Supersymmetry Variations Research Articles

The (1,0) tensor and hypermultiplets in loop space

We show that the (1,0) tensor and hypermultiplet supersymmetry variations can be uplifted to loop space. Upon dimensional reduction we make contact with abelian five-dimensional super Yang-Mills, which has a nonabelian generalization that we subsequently uplift back to loop space where we conjecture a nonabelian generalization of the (1,0) supersymmetry variations and demonstrate their on-shell closure.

Rounding out the story of higher derivative consistent truncations

At the two-derivative order, the group manifold reduction of heterotic supergravity on S3 results in a half-maximal 7D gauged supergravity coupled to three vector multiplets, and a further truncation can be taken to remove the vector multiplets. We demonstrate that this truncation remains consistent at the four-derivative level; we do so both by analysis of the equations of motion and the supersymmetry variations.

Supersymmetry of the D3/D5 defect field theory

Four-dimensional mathcal{N} = 4 super Yang-Mills, with a codimension-one defect breaking half of the supersymmetry, arises as the field theory description of the D3/D5 intersection in the holographic limit. This is one of the earliest, most extensively studied, and commonly used systems in holography. In this note we give the full R-symmetry-covariant supersymmetry variations for this system. We also provide the supercurrents and compute the algebra of the corresponding supercharges, obtaining the full set of central charges. We show that magnetically charged finite-energy field configurations preserving half of the supersymmetry are solutions to a new form of the extended Bogomolny equations, in which the defect fields play the role of jumping data for the Nahm-like part of the equations. In the appendices, we explain the connection between our results and the superspace-based formulations in the literature.

Janus and RG-flow interfaces in three-dimensional gauged supergravity

In this paper, we construct Janus-type solutions of three-dimensional gauged supergravity with sixteen supersymmetries. We find solutions which correspond to interfaces between the same CFT on both sides, as well as RG-flow interfaces between CFTs with different numbers of supersymmetries and central charges. The solutions are obtained by solving the flow equations derived from the supersymmetry variations, and they preserve some fraction of the supersymmetries of the AdS3 vacua.

Non-relativistic ten-dimensional minimal supergravity

We construct a non-relativistic limit of ten-dimensional mathcal{N} = 1 supergravity from the point of view of the symmetries, the action, and the equations of motion. This limit can only be realized in a supersymmetric way provided we impose by hand a set of geometric constraints, invariant under all the symmetries of the non-relativistic theory, that define a so-called ‘self-dual’ Dilatation-invariant String Newton-Cartan geometry. The non-relativistic action exhibits three emerging symmetries: one local scale symmetry and two local conformal supersymmetries. Due to these emerging symmetries the Poisson equation for the Newton potential and two partner fermionic equations do not follow from a variation of the non-relativistic action but, instead, are obtained by a supersymmetry variation of the other equations of motion that do follow from a variation of the non-relativistic action. We shortly discuss the inclusion of the Yang-Mills sector that would lead to a non-relativistic heterotic supergravity action.

Novel 3D supersymmetric massive Yang-Mills theory

We construct three-dimensional, $\mathcal{N}=1$ off-shell supersymmetric massive Yang-Mills (YM) theory whose YM equation is ``third-way'' consistent. This means that the field equations of this model do not come from variation of a local action without additional fields, yet the gauge-covariant divergence of the YM equation still vanishes on shell. To achieve this, we modify the massive Majorana spinor equation so that its supersymmetry variation gives a modified YM equation whose bosonic part coincides with the third-way consistent pure massive YM model.

