We reconsider the one loop divergence of ${\cal N}=8$ supergravity in four dimensions. We compute the finite effective potential of ${\cal N}=8$ anti-deSitter supergravity and interpret it as logarithmic running of the cosmological constant. We find that quantum inequivalence between fields that are classically dual is due to boundary modes in AdS$_4$. Some subtleties are traced to the difference between the Euler characteristic of global and thermal AdS$_4$.
We compute the coefficient of the potential three-loop divergence in pure N=4 supergravity and show that it vanishes, contrary to expectations from symmetry arguments. The recently uncovered duality between color and kinematics is used to greatly streamline the calculation. We comment on all-loop cancellations hinting at further surprises awaiting discovery at higher loops.
Using Schrödinger functional methods, we show that in the ${\cal N}=4$ SYM/Type IIB Supergravity correspondence the renormalisation of the boundary Newton and gravitational constants arising from bulk fields cancels when we sum over all the Kaluza-Klein modes of Supergravity. This accords with the expected finiteness of ${\cal N}=4$ SYM, and it is expected that other renormalisations cancel in a similar way.
We investigate the occurrence of divergences in maximal supergravity in various dimensions from the point of view of supersymmetry constraints on the U-duality invariant threshold functions defining the higher derivative couplings in the effective action. Our method makes use of tensorial differential equations that constrain these couplings. We study in detail the Fourier modes and wave-front sets of the associated automorphic functions and find that they are always associated to special nilpotent orbits.
We determine the minimum dimension at which divergences first occur in half-maximal 16-supercharge supergravity at three loops. For the four-point amplitudes, we find that the critical dimension is D=14/3 and give the explicit form of the divergence. We also give the divergence of an additional piece that gives us access to theories with higher degrees of supersymmetry, in particular 20- and 32-supercharge supergravities. We give the divergences of half-maximal supergravity when matter vector multiplets are included as well. Explicit forms of various divergences at one and two loops are also g…
We review the question of quantum consistency of N=4 conformal supergravity in 4 dimensions. The UV divergences and anomalies of the standard ("minimal") conformal supergravity where the complex scalar $ϕ$ is not coupled to the Weyl graviton kinetic term can be cancelled by coupling this theory to N=4 super Yang-Mills with gauge group of dimension 4. The same turns out to be true also for the "non-minimal" N=4 conformal supergravity with the action (recently constructed in arXiv:1609.09083) depending on an arbitrary holomorphic function $f(ϕ)$. The special case of the "non-minimal" conformal s…
We demonstrate how duality invariance of the low energy expansion of the four-graviton amplitude in type II string theory determines the precise coefficients of multiloop logarithmic ultraviolet divergences of maximal supergravity in various dimensions. This is illustrated by the explicit moduli-dependence of terms of the form D^{2k}R^4, with k<= 3, in the effective action. Furthermore, we show that in the supergravity limit the perturbative contributions are swamped by an accumulation of non-perturbative effects of zero-action instantons.
We consider super Yang-Mills Theory in $N=1$ conformal supergravity. Using the background field method and the Feddeev-Popov procedure, the quantized action of the theory is presented. Its one-loop effective action is studied using the heat kernel method. We shall develop a non-iterative scheme, generalizing the non-supersymmetric case, to obtain the super heat kernel coefficients. In particular, the first three coefficients, which govern the one-loop divergence, will be calculated. We shall also demonstrate how to schematically derive the higher order coefficients. The method presented here c…
We argue that N=4 supergravity is 3-loop UV finite because the relevant supersymmetric candidate counterterm is known to be SL(2, R)x SO(6) invariant, which violates the Noether-Gaillard-Zumino current conservation. Analogous arguments, based on the universality properties of groups of type E7, also apply to N=5,6,8 in 4,5,7 loops, respectively, since the 1/N BPS invariants break duality symmetry between Bianchi identities and quantum corrected vector field equations.
