Newton-Cartan supergravity with torsion and Schrödinger supergravity

The thing worth your time here is the machinery, not the final multiplet. Bergshoeff, Rosseel and Zojer build a non-relativistic analogue of the superconformal tensor calculus: they gauge the d=3, N=2 Schrödinger superalgebra, solve for the dependent gauge fields via conventional constraints, and then construct two compensator multiplets (a scalar and a truncated vector) as non-relativistic limits of relativistic ones. That gives them old minimal and new minimal torsionfull Newton-Cartan supergravity. The structural point that makes the whole exercise work is eq. (75): the spatial part b_a of the dilatation gauge field is proportional to torsion and, unlike the relativistic case, cannot be shifted away by a special conformal transformation. That is a genuine insight, and it explains why the superconformal route naturally lands on torsionfull geometry rather than torsionless geometry plus an option. The truncation in section 5 is a real external benchmark: setting b_a = 0 and matching to the off-shell theory of [27] via eqs. (87)-(88) ties the new construction to prior work rather than leaving it floating. The transformation rules and constraint equations are written out in enough detail that a dedicated reader can check them, which is more than many papers in this genre offer. The soft spots are real but modest. The constraint chain (29) terminates on the assertion that (28) is inert under all symmetries, and the Bianchi identities (30) are used as inputs rather than derived; both are checkable from (38)-(39) and (14), (16), (20), but neither is displayed. The stress-test note is right that this is more load-bearing than the acknowledged δf_μ gap, though I would not call it a flaw so much as a verification burden that the paper's own conventions make tractable. The bigger practical limitation is that there is no action and no equation of motion; the authors say so in the conclusions and explain that a second compensator with non-trivial central charge would be needed. That is honest, and it means the paper is a construction paper, not a tool you can immediately compute with. Who should read it: people working on Newton-Cartan geometry, non-relativistic holography, and supersymmetric backgrounds for condensed-matter systems. If you work in that area, this is the reference for torsionfull NC supergravity and you will cite it. It deserves a serious referee; the algebra is explicit enough that a careful report can pin down the constraint-chain question without the authors having to redo everything.

The paper's central claim is that the constraint set (21)-(22) is consistent and that the chain (29) exhausts its consequences, so that the resulting old/new minimal multiplets are genuine off-shell formulations. The chain terminates at (28): (3/4)ε^{ab}R_{µν ab}(J) = R_{µν}(R), which the paper asserts in section 2.2 'is inert under all symmetries.' That assertion is the linchpin: if (28) is not inert, its variation generates a further constraint not contained in (29), so the set is incomplete and the off-shell closure claim fails. The derivation also invokes the Bianchi identities (30) — R_ab(D)=0, R_{0[ab]}(G)=0, R_{abc}(G)=2R_{0[ab]c}(J), R_ab(S)=0 — as inputs, but does not derive them from the constraints. Both steps are checkable by direct computation from the transformation rules (38)-(39) and the curvature expressions (14), (16), (20). The reader flags the missing fermionic variation of f_µ; that gap is acknowledged in section 2.3 and the paper argues it is not needed for closure on independent fields, which is plausible because f_µ is dependent (eq. 37) and enters the closure parameters only through λ_K. The sharper unverified step is the inertness of (28), not δf.

The reader's verdict of CONDITIONAL with medium correctness risk is reasonable for a paper whose central claim is a consistency statement checked by lengthy algebra that is not fully displayed. I agree with the reader that the consistency of the constraint system is the load-bearing concern, but I would redirect it. The reader's weakest_assumption emphasizes the missing fermionic variation of f_µ. That gap is explicitly acknowledged in section 2.3, and the paper's argument that it is unnecessary for closure on independent fields is plausible: f_µ is a dependent field given by eq. (37), and it enters the closure parameters (41) only through λ_K, which acts on b; the left-hand side of the commutator on b is built from δb (eq. 32), which does not contain f_µ directly. So the δf gap is probably benign. The sharper and less-discussed soft spot is the assertion that (28) is inert under all symmetries, together with the Bianchi identities (30) used as inputs. These are the steps that guarantee the chain (29) terminates; if either fails, the constraint set is incomplete and the off-shell claim is undermined. This is a genuine, checkable gap rather than a disagreement with consensus. The paper does have independent support: the truncation to zero torsion in section 5 reproduces the off-shell theory of [27] via eqs. (87)-(88), which is a real external benchmark, and the structural observation that b_a is proportional to torsion (eq. 75) is a genuine insight. Those strengthen the case but do not substitute for the constraint-chain check. My concern does not move the verdict; it sharpens where the residual risk sits, so UNCHANGED is appropriate.

The paper provides a systematic framework for constructing torsionfull Newton–Cartan supergravity with matter, extending earlier on-shell and torsionless constructions. The explicit transformation rules, the constraint analysis, and the truncation check against [27] are valuable. The conceptual point that torsion arises from the spatial components of the dilatation gauge field, which cannot be shifted away by a special conformal transformation, is a genuine difference from the relativistic case and is likely to be useful for future work on non-relativistic supergravity and holography.

The paper asks whether non-relativistic supergravity — gravity whose geometry is Newton–Cartan rather than Einstein — can be constructed systematically with supersymmetry and with torsion, instead of only by taking careful limits of relativistic supergravity. The authors gauge the three-dimensional $N=2$ Schrödinger superalgebra to obtain a 'superconformal' Schrödinger supergravity, then import the relativistic superconformal tensor calculus: couple that multiplet to a compensator matter multiplet and gauge-fix the extra symmetries (dilatations, the special conformal boost, $S$-supersymmetry). Two compensators — a scalar multiplet and a truncated vector multiplet — produce two off-shell formulations the authors call old minimal and new minimal. The payoff is that torsion appears automatically, because the spatial part $b_a$ of the dilatation gauge field is a dependent field rather than a Stückelberg field that can be shifted away, so the superconformal route lands on torsionfull Newton–Cartan supergravity rather than the torsionless version known from earlier work.

