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Higher-derivative interactions and transformation rules of the fields in the effective field theories of the massless string states are strongly constrained by space-time symmetries and dualities. Here we use an exact formulation of ten dimensional ${\cal N}=1$ supergravity coupled to Yang-Mills with manifest T-duality symmetry to construct the first order $\alpha'$-corrections of the heterotic string effective action. The theory contains a supersymmetric and T-duality covariant generalization of the Green-Schwarz mechanism that determines the modifications to the leading order supersymmetry transformation rules of the fields. We compute the resulting field-dependent deformations of the coefficients in the supersymmetry algebra and construct the invariant action, with up to and including four-derivative terms of all the massless bosonic and fermionic fields of the heterotic string spectrum., Comment: 58 pages; v2: fixed typos, version to appear in JHEP

A new twistor field formulation of a model of a massless infinite spin particle is proposed. A twistor infinite spin field is found, and its helicity decomposition is obtained. Twistorial equations of motion for infinite spin fields are derived in the cases of integer and half-integer helicities. The infinite integer spin field and infinite half-integer spin field are shown to form an N = 1 infinite spin supermultiplet. The corresponding supersymmetry transformations are presented. It is proved that the supersymmetry algebra is closed on-shell.

We introduce a formalism for describing holomorphic blocks of 3d quiver gauge theories using networks of Ding-Iohara-Miki algebra intertwiners. Our approach is very direct and gives an explicit identification of the blocks with Dotsenko-Fateev type integrals for q-deformed quiver W-algebras. We also explain how quiver theories corresponding to Dynkin diagrams of superalgebras arise, write down the corresponding partition functions and W-algebras, and explain the connection with supersymmetric Macdonald-Ruijsenaars commuting Hamiltonians., 22 pages, v2: typos corrected, references added, v3: more typos corrected

We consider supersymmetry in five dimensions, where the fermionic parameters are a 2-form under SL(5). 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 ${\scr B}(E_4)$, the tensor hierarchy algebra $S(E_4)$ and the exceptional superalgebra $E(5,10)$. A theorem relating ${\scr B}(E_4)$ and $E(5,10)$ to all levels is given., Comment: 14 pp. v2: minor changes, added refs

We study string theory on the pp-wave geometry obtained by taking the Penrose limit around a certain null geodesic of the non-supersymmetric Schrodinger background. We solve for the spectrum of bosonic excitations and find compelling agreement with the dispersion relation of the giant magnons in the Schrodinger background obtained previously in arXiv:1712.03091. Inspired by the pp-wave spectrum we conjecture an exact in the t'Hooft coupling dispersion relation for the magnons in the original Schrodinger background. We show that the pp-wave background admits exactly 16 Killing spinors. We use the explicit form of the latter in order to derive the supersymmetry algebra of the background which explicitly depends on the deformation parameter. Its bosonic subalgebra is of the Newton-Hooke type., Comment: 35 pages; v2: Minor changes, JHEP version

Massless spinning particle model that interacts with electromagnetic and antisymmetric gauge fields in anti-de Sitter space-time is considered as a constrained Hamiltonian system. d-dimensional anti-de Sitter space-time is realized as a real projective manifold parametrized by the homogeneous coordinates. Classical constraints that generate in the presence of interactions minimal world-line supersymmetry algebra extended by the dilatations of the ambient-space homogeneous coordinates are found. Various representations of the Lagrangian of the spinning particle are obtained. Dirac quantization is shown to produce first- and second-order equations for the wave function of the spinning particle that are presented in the homogeneous, inhomogeneous and intrinsic coordinates of $$AdS_d$$ .

We study the unitary representations of the non-compact real forms of the complex Lie superalgebra $${\mathfrak{sl}({\rm n}|{\rm m})}$$ . Among them, only the real form $${\mathfrak{su}({\rm p}, {\rm q}|{\rm m})}$$ with $${({\rm p} + {\rm q}= {\rm n)}}$$ admits nontrivial unitary representations, and all such representations are of the highest-weight type (or the lowest-weight type). We extend the standard oscillator construction of the unitary representations of non-compact Lie superalgebras over standard Fock spaces to generalised Fock spaces which allows us to define the action of oscillator determinants raised to non-integer powers. We prove that the proposed construction yields all the unitary representations including those with continuous labels. The unitary representations can be diagrammatically represented by non-compact Young diagrams. We apply our general results to the physically important case of four-dimensional conformal superalgebra $${\mathfrak{su}(2,2|4)}$$ and show how it yields readily its unitary representations including those corresponding to supermultiplets of conformal fields with continuous (anomalous) scaling dimensions.