The ultraviolet divergences of amplitude diagrams in maximal supergravity are investigated using the pure spinor superfield formalism in maximal supergravity, with maximally supersymmetric Yang-Mills theory for reference. We comment on the effects of the loop regularisation in relation to the actual absence of high powers (within the degrees of freedom) of the non-minimal variable r. The absence affects previous results of the field theory description, which is examined more closely (with a new b-ghost) with respect to the limit on the dimension for finiteness of the theory, dependent on the n…
We show that the double-copy structure of gravity forbids divergences in pure half-maximal (16 supercharge) supergravity at four and five points at one loop in D<8 and at two loops in D<6. We link the cancellations that render these supergravity amplitudes finite to corresponding ones that eliminate forbidden color factors from the divergences of pure nonsupersymmetric Yang-Mills theory. The vanishing of the two-loop four-point divergence in D=5 half-maximal supergravity is an example where a valid counterterm satisfying the known symmetries exists, yet is not present. We also give explicit fo…
Using the duality between color and kinematics, we construct the two- and three-loop amplitudes of half-maximal supergravity with matter multiplets and show that new divergences occur in D=4 and D=5. Bossard, Howe and Stelle have recently conjectured the existence of 16-supercharge off-shell harmonic superspaces in order to explain the ultraviolet finiteness of pure half-maximal supergravity with no matter multiplets in D=4 at three loops and in D=5 at two loops. By assuming the required superspace exists in D=5, they argued that no new divergences should occur at two loops even with the addit…
We apply the heat kernel method (using Avramidi's non-recursive technique) to the study of the effective action of chiral matter in a complex representation of an arbitrary gauge sector coupled to background U(1) supergravity. This generalizes previous methods, which restricted to 1) real representations of the gauge sector in traditional Poincaré supergravity or 2) vanishing supergravity background. In this new scheme, we identify a classical ambiguity in these theories which mixes the supergravity U(1) with the gauge U(1). At the quantum level, this ambiguity is maintained since the effectiv…
The ultraviolet divergences of amplitude diagrams in maximal supergravity are characterised by a first possible divergence at seven loops for the 4-point amplitude (logarithmic) and, in its absence, at eight loops. We revisit the pure spinor superfield theory results of [arXiv:1412.5983], stating the absence of the divergence originating in the 4-point 7-loop amplitude as well as those of more than seven loops. The analysis, performed in terms of the one-particle irreducible loop structures giving rise to the divergences, is extended, especially with respect to the limits on the dimension for …
We demonstrate that pure N=4 supergravity is ultraviolet divergent at four loops. The form of the divergence suggests that it is due to the rigid U(1) duality-symmetry anomaly of the theory. This is the first known example of an ultraviolet divergence in a pure ungauged supergravity theory in four dimensions. We use the duality between color and kinematics to construct the integrand of the four-loop four-point amplitude, whose ultraviolet divergence is then extracted by standard integration techniques.
Nonrenormalizable couplings in supergravity-coupled supersymmetric theories can give rise to power-law divergences that destabilize the weak-scale hierarchy. For the case of the standard-model gauge group, the problem can arise in theories with \321 gauge-singlet chiral superfields. The minimal supersymmetric standard model is free from such destabilizing divergences.
Concrete calculations have pointed out that amplitudes in perturbative gravity exhibit unanticipated cancellations taming their ultraviolet behaviour independently of supersymmetry. Similar ultraviolet behaviour of N=4 super-Yang-Mills and N=8 maximal supergravity has explicitly been observed until three loops. These cancellations can be connected to two manifest features of gravitational theories: firstly gauge invariance from diffeomorphism symmetries and secondly that amplitudes are colourless and exhibits crossing symmetry. We will give a simple physical explanation of the cancellations ex…
Before turning to the new result that D=11 supergravity is 2-loop nonrenormalizable, we give a very brief history of the ultraviolet problems of ordinary quantum gravity and of supergravities in general D.
At low energies, interactions of massless particles in type II strings compactified on a torus $T^d$ are described by an effective Wilsonian action $\mathcal{S}(Λ)$, consisting of the usual supergravity Lagrangian supplemented by an infinite series of higher-derivative vertices, including the much studied $\nabla^{4p+6q} \mathcal{R}^4$ gravitational interactions. Using recent results on the asymptotics of the integrands governing four-graviton scattering at genus one and two, I determine the $Λ$-dependence of the coefficient of the above interaction, and show that the logarithmic terms appeari…
The invariants in D=4, N=4 supergravity are discussed up to the three-loop order (where one expects a general R^4 structure). Because there is an anomaly in the rigid SL(2,R) symmetry of this theory, the analysis of possible restrictions on three-loop divergences due to duality needs careful treatment. We show that this anomalous symmetry is still strong enough at the three-loop order to require duality invariance of candidate counterterms. Provided one makes the additional assumption that there exists a full 16-supercharge off-shell formulation of the theory, counterterms at L \ge 2 loops wou…
We study the stability of the gauge hierarchy in hidden-sector supergravity theories. We show that a destabilizing tadpole can appear if a theory has a gauge- and global-symmetry singlet with renormalizable couplings to the visible fields. We also find a quadratically divergent two-loop contribution to the supersymmetric effective potential. This term illustrates the difficulty of using the "LHC mechanism" to control quadratic divergences in theories with Planck-scale vevs.