What carries the argument. The load-bearing device is the chain of conventional and un-conventional curvature constraints, eq. (29), together with the explicit solutions for the dependent gauge fields $~omega_~mu{}^{ab}$, $~omega_~mu{}^a$, $b_a$, $f_~mu$ and $~varphi_~mu$ in eqs. (31)–(37). These constraints convert the gauged Schrödinger algebra into a geometry: solving $R_{~mu~nu}{}^a(P)=0$ and $R_{~mu~nu}(Z)=0$ fixes the spin and boost connections, $R_{a0}(H)=0$ fixes $b_a$ and hence the torsion, and the gravitino constraints fix $~varphi_~mu$. The second device is the compensator mechanism transplanted from the relativistic calculus: a matter multiplet whose scalar and spinor are gauge-fixed to constants removes dilatations, $R$-symmetry and $S$-supersymmetry, leaving an off-shell Bargmann supergravity multiplet. The third device is the truncation to zero torsion, $b_a=0$, which turns a conventional constraint into a genuine one and is shown to reproduce the known torsionless off-shell theory, with $r_~mu = -~frac{3}{w}~tau_~mu S$ up to a constant.

Core claim. The central claim is that a non-relativistic analogue of the superconformal tensor calculus exists and works: gauging the $d=3$, $N=2$ Schrödinger superalgebra yields a Weyl-type multiplet $ (b, r_~mu, ~tau_~mu, e_~mu{}^a, m_~mu, ~psi_{~mu+}, ~psi_{~mu-}) $ whose conventional curvature constraints can be solved for the spin connection, the Galilean boost connection, the $S$-supersymmetry gauge field $~varphi_~mu$ and the special conformal gauge field $f_~mu$, with the constraint chain of eq. (29) closing and the commutator algebra of two supersymmetries reproducing general coordinate, rotation, boost, central charge, dilatation, special conformal, $Q_~pm$, $S$ and $R$-symmetry transformations with the parameters listed in eq. (41). Coupling this multiplet to a scalar compensator and gauge-fixing $~phi_1=1$, $~phi_2=0$, $~chi_+=0$ gives an 'old minimal' off-shell multiplet $ (~tau_~mu, e_~mu{}^a, m_~mu, r_~mu, ~psi_{~mu~pm}, ~chi_-, F_1, F_2) $; coupling instead to a truncated vector compensator and gauge-fixing $~varphi=1$, $~lambda=0$ gives a 'new minimal' multiplet $ (~tau_~mu, e_~mu{}^a, m_~mu, r_~mu, ~psi_{~mu~pm}, S) $. In both cases the resulting geometry carries torsion, since $b_a = ~frac{1}{2} e_~mu{}^a ~tau{}^~nu (2~partial_{[~mu}~tau_{~nu]} - ~frac{1}{2} ~bar{~psi}_{[~mu+}~gamma_0 ~psi_{~nu]+})$ is a dependent quantity that cannot be removed by a special conformal transformation, unlike the relativistic case where the whole dilatation gauge field is Stückelberg.

The premise it rests on. Everything rests on the claim that the constraint set of eqs. (21)–(26) is consistent and that the supersymmetry commutator closes on the independent fields, and the authors themselves state in section 2.3 that they did not derive the fermionic variation of the dependent special conformal gauge field $f_~mu$, arguing it is not needed for that closure check.

Off-shell torsionfull Newton–Cartan supergravity multiplets with a definite field content exist in three dimensions, so non-relativistic supersymmetric field theories can in principle be coupled to arbitrary torsionfull Newton–Cartan backgrounds rather than only to flat or torsionless ones.

The relativistic distinction between old minimal and new minimal supergravity survives the non-relativistic construction: the choice of compensator (scalar versus vector) is a real choice of off-shell formulation, not an artifact of how the limit is taken.

Setting the torsion to zero is shown to be a consistent truncation that lands exactly on the previously known torsionless off-shell multiplet, with the $R$-symmetry gauge field then determined by the clock one-form and the auxiliary scalar $S$.

Because the torsion is carried by a dependent field component rather than an independent one, the torsionfull and torsionless theories share the same independent field content in the new minimal case.

The construction provides a template for extending localization-style computations of exact results to non-relativistic supersymmetric theories on curved backgrounds, which the authors list as motivation.

If torsion is always carried by the dependent $b_a$ rather than by an independent field, then 'torsionfull' and 'torsionless' Newton–Cartan supergravities may differ only by which constraint is imposed, not by their degrees of freedom — which would make the two labels less distinct than the terminology suggests.

The authors note that going on-shell with torsion likely requires a second compensator whose scalar transforms non-trivially under central charge, since the scalar and vector multiplets used here both have vanishing central charge weight; that suggests the multiplets derived here are kinematics, and an action principle needs an enlarged field content.

Where both routes apply, the gauging construction here and the non-relativistic limit procedure used previously should agree, so any future disagreement between the two would be a sharp check on one of them.

The algebra used is described as existing only in odd dimensions, so a four-dimensional analogue probably cannot be reached by the same gauging and would need the limiting route the authors propose instead.

This review was written by a machine. No person has reviewed this paper on Pith yet. A scientist listed on Pith can review it for a fee, and their review is published here whatever it concludes.

Request a human review