We review a variety of mathematical structures underlying supersymmetry, ranging from the description of gauge theories and gravity in Superspace to the combinatorial and graph-theoretic frameworks that capture the algebraic content of its representations. The discussion revisits how geometric notions commonly associated with manifestly supersymmetric gauge theories can emerge rather than being assumed a priori. This exploration is followed by an extensive treatment of supersymmetry representations through the lens of combinatorics, higher-dimensional polytopes and graph theory. The synthesis of these perspectives shows that supersymmetry, far from being confined to a single language, builds bridges to seemingly unrelated domains of mathematics and finds expression in multiple guises, each offering complementary insights into its structure.

We show that the 3d 3d Born-Infeld theory can be generated via an irrelevant deformation of the free Maxwell theory. The deforming operator is constructed from the energy-momentum tensor and includes a novel non-analytic contribution that resembles root- T \ overline{T} ToverlineT . We find that a similar operator deforms a free scalar into the scalar sector of the Dirac-Born-Infeld action, which describes transverse fluctuations of a D-brane, in any dimension. We also analyse trace flow equations and obtain flows for subtracted models driven by a relevant operator. In 3d 3d , the irrelevant deformation can be made manifestly supersymmetric by presenting the flow equation in \mathcal{N} = 1 𝒩=1 superspace, where the deforming operator is built from supercurrents. We demonstrate that two supersymmetric presentations of the D2-brane effective action, the Maxwell-Goldstone multiplet and the tensor-Goldstone multiplet, satisfy superspace flow equations driven by this supercurrent combination. To do this, we derive expressions for the supercurrents in general classes of vector and tensor/scalar models by directly solving the superspace conservation equations and also by coupling to \mathcal{N} = 1 𝒩=1 supergravity. As both of these multiplets exhibit a second, spontaneously broken supersymmetry, this analysis provides further evidence for a connection between current-squared deformations and nonlinearly realized symmetries.

We present a nongeometric derivation of N 1/4 1 Super Yang-Mills by focusing on the consistency of interactions that extend the free vector supermultiplet rather than assuming gauge invariance under extended symmetries. By utilizing a superspace first-order description, the theory is given in closed form as a third-order polynomial, which includes a single cubic interaction term instead of an infinite series, thus eliminating the need for a special gauge. The geometrical interpretation of the theory emerges, as opposed to being presupposed.

We investigate the contribution of higher spin particles in the signal of direct detection searches for dark matter. We consider a bosonic or fermionic higher spin dark matter (HSDM) candidate which interacts with the Standard Model via a dark U(1) mediator. For a particular subclass of interactions, spin-polarized targets may be used for spin determination: The angular dependence of scatterings can distinguish integer (spin-s) vs. half-integer (spin-s+1/2), while the recoil energy dependence of the signal determines s. We consider also the signal of a supersymmetric higher spin dark sector, which suggests a characteristic signal (“SUSY Rilles”) for directional direct detection.

The exploration of scalar field theories that exhibit Carroll and Galilei symmetries has attracted a lot of attention. In this paper, we generalize these studies to fermionic field theories and construct consistent electric and magnetic descriptions of Carrollian and Galilean spin $\tfrac{1}{2}$ fermions. We showcase various methods that offer complementary perspectives into the limiting process of the underlying relativistic theories. Moreover, we extend our study to $\mathcal{N}=1$ off-shell supersymmetric field theories in four dimensions. By introducing suitable Grassmann-analyticity conditions, we formulate the corresponding super-Carrollian and super-Galilean theories. These theories combine the established Carroll/Galilei scalars with the Carroll/Galilei fermions and a range of auxiliary fields into supermultiplets.

We consider the superspace BRST and BV description of 4D,N=1 super-Maxwell theory and its non-abelian generalization Super Yang–Mills. By fermionizing the superspace gauge transformation of the gauge superfields, we define the nilpotent superspace BRST symmetry transformation (𝓈). After introducing an appropriate set of anti-superfields and defining the superspace antibracket, we use it to construct the BV-BRST nilpotent differential operator (s) in terms of superspace covariant derivatives. The anti-superfield independent terms of s provide a superspace generalization of the Koszul–Tate resolution (δ). In the linearized limit, the set of superspace differential operators that appear in s satisfy a nonlinear algebra which can be used to construct a BRST charge Q, without requiring pure spinor variables. Q acts on the Hilbert space of superfield states, and its cohomology generates the expected superspace equations of motion.