S -duality and loop operators in canonical formalism

We study the gauge invariant 't Hooft operator in canonical formalism for Yang-Mills theory as well as the $\mathcal{N}=4$ super-Yang-Mills theory with the gauge group $U(N)$. It is shown that the spectrum of the 't Hooft operator labeled by the arbitrary irreducible representation of the gauge group is the same as the spectrum of the Wilson operator labeled by the same representation. So it is possible to construct a unitary operator $S$ making the two kinds of loop operators transformed into each other. S-duality transformation could be realized by the operator $S$. We compute the supersymmetry variations of the loop operators with the fermionic couplings turned off. The result is consistent with the expectation that the action of $S$ should make supercharges transform with a $U(1{)}_{Y}$ phase.

Minimal D = 4 mathcal{N} = 2 supergravity from D = 11: an M-theory free lunch

ABSTRACTWe present a new consistent truncation of D = 11 supergravity to D = 4 mathcal{N} = 2 minimal gauged supergravity, on the seven-dimensional internal Riemannian space corresponding to the most general class of D = 11 solutions with an AdS4 factor and mathcal{N} = 2 supersymmetry. A truncation ansatz is proposed and its consistency checked at the level of the D = 11 Bianchi identity, bosonic equations of motion, and supersymmetry variations of the gravitino. The general class includes an mathcal{N} = 2 AdS4 solution dual to the conformal, low-energy physics phase corresponding to a mass deformation of the M2-brane field theory. A consistent truncation recently constructed on this particular geometry is recovered from our formalism.

SCFT/VOA correspondence via \u03a9-deformation

We investigate an alternative approach to the correspondence of four­ dimensional mathcal{N} = 2 superconformal theories and two-dimensional vertex operator algebras, in the framework of the Ω-deformation of supersymmetric gauge theories. The two­dimensional Ω-deformation of the holomorphic-topological theory on the product four­ manifold is constructed at the level of supersymmetry variations and the action. The supersymmetric localization is performed to achieve a two-dimensional chiral CFT. The desired vertex operator algebra is recovered as the algebra of local operators of the resulting CFT. We also discuss the identification of the Schur index of the mathcal{N} = 2 superconformal theory and the vacuum character of the vertex operator algebra at the level of their path integral representations, using our Ω-deformation point of view on the correspondence.

The non-Abelian tensor multiplet

We assume the existence of a background vector field that enables us to make an ansatz for the superconformal transformations for the non-Abelian 6d (1, 0) tensor multiplet. Closure of supersymmetry on generators of the conformal algebra and the R-symmetry, requires that the vector field is Abelian, has scaling dimension minus one and that the supersymmetry parameter as well as all the fields in the tensor multiplet have vanishing Lie derivatives along this vector field. We couple the tensor multiplet to an adjoint hypermultiplet, and present a Lagrangian for the combined system that has enhanced (2, 0) superconformal symmetry. We also obtain the off-shell supersymmetry variations for both the tensor and the hypermultiplets.

The action of the (free) theory in six spacetime dimensions

The action of the free theory in six spacetime dimensions is explicitly constructed. The variables of the variational principle are prepotentials adapted to the self-duality conditions on the fields. The (3, 1) supersymmetry variations are given and the invariance of the action is verified. The action is first-order in time derivatives. It is also Poincaré invariant but not manifestly so, just like the Hamiltonian action of more familiar relativistic field theories.

A geometric formulation of supersymmetry

The scalar fields of supersymmetric models are coordinates of a geometric space. We propose a formulation of supersymmetry that is covariant with respect to reparametrizations of this target space. Employing chiral multiplets as an example, we introduce modified supersymmetry variations and redefined auxiliary fields that transform covariantly under reparametrizations. The resulting action and transformation laws are manifestly covariant and highlight the geometric structure of the supersymmetric theory. The covariant methods are developed first for general theories (not necessarily supersymmetric) whose scalar fields are coordinates of a Riemannian target space.