We start developing a formalism which allows to construct supersymmetric theories systematically across space-time signatures. Our construction uses a complex form of the supersymmetry algebra, which is obtained by doubling the spinor representation. This allows one to partially disentangle the Lorentz and R-symmetry group and generalizes symplectic Majorana spinors. For the case where the spinor representation is complex-irreducible, the R-symmetry only acts on an internal multiplicity space, and we show that the connected groups which occur are SO(2), SO(1,1), SU(2) and SU(1,1). As an application we construct the off-shell supersymmetry transformations and supersymmetric Lagrangians for five-dimensional vector multiplets in arbitrary signature (t,s). In this case the R-symmetry groups are SU(2) or SU(1, 1), depending on whether the spinor representation carries a quaternionic or para-quaternionic structure. In Euclidean signature the scalar and vector kinetic terms differ by a relative sign, which is consistent with previous results in the literature and shows that this sign flip is an inevitable consequence of the Euclidean supersymmetry algebra.

In this text we combine the notions of supergeometry and supersymmetry. We construct a special class of supermanifolds whose reduced manifolds are (pseudo) Riemannian manifolds. These supermanifolds allow us to treat vector fields on the one hand and spinor fields on the other hand as equivalent geometric objects. This is the starting point of our definition of {supersymmetric Killing structures}. The latter combines subspaces of vector fields and spinor fields, provided they fulfill certain field equations. This naturally leads to a superalgebra that extends the supersymmetry algebra to the case of non-flat reduced space. We examine in detail the additional terms that enter into this structure and we give a lot of examples.

Given the success of the deconstruction program in obtaining ghost-free massive gravity from 5-D Einstein gravity, we propose a modification of the deconstruction procedure that incorporates supersymmetry at the linear level. We discuss the relevant limits of a conjectured interacting theory of a massive spin 2 supermultiplet, and determine the linear theory to be the N=1 Zinoviev theory, a supersymmetric extension of Fierz-Pauli theory. We develop a family of 1-site deconstruction procedures for fermionic fields (yielding Dirac and Majorana mass terms). The deconstruction procedure appropriate for giving fermions a Dirac mass is found to preserve half of the supersymmetry of the 5-D theory. We explicitly check this by deconstructing 5-D N=2 super-Maxwell theory down to 4-D N=1 super-Proca theory, and deconstructing linear 5-D N=2 supergravity down to 4-D N=1 Zinoviev theory, and derive the full 4-D supersymmetry algebras and Stueckelberg symmetries from the 5-D superalgebras and gauge symmetries, respectively. We conjecture that this procedure should admit a generalization to fully non-linear theories., Comment: 37 pages. Typos corrected, references and points of clarification added

In $\mathcal N \geq 2$ superconformal Chern-Simons-matter theories we construct the infinite family of Bogomol'nyi-Prasad-Sommerfield (BPS) Wilson loops featured by constant parametric couplings to scalar and fermion matter, including both line Wilson loops in Minkowski spacetime and circle Wilson loops in Euclidean space. We find that the connection of the most general BPS Wilson loop cannot be decomposed in terms of double-node connections. Moreover, if the quiver contains triangles, it cannot be interpreted as a supermatrix inside a superalgebra. However, for particular choices of the parameters it reduces to the well-known connections of 1/6 BPS Wilson loops in Aharony-Bergman-Jafferis-Maldacena (ABJM) theory and 1/4 BPS Wilson loops in $\mathcal N = 4$ orbifold ABJM theory. In the particular case of $\mathcal N = 2$ orbifold ABJM theory we identify the gravity duals of a subset of operators. We investigate the cohomological equivalence of fermionic and bosonic BPS Wilson loops at quantum level by studying their expectation values, and find strong evidence that the cohomological equivalence holds quantum mechanically, at framing one. Finally, we discuss a stronger formulation of the cohomological equivalence, which implies non-trivial identities for correlation functions of composite operators in the defect CFT defined on the Wilson contour and allows to make novel predictions on the corresponding unknown integrals that call for a confirmation., Comment: 53 pages, 12 figures