On-shell Pauli-Villars regularization of the one-loop divergences of supergravity theories is used to study the anomaly structure of supergravity and the cancellation of field theory anomalies under a $U(1)$ gauge transformation and under the T-duality group of modular transformations in effective supergravity theories with three Kähler moduli $T^i$ obtained from orbifold compactification of the weakly coupled heterotic string. This procedure requires constraints on the chiral matter representations of the gauge group that are consistent with known results from orbifold compactifications. Paul…
The invariants in half-maximal supergravity theories in D=4,5 are discussed in detail up to dimension eight (e.g. R^4). In D=4, owing to the anomaly in the rigid SL(2,R) duality symmetry, the restrictions on divergences need careful treatment. In pure N=4 supergravity, this anomalous symmetry still implies duality invariance of candidate counterterms at three loops. Provided one makes the additional assumption that there exists a full 16-supercharge off-shell formulation of the theory, counterterms at L>1 loops would also have to be writable as full-superspace integrals. At the three-loop orde…
We derive a new non-singular tree-level KLT relation for the $n=5$-point amplitudes, with manifest $2(n-2)!$ symmetry, using information from one-loop amplitudes and IR divergences, and speculate how one might extend it to higher n-point functions. We show that the subleading-color ${\cal N}=4$ SYM 5-point amplitude has leading IR divergence of $1/ε$, which is essential for the applications of this paper. We also propose a relation between the subleading-color ${\cal N}=4$ SYM and ${\cal N}=8$ supergravity 1-loop 5-point amplitudes, valid for the IR divergences and possibly for the whole ampli…
Recent calculations have confirmed the belief that the ultraviolet behaviour of N-extended supergravities improves with increasing N. We give a comprehensive review.
We examine the stability of the mass hierarchy in hidden-sector supergravity theories. We show that a quadratically divergent tadpole can appear at two loops, even in minimal supergravity theories, provided the theory has a gauge- and global-symmetry singlet with renormalizable couplings to the visible fields. This tadpole can destabilize the hierarchy. We also find a quadratically divergent two-loop contribution to the field-dependent vacuum energy. This result casts doubt on the efficacy of the "LHC mechanism" for controlling quadratic divergences. We carry out the two-loop calculation in a …
The $D^4\mathcal{R}^4$ and $D^6\mathcal{R}^4$ coefficient functions in the effective action of type II string theory compactified on $T^d$ contain terms of the form $\mathcal{E}_1 {\rm ln}g_d$ and $\mathcal{E}_2 ({\rm ln}g_d)^2$ in specific dimensions, where $g_d$ is the T--duality invariant string coupling, and $\mathcal{E}_1$ and $\mathcal{E}_2$ are U--duality invariant coefficient functions. We derive these non--analytic terms from nested ultraviolet divergences in two and three loop maximal supergravity. For the $D^4\mathcal{R}^4$ coupling, the contribution involves $\mathcal{E}_{\mathcal{…
We establish a direct correspondence between certain higher-rank p-form Chern-Simons topological type theories in the bulk of a manifold with boundary and particular sectors of supergravity models on the boundary, provided that certain boundary conditions are satisfied. The cases we investigate include eleven-dimensional supergravity and both of the type II theories in ten dimensions.
