Superspace Methods Research Articles

BRST symmetry and the convolutional double copy

Motivated by the results of Anastasiou et al., we consider the convolutional double copy for BRST and anti-BRST covariant formulations of gravitational and gauge theories in more detail. We give a general BRST and anti-BRST invariant formulation of linearised mathcal{N} = 0 supergravity using superspace methods and show how this may be obtained from the square of linearised Yang-Mills theories. We demonstrate this relation for the Schwarzschild black hole and the ten-dimensional black string solution as two concrete examples.

Super Hamiltonian in superspace for incommensurate superlattices and quasicrystals

Infinite quasiperiodic arrangements in space, such as quasicrystals, are typically described as projections of higher-dimensional periodic lattices onto the physical dimension. The concept of a reference higher-dimensional space, called a superspace, has proved useful in relation to quasiperiodic systems. Although some quantum-mechanical systems in quasiperiodic media have been shown to admit quasiperiodic states, any sort of general Hamiltonian formalism in superspace is lacking to this date. Here, we show how to extend generic quantum-mechanical Hamiltonians to higher dimensions in such a way that eigenstates of the original Hamiltonian are obtained as projections of the Hamiltonian in superspace, which we call the super Hamiltonian. We apply the super Hamiltonian formalism to a simple, yet realistic one-dimensional quantum particle in a quasiperiodic potential without the tight-binding approximation, and obtain continuously labelled eigenstates of the system corresponding to a continuous spectrum. All states corresponding to the continuum are quasiperiodic. We also obtain the Green’s functions for continuum states in closed form and, from them, the density of states and local density of states, and scattering states off defects and impurities. The closed form of this one-dimensional Green’s function is equally valid for any continuum state in any one-dimensional single-particle quantum system admitting continuous spectrum. With the basis set we use, which is periodic in superspace, and therefore quasiperiodic in physical space, we find that Anderson-localised states are also quasiperiodic if distributional solutions are admitted, but circumvent this difficulty by generalising the superspace method to open boundary conditions. We also obtain an accurate estimate of the critical point where the ground state of the system changes from delocalised to Anderson localised, and of the critical exponent for the effective mass. Finally, we calculate, within the superspace formalism, topological edge states for the semi-infinite system, and observe that these exist, in the delocalised phase, within all spectral gaps we have been able to resolve. Our formalism opens up a plethora of possibilities for studying the physics of electrons, atoms or light in quasicrystalline and other aperiodic media.

-graded supersymmetry: 2-d sigma models

We propose a natural -graded generalisation of d = 2, supersymmetry and construct a -space realisation thereof. Due to the grading, the supercharges close with respect to, in the classical language, a commutator rather than an anticommutator. This is then used to build classical (linear and non-linear) sigma models that exhibit this novel supersymmetry via mimicking standard superspace methods. The fields in our models are bosons, right-handed and left-handed Majorana–Weyl spinors, and exotic bosons. The bosons commute with all the fields, the spinors belong to different sectors that cross commute rather than anticommute, while the exotic boson anticommute with the spinors. As a particular example of one of the models, we present a ‘double-graded’ version of supersymmetric sine-Gordon theory.

Mathcal{N}=2 SYK model in the superspace formalism

We use superspace methods to study an SYK-like model with mathcal{N}=2 supersymmetry in one dimension, and an analog of this model in two dimensions. We find the four-point function as an expansion in the basis of eigenfunctions of the Casimir of su(1, 1|1). We also find retarded kernels and Lyapunov exponents for both cases.

Forms and algebras in (half-)maximal supergravity theories

The forms in D-dimensional (half-)maximal supergravity theories are discussed for 3 $\leq$ D $\leq$ 11. Superspace methods are used to derive consistent sets of Bianchi identities for all the forms for all degrees, and to show that they are soluble and fully compatible with supersymmetry. The Bianchi identities determine Lie superalgebras that can be extended to Borcherds superalgebras of a special type. It is shown that any Borcherds superalgebra of this type gives the same form spectrum, up to an arbitrary degree, as an associated Kac-Moody algebra. For maximal supergravity up to D-form potentials, this is the very extended Kac-Moody algebra E11. It is also shown how gauging can be carried out in a simple fashion by deforming the Bianchi identities by means of a new algebraic element related to the embedding tensor. In this case the appropriate extension of the form algebra is a truncated version of the so-called tensor hierarchy algebra.

