Quantum Effects in Supersymmetric Galilean Yang-Mills

Abstract: We study the quantum dynamics of supersymmetric Galilean Yang-Mills (SGYM), with a focus on three dimensions where there is a classical scale symmetry. This is believed to be the worldvolume theory describing D2-branes within type IIA non-relativistic string theory and arises as a non-relativistic limit of maximally supersymmetric (relativistic) Yang-Mills. Unlike the relativistic case, we find that SGYM only admits perturbative dynamical excitations on the Coulomb branch. We show that in three dimensions the classical scale symmetry is broken at one-loop due to logarithmic divergences. These cannot be regulated without introducing additional marginal deformations, and we present evidence that the theory is asymptotically free once these are included. In particular our analysis suggests that three-dimensional SGYM flows to a strongly coupled non-relativistic M2-brane theory.

• The paper shows that perturbative dynamics in three-dimensional supersymmetric Galilean Yang–Mills exist only on the Coulomb branch, where off-diagonal fields acquire Schrödinger-type propagators while Cartan modes remain nondynamical.
• The paper finds that one-loop logarithmic divergences break classical scale invariance and cannot be absorbed into operators in the original action, forcing new classically marginal deformations for consistent quantization.
• The paper identifies a one-loop beta function, μ dẽ/dμ = −ẽ³/π, indicating asymptotic freedom and suggesting an infrared flow toward a strongly coupled non-relativistic M2-brane theory, although full supersymmetric renormalizability remains unresolved.

Overview

The paper studies the quantum dynamics of supersymmetric Galilean Yang-Mills (SGYM), a non-Lorentzian gauge theory obtained either as the null reduction of maximally supersymmetric Yang-Mills in $(d+2)$ spacetime dimensions or as the non-relativistic string limit of D-brane worldvolume theories in type II string theory (2608.18843). The central focus is the three-dimensional theory, which is classically invariant under the Schrödinger algebra with $z=2$ Lifshitz scaling and is conjectured to be the worldvolume theory of D2-branes within non-relativistic type IIA string theory. The main results are threefold: perturbative dynamics exist only on the Coulomb branch; the classical scale symmetry is broken at one loop by logarithmic divergences that cannot be absorbed into counterterms for operators already present in the action; and once the required additional marginal deformations are included, evidence emerges for asymptotic freedom, suggesting an infrared flow toward a strongly coupled non-relativistic M2-brane theory.

Classical structure and symmetries

The action contains a gauge field $A_\mu$, adjoint scalars $\{X, Y^A\}$, and Majorana-Weyl spinors projected by $\Gamma_{0X}$, with interactions of the schematic form $D_0 X D_0 X - 2D_iX F_{0i} - \tfrac12 F_{ij}F_{ij}$ plus commutator couplings. Two derivations are given: a rescaling limit of relativistic SYM mimicking the non-relativistic string limit, and null reduction of $(d+2)$-dimensional MSYM. The energy functional has positive-definite bosonic terms, so the classical theory is well-defined despite its unconventional derivative structure.

A key structural feature is that kinetic terms for all components arise only through couplings to $X$. Expanding about a Coulomb branch vacuum $X = v\sigma$, only components not commuting with $\sigma$ acquire Schrödinger-type propagators of mass $z=2$0; components along the Cartan direction are non-dynamical and mediate instantaneous interactions. Consequently, the theory admits no perturbative excitations away from the Coulomb branch, a point the authors state plainly: recovering the full Schrödinger symmetry by setting $z=2$1 would require control of the strongly coupled regime, since the invariant coupling $z=2$2 diverges there.

On the symmetry side, the Coulomb branch boundary condition on $z=2$3 breaks dilatations and special conformal transformations except in $z=2$4, where the full Schrödinger algebra survives together with an $z=2$5 R-symmetry and 24 supercharges inherited from the superconformal symmetry of four-dimensional $z=2$6 SYM. The supersymmetry algebra closes only up to gauge transformations, with closure on $z=2$7 requiring the field's own equation of motion — a standard pattern for such constructions.

One-loop quantization

Quantization proceeds via the background field method in 't Hooft–Feynman gauge, with gauge-fixing function $z=2$8. A technical subtlety specific to first-order-in-time systems is handled by a temporal point-splitting prescription on the Schrödinger propagator $z=2$9, which enforces vanishing of equal-time contractions and of loop-energy integrals with all propagators aligned along the loop momentum. This identity eliminates most candidate diagrams: the one-, two-, and three-point functions vanish identically at one loop, with power-law divergences cancelling through supersymmetric matching between bosonic and fermionic degrees of freedom.

