Superfield Formalism Research Articles

Solitonic ground state in supersymmetric theory in background

A solitonic ground state called a chiral soliton lattice (CSL) is realized in a supersymmetric theory with background magnetic field and finite chemical potential. To this end, we construct, in the superfield formalism, a supersymmetric chiral sine-Gordon model as a neutral pion sector of a supersymmetric two-flavor chiral Lagrangian with a Wess-Zumino-Witten term. The CSL ground state appears in the presence of either a strong magnetic field and/or large chemical potential, or a background fermionic condensate in the form of a fermion bilinear consisting of the gaugino and a superpartner of a baryon gauge field.

Z22-graded supersymmetry via superfield on minimal Z22-superspace

Abstract A superfield formalism for the minimal Z2 2-graded version of supersymmetry is developed. This is done by using the recently introduced definition of integration on the minimal Z2 2-superspace. It is shown that one may construct Z2 2-supersymmetric action by the procedure similar to the standard supersymmetry. 
However, the Lagrangian obtained has very general interaction terms, which give rise to a Z2 2-graded extension of many known theories defined in two-dimensional spacetime. 
As an illustration, we will give a Z2 2-extension of the sine-Gordon model different from the one already discussed in the literature.

[formula omitted] higher-spin supercurrents

We have constructed conserved N=2 higher-spin supercurrents for an arbitrary integer spin through the unconstrained superfield formulation of 4D,N=2 massless higher-spin theories in the harmonic superspace. The obtained supercurrents are gauge-invariant and determine consistent cubic interactions in the gauge superfield × supercurrent form for massless spin s1 and two massless spin s2N=2 supermultiplets. Such interactions and N=2 supercurrents exist only for s1≥2s2. These supercurrents are the N=2 supersymmetric extension of Berends-Burgers-van Dam higher-spin currents and generalize the linearized Bel-Robinson tensor. The relevant N=2 supercurrents can be considered as the descendants of the N=2 principle supercurrent, which has a simple universal structure. An important feature of our outcomes is that, ultimately, all N=2 supercurrents are built from the N=2 superfield strengths, which are constructed from the unconstrained analytical higher-spin prepotentials.

Canonical Supermultiplets and Their Koszul Duals

The pure spinor superfield formalism reveals that, in any dimension and with any amount of supersymmetry, one particular supermultiplet is distinguished from all others. This “canonical supermultiplet” is equipped with an additional structure that is not apparent in any component-field formalism: a (homotopy) commutative algebra structure on the space of fields. The structure is physically relevant in several ways; it is responsible for the interactions in ten-dimensional super Yang–Mills theory, as well as crucial to any first-quantised interpretation. We study the L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_\\infty $$\\end{document} algebra structure that is Koszul dual to this commutative algebra, both in general and in numerous examples, and prove that it is equivalent to the subalgebra of the Koszul dual to functions on the space of generalised pure spinors in internal degree greater than or equal to three. In many examples, the latter is the positive part of a Borcherds–Kac–Moody superalgebra. Using this result, we can interpret the canonical multiplet as the homotopy fiber of the map from generalised pure spinor space to its derived replacement. This generalises and extends work of Movshev–Schwarz and Gálvez–Gorbounov–Shaikh–Tonks in the same spirit. We also comment on some issues with physical interpretations of the canonical multiplet, which are illustrated by an example related to the complex Cayley plane, and on possible extensions of our construction, which appear relevant in an example with symmetry type G2×A1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G_2 \ imes A_1$$\\end{document}.

