Integral invariants in maximally supersymmetric Yang-Mills theories

TL;DR

This work classifies integral invariants for maximally supersymmetric Yang-Mills theories in dimensions $4\le D\le 10$, revealing three short invariants beyond the action: the single- and double-trace $F^4$ invariants and the double-trace $d^2\mathrm{tr}^2 F^4$. The authors develop the ectoplasm framework to construct closed super-$D$-forms and analyze their cohomology, showing the $F^4$ invariants are CS-type in $D=9,10$ while $d^2\mathrm{tr}^2 F^4$ is a strictly gauge-invariant ten-form whose cocycle matches that of the action. Dimensional reduction demonstrates how these invariants descend to lower dimensions, where the $F^4$ invariants become gauge-invariant super-$D$-forms, and the $d^2\mathrm{tr}^2 F^4$ invariant remains expressible in a gauge-invariant form. In the algebraic renormalisation framework, CS-type invariants are protected, while the $d^2\mathrm{tr}^2 F^4$ invariant is not, and the finite behavior of some multi-loop sectors (e.g., $D=6$) remains a subtle open issue.

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.