The supersymmetry invariance of flat supergravity (i.e., supergravity in the absence of any internal scale in the Lagrangian) in four dimensions on a manifold with non-trivial boundary is explored. Using a geometric approach we find that the supersymmetry invariance of the Lagrangian requires to add appropriate boundary terms. This is achieved by considering additional gauge fields to the boundary without modifying the bulk Lagrangian. We also construct an enlarged supergravity model from which, in the vanishing cosmological constant limit, flat supergravity with a non-trivial boundary emerges…
${{\cal N}=4}$ supergravity is understood to contain a $U(1)$ anomaly which manifests itself via the nonvanishing of loop-level scattering amplitudes that violate a tree-level charge conservation rule. In this letter we provide detailed evidence that at one loop such anomalous amplitudes can be set to zero by the addition of a finite local counterterm. We show that the same counterterm also cancels evanescent contributions which play an important role in the analysis of ultraviolet divergences in dimensionally regularized gravity. These cancellations call for a reanalysis of the four-loop ultr…
A new action for eleven dimensional supergravity on a manifold with boundary is presented. The action is a possible low energy limit of $M$-theory. Previous problems with infinite constants in the action are overcome and a new set of boundary conditions relating the behaviour of the supergravity fields to matter fields are obtained. One effect of these boundary conditions is that matter fields generate gravitational torsion.
This paper studies the semiclassical approximation of simple supergravity in Riemannian four-manifolds with boundary, within the framework of $ζ$-function regularization. The massless nature of gravitinos, jointly with the presence of a boundary and a local description in terms of potentials for spin ${3\over 2}$, force the background to be totally flat. First, nonlocal boundary conditions of the spectral type are imposed on spin-${3\over 2}$ potentials, jointly with boundary conditions on metric perturbations which are completely invariant under infinitesimal diffeomorphisms. The axial gauge-…
This paper considers eleven dimensional supergravity on a manifold with boundary and the theories related to heterotic $M$-theory, in which the matter is confined to the boundary. New low energy actions and boundary conditions on supergravity fields are derived. Previous problems with infinite constants in the action are overcome. The new boundary conditions are shown to be consistent with supersymmetry, and their role in the ten dimensional reduction and gaugino condensation is briefly discussed.
A brief review is presented of motivations and results in one-loop supergravity on manifolds with boundary, whose consideration is suggested by quantum cosmology.
This paper studies the role of the axial gauge in the semiclassical analysis of simple supergravity about the Euclidean four-ball, when non-local boundary conditions of the spectral type are imposed on gravitino perturbations at the bounding three-sphere. Metric perturbations are instead subject to boundary conditions completely invariant under infinitesimal diffeomorphisms. It is shown that the axial gauge leads to a non-trivial cancellation of ghost-modes contributions to the one-loop divergence. The analysis, which is based on zeta-function regularization, provides a full $ζ(0)$ value which…
We begin with the simplest possible introduction to supergravity. Then we discuss its spin 3/2 stress tensor; these results are new. Next, we discuss boundary conditions on fields and boundary actions for N=1 supergravity. Finally, we discuss new boundary contributions to the mass and central charge of monopoles in N=4 super Yang-Mills theory. All models are in 3+1 dimensions.
We derive the complete orbit of boundary conditions for supergravity models which is closed under the action of all local symmetries of these models, and which eliminates spurious field equations on the boundary. We show that the Gibbons-Hawking boundary conditions break local supersymmetry if one imposes local boundary conditions on all fields. Nonlocal boundary conditions are not ruled out. We extend our analysis to BRST symmetry and to the Hamiltonian formulation of these models.