N=8 invariant interaction of dynamical and semidynamical N=4 multiplets

We present a new model of N=8 mechanics with semidynamic supermultiplets. The model is constructed as an interaction of N=4 supermultiplets which carry an implicit N=4 supersymmetry. The initial field content consists of three dynamical (1,4,3) multiplets: one bosonic and two fermionic. To ensure implicit N=4 supersymmetry, we introduce the superfields describing three semidynamical (4,4,0) multiplets: one fermionic and two bosonic. To avoid the second-order Lagrangian for fermions from the fermionic (1,4,3) multiplets, we convert their velocities into new auxiliary fields. After conversion, these multiplets turn into semidynamical mirror (4,4,0) multiplets without noncanonical terms in the N=8 Lagrangian at the component level. The final N=8 multiplet content is (1,8,7)⊕(8,8,0). As a first step to the ultimate N=4 superfield formulation of the model, we recall a natural description of the standard and mirror (4,4,0) multiplets in the framework of N=4,d=1 biharmonic superspace. Published by the American Physical Society 2024

The Derived Pure Spinor Formalism as an Equivalence of Categories

We construct a derived generalization of the pure spinor superfield formalism and prove that it exhibits an equivalence of dg-categories between multiplets for a supertranslation algebra and equivariant modules over its Chevalley-Eilenberg cochains. This equivalence is closely linked to Koszul duality for the supertranslation algebra. After introducing and describing the category of supermultiplets, we define the derived pure spinor construction explicitly as a dg-functor. We then show that the functor that takes the derived supertranslation invariants of any supermultiplet is a quasi-inverse to the pure spinor construction, using an explicit calculation. Finally, we illustrate our findings with examples and use insights from the derived formalism to answer some questions regarding the ordinary (underived) pure spinor superfield formalism.

A Minimal b Ghost

AbstractThe b ghost, or b operator, used for fixing Siegel gauge in the pure spinor superfield formalism, is a composite operator of negative ghost number, satisfying , where q is the pure spinor differential (BRST operator). It is traditionally constructed using non‐minimal variables. However, since all cohomology has minimal representatives, it seems likely that there should be versions of physically meaningful operators, also with negative ghost number, using only minimal variables. The purpose of this letter is to demonstrate that this statement holds by providing a concrete construction in super‐Yang–Mills theory, and to argue that it is a general feature in the pure spinor superfield formalism.

Constraints in the BV formalism: Six-dimensional supersymmetry and its twists

We formulate the abelian six-dimensional N=(2,0) theory perturbatively, in a generalization of the Batalin–Vilkovisky formalism. Using this description, we compute the holomorphic and non-minimal twists at the perturbative level. This calculation hinges on the existence of an L∞ action of the supersymmetry algebra on the abelian tensor multiplet, which we describe in detail. Our formulation appears naturally in the pure spinor superfield formalism, but understanding it requires developing a presymplectic generalization of the BV formalism, inspired by Dirac's theory of constraints. The holomorphic twist consists of symplectic-valued holomorphic bosons from the N=(1,0) hypermultiplet, together with a degenerate holomorphic theory of holomorphic one-forms from the N=(1,0) tensor multiplet, which can be seen to describe the infinitesimal intermediate Jacobian variety. We check that our formulation and our results match with known ones under various dimensional reductions, as well as comparing the holomorphic twist to Kodaira–Spencer theory. Matching our formalism to five-dimensional Yang–Mills theory after reduction leads to some issues related to electric–magnetic duality; we offer some speculation on a nonperturbative resolution.

Maximally Twisted Eleven-Dimensional Supergravity

We perform the maximal twist of eleven-dimensional supergravity. This twist is partially topological and exists on manifolds of G_2 times SU(2) holonomy. Our derivation starts with an explicit description of the Batalin–Vilkovisky complex associated to the three-form multiplet in the pure spinor superfield formalism. We then determine the L_infty module structure of the supersymmetry algebra on the component fields. We twist the theory by modifying the differential of the Batalin–Vilkovisky complex to incorporate the action of a scalar supercharge. We find that the resulting free twisted theory is given by the tensor product of the de Rham and Dolbeault complexes of the respective G_2 and SU(2) holonomy manifolds as conjectured by Costello.