3D supergravity from wrapped M5-branes

Through consistent Kaluza-Klein reduction, we construct 3D $\mathcal{N} =2$ gauged supergravities corresponding to twisted compactifications of M5-branes on a product of constant curvature Riemann surfaces, including K\"{a}hler-Einstein four-manifolds. We extend the reduction to fermionic supersymmetry variations in order to determine the 3D Killing spinor equations and classify all timelike supersymmetric solutions. As a by-product, we identify an infinite class of new supersymmetric warped $AdS_3$ (G\"odel) and warped $d S_3$ solutions. Moreover, we show that the superpotential $T$ encodes the central charge and R symmetry of the dual $\mathcal{N} = (0,2)$ SCFTs in the large $N$ limit. We demonstrate that the R symmetry matches the canonical $U(1)$ isometry from existing classifications of supersymmetric $AdS_3$ solutions to 11D supergravity with $\mathcal{N} = (0,2)$ supersymmetry.

All supersymmetric solutions of 3D U(1)3 gauged supergravity.

D3-branes wrapping constant curvature Riemann surfaces give rise to 2D N = (0,2) SCFTs, where the superconformal fixed-points are mapped to vacua of 3D N =2 U(1)^3 gauged supergravity. In this work we determine the fermionic supersymmetry variations of the theory and present all supersymmetric solutions. For spacetimes with a timelike Killing vector, we identify new timelike warped AdS_3 (G\"odel) and timelike warped dS_3 critical points. We outline the construction of numerical solutions interpolating between fixed-points, demonstrate that these flows are driven by an irrelevant scalar operator in the SCFT and identify the inverse of the superpotential as a candidate c-function. We further classify all spacetimes with a null Killing vector, in the process producing loci in parameter space where null-warped AdS_3 vacua with Schrodinger z=2 symmetry exist. We construct non-supersymmetric spacelike warped AdS_3 geometries based on D3-branes.

A profusion of 1/2 BPS Wilson loops in N = 4 $$ \mathcal{N}=4 $$ Chern-Simons-matter theories

We initiate the study of $1/2$ BPS Wilson loops in $\mathcal{N}=4$ Chern-Simons-matter theories in three dimensions. We consider a circular or linear quiver with Chern-Simons levels $k$, $-k$ and $0$, and focus on loops preserving one of the two $SU(2)$ subgroups of the $R$-symmetry. In the cases with no vanishing Chern-Simons levels, we find a pair of Wilson loops for each pair of adjacent nodes on the quiver connected by a hypermultiplet (nodes connected by twisted hypermultiplets have Wilson loops preserving another set of supercharges). We expect this classical pairwise degeneracy to be lifted by quantum corrections. In the case with nodes with vanishing Chern-Simons terms connected by twisted hypermultiplets, we find that the usual $1/4$ BPS Wilson loops are automatically enlarged to $1/2$ BPS, as happens also in 3-dimensional Yang-Mills theory. When the nodes with vanishing Chern-Simons levels are connected by untwisted hypermultiplets, we do not find any Wilson loops coupling to those nodes which are classically invariant. Rather, we find several loops whose supersymmetry variation, while non zero, vanishes in any correlation function, so is weakly zero. We expect only one linear combination of those Wilson loops to remain BPS when quantum corrections are included. We analyze the M-theory duals of those Wilson loops and comment on their degeneracy. We also show that these Wilson loops are cohomologically equivalent to certain $1/4$ BPS Wilson loops whose expectation value can be evaluated by the appropriate localized matrix model.

Supersymmetry with non-geometric fluxes, or a β-twist in Generalized Geometry and Dirac operator

We study ten-dimensional supersymmetric vacua with NSNS non-geometric fluxes, in the framework of β-supergravity. We first provide expressions for the fermionic supersymmetry variations. Specifying a compactification ansatz to four dimensions, we deduce internal Killing spinor equations. These supersymmetry conditions are then reformulated in terms of pure spinors, similarly to standard supergravity vacua admitting an SU(3)×SU(3) structure in Generalized Complex Geometry. The standard d–H∧ acting on the pure spinors is traded for a generalized Dirac operator $$ \mathcal{D} $$ , depending here on the non-geometric fluxes. Rewriting it with an exponential of the bivector β leads us to discuss the geometrical characterisation of the vacua in terms of a β-twist, in analogy to the standard twist by the b-field. Thanks to $$ \mathcal{D} $$ , we also propose a general expression for the superpotential to be obtained from standard supergravities or β-supergravity, and verify its agreement with formulas of the literature. We finally comment on the Ramond-Ramond sector, and discuss a possible relation to intermediate or dynamical SU(2) structure solutions.