In this work, we build the most general boundary conditions for $\mathcal N=\left(1,1\right)$ and $\mathcal N=\left(2,2\right)$ extended supergravity. We show that for the loosest set of boundary conditions, their asymptotic symmetry algebras are two copies of the $\mathfrak{osp}\left(1\mid 2\right)_k$ and $\mathfrak{osp}\left(2\mid 2\right)_k$ algebra, respectively. Then, we restrict the gauge fields on the boundary conditions in order to obtain supersymmetric extensions of the Brown-Henneaux and Troessaert boundary conditions. Furthermore, for $\mathcal N=1$ and $\mathcal N=2$ (extended) sup…
Boundary conditions in supergravity on a manifold with boundary relate the bulk gravitino to the boundary supercurrent, and the normal derivative of the bulk metric to the boundary energy-momentum tensor. In the 3D N=1 setting, we show that these boundary conditions can be stated in a manifestly supersymmetric form. We identify the Extrinsic Curvature Tensor Multiplet, and show that boundary conditions set it equal to (a conjugate of) the boundary supercurrent multiplet. Extension of our results to higher-dimensional models (including the Randall-Sundrum and Horava-Witten scenarios) is discuss…
The standard formulation of the AdS/CFT correspondence is incomplete since it requires adding to a supergravity action some a priori unknown boundary terms. We suggest a modification of the correspondence principle based on the Hamiltonian formulation of the supergravity action, which does not require any boundary terms. Then all the boundary terms of the standard formulation naturally appear by passing from the Hamiltonian version to the Lagrangian one. As examples the graviton part of the supergravity action on the product of $AdS_{d+1}$ with a compact Einstein manifold $\cal E$ and fermions…
We consider N=4 SU(2)xU(1) gauged supergravity on asymptotically-AdS_5 backgrounds. By a near-boundary analysis we determine the boundary-dominant components of the bulk fields from their partially gauge-fixed field equations. Subdominant components are projected out in the boundary limit and we find a reduced set of boundary fields, constituting the N=2 Weyl multiplet. The residual bulk symmetries are found to act on the boundary fields as four-dimensional diffeomorphisms, N=2 supersymmetry and (super-)Weyl transformations. This shows that the on-shell N=4 supergravity multiplet yields the N=…
Bulk supergravity on a manifold with boundary must be supplemented by boundary conditions that preserve local supersymmetry. This "downstairs" picture has certain advantages over the equivalent "upstairs" picture, expressed in terms of orbifolds. In particular, Scherk-Schwarz supersymmetry breaking can be described much more simply in the downstairs picture. Nevertheless, physics on the fundamental domain can always be lifted upstairs, so long as fields are allowed to be discontinuous across the boundary. In this talk we apply these considerations to five-dimensional supergravity in a warped R…
We use the duality between color and kinematics to simplify the construction of the complete four-loop four-point amplitude of N=4 super-Yang-Mills theory, including the nonplanar contributions. The duality completely determines the amplitude's integrand in terms of just two planar graphs. The existence of a manifestly dual gauge-theory amplitude trivializes the construction of the corresponding N=8 supergravity integrand, whose graph numerators are double copies (squares) of the N=4 super-Yang-Mills numerators. The success of this procedure provides further nontrivial evidence that the dualit…
The two-dimensional super-BMS$_{3}$ invariant theory dual to three-dimensional asymptotically flat $\mathcal{N}=1$ supergravity is constructed. It is described by a constrained or gauged chiral Wess-Zumino-Witten action based on the super-Poincaré algebra in the Hamiltonian, respectively the Lagrangian formulation, whose reduced phase space description corresponds to a supersymmetric extension of flat Liouville theory.
This paper constructs the reduction of heterotic $M$-theory in eleven dimensions to a supergravity model on a manifold with boundary in five dimensions using a Calabi-Yau three-fold. New results are presented for the boundary terms in the action and for the boundary conditions on the bulk fields. Some general features of dualisation on a manifold with boundary are used to explain the origin of some topological terms in the action. The effect of gaugino condensation on the fermion boundary conditions leads to a `twist' in the chirality of the gravitino which can provide an uplifting mechanism i…
In this paper, we present a systematic analysis of eleven-dimensional supergravity on a manifold with boundary, which is believed to be relevant to the strong coupling limit of the $E_8\times E_8$ heterotic string. Gauge and gravitational anomalies enter at a very early stage, and require a refinement of the standard Green-Schwarz mechanism for their cancellation. This uniquely determines the gauge group to be a copy of $E_8$ for each boundary component, fixes the gauge coupling constant in terms of the gravitational constant, and leads to several striking new tests of the hypothesis that ther…
Using the method of maximal cuts, we obtain a form of the three-loop four-point scattering amplitude of N=8 supergravity in which all ultraviolet cancellations are made manifest. The Feynman loop integrals that appear have a graphical representation with only cubic vertices, and numerator factors that are quadratic in the loop momenta, rather than quartic as in the previous form. This quadratic behavior reflects cancellations beyond those required for finiteness, and matches the quadratic behavior of the three-loop four-point scattering amplitude in N=4 super-Yang-Mills theory. By direct integ…
We consider the bosonic and fermionic symmetries of five-dimensional Maxwell- and Yang-Mills-Einstein supergravity theories on a spacetime with boundaries (isomorphic to M x S1/Z2). Due to the appearance of the "Chern-Simons" term, the classical action is not generally invariant under gauge and supersymmetries. Once bulk vector fields are allowed to propagate on the boundaries, there is an "inflow" governed by the rank-3 symmetric tensor that defines the five-dimensional theories. We discuss the requirements that invariance of the action imposes on new matter content and boundary conditions.