Li filtrations of SUSY vertex algebras

Any vertex algebra has a canonical decreasing filtration, called the Li filtration, whose associated graded space has a natural structure of a vertex Poisson algebra. In this note, we introduce an analogous filtration for any SUSY vertex algebra, which was introduced by Heluani and Kac as a superfield formalism for supersymmetric vertex algebras. We prove that the associated graded superspace of our filtration has a structure of a SUSY vertex Poisson algebra. We also introduce and discuss related notions, such as Zhu’s \(C_2\)-Poisson superalgebras, associated superschemes, and singular supports, for SUSY vertex algebras.

Perspectives on the pure spinor superfield formalism

In this note, we study, formalize, and generalize the pure spinor superfield formalism from a rather nontraditional perspective. To set the stage, we review the notion of a multiplet for a general super Lie algebra, working in the context of the BV and BRST formalisms. Building on this, we explain how the pure spinor superfield formalism can be viewed as constructing a supermultiplet out of the input datum of an equivariant graded module over the ring of functions on the nilpotence variety. We use the homotopy transfer theorem and other computational techniques from homological algebra to relate these multiplets to more standard component-field formulations. Physical properties of the resulting multiplets can then be understood in terms of algebrogeometric properties of the nilpotence variety. We illustrate our discussion with many examples in various dimensions.

UV divergences, RG equations and high energy behaviour of the amplitudes in the Wess-Zumino model with quartic interaction

We analyse the UV divergences for the scattering amplitudes in the Wess-Zumino SUSY model with the quartic superpotential. We argue that the only divergent amplitudes are those with four external legs. Within the superfield formalism, we calculate the corresponding Feynman diagrams and evaluate their leading divergences up to 4 loop order of PT. Then we construct recurrence relations that connect the leading UV divergences in subsequent orders of perturbation theory. These recurrence relations allow us to calculate the leading divergences in a pure algebraic way starting from the one loop contribution. We check that the obtained relations correctly reproduce the lower order diagrams evaluated explicitly. At last, we convert the recurrence relations into the RG equations that have integro-differential form. Solving these equations for a particular sequence of diagrams, we find out the high energy behaviour of the amplitude. We then argue that the full amplitude has a similar behaviour with the key feature of the existence of a pole in the s-channel corresponding to a state with a mass ∼ 1/g, where g is the original dimensionfull coupling of the theory. We find out the this state is actually a ghost one similar to the Landau pole in scalar theory.

The coupling flow of mathcal{N} = 4 super Yang-Mills theory

We offer a novel perspective on mathcal{N} = 4 supersymmetric Yang-Mills (SYM) theory through the framework of the Nicolai map, a transformation of the bosonic fields that allows one to compute quantum correlators in terms of a free, purely bosonic functional measure. Generally, any Nicolai map is obtained through a path-ordered exponential of the so-called coupling flow operator. The latter can be canonically constructed in any gauge using an mathcal{N} = 1 off-shell superfield formulation of mathcal{N} = 4 SYM, or alternatively through dimensional reduction of the result from mathcal{N} = 1 D = 10 SYM, in which case we need to restrict to the Landau gauge. We propose a general theory of the mathcal{N} = 4 coupling flow operator, arguing that it exhibits an ambiguity in form of an R-symmetry freedom given by the Lie algebra mathfrak{su} (4). This theory incorporates our two construction approaches as special points in mathfrak{su} (4) and defines a broad class of Nicolai maps for mathcal{N} = 4 SYM.

Dualization between supersymmetric self-dual and topologically massive models coupled to matter in four-dimensional superspace

Using the superfield formalism and the master action approach, we prove, both at the classical and quantum levels, the dual equivalence between four-dimensional supersymmetric self-dual and topologically massive models coupled to dynamical matter.

Path Integral Quantization of Doubly Supersymmetric Model

The Hamilton-Jacobi formalism is used to discuss the path integral quantization of the double supersymmetric models with the spinning superparticle in the component and superfield form. The equations of motion are obtained as total differential equations in many variables. The equations of motion are integrable, and the path integral is obtained as an integration over the canonical phase space coordinates.

