Friedemann Brandt
The primitive elements of the supersymmetry algebra cohomology as defined in a previous paper are computed for standard supersymmetry algebras in four and five dimensions, for all signatures of the metric and any number of supersymmetries.
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The primitive elements of the supersymmetry algebra cohomology as defined in a companion paper are computed exhaustively for standard supersymmetry algebras in dimensions D=2 and D=3, for all signatures (t,D-t) and all numbers N of sets of supersymmetries.
An Adinkra is a class of graphs with certain signs marking its vertices and edges, which encodes off-shell representations of the super Poincaré algebra. The markings on the vertices and edges of an Adinkra are cochains for cubical cohomology. This article explores the cubical cohomology of Adinkras, treating these markings analogously to characteristic classes on smooth manifolds.
The paper concerns standard supersymmetry algebras in diverse dimensions, involving bosonic translational generators and fermionic supersymmetry generators. A cohomology related to these supersymmetry algebras, termed supersymmetry algebra cohomology, and corresponding "primitive elements" are defined by means of a BRST-type coboundary operator. A method to systematically compute this cohomology is outlined and illustrated by simple examples.
The primitive elements of the supersymmetry algebra cohomology as defined in previous work are derived for standard supersymmetry algebras in dimensions D=5,...,11 for all signatures of the related Clifford algebras of gamma matrices and all numbers of supersymmetries. The results are presented in a uniform notation along with results of previous work for D=4, and derived by means of dimensional extension from D=4 up to D=11.
The cohomology of a compact Kaehler (resp. hyperKaehler) manifold admits the action of the Lie algebra so(2,1) (resp. so(4,1)). In this paper we show, following an idea of Witten, how this action follows from supersymmetry, in particular from the symmetries of certain supersymmetric sigma models. In addition, many of the fundamental identities in Hodge-Lefschetz theory are also naturally derived from supersymmetry.
We present a complete classification, at the classical level, of the observables of topological Yang-Mills theories with an extended shift supersymmetry of N generators, in any space-time dimension. The observables are defined as the Yang-Mills BRST cohomology classes of shift supersymmetry invariants. These cohomology classes turn out to be solutions of an N-extension of Witten's equivariant cohomology. This work generalizes results known in the case of shift supersymmetry with a single generator.
We introduce new finite-dimensional cohomologies on symplectic manifolds. Each exhibits Lefschetz decomposition and contains a unique harmonic representative within each class. Associated with each cohomology is a primitive cohomology defined purely on the space of primitive forms. We identify the dual currents of lagrangians and more generally coisotropic submanifolds with elements of a primitive cohomology, which dualizes to a homology on coisotropic chains.
We show that the exterior derivative operator on a symplectic manifold has a natural decomposition into two linear differential operators, analogous to the Dolbeault operators in complex geometry. These operators map primitive forms into primitive forms and therefore lead directly to the construction of primitive cohomologies on symplectic manifolds. Using these operators, we introduce new primitive cohomologies that are analogous to the Dolbeault cohomology in the complex theory. Interestingly, the finiteness of these primitive cohomologies follows directly from an elliptic complex. We calcul…
We study the homology and cohomology groups of super Lie algebra of supersymmetries and of super Poincare Lie algebra in various dimensions. We give complete answers for (non-extended) supersymmetry in all dimensions $\leq 11$. For dimensions $D=10,11$ we describe also the cohomology of reduction of supersymmetry Lie algebra to lower dimensions. Our methods can be applied to extended supersymmetry algebra.
We present an equivariant extension of the Thom form with respect to a vector field action, in the framework of the Mathai-Quillen formalism. The associated Topological Quantum Field Theories correspond to twisted $N=2$ supersymmetric theories with a central charge. We analyze in detail two different cases: topological sigma models and non-abelian monopoles on four-manifolds.
The field equations of the auxiliary fields are nonlinear and free of derivatives. Hence, it is argued, a Legendre transform to generate the 1PI Generating Functionals is not correct for the auxiliary fields. A corrected formulation of the BRS symmetry must be constructed by integrating the auxiliary fields. This necessarily destroys manifest supersymmetry. It also generates a new "Physical BRS-ZJ" cohomology problem. The resulting "Physical BRS-ZJ cohomology" is then described. The cohomology contains a rich spectrum of potential supersymmetry anomalies in certain composite operators that hav…
The application and extension of well-known BRST cohomological methods to supersymmetric field theories are discussed. The focus is on the emergence and particular features of supersymmetry algebra cohomology in this context. In particular it is discussed and demonstrated that supersymmetry algebra cohomology emerges within the cohomological analysis of standard supersymmetric field theories whether or not the commutator algebra of the symmetry transformations closes off-shell.