When quantum supergravity is studied on manifolds with boundary, one may consider local boundary conditions which fix on the initial surface the whole primed part of tangential components of gravitino perturbations, and fix on the final surface the whole unprimed part of tangential components of gravitino perturbations. This paper studies such local boundary conditions in a flat Euclidean background bounded by two concentric 3-spheres. It is shown that, as far as transverse-traceless perturbations are concerned, the resulting contribution to $ζ(0)$ vanishes when such boundary data are set to z…
It is shown that the one-loop quadratic divergences of standard supergravity can be regulated by the introduction of heavy Pauli-Villars fields belonging to chiral and abelian gauge multiplets. The resulting one-loop correction can be interpreted as a renormalization of the Kähler potential. Regularization of the dilaton couplings to the Yang-Mills sector requires special care, and may shed some light on chiral/linear multiplet duality of the dilaton supermultiplet.
We investigate the graviton propagator in the type IIB supergravity background which is dual to 4 dimensional noncommutative gauge theory. We assume that the boundary is located not at the infinity but at the noncommutative scale where the string frame metric exhibits the maximum. We argue that the Neumann boundary condition is the appropriate boundary condition to be adopted at the boundary. We find that the graviton propagator behaves just as that of the 4 dimensional massless graviton. On the other hand, the non-analytic behaviors of the other Kaluza-Klein modes are not significantly affect…
We give a complete account of the Schrödinger representation approach to the calculation of the Weyl anomaly of ${\cal N}=4$ SYM from the AdS/CFT correspondence. On the AdS side, the $1/N^2$ correction to the leading order result receives contributions from all the fields of Type IIB Supergravity, the contribution of each field being given by a universal formula. The correct matching with the CFT result is thus a highly non-trivial test of the correspondence.
We discuss local supercurrents as sources of the super-Einstein equations in the superconformal approach in the old and new minimal (auxiliary fields) formulation. Modifications of the Ward identity giving the covariant divergence of the Einstein multiplet are considered in presence of a Fayet-Iliopoulos term. Curvature multiplets can be used as alternative to the gravitino variation in the search for rigid supersymmetric curved backgrounds.
We examine the boundary behaviour of the gauged N=(2,0) supergravity in D=3 coupled to an arbitrary number of scalar supermultiplets which parametrize a Kahler manifold. In addition to the gravitational coupling constant, the model depends on two parameters, namely the cosmological constant and the size of the Kahler manifold. It is shown that regular and irregular boundary conditions can be imposed on the matter fields depending on the size of the sigma model manifold. It is also shown that the super AdS transformations in the bulk produce the transformations of the N=(2,0) conformal supergra…
We use the recently developed generalized double-copy construction to obtain an improved representation of the five-loop four-point integrand of $N = 8$ supergravity whose leading ultraviolet behavior we analyze using state of the art loop-integral expansion and reduction methods. We find that the five-loop critical dimension where ultraviolet divergences first occur is $D_c=24/5$, corresponding to a $D^8 R^4$ counterterm. This ultraviolet behavior stands in contrast to the cases of four-dimensional $N = 4$ supergravity at three loops and $N = 5$ supergravity at four loops whose improved ultra…
Three-dimensional $\mathcal{N}=4$ supersymmetric field theories admit a natural class of chiral half-BPS boundary conditions that preserve $\mathcal{N}=(0,4)$ supersymmetry. While such boundary conditions are not compatible with topological twists, deformations that define boundary conditions for the topological theories were recently introduced by Costello and Gaiotto. Not all $\mathcal{N}=(0,4)$ boundary conditions admit such deformations. We revisit this construction, working directly in the setting of the holomorphically twisted theory and viewing the topological twists as further deformat…
In this paper we perform, in the spirit of the holographic correspondence, a particular asymptotic limit of N=2, AdS_4 supergravity to N=2 supergravity on a locally AdS_3 boundary. Our boundary theory enjoys OSp(2|2) x SO(1,2) invariance and is shown to contain the D=3 super-Chern Simons OSp(2|2) theory considered in [Alvarez:2011gd] and featuring "unconventional local supersymmetry". The model constructed in that reference describes the dynamics of a spin-1/2 Dirac field in the absence of spin 3/2 gravitini and was shown to be relevant for the description of graphene, near the Dirac points, f…
