### Speaker

Dr
Daniel Ferrante
(Brown University)

### Description

Lately, it is becoming increasingly clear that extending the
Feynman Path Integral into the Complex domain yields desirable
properties. A first hint in support of such construction can be
seen from the connection between SUSY Quantum Mechanics and the
Langevin dynamics: analytically continuing the Langevin leads into
different SUSY Quantum Mechanical systems (which share the same
algebra of observables). Secondly, new results by E. Witten have
brought forward new results in 3-dimensional Chern-Simons theory
(in the form of Complex geometries) and Super Yang-Mills theory in
four dimensions, as well as the relation between Khovanov homology
and systems of branes.
Given a system of D0-branes, it is possible to understand it in terms
of a Fourier Transform. As such, we can extend this system into the
Complex plane, reinterpreting the Path Integral as a Fourier Transform
over a certain integration cycle, which results in a Fractional Fourier
Transform. This can be further understood in terms of the Phase Space
of this system, where the Fractional Fourier Transform is related to the
Wigner function. In this way, we realize that the label of our Fractional
Fourier Transform, which is the Path Integral quantizing the system, acts
as the parameter determining the vacuum state. Therefore, the allowed
values of this label, for which the Path Integral converges,
determine the quantum phases of the system. This can be immediately
extended to Matrix models and Lie algebra-valued ones
(known as Group Field Theory): the same results hold, so long as
certain properties of the Action are satisfied, guaranteeing the convergence
of the Path Integral.
These results can be dimensionally extended to systems of Dp-branes,
showing some relations with the Geometric Langlands Duality and
Mirror Symmetry. Furthermore, they can also be understood in terms of
coherent state quantization, which opens a window into quantum
tomography, and quantum chaos.

### Summary

The Path Integral will be extended into the Complex plane and presented from the point of view of the Fractional Fourier Transform.

### Primary author

Dr
Daniel Ferrante
(Brown University)

### Co-author

Prof.
Gerald Guralnik
(Brown University)