We examine the BRS cohomology of chiral matter in $N=1$, $D=4$ supersymmetry to determine a general form of composite superfield operators which can suffer from supersymmetry anomalies. Composite superfield operators $\Y_{(a,b)}$ are products of the elementary chiral superfields $S$ and $\ov S$ and the derivative operators $D_\a$, $\ov D_{\dot \b}$ and $\pa_{\a \dot \b}$. Such superfields $\Y_{(a,b)}$ can be chosen to have `$a$' symmetrized undotted indices $\a_i$ and `$b$' symmetrized dotted indices $\dot \b_j$. The result derived here is that each composite superfield $\Y_{(a,b)}$ is subject…
We analyse the non-abelian algebra and the supersymmetric cohomology associated to the local and non-local conserved charges of N=1 SKdV under Poisson brackets. We then consider the breaking of the supersymmetry and obtain an integrable model in terms of Clifford algebra valued fields. We discuss the remaining conserved charges of the new system and the stability of the solitonic solutions.
We analyse the BRST cohomology of the Ramond sector of heterotic ambitwistor theory. We also write the free string field theory action and compute the supersymmetry transformations.
This paper is the second paper in a series of four papers that introduce cybersusy, which is a new method for analyzing supersymmetry breaking in the standard supersymmetric model (SSM). The first paper was a summary of the results and the three next papers set out the details. In this second paper, we derive the full BRS operator and action for the general massless Wess-Zumino chiral supersymmetry action. This includes the source terms which bring in the equations of motion. The auxiliary field is integrated, which removes manifest supersymmetry, but which allows the Legendre transform to ope…
The local cohomology of an extended BRST differential which includes global N=1 supersymmetry and Poincare transformations is completely and explicitly computed in four-dimensional supersymmetric gauge theories with super-Yang-Mills multiplets, chiral matter multiplets and linear multiplets containing 2-form gauge potentials. In particular we determine to first order all N=1 supersymmetric and Poincare invariant consistent deformations of these theories that preserve the N=1 supersymmetry algebra on-shell modulo gauge transformations, and all Poincare invariant candidate gauge and supersymmetr…
Unleash $Q$! Cohomology, Localization, and Interpolation in Parisi-Sourlas Supersymmetry
本研究通过Q-精确性解释维度约简，提供新的超对称局部化证明，并扩展卡迪插值证明至任意具有巴黎-苏拉斯超对称性的理论。
Supersymmetry bicomplex of pure spinor AdS background
本文通过BRST算子与李上同调微分构成的双复形及其谱序列，研究AdS超对称代数在AdS₅×S⁵无穷小形变表示上的结构，特别澄清了鬼数三零模的结构。
The established results concerning the BRS cohomology of supersymmetric theories in four space-time dimensions are briefly reviewed. The current status of knowledge concerning supersymmetry anomalies and the possibility that supersymmetry breaks itself through anomalies in local composite operators is then discussed. It turns out that the simplest allowable supersymmetry anomalies occur only in conjunction with the spontaneous breaking of gauge symmetry. A simple example of such a possible supersymmetry anomaly is presented.
Supersymmetry might be broken, in the real world, by anomalies that affect composite operators, while leaving the action supersymmetric. New constraint equations that govern the composite operators and their anomalies are examined. It is shown that the supersymmetric standard model has special properties that allow simple and physically interesting solutions to the constraint equations.
Supersymmetry transformations are a kind of square root of spacetime translations. The corresponding Lie superalgebra always contains the supertranslation operator $ δ= c^α σ^μ_{α\dot β} {\overline c}^{\dot β} (ε^μ)^† $. We find that the cohomology of this operator depends on a spin-orbit coupling in an SU(2) group and has a quite complicated structure. This spin-orbit type coupling will turn out to be basic in the cohomology of supersymmetric field theories in general.
Pure spinor cohomology has been used to describe maximally supersymmetric theories, like D=10 super-Yang-Mills and D=11 supergravity. Supersymmetry closes on-shell in such theories, and the fields in the cohomology automatically satisfy the equations of motion. In this paper, we investigate the corresponding structure in a model with off-shell supersymmetry, N=1 super-Yang-Mills theory in D=6. Here, fields and antifields are obtained as cohomologies in different complexes with respect to the BRST operator Q. It turns out to be natural to enlarge the pure spinor space with additional bosonic va…
This is the third paper in a series of four papers which introduce cybersusy, which is a new mechanism for supersymmetry breaking in the supersymmetric standard model (SSM). In this paper we display some solutions to the constraint equations of BRS cohomology in the SSM. In particular we discuss the leptonic dotspinor pseudosupermultiplets that were used in Cybersusy I to calculate leptonic supersymmetry breaking in the supersymmetric standard model. We also introduce examples of hadronic dotspinor pseudosupermultiplets that will induce baryonic supersymmetry breaking in the SSM for the baryon…
We study the homology and cohomology groups of super Lie algebra of supersymmetries and of super Poincare algebra. We discuss in detail the calculation in dimensions D=10 and D=6. Our methods can be applied to extended supersymmetry algebra and to other dimensions.
