# Mathematical-Physics day

When: Wednesday 27 of November

Where: Auditório do IME, 1th Floor, Bloco G, Campus do Gragoatá

Speakers:

- Ákos Nagy (Duke University)
- Antônio Duarte (UFF)
- Daniel Fadel (Unicamp)
- Luis Oxman (UFF)
- Rodrigo Sobreiro (UFF)

Schedule:

- 10:00 - 11:00 : Daniel Fadel
- 11:30 - 12:30 : Luis Oxman

Lunch

- 14:00 - 15:00: Antônio Duarte
- 15:15 - 16:15: Ákos Nagy
- 16:45 - 17:45: Rodrigo Sobreiro

Titles and Abstracts:

- Ákos Nagy

Title: Majorana fermions in Jackiw–Rossi type theories.

Abs: In this talk I will introduce the Jackiw-Rossi equation in 2 dimensions. This is a Dirac-type equation with a conjugate linear perturbation, thus solutions are interpreted as Majorana spinors. Physically, they model electronic excitations on topological superconductors, and the perturbation is governed by the superconducting order parameter.

My main result is a geometric description of the low energy eigenspinors. In particular, I will show that ground states (elements of the kernel of the Jackiw-Rossi operator) concentrate around zeros of the order parameter (at "the vortex").

- Antônio Duarte

Title: Gauge copies and infrared Yang-Mills

Abs: The standard continuum path integral quantization of gauge theories requires the introduction of a gauge-fixing procedure in general. Since 1978 it is known that the usual gauge-fixing prescription due to Faddeev and Popov is plagued by spurious configurations, the so-called Gribov copies. In this talk, I will review a particular framework to deal with such configurations, the refined Gribov-Zwanziger scenario. The fate of BRST symmetry and the consequences of the removal of gauge copies to the description of gluon confinement will be discussed.

- Daniel Fadel

Title: The behavior of sequences of large mass monopoles on AC 3-manifolds.

Abs: Given a noncompact, oriented, Riemannian 3-manifold X, an SU(2) monopole on X is a solution to the so-called Bogomolnyi equation: a first order PDE for pairs consisting of a connection and a Higgs field on the (trivial) SU(2)-bundle over X; it describes a particular kind of critical point of the SU(2) Yang-Mills-Higgs energy functional, and can be further obtained by dimensional reduction from the (anti)-self-dual instanton equation in 4 dimensions.

In this talk I will consider monopoles on an asymptotically conical 3-manifold with one end. In this case the connected components of the moduli space of monopoles are labeled by an integer called the charge, and I will consider the problem of the limiting behavior of sequences of monopoles with fixed charge k and infinitely large masses (or, equivalently, whose sequence of Yang-Mills-Higgs energies is unbounded). I will explain that the limiting behavior of such monopoles is characterized by energy concentration along a finite subset Z of X, consisting of at most k points at which the zeros of the Higgs fields accumulate, and that outside Z the sequence converges uniformly with all derivatives . This is joint work with Gonçalo Oliveira.

- Luis Oxman

Title: 4d ensembles of percolating center vortices and chains

Abs: Ensembles of magnetic defects successfully explain many properties of confinement and are strongly believed to capture the (infrared) YM path-integral measure. In this work, we motivate and propose a measure to compute center element averages over vortices and chains (with non-Abelian d.o.f. and monopole fusion). When center vortices percolate and monopoles condense, we suggest that the average is captured by a saddle point and collective modes in a YMH model. In this manner, flux tubes with N-ality and Lüscher terms are accommodated in an ensemble picture.

- Rodrigo Sobreiro

Title: Remarks on the quantum properties of topological Yang-Mills theory

Abs: We explore the BRST quantization method for the so called topological Yang-Mills (TYM) theory. TYM theory is known to be equivalent to the Donaldson-Witten topological theory, at least at the level of observables. By employing the technique of algebraic renormalization and Feynman diagrams we show that the theory is renormalizable to all orders in perturbation theory and free of radiative corrections. Moreover, we discuss the Gribov problem in such theories and argue the issue of Gribov copies may be irrelevant to TYM theory. The content of the talk can be found in arXiv:1707.06666, arXiv:1805.01850, arXiv:1807.01517, and arXiv:1907.05460.