Twisted 3D $\mathcal {N}=4$N=4 supersymmetric YM on deformed $\mathbb {A}_{3}^{\ast }$A3* lattice

We study a class of twisted 3D \documentclass[12pt]{minimal}\begin{document}$\mathcal {N}=4$\end{document}N=4 supersymmetric Yang-Mills (SYM) theory on particular 3-dimensional lattice \documentclass[12pt]{minimal}\begin{document}$\mathbb {L}_{3D}$\end{document}L3D formally denoted as \documentclass[12pt]{minimal}\begin{document}$\mathcal {L}_{3D}^{su_{3}\times u_{1}}$\end{document}L3Dsu3×u1 and given by non-trivial fibration \documentclass[12pt]{minimal}\begin{document}$\mathcal {L}_{1D}^{u_{1}}\times \mathcal {L}_{2D}^{su_{3}}$\end{document}L1Du1×L2Dsu3 with base \documentclass[12pt]{minimal}\begin{document}$\mathcal {L}_{2D}^{su_{3}}=\mathbb {A}_{2}^{\ast }$\end{document}L2Dsu3=A2*, the weight lattice of SU(3). We first, develop the twisted 3D \documentclass[12pt]{minimal}\begin{document}$\mathcal {N}=4$\end{document}N=4 SYM in continuum by using superspace method where the scalar supercharge Q is manifestly exhibited. Then, we show how to engineer the 3D lattice \documentclass[12pt]{minimal}\begin{document}$\mathcal {L}_{3D}^{su_{3}\times u_{1}}$\end{document}L3Dsu3×u1 that host this theory. After that we build the lattice action \documentclass[12pt]{minimal}\begin{document}$\mathcal {S}_{latt}$\end{document}Slatt invariant under the following three points: (i) U(N) gauge invariance, (ii) BRST symmetry, (iii) the \documentclass[12pt]{minimal}\begin{document}$\mathbb {S} _{3}$\end{document}S3 point group symmetry of \documentclass[12pt]{minimal}\begin{document}$\mathcal {L}_{3D}^{su_{3}\times u_{1}}$\end{document}L3Dsu3×u1. Other features such as reduction to twisted 2D supersymmetry with 8 supercharges living on \documentclass[12pt]{minimal}\begin{document}$\mathbb {L}_{2D}\equiv \mathcal {L}_{2D}^{su_{2}\times u_{1}}$\end{document}L2D≡L2Dsu2×u1, the extension to twisted maximal 5D SYM with 16 supercharges on lattice \documentclass[12pt]{minimal}\begin{document}$\mathbb {L}_{5D}\equiv \mathcal {L}_{5D}^{su_{4}\times u_{1}}$\end{document}L5D≡L5Dsu4×u1 as well as the relation with known results are also given.

The leading-order dressing phase in ABJM theory

We study the planar asymptotic dilatation operator of ABJM theory in the SU(2) x SU(2) sector up to eight loops. Combining Bethe Ansatz techniques and N = 2 superspace methods, we are able to fix all the coefficients appearing in the maximal-reshuffling terms. In particular, we can directly compute from Feynman diagrams the leading order coefficient of the dressing phase and find an agreement with the relation conjectured by Gromov and Vieira between the ABJM and N = 4 SYM phase factor.

Geometry and harmonic superspace: Some recent progress

This contribution follows the talk, given by F. Delduc at the conference SQS'2011 in Dubna, Russia (July 18-23, 2011). To a considerable extent it is a summary of known facts about the links between geometry and extended supersymmetry in d=1 mechanics, with emphasis on the harmonic superspace method created in Dubna in the 80's. Some recent developments based on ref. [1] are also presented.

A Superspace Method for Discrete-Time Bilinear Model Identification by Interaction Matrices

This paper presents a method for discrete-time bilinear state-space model identification from a single set of input–output data. The initial state can be unknown. By extending the interaction matrix formulation from linear to bilinear state-space models, the bilinear system state is expressed in terms of input–output measurements. This relationship is used to convert the bilinear model to an Equivalent Linear Model (ELM) which can be identified by any linear identification method such as a superspace method presented here. The bilinear state-space model is then extracted from the identified ELM.

Component reduction in $ \mathcal{N} = 2 $ supergravity: the vector, tensor, and vector-tensor multiplets

Recent advances in curved N=2 superspace methods have rendered the component reduction of superspace actions more feasible than in the past. In this paper, we consider models involving both vector and tensor multiplets coupled to supergravity and demonstrate explicitly how component actions may be efficiently obtained. In addition, tensor multiplets coupled to conformal supergravity are considered directly within projective superspace, where their formulation is most natural. We then demonstrate how the inverse procedure -- the lifting of component results to superspace -- can simplify the analysis of complicated multiplets. We address the off-shell N=2 vector-tensor multiplet coupled to conformal supergravity with a central charge and demonstrate explicitly how its constraints and Lagrangian can be written in a simpler way using superfields.

