Quantum field theory is a highly successful and experimentally well-tested framework for describing elementary particles and the four fundamental forces of nature. In quantum field theory, the primary objects of interest are scattering amplitudes. These amplitudes are plagued by ultraviolet divergences at high energies. Enhancing the symmetry in the theory improves the ultraviolet properties. Supersymmetric quantum field theories are of great interest in this context as they exhibit dramatically improved ultraviolet behavior. The theme of my current research, thus, focuses on studying the aspects of maximally supersymmetric theories, particularly concerning N=4 super Yang-Mills theory in four dimensions.

Supersymmetry without anti-commuting variables

Pure supersymmetric Yang-Mills theories exist in D = 3,4,6 and 10 dimensions. The maximally extended N=4; D=4 theory is ultraviolet finite and conformally invariant. It is related under the AdS/CFT correspondence to type II superstring theory compactified on a 5-sphere to an asymptotically anti-de Sitter space-time. All these links motivate us to investigate supersymmetric Yang-Mills theories and their properties from every possible angle.

Supersymmetric theories can be formulated without the use of anti-commuting variables. In this approach, supersymmetric gauge theories are characterized by a Nicolai map – a transformation of the bosonic fields such that the Jacobian determinant of the transformation exactly cancels against the product of the Matthews-Salam-Seiler (MSS) and Faddeev-Popov determinants. The formalism avoids any use of anti-commuting objects thus offering an alternate perspective on the physics of gauge theories.

We took up an old thread of development concerning the characterization of supersymmetric theories without any use of anticommuting variables. Our special focus was on the formulation of supersymmetric Yang-Mills theories, extending previous results beyond D = 4 dimensions.

We derived for the first time the map up to the second order in the coupling constant and extended it to all pure supersymmetric Yang-Mills theories. The above construction makes use of a method called R prescription. As a new result, we recovered the classical result that interacting pure supersymmetric Yang-Mills theories can exist only in space-time dimensions D = 3, 4, 6, 10. The above result is quite remarkable in the sense that we can entirely formulate supersymmetric Yang-Mills theories in these space-time dimensions without using anti-commuting variables.
We also extended the map and the framework itself to the third order in the coupling constant by deriving an explicit formula for the transformation. The Feynman-like graphical approach was outlined using tree diagrams, which in principle allow one to extend the construction to all orders in the coupling constant.
We addressed the uniqueness of the Nicolai map by finding a new transformation, also to the third order in the coupling constant, but valid exclusively in six dimensions. We arrived at the map by starting with an educated guess. This new map is simpler than the one obtained using the R operator thus highlighting a potential ambiguity in the formalism.

I plan to study and understand how the symmetries in the theory are reflected in this new `mapped’ formalism. This approach yields new insights beyond those obtained using the traditional formulation: specifically for objects of interest to a physicist, like scattering amplitudes and correlation functions

Scattering amplitudes and Soft theorems

Scattering amplitudes in a quantum field theory carry the physical information in the theory. The light-cone gauge is ideal for the study of the scattering amplitudes. It focuses solely on physical degrees of freedom because it is not manifestly covariant and local. The spinor helicity variables which are the basic building block of the scattering amplitudes emerge naturally in this gauge.

In gauge theories and gravity, the scattering amplitudes of any number of external particles under the soft limit reduce to amplitudes with one less number of external particles times a universal soft factor. The inverse soft method in recent years has been developed to construct amplitudes in gauge theory and gravity. The idea is to build higher-order amplitudes by multiplying the lower-order amplitudes with a universal factor associated with the emission of a soft boson.

The universal factors associated with the emission of a soft boson in Yang-Mills theory and gravity are extracted from the respective light-cone actions. We used these soft factors to construct higher-point interaction vertices for gauge theory and gravity. We extended the idea of the inverse soft method for higher-spin theories. As a new result, we first proved that the universal soft factor associated with the emission of a higher-spin boson is not gauge invariant. We then argue how the gauge invariance of the soft factor is related to the mutual consistency of different BCFW constructions of four-particle amplitudes and show that the higher-spin theories are not constructible.
N=8 supergravity is a maximally supersymmetric theory of gravity; it is possibly ultraviolet finite to all orders. The proof for the perturbative finiteness for N=4 super Yang-Mills (by power counting) exists only in the light-cone gauge. We studied the soft theorems in the light-cone superspace for maximally supersymmetric theories and constructed the quartic vertex for N=8 supergravity. The next step in this program is to analyze the finiteness properties of N=8 supergravity in the light-cone superspace.

Higher spins as a quadratic form

The light-cone Hamiltonian of pure and maximally supersymmetric Yang-Mills theories and gravity can be expressed as quadratic forms Simple structures like quadratic forms are interesting because they often signal the presence of a hidden symmetry or (and) produce considerable mathematical simplifications in the way we formulate these theories. We have shown that higher-spin theories with and without supersymmetry can be written as a quadratic form.

This “complete square” structure, highly reminiscent of the Nicolai map, is likely related to a light-cone realization of T_g. We aim to understand these structures further and make this relationship concrete.