We show that there are no smooth warped AdS$_3$ solutions in 10- and 11-dimensional supergravities which preserve strictly more than 16 supersymmetries and have internal space a compact without boundary manifold., Comment: 50 pages

The modular properties of the simple vertex operator superalgebra associated with the affine Kac–Moody superalgebraosp(1|2) at level −5 4 are investigated. After classifying the relaxed highest-weight modules over this vertex operator superalgebra, the characters and supercharacters of the simple weight modules are computed and their modular transforms are determined. This leads to a complete list of the Grothendieck fusion rules by way of a continuous superalgebraic analog of the Verlinde formula. All Grothendieck fusion coefficients are observed to be non-negative integers. These results indicate that the extension to general admissible levels will follow using the same methodology once the classification of relaxed highest-weight modules is completed.

We prove a determinant formula for a parabolic Verma module of a Lie superalgebra, previously conjectured by the second author. Our determinant formula generalizes the previous results of Jantzen for a parabolic Verma module of a (non-super) Lie algebra, and of Kac concerning a (non-parabolic) Verma module for a Lie superalgebra. The resulting formula is expected to have a variety of applications in the study of higher-dimensional supersymmetric conformal field theories. We also discuss irreducibility criteria for the Verma module., 24 pages

We compute the $L_\infty$-theoretic dimensional reduction of the F1/D$p$-brane super $L_\infty$-cocycles with coefficients in rationalized twisted K-theory from the 10d type IIA and type IIB super Lie algebras down to 9d. We show that the two resulting coefficient $L_\infty$-algebras are naturally related by an $L_\infty$-isomorphism which we find to act on the super $p$-brane cocycles by the infinitesimal version of the rules of topological T-duality and inducing an isomorphism between $K^0$ and $K^1$, rationally. In particular this is a derivation of the Buscher rules for RR-fields (Hori's formula) from first principles. Moreover, we show that these $L_\infty$-algebras are the homotopy quotients of the RR-charge coefficients by the "T-duality Lie 2-algebra". We find that the induced $L_\infty$-extension is a gerby extension of a 9+(1+1) dimensional (i.e. "doubled") T-duality correspondence super-spacetime, which serves as a local model for T-folds. We observe that this still extends, via the D0-brane cocycle of its type IIA factor, to a 10+(1+1)-dimensional super Lie algebra. Finally we observe that this satisfies expected properties of a local model space for F-theory elliptic fibrations., 44 pages; v2: added more discussion of double dimensional reduction via cyclic L-infinity cohomology; v3: added derivation of Buscher rules for RR-fields, expanded on role of curved L-infinity algebras in double dimensional reduction

In this paper we introduce the p -deformed fermion algebra as a kind of generalization of the ordinary fermion algebra. We discuss the p -fermion algebra and its representation. We also find the generalized Pauli matrices related to the p -fermion algebra. As applications we discuss the thermodynamics in the canonical ensemble of the p -fermion and the p -fermion gas model.

The basic first-order differential operators of spin geometry that are Dirac operator and twistor operator are considered. Special types of spinors defined from these operators such as twistor spinors and Killing spinors are discussed. Symmetry operators of massless and massive Dirac equations are introduced and relevant symmetry operators of twistor spinors and Killing spinors are constructed from Killing-Yano (KY) and conformal Killing-Yano (CKY) forms in constant curvature and Einstein manifolds. The squaring map of spinors gives KY and CKY forms for Killing and twistor spinors respectively. They constitute a graded Lie algebra structure in some special cases. By using the graded Lie algebra structure of KY and CKY forms, extended Killing and conformal superalgebras are constructed in constant curvature and Einstein manifolds., Comment: 7 pages, published version

