Supersymmetric marginal deformations yielding a renormalizable SGYM theory

Determine whether supersymmetry-preserving marginal deformations of the three-dimensional supersymmetric Galilean Yang–Mills theory can render it renormalizable and simultaneously preserve Schrödinger symmetry at the quantum level.

Background

One-loop four-point functions exhibit logarithmic divergences that require additional marginal operators beyond those in the undeformed action. The paper studies a restricted class of deformations inherited from a nine-dimensional theory, but finds that this class does not appear sufficient to accommodate all required counterterms while maintaining the desired properties.

The unresolved problem is to analyze the broader space of genuinely three-dimensional marginal terms, determine their supersymmetric completions, and identify whether any resulting theory is both renormalizable and quantum-mechanically Schrödinger invariant.

References

A question that should be dealt with is the fate of supersymmetry-preserving marginal deformations in the three-dimensional theory that render the theory renormalizable. As mentioned previously, it does not appear that one can find a well-defined nine-dimensional supersymmetric extension whose reduction to three dimensions is free of logarithmic divergences, and the structure of divergences means such an approach will not encompass all required counterterms. A more complete analysis of the possible three-dimensional terms that can be added is therefore desirable, though obviously far more complex; the hope would be that within this set lies a theory in which the Schr"odinger symmetry is preserved at the quantum level.

— Quantum Effects in Supersymmetric Galilean Yang-Mills  (2608.18843 - Lambert et al., 19 Aug 2026) in Section 6, “Conclusion and Outlook”

This is a supersymmetric Chern-Simons matter theory which (like three-dimensional SGYM) is classically invariant under the Schr"odinger algebra. All 24 generators of the original theory's superconformal transformations\footnote{Enhanced to 32 supercharges for the non-Lorentzian limit of BLG.} are also (classically) preserved by the limit, and it would be fascinating to see whether this holds once quantum corrections are included. Additionally, it is believed that relativistic three-dimensional $U(N)$ $#1{N}=8$ SYM flows to $U(N)1 \times U(N){-1}$ ABJM in the IR, which for the $N=2$ case considered in this paper is the BLG theory. Whether such a connection holds between the non-Lorentzian limits of the two theories is an intriguing one.

— Quantum Effects in Supersymmetric Galilean Yang-Mills  (2608.18843 - Lambert et al., 19 Aug 2026) in Section 6, “Conclusion and Outlook”