After defining cohomologically higher order BRST and anti-BRST operators for a compact simple algebra {\cal G}, the associated higher order Laplacians are introduced and the corresponding supersymmetry algebra $Σ$ is analysed. These operators act on the states generated by a set of fermionic ghost fields transforming under the adjoint representation. In contrast with the standard case, for which the Laplacian is given by the quadratic Casimir, the higher order Laplacians $W$ are not in general given completely in terms of the Casimir-Racah operators, and may involve the ghost number operator. …
The BRS Cohomology of the Wess Zumino Chiral Scalar supersymmetric model with exotic pairs and exotic triplets (E2)
本文利用谱序列方法研究Wess-Zumino超对称模型中引入赝场后的BRS上同调，发现零维产生“奇异对”不变量与超对称反常、一维产生“奇异三重态”，并考察了低自旋低维情形下的约束与结构。
The forms of the invariant primitive tensors for the simple Lie algebras A_l, B_l, C_l and D_l are investigated. A new family of symmetric invariant tensors is introduced using the non-trivial cocycles for the Lie algebra cohomology. For the A_l algebra it is explicitly shown that the generic forms of these tensors become zero except for the l primitive ones and that they give rise to the l primitive Casimir operators. Some recurrence and duality relations are given for the Lie algebra cocycles. Tables for the 3- and 5-cocycles for su(3) and su(4) are also provided. Finally, new relations invo…
The BRST cohomology of the N=2 supersymmetric Yang-Mills theory in four dimensions is discussed by making use of the twisted version of the N=2 algebra. By the introduction of a set of suitable constant ghosts associated to the generators of N=2, the quantization of the model can be done by taking into account both gauge invariance and supersymmetry. In particular, we show how the twisted N=2 algebra can be used to obtain in a straightforward way the relevant cohomology classes. Moreover, we shall be able to establish a very useful relationship between the local gauge invariant polynomial $trϕ…
The renormalization of quantum field theory twists the antipode of a noncocommutative Hopf algebra of rooted trees, decorated by an infinite set of primitive divergences. The Hopf algebra of undecorated rooted trees, ${\cal H}_R$, generated by a single primitive divergence, solves a universal problem in Hochschild cohomology. It has two nontrivial closed Hopf subalgebras: the cocommutative subalgebra ${\cal H}_{\rm ladder}$ of pure ladder diagrams and the Connes-Moscovici noncocommutative subalgebra ${\cal H}_{\rm CM}$ of noncommutative geometry. These three Hopf algebras admit a bigrading by …
We give a very simple derivation of the forms of $N=2,D=10$ supergravity from supersymmetry and $SL(2,\bbR)$ (for IIB). Using superspace cohomology we show that, if the Bianchi identities for the physical fields are satisfied, the (consistent) Bianchi identities for all of the higher-rank forms must be identically satisfied, and that there are no possible gauge-trivial Bianchi identities ($dF=0$) except for exact eleven-forms. We also show that the degrees of the forms can be extended beyond the spacetime limit, and that the representations they fall into agree with those predicted from Borche…
It has long been appreciated that superalgebras with bosonic and fermionic generators additional to those in the super-Poincare algebra underlie p-brane and D-brane actions in superstring theory. These algebras have been revealed via "bottom up" approaches, involving consideration of Noether charges, and by "top down" approaches, involving the construction of manifestly supersymmetry invariant Wess-Zumino actions. In this paper, we give an alternative derivation of these algebras based on integrability of supersymmetry transformations assigned to fields in order to solve a cohomology problem r…
We investigate the super-de Rham complex of five-dimensional superforms with $N=1$ supersymmetry. By introducing a free supercommutative algebra of auxiliary variables, we show that this complex is equivalent to the Chevalley-Eilenberg complex of the translation supergroup with values in superfields. Each cocycle of this complex is defined by a Lorentz- and iso-spin-irreducible superfield subject to a set of constraints. Restricting to constant coefficients results in a subcomplex in which components of the cocycles are coboundaries while the constraints on the defining superfields span the co…
We develop BRST quantization of gauge theories with a soft gauge algebra on spaces with asymptotic boundaries. The asymptotic boundary conditions are imposed on background fields, while quantum fluctuations about these fields are described in terms of quantum fields that vanish at the boundary. This leads us to construct a suitable background field formalism that is generally applicable to soft gauge algebras, and therefore to supergravity. We define a nilpotent BRST charge that acts on both the background and the quantum fields, as well as on the background and quantum ghosts. When the backgr…
We explore two-dimensional sigma models with (0,2) supersymmetry through their chiral algebras. Perturbatively, the chiral algebras of (0,2) models have a rich infinite-dimensional structure described by the cohomology of a sheaf of chiral differential operators. Nonperturbatively, instantons can deform this structure drastically. We show that under some conditions they even annihilate the whole algebra, thereby triggering the spontaneous breaking of supersymmetry. For a certain class of Kähler manifolds, this suggests that there are no harmonic spinors on their loop spaces and gives a physica…
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