An off-shell representation of supersymmetry is a representation of the super Poincare algebra on a dynamically unconstrained space of fields. We describe such representations formally, in terms of the fields and their spacetime derivatives, and we interpret the physical concept of engineering dimension as an integral grading. We prove that formal graded off-shell representations of one-dimensional N-extended supersymmetry, i.e., the super Poincare algebra $\mathfrak {p}^{1|N}$ , correspond to filtered Clifford supermodules over Cl(N). We also prove that formal graded off-shell representations of two-dimensional (p,q)-supersymmetry, i.e., the super Poincare algebra $\mathfrak {p}^{1,1|p,q}$ , correspond to bifiltered Clifford supermodules over Cl(p + q). Our primary tools are Rees superalgebras and Rees supermodules, the formal deformations of filtered superalgebras and supermodules, which give a one-to-one correspondence between filtered spaces and graded spaces with even degree-shifting injections. This generalizes the machinery used by Gerstenhaber to prove that every filtered algebra is a deformation of its associated graded algebra. Our treatment extends the notion of Rees algebras and modules to filtrations which are compatible with a supersymmetric structure. We also describe the analogous constructions for bifiltrations and bigradings.

A bosonization of the quantum affine superalgebra $U_q(\widehat{sl}(M|N))$ is presented for an arbitrary level $k \in {\bf C}$. Screening operators that commute with $U_q(\widehat{sl}(M|N))$ are presented for the level $k \neq -M+N$., LaTEX, 37 pages

We show the impossibility to obtain the D’auria–Fre-type superalgebras that allow for an underlying gauge theoretical structure of D = 11 supergravity from the superalgebra osp(1|32)⊕osp(1|32)−, by means of aWeimar-Woods contraction.

We discuss a linearization procedure of nonlinear supersymmetry (NLSUSY) based on the closure of the commutator algebra for variations of functionals of Nambu-Goldstone fermions and their derivative terms under NLSUSY transformations in Volkov-Akulov NLSUSY theory. In the case of a set of bosonic and fermionic functionals, which leads to (massless) vector linear supermultiplets, we explicitly show that general linear SUSY transformations of basic components defined from those functionals are uniquely determined by examining the commutation relation in the NLSUSY theory., 13 pages; references and discussions added

We determine the general structure of quantum anomalies for the $R$-multiplet of four dimensional $\mathcal{N}=1$ supersymmetric quantum field theories in the presence of background fields for an arbitrary number of Abelian flavor multiplets. By solving the Wess-Zumino consistency conditions for off-shell new minimal supergravity in four dimensions with an arbitrary number of Abelian vector multiplets, we compute the anomaly in the conservation of the supercurrent to leading non trivial order in the gravitino and vector multiplet fermions. We find that both $R$-symmetry and flavor anomalies necessarily lead to a supersymmetry anomaly, thus generalizing our earlier results to non superconformal theories with Abelian flavor symmetries. The anomaly in the conservation of the supercurrent leads to an anomalous transformation for the supercurrent under rigid supersymmetry on bosonic backgrounds that admit new minimal Killing spinors. The resulting deformation of the supersymmetry algebra has implications for supersymmetric localization computations on such backgrounds., Comment: 34 pages; v2: generalized analysis of flavor supersymmetry anomalies; references added

We find new homogeneous r matrices containing supercharges, and use them to find new backgrounds of Yang-Baxter deformed superstrings. We obtain these as limits of unimodular inhomogeneous r matrices and associated deformations of AdS2 x S2 x T6 and AdS5 x S5. Our r matrices are jordanian, but also unimodular, and lead to solutions of the regular supergravity equations of motion. In general our deformations are equivalent to particular non-abelian T duality transformations. Curiously, one of our backgrounds is also equivalent to one produced by TsT transformations and an S duality transformation., v3, 16 pages, typos corrected, final version submitted to SciPost