Comments of [formula omitted]-supersymmetry in superfield formalism

We investigate superfield formulation of the minimal Z22-supersymmetry. It is shown that the integrability on Z22-superspace guarantees the invariance of action. Then we present two superfields which carry distinct irreducible representations of the Z22-supersymmetry algebra. One of them gives integrable Lagrangian and the other does not. We also show that integrable superfields with different Z22-degree also carry irreducible representations and they give invariant actions. To perform this analysis, the representation theory of the minimal Z22-supersymmetry algebra is studied in some detail.

Unconstrained off-shell superfield formulation of 4D, mathcal{N} = 2 supersymmetric higher spins

We present, for the first time, the complete off-shell 4D, mathcal{N} = 2 superfield actions for any free massless integer spin s ≥ 2 fields, using the mathcal{N} = 2 harmonic super-space approach. The relevant gauge supermultiplet is accommodated by two real analytic bosonic superfields {h}_{alpha left(s-1right)dot{alpha}left(s-1right)}^{++} , {h}_{alpha left(s-2right)dot{alpha}left(s-2right)}^{++} and two conjugated complex analytic spinor superfields {h}_{alpha left(s-1right)dot{alpha}left(s-1right)}^{+3} , {h}_{alpha left(s-2right)dot{alpha}left(s-1right)}^{+3} , where α(s) := (α1. . . αs), dot{alpha} (s) := ( dot{alpha} 1. . . dot{alpha} s). Like in the harmonic superspace formulations of mathcal{N} = 2 Maxwell and supergravity theories, an infinite number of original off-shell degrees of freedom is reduced to the finite set (in WZ-type gauge) due to an infinite number of the component gauge parameters in the analytic superfield parameters. On shell, the standard spin content (s,s−1/2,s−1/2,s−1) is restored. For s = 2 the action describes the linearized version of “minimal” mathcal{N} = 2 Einstein supergravity.

Linearized supergravity with a dynamical preferred frame

We study supersymmetric extension of the Einstein-aether gravitational model where local Lorentz invariance is broken down to the subgroup of spatial rotations by a vacuum expectation value of a timelike vector field called aether. Embedding aether into a chiral vector superfield, we construct the most general action which describes dynamics of linear perturbations around the Lorentz-violating vacuum and is invariant under the linearized supergravity transformations. The analysis is performed both in the off-shell non-minimal superfield formulation of supergravity and in the “on-shell” approach invoking only physical component fields. The resulting model contains a single free coupling, in addition to the standard supergravity parameters. The spectrum of physical excitations features an enhanced on-shell gravity multiplet comprising four states with helicities 2, 3/2, 3/2 and 1 propagating with superluminal velocity. The remaining excitations propagate with the speed of light. We outline the observational constraints on the model following from its low-energy phenomenology.

SL(5) Supersymmetry

AbstractWe consider supersymmetry in five dimensions, where the fermionic parameters are a 2‐form under . Supermultiplets are investigated using the pure spinor superfield formalism, and are found to be closely related to infinite‐dimensional extensions of the supersymmetry algebra: the Borcherds superalgebra , the tensor hierarchy algebra and the exceptional superalgebra E(5, 10). A theorem relating and E(5, 10) to all levels is given.

BRST and Superfield Formalism—A Review

This article, which is a review with substantial original material, is meant to offer a comprehensive description of the superfield representations of BRST and anti-BRST algebras and their applications to some field-theoretic topics. After a review of the superfield formalism for gauge theories, we present the same formalism for gerbes and diffeomorphism invariant theories. The application to diffeomorphisms leads, in particular, to a horizontal Riemannian geometry in the superspace. We then illustrate the application to the description of consistent gauge anomalies and Wess–Zumino terms for which the formalism seems to be particularly tailor-made. The next subject covered is the higher spin YM-like theories and their anomalies. Finally, we show that the BRST superfield formalism applies as well to the N=1 super-YM theories formulated in the supersymmetric superspace, for the two formalisms go along with each other very well.