In physics, an "anomaly" is the failure of a classical symmetry at
the quantum level. Anomalies play a key role in assessing the
consistency of a quantum field theory, and link up with cohomology
in mathematics, a general tool by which mathematicians understand
whether a desired construction is possible. In this informal series
of talks, we aim to understand what physicists mean by an "anomaly"
and their mathematical interpretation.
We define a gerbe, and show gerbes can be "transgressed" to give
line bundles over loop space. Trivial gerbes give trivial bundles
on loop space, whose sections are thus mere functions. Any compact,
simply connected Lie group comes with a god-given gerbe whose
curvature is the canonical invariant 3-form. Restricting this gerbe
to certain submanifolds, we get trivial gerbes who thus transgress
to trivial line bundles, "cancelling" the anomaly of a nontrivial
line bundle.
We briefly review spin foam state sums for triangulated
manifolds and motivate the introduction of state sums based on
2-groups. We describe 2-BF gauge theories and the construction of
the corresponding path integrals (state sums) in the case of
Poincaré 2-group.
References
J. F. Martins and A. Mikovic, Lie crossed modules and
gauge-invariant actions for 2-BF theories, Adv. Theor. Math.
Phys. 15 (2011) 1059,
arxiv:1006.0903
A. Mikovic and M. Vojinovic, Poincaré 2-group and quantum
gravity, Class. Quant. Grav. 291 (2012) 165003, arxiv:1110.4694
We will introduce the notion of stable isomorphism for gerbes, and
talk about how stable isomorphism classes are in one-to-one
correspondence with Deligne cohomology classes. We define WZW
branes and discuss how the basic gerbe on a group trivializes when
restricted to the brane.
This series of lectures will be a gentle introduction to quantum
field theory for mathematicians. In our first lecture, we give a
lightning introduction to quantum mechanics and discuss the
simplest quantum system: the harmonic oscillator. We then sketch
how this system is used to quantize the free scalar field.
We continue our gentle introduction to quantum field theory for
mathematicians. We discuss the Klein-Gordon equation, and how it
decomposes into oscillators. We quantize this system by quantizing
the oscillators, obtaining the free scalar field, the simplest
quantum field there is.
Last time, we talked about quantization of the free scalar field by
replacing the modes of the field by quantum oscillators. Now, we
put this field into the form used by physicists, and talk about the
Wightman axioms, which allow a rigorous treatment of free fields.
Several exceptional geometric structures in dimensions 5, 6, and 7
are related in a striking panorama grounded in the algebra of the
octonions and split octonions. Considering strictly nearly Kähler
structures in dimension 6 leads to prolonging the Killing-Yano (KY)
equation in this dimension, and the solutions of the prolonged
system define a holonomy reduction to a group of exceptional type
of a natural rank-7 vector bundle, which can in turn be
realized as the tangent bundle of a pseudo-Riemannian manifold,
which hence relates this construction to exceptional metric
holonomy. In the richer case of indefinite signature, a suitable
solution of the KY equation can degenerate along a (hence
5-dimensional) hypersurface , in which case it partitions
the underlying manifold into a union of three submanifolds and
induces an exceptional geometric structure on each. On the two open
manifolds (which have common boundary ), defines
asymptotically hyperbolic nearly Kähler and nearly para-Kähler
structures. On itself, determines a generic
-plane field, the type of structure whose equivalence problem
Cartan investigated in his famous Five Variables paper. The
conformal structure this plane field induces via Nurowski's
construction is a simultaneous conformal infinity for the nearly
(para-)Kähler structures.
This project is a collaboration with Rod Gover and Roberto
Panai.
Jarvis and Meekin have shown that the classical Fermat equation
\(x^p + y^p = z^p\) has no non-trivial solutions over
\(\mathbb{Q}(\sqrt{2})\). This is the only result available over
number fields. Two major obstacles to attack the equation over
other number fields are the modularity of the Frey curves and the
existence of newforms in the spaces obtained after level
lowering.
In this talk, we will describe how we deal with these
obstacles, using recent modularity lifting theorems and level
lowering. In particular, we will solve the equation for infinitely
many real quadratic fields.
We describe Baez and Dolan's cobordism hypothesis - a deep
connection between topological quantum field theory, higher
categories, and manifolds. Physically, this encodes the idea that
quantum field theories, even "topological" ones, should be local:
no matter how we cut up the spacetime on which they are defined in
order to perform the path integral, the net result must be the
same. Recently, this hypothesis was formulated and proved by Jacob
Lurie using the tools of homotopy theory. We describe the version
of the hypothesis he proved. Finally, we touch on Freed, Hopkins,
Lurie and Teleman's recent work on Chern-Simons theory, and on Urs
Schreiber's ideas for using Lurie's toolkit in full-fledged quantum
field theory.
I will present some recent results on the dimensional extension of
pseudo-differential operators. Using this formalism it is possible
to generalize the standard Weyl quantization and obtain, in a
systematic way, several phase space (operator) representations of
quantum mechanics. I will present the Schrodinger and the Moyal
phase space representations and discuss some of their properties,
namely in what concerns the relation with deformation quantization.
The phase space given by the cotangent bundle of a Lie group appears in the context of several models for physical systems. In quantum mechanics on the Euclidean space, the standard Fourier transform gives a unitary map between the position representation — functions on the configuration space — and the momentum representation — functions on the corresponding cotangent space. That is no longer the case for systems whose configuration space is a more general Lie group. In this talk I will introduce a notion of Fourier transform that extends this duality to arbitrary Lie groups.
We will give an overview of the renormalization procedure in Quantum Field Theory. The emphasis will be on the general idea of constructing a finite QFT from the one plagued by divergencies, in the standard perturbative approach, and discussing the uniqueness of the resulting QFT. The lecture does not assume much background knowledge in QFT, and should be accessible to a wide audience.