-theory of topologial spaces. It will then pass on to the
concept of twists and twisted K-theory, an important variant of the
classical notion. It should be noted that the set of isomorphism classes
of twists is canonically isomorphic to the third cohomology of the space
(with integer coefficients), but twists themselves are a more subtle
concept, involving also some basic ideas from (higher) category theory
which will will discuss as well.
We will the introduce the concept of T-duality and study when two pair
(E,B) and (F,C) are dual to each other, and how many T-duals there are.
The main results in this direction: the introduction of different
descriptions of T-duality and the proof that they are equivalent, and
the classification of T-duality pairs.
We will construct the T-duality transformation from twisted K-theory of
E to twisted K-theory of F and prove that it is indeed an isomorphism.
We will also try to cover more general and more sophisticated
situations, e.g. situations where the bundles have higher dimensional fibers or have singularities.