Dp-branes acquire effective nonlinear descriptions whose bosonic part is related to the Born-Infeld action. This nonlinearity has been proven to be a consequence of the partial N=2→N=1 supersymmetry breaking, originating from the solitonic nature of the branes. In this work, we focus on the effective descriptions of D2-branes. Using the Goldstone multiplet interpretation of the action and the method of nilpotent N=2 superfields, we construct the 3D, N=1 superspace effective action which makes the first supersymmetry manifest and realizes the second, spontaneously broken, supersymmetry nonlinearly. We show that there are two such supersymmetric extensions of the 3D Born-Infeld action which correspond to the dynamics of the 3D Maxwell-Goldstone multiplet and the 3D projection of the Tensor-Goldstone multiplet respectively. Moreover, we demonstrate that these results are derived by applying the constrained superfield approach on the N=2,D=3 vector and chiral multiplets after expanding them around a nontrivial vacuum. We find that these two descriptions are related by a duality transformation which results in the inversion of a dimensionless parameter. For both descriptions we derive the explicit bosonic and fermionic parts of the 3D super Born-Infeld action. Finally, consider the deformation of the Maxwell-Goldstone superspace action by the characteristic Chern-Simons-like, gauge invariant, mass term.

A new description of free massless superfields of arbitrary superspin Y (Y > 1/2) is proposed. Following the first-order philosophy, we relax some of the properties (reality, gauge redundancy) of the unconstrained higher spin prepotentials and we construct first and half order invariants quantities. These are used to write trivially invariant actions. Additional auxiliary superfields that play the role of spin connections are used to enforce a new local symmetry that restores the degrees of freedom.

An explicit form for the Lagrangian of a massive arbitrary half-integer super-spin Y = s + 1 / 2 supermultiplet is obtained in 4D , N $$ \mathcal{N} $$ = 1 superspace. This is accomplished by the introduction of a tower of pairs of auxiliary superfields of increasing rank which are required to vanish on-shell for free theories. In the massless limit almost all auxiliary super- fields decouple except one, which plays the role of compensator as required by the emergent gauge redundancy of the Lagrangian description of the massless theory. The number of off-shell degrees of freedom carried by the theory is 8 3 $$ \frac{8}{3} $$ ( s + 1)(4 s 2 + 11 s + 3). For s = 1 our results are in agreement with those obtained in [ 1 ].

We investigate cubic interactions between a chiral superfield and higher spin superfields corresponding to irreducible representations of the 4 D , N = 1 super-Poincaré algebra. We do this by demanding an invariance under the most general transformation, linear in the chiral superfield. Following Noether’s method we construct an infinite tower of higher spin supercurrent multiplets which are quadratic in the chiral superfield and include higher derivatives. The results are that a single, massless, chiral superfield can couple only to the half-integer spin supermultiplets ( s + 1 , s + 1 / 2 ) and for every value of spin there is an appropriate improvement term that reduces the supercurrent multiplet to a minimal multiplet which matches that of superconformal higher spins. On the other hand a single, massive, chiral superfield can couple only to higher spin supermultiplets of type ( 2 l + 2 , 2 l + 3 / 2 ) (only odd values of s, s = 2 l + 1 ) and there is no minimal multiplet. Furthermore, for the massless case we discuss the component level higher spin currents and provide explicit expressions for the integer and half-integer spin conserved currents together with a R-symmetry current.

We consider a four dimensional generalized Wess-Zumino model formulated in terms of an arbitrary Kähler potential \( \mathcal{K}\left(\varPhi, \overline{\varPhi}\right) \) and an arbitrary chiral superpotential \( \mathcal{W}\left(\varPhi \right) \). A general analysis is given to describe the possible interactions of this theory with external higher spin gauge superfields of the (s + 1, s + 1/2) supermultiplet via higher spin supercurrents. It is shown that such interactions do not exist beyond supergravity (s ≥ 2) for any \( \mathcal{K} \) and \( \mathcal{W} \). However, we find three exceptions, the theory of a free massless chiral, the theory of a free massive chiral and the theory of a free chiral with linear superpotential. For the first two, the higher spin supercurrents are known and for the third one we provide the explicit expressions. We also discuss the lower spin supercurrents. As expected, a coupling to (non-minimal) supergravity (s = 1) can always be found and we give the generating supercurrent and supertrace for arbitrary \( \mathcal{K} \) and \( \mathcal{W} \). On the other hand, coupling to the vector supermultiplet (s = 0) is possible only if \( \mathcal{K} = \mathcal{K}\left(\overline{\varPhi}\varPhi \right) \) and \( \mathcal{W}=0 \).

We give an explicit superspace construction of higher spin conserved supercurrents built out of 4D, \( \mathcal{N}=1 \) massless supermultiplets of arbitrary spin. These supercurrents are gauge invariant and generate a large class of cubic interactions between a massless supermultiplet with superspin Y1 = s1 + 1/2 and two massless supermultiplets of arbitrary superspin Y2. These interactions are possible only for s1 ≥ 2Y2. At the equality, the supercurrent acquires its simplest form and defines the supersymmetric, higher spin extension of the linearized Bel-Robinson tensor.