Supersymmetry and non-Abelian T-duality in type II supergravity

We study the effect of T-duality on supersymmetry in the context of type II supergravity. For both U(1) Abelian and SU(2) non-Abelian T-duality, we demonstrate that the supersymmetry variations after T-duality are related to the variations before T-duality through the Kosmann spinorial Lie derivative, which vanishes when the Killing spinors are independent of the T-duality directions. As a byproduct of our analysis, we present closed expressions for SU(2) T-duality in a class of spacetimes with diagonal Bianchi IX symmetry and comment on specific examples of T-dual geometries, including a novel AdS3 geometry with large superconformal symmetry.

Supergravity as generalised geometry II: E d(d) × ℝ+ and M theory

We reformulate eleven-dimensional supergravity, including fermions, in terms of generalised geometry, for spacetimes that are warped products of Minkowski space with a d-dimensional manifold M with d ≤ 7. The reformulation has an E d(d) × ℝ+ structure group and it has a local $ {{\widetilde{H}}_d} $ symmetry, where $ {{\widetilde{H}}_d} $ is the double cover of the maximally compact subgroup of E d(d). The bosonic degrees for freedom unify into a generalised metric, and, defining the generalisked analogue D of the Levi-Civita connection, one finds that the corresponding equations of motion are the vanishing of the generalised Ricci tensor. To leading order, we show that the fermionic equations of motion, action and supersymmetry variations can all be written in terms of D. Although we will not give the detailed decompositions, this reformulation is equally applicable to type IIA or IIB supergravity restricted to a (d − 1)-dimensional manifold. For completeness we give explicit expressions in terms of $ {{\widetilde{H}}_4} $ = Spin(5) and $ {{\widetilde{H}}_7} $ = SU(8) representations for d = 4 and d = 7.

Super Weyl invariance: BPS equations from heterotic worldsheets

It is well-known that the beta functions on a string worldsheet correspond to the target space equations of motion, e.g. the Einstein equations. We show that the BPS equations, i.e. the conditions of vanishing supersymmetry variations of the space-time fermions, can be directly derived from the worldsheet. To this end we consider the RNS-formulation of the heterotic string with (2,0) supersymmetry, which describes a complex torsion target space that supports a holomorphic vector bundle. After a detailed account of its quantization and renormalization, we establish that the cancellation of the Weyl anomaly combined with (2,0) finiteness implies the heterotic BPS conditions: At the one loop level the geometry is required to be conformally balanced and the gauge background has to satisfy the Hermitean Yang-Mills equations.

Supersymmetric Chern-Simons theory in the presence of a boundary

In this paper we analyse super-Chern-Simons theory in $\mathcal{N} =1$ superspace formalism, in the presence of a boundary. We modify the Lagrangian for the Chern-Simons theory in such a way that it is supersymmetric even in the presence of a boundary. Also, even though the Chern-Simons theory is not gauge invariant in the presence of a boundary, if it is suitable coupled to a gauged Wess-Zumino-Witten model, then the resultant theory can be made gauge invariant. Thus, by suitably adding extra boundary degrees of freedom, the gauge and supersymmetry variations of the boundary theory exactly cancel the boundary terms generated by the variations of the bulk Chern-Simons theory. We also discuss how this can be applied to the ABJM model in $\mathcal{N} =1$ superspace, and we then describe the BRST and anti-BRST symmetries of the resultant gauge invariant supersymmetric theory.