Harmonic Superfields in N=4 Supersymmetric Quantum Mechanics

This is a brief survey of applications of the harmonic superspace methods to the models of N=4 supersymmetric quantum mechanics (SQM). The main focus is on a recent progress in constructing SQM models with couplings to the background non-Abelian gauge fields. Besides reviewing and systemizing the relevant results, we present some new examples and make clarifying comments.

Modes vs. modulations: Symmetry-mode analysis of commensurate modulated structures compared with the superspace method

We know from Landau theory that the natural language to deal with distorted structures is the one of symmetry-adapted modes. A symmetry-mode decomposition is both useful for investigating the mechanisms responsible of these phases and for pure crystallographic purposes. Some symmetry-mode components are usually dominant, introducing a parameter hierarchy or reducing in practice the crystallographic degrees of freedom. Several computer tools are now freely available allowing structure refinements directly using symmetry-mode collective coordinates. Alternatively, the superspace formalism is a very efficient method for similar purposes. By means of several examples we compare the two approaches.

FRAMES FOR SUPERSYMMETRY

We explain how the redefinitions of supermultiplet component fields, comprising what we call "frame shifts," can be used in conjuction with the graphical technology of multiplet Adkinras to render manifest the reducibility of off-shell representations of supersymmetry. This technology speaks to possibility of organizing multiplet constraints in a way which complements and extends the possibilities afforded by superspace methods.

On black hole effective potential in 6D/7D [formula omitted] supergravity

Using the harmonic superspace method and the duality between real and complex representations of hypermultiplets, we compute the explicit scalar field expression of the quaternionic metric G m n ( φ ) of the moduli space SO ( 4 , k ) × SO ( 1 , 1 ) SO ( 4 ) × SO ( k ) of 6D N = 2 supergravity with generic k Maxwell supermultiplets. The obtained metric includes the particular case k = 20 associated with 10D type IIA superstring on K3. Uplifting to 7D N = 2 supergravity is described and aspects of 6D/7D black attractor effective potentials are studied.

Field Strengths of Linearized 5D,N= 1 Superfield Supergravity On a 3-Brane

Recently a description of linearized 5D supergravity in 4D, N =1 superspace was presented. By analyzing the on-shell component Lagrangian, this description was proven to be Lorentz invariant in five dimensions. This paper describes a geometric formulation of the 5D supergravity in 4D, N =1 superspace. Using this new formalism, the on-shell structure of the previous description is verified from purely superspace methods. This geometric description provides a connection to dimensionally reduced manifestly supersymmetric 5D supergravity.

BMN Operators for Script N = 1 superconformal Yang-Mills theories and associated string backgrounds

We study a class of near-BPS operators for a complex 2-parameter family of N=1 superconformal Yang-Mills theories that can be obtained by a Leigh-Strassler deformation of N=4 SYM theory. We identify these operators in the large N and large R-charge limit and compute their exact scaling dimensions using N=1 superspace methods. From these scaling dimensions we attempt to reverse-engineer the light-cone worldsheet theory that describes string propagation on the Penrose limit of the dual geometry.

Systematics of quarter BPS operators in Script N = 4 SYM

A systematic construction is presented of 1/4 BPS operators in N=4 superconformal Yang-Mills theory, using either analytic superspace methods or components. In the construction, the operators of the classical theory annihilated by 4 out of 16 supercharges are arranged into two types. The first type consists of those operators that contain 1/4 BPS operators in the full quantum theory. The second type consists of descendants of operators in long unprotected multiplets which develop anomalous dimensions in the quantum theory. The 1/4 BPS operators of the quantum theory are defined to be orthogonal to all the descendant operators with the same classical quantum numbers. It is shown, to order $g^2$, that these 1/4 BPS operators have protected dimensions.

Perturbative computation of glueball superpotentials for SO(N) and USp(N)

We use the superspace method of hep-th/0211017 to prove the matrix model conjecture for N=1 USp(N) and SO(N) gauge theories in four dimensions. We derive the prescription to relate the matrix model to the field theory computations. We perform an explicit calculation of glueball superpotentials. The result is consistent with field theory expectations.

OPEs and 3-point correlators of protected operators in N=4 SYM

Two- and three-point correlation functions of arbitrary protected operators are constructed in N=4 SYM using analytic superspace methods. The OPEs of two chiral primary multiplets are given. It is shown that the n-point functions of protected operators for n⩽4 are invariant under U(1) Y and it is argued that this implies that the two- and three-point functions are not renormalised. It is shown explicitly how unprotected operators can be accommodated in the analytic superspace formalism in a way which is fully compatible with analyticity. Some new extremal correlators are exhibited.

Supersymmetry with a ghost time

The progress brought to the study of chiral fermions and gauge theories by quantization methods with a bulk time suggests their usefulness in supersymmetric theories. Using superspace methods, we show how an explicitly supersymmetric version of such quantization methods may be given.