Integral invariants in maximally supersymmetric Yang-Mills theories

TL;DR

This paper explores integral invariants in maximally supersymmetric Yang-Mills theories across dimensions 4 to 10, identifying special invariants and their cohomological properties with implications for ultraviolet divergences.

Contribution

It identifies three special invariants in all dimensions and analyzes their cohomological properties and implications for renormalization in supersymmetric Yang-Mills theories.

Findings

Identified three special invariants in all dimensions.

Analyzed cohomological properties of super-ten-forms.

Discussed implications for ultraviolet divergences.

Abstract

Integral invariants in maximally supersymmetric Yang-Mills theories are discussed in spacetime dimensions $4\leq D\leq 10$ for $SU(k)$ gauge groups. It is shown that, in addition to the action, there are three special invariants in all dimensions. Two of these, the single- and double-trace $F^4$ invariants, are of Chern-Simons type in $D=9,10$ and BPS type in $D\leq 8$, while the third, the double-trace of two derivatives acting on $F^4$, can be expressed in terms of a gauge-invariant super-$D$-form in all dimensions. We show that the super-ten-forms for $D=10$ $F^4$ invariants have interesting cohomological properties and we also discuss some features of other invariants, including the single-trace $d^2 F^4$, which has a special form in $D=10$. The implications of these results for ultra-violet divergences are discussed in the framework of algebraic renormalisation.