The Newlander-Nirenberg theorem says that a necessary and sufficient condition for the complex coordinates associated with a given almost complex structure tensor $I_M{}^N$ to exist is the vanishing of the Nijenhuis tensor ${\cal N}_{MN}{}^K$. In the first part of the paper, we give a simple explicit proof of this fact. In the second part, we discuss a supersymmetric interpretation of this theorem. ${\it (i)}$ The condition ${\cal N}_{MN}{}^K = 0$ is necessary for a certain $N=1$ supersymmetric mechanical sigma models to enjoy $N=2$ supersymmetry. ${\it (ii)}$ The sufficiency of this condition for the existence of complex coordinates implies that the representation of the supersymmetry algebra realized by the superfields associated with all the real coordinates and their superpartners can be presented as a direct sum of d irreducible representations (d is the complex dimension of the manifold)., Comment: 14 pages; final version to appear in the Proceedings of the XIII Int. Workshop on Lie Theory and Its Applications in Physics (Varna, June 2019)

Operators transform anomalously under the symmetry in the presence of quantum anomalies. We study this aspect of the super-Weyl anomaly in $\mathcal N=(1,1)$ superconformal field theories (SCFTs), in the context of AdS/CFT. In particular, we carry out holographic renormalization for $(1,1)$ pure AdS$_3$ supergravity that is supposed to be a gravity dual of the $\mathcal N=(1,1)$ SCFT, and derive holographic superconformal Ward identities with corresponding anomalies. We show that the obtained super-Weyl anomaly of the $\mathcal N=(1,1)$ SCFT induces a quantum correction term in the transformation law of the supercurrent under the rigid supersymmetry. We demonstrate that the correction term does not affect the $\mathcal N=(1,1)$ rigid supersymmetry algebra on two-sphere., 17 pages

We solve the Wess-Zumino consistency conditions of $\mathcal{N}=1$ off-shell conformal supergravity in four dimensions and determine the general form of the superconformal anomalies for arbitrary $a$ and $c$ anomaly coefficients to leading non trivial order in the gravitino. Besides the well known Weyl and $R$-symmetry anomalies, we compute explicitly the fermionic $\mathcal{Q}$- and $\mathcal{S}$-supersymmetry anomalies. In particular, we show that $\mathcal{Q}$-supersymmetry is anomalous if and only if $R$-symmetry is anomalous. The $\mathcal{Q}$- and $\mathcal{S}$-supersymmetry anomalies give rise to an anomalous supersymmetry transformation for the supercurrent on curved backgrounds admitting Killing spinors, resulting in a deformed rigid supersymmetry algebra. Our results may have implications for supersymmetric localization and supersymmetry phenomenology. Analogous results are expected to hold in dimensions two and six and for other supergravity theories. The present analysis of the Wess-Zumino consistency conditions reproduces the holographic result of arxiv:1703.04299 and generalizes it to arbitrary $a$ and $c$ anomaly coefficients., 13+13 pages; v2: minor corrections and improvements; references added; v3: further minor typos corrected; version published in JHEP

We study partial supersymmetry breaking from ${\cal N}=2$ to ${\cal N}=1$ by adding non-linear terms to the ${\cal N}=2$ supersymmetry transformations. By exploiting the necessary existence of a deformed supersymmetry algebra for partial breaking to occur, we systematically use ${\cal N}=2$ projective superspace with central charges to provide a streamlined setup. For deformed ${\cal O}(2)$ and ${\cal O}(4)$ hypermultiplets, besides reproducing known results, we describe new models exhibiting partial supersymmetry breaking with and without higher-derivative interactions., 49 pages, v2

We explicitly construct a supersymmetric $so(n)$ spin-Calogero model with an arbitrary even number $\cal N$ of supersymmetries. It features $\frac{1}{2}{\cal N}n(n{+}1)$ rather than ${\cal N}n$ fermionic coordinates and a very simple structure of the supercharges and the Hamiltonian. The latter, together with additional conserved currents, form an $osp({\cal N}|2)$ superalgebra. We provide a superspace description for the simplest case, namely ${\cal N}{=}2$ supersymmetry. The reduction to an $\cal N$-extended supersymmetric goldfish model is also discussed., 10 pages, v2: additional remark in Conclusions, typos corrected, matches published version

We explain how to perform topological twisting of supersymmetric field theories in the language of factorization algebras. Namely, given a supersymmetric factorization algebra with a choice