The first genuine quantum corrections appear in four-point functions. For charged external gauge fields $A_\mu$0, the linearly divergent pieces cancel but a logarithmic divergence remains:

$A_\mu$1

Crucially, this index structure cannot be reproduced by renormalizing any quartic term already present in the bare action while preserving gauge invariance. The analogous scalar four-point function yields a divergence proportional to the existing $A_\mu$2 tensor structure, but the coefficients of those terms are tied by $A_\mu$3 covariance to neutral-sector terms that provably receive no quantum corrections; hence these counterterms also cannot be absorbed within the original action. The conclusion is that the bare SGYM action is not renormalizable as written: consistent quantization forces the inclusion of additional classically marginal operators built from powers of the dimensionless scalar $A_\mu$4. The authors note explicitly that their result relies on the temporal point-splitting regulator, and that checking regulator-independence is left open.

Marginal deformations and asymptotic freedom

Rather than enumerating higher-point functions, the authors study a class of deformations descending from eight spatial dimensions:

$A_\mu$5

which is positive-definite for positive $A_\mu$6. Focusing on the logarithmic divergence of the charged scalar four-point function — which depends only on bosonic couplings and is therefore insensitive to the unknown supersymmetric completion — the deformation modifies the divergence to a factor $A_\mu$7, which vanishes exactly at $A_\mu$8. This is a striking result: a specific deformation can cancel the leading logarithmic divergence outright.

However, consistency requirements complicate the picture. Matching the deformed supersymmetry algebra's closure constraints with the equations of motion requires extra $A_\mu$9 terms; once these are included, the divergence becomes a perfect square with no root for positive $\{X, Y^A\}$0, though its sign preserves the beta-function structure. Additional three-dimensional potential terms can restore cancellation at $\{X, Y^A\}$1, but the $\{X, Y^A\}$2 term needed lacks a known supersymmetric completion without introducing a $\{X, Y^A\}$3-quadratic term that would alter the propagator structure and likely break supersymmetry. The paper thus identifies a concrete tension: supersymmetry, boundedness of the energy, and UV-finiteness cannot apparently be satisfied simultaneously within this nine-dimensional reduction scheme.

Working directly with the Abelian effective theory and promoting the quartic coupling to run, dimensional regularization gives the one-loop beta function

$\{X, Y^A\}$4

i.e., asymptotic freedom in the invariant coupling $\{X, Y^A\}$5. The authors qualify this carefully: verifying UV behavior of all couplings introduced by the full supersymmetric deformation set remains to be done, though they express hope that supersymmetry guarantees it.

Limitations and open questions

Several caveats are stated in the paper itself. All explicit computations use $\{X, Y^A\}$6; extension to general gauge group and large $\{X, Y^A\}$7 — essential for holographic applications — awaits a more covariant formulation of the Coulomb-branch perturbation theory. It is possible the observed divergences are subleading in $\{X, Y^A\}$8 and invisible in the supergravity dual, which would reconcile quantum symmetry breaking with the classical scaling symmetry of the geometry. The regulator-dependence of the four-point divergences is unverified. Whether the nine-dimensional deformation class suffices for renormalizability is doubtful given the differing index structures of the gauge and scalar divergences, and a complete three-dimensional analysis is outstanding. Finally, the backreacted D2-brane solution of non-relativistic IIA supergravity preserves only 8 Killing spinors, suggesting the deformation required on the gravity side must break the $\{X, Y^A\}$9 supersymmetry — a mismatch whose field-theory counterpart is unresolved.

Conclusion

This work provides the first quantization of a non-Abelian supersymmetric Galilean gauge theory. Its principal findings — the necessity of the Coulomb branch for perturbative dynamics, the one-loop breaking of classical scale invariance via logarithmic divergences requiring new marginal operators, and asymptotic freedom pointing toward a strongly coupled non-relativistic M2-brane fixed point — establish SGYM as a controlled laboratory for quantum effects in non-Lorentzian field theory. The analogy with two-dimensional sigma models, where dimensionless scalars generate rich families of marginal deformations constrained by conformal invariance, suggests that background configurations cancelling loop divergences may play a distinguished role in the string-theoretic embedding. The relation to the M2-brane limit of BLG/ABJM, supported both by the spacetime embedding via circle reduction and by the expected IR flow, poses a concrete question the paper leaves open: whether the non-Lorentzian limits of three-dimensional SYM and ABJM are connected by an RG flow, mirroring their relativistic counterparts.