We continue the program of constructing cubic interactions between matter and higher spin supermultiplets. In this work we consider a complex linear superfield and we find that it can have cubic interactions only with supermultiplets with propagating spins j = s + 1, j = s + 1/2 for any non-negative integer s (half-integer superspin super-multiplets). We construct the higher spin supercurrent and supertrace, these compose the canonical supercurrent multiplet which generates the cubic interactions. We also prove that for every s there exist an alternative minimal supercurrent multiplet, with vanishing supertrace. Furthermore, we perform a duality transformation in order to make contact with the corresponding chiral theory. An interesting result is that the dual chiral theory has the same coupling constant with the complex linear theory only for odd values of s, whereas for even values of s the coupling constants for the two theories have opposite signs. Additionally we explore the component structure of the supercurrent multiplet and derive the higher spin currents. We find two bosonic currents for spins j = s and j = s + 1 and one fermionic current for spin j = s + 1/2.

We present a new method of deriving the off-shell spectrum of supergravity and massless 4D, \( \mathcal{N} \) = 1 higher spin multiplets without the need of an action and based on a set of natural requirements: (a.) existence of an underlying superspace description, (b.) an economical description of free, massless, higher spins and (c.) equal numbers of bosonic and fermionic degrees of freedom. We prove that for any theory that respects the above, the fermionic auxiliary components come in pairs and are gauge invariant and there are two types of bosonic auxiliary components. Type (1) are pairs of a (2, 0)-tensor with real or imaginary (1, 1)-tensor with non-trivial gauge transformations. Type (2) are singlets and gauge invariant. The outcome is a set of Diophantine equations, the solutions of which determine the off-shell spectrum of supergravity and massless higher spin multiplets. This approach provides (i ) a classification of the irreducible, supersymmetric, representations of arbitrary spin and (ii ) a very clean and intuitive explanation to why some of these theories have more than one formulations (e.g. the supergravity multiplet) and others do not.

In the framework of linearized non-minimal supergravity (20/20), we present the embedding of the R+R2 model and we analyze its field spectrum. As usual, the auxiliary fields of the Einstein theory now become propagating, giving rise to additional degrees of freedom, which organize themselves into on-shell irreducible supermultiplets. By performing the analysis both in component and superspace formulations we identify the new supermultiplets. On top of the two massive chiral superfields reminiscent of the old-minimal supergravity embedding, the spectrum contains also a consistent physical, massive, vector supermultiplet and a tachyonic ghost, massive, vector supermultiplet.

Higher Super-Spins, are the irreducible representations of the Super-Poincaré group. We study the representation theory of this group in 4D, N = 1 and demonstrate the off-shell Superspace realization of these theories. On the one hand, for the massless case, using gauge symmetry as a guide we can describe the arbitrary (integer or half-integer) superhelicity system. On the other hand for the massive case the general superspin case is still an open problem. However we would like to report on some recent progress towards that direction. We complete the picture by presenting various aspects of the Higher Super-Spin theories, such as the off-shell spectrum and degrees of freedom for the arbitrary superhelicity and how to construct N = 2 theories out of the N = 1 theories.

We present a new theory of free massive superspin Y = 3/2 irreducible representation of the 4D, \( \mathcal{N} \) = 1 Super-Poincaré group, which has linearized non-minimal supergravity (superhelicity Y = 3/2) as it’s massless limit. The new results will illuminate the underlying structure of auxiliary superfields required for the description of higher massive superspin systems.

We present an alternative method of exploring the component structure of an arbitrary super-helicity (integer Y = s, or half odd integer Y = s+1/2 for any integer s) irreducible representation of the Super-Poincaré group. We use it to derive the component action and the SUSY transformation laws. The effectiveness of this approach is based on the equations of motion and their properties, like the Bianchi identities. These equations are generated by the superspace action when it is expressed in terms of prepotentials. For that reason we reproduce the superspace action for arbitrary superhelicity, using unconstrained superfields. The appropriate, to use, superfields are dictated by the representation theory of the group and the requirement that there is a smooth limit between the massive and massless case.

bstract We present an alternative method of exploring the component structure of an arbitrary super-helicity (integer Y = s , or half odd integer Y = s +1 / 2 for any integer s ) irreducible representation of the Super-Poincaré group. We use it to derive the component action and the SUSY transformation laws. The effectiveness of this approach is based on the equations of motion and their properties, like the Bianchi identities. These equations are generated by the superspace action when it is expressed in terms of prepotentials. For that reason we reproduce the superspace action for arbitrary superhelicity, using unconstrained superfields. The appropriate, to use, superfields are dictated by the representation theory of the group and the requirement that there is a smooth limit between the massive and massless case.

We present an alternative method of exploring the component structure of an arbitrary super-helicity (integer Y = s, or half odd integer Y = s+1/2 for any integer s) irreducible representation of the Super-Poincaré group. We use it to derive the component action and the SUSY transformation laws. The effectiveness of this approach is based on the equations of motion and their properties, like the Bianchi identities. These equations are generated by the superspace action when it is expressed in terms of prepotentials. For that reason we reproduce the superspace action for arbitrary superhelicity, using unconstrained superfields. The appropriate, to use, superfields are dictated by the representation theory of the group and the requirement that there is a smooth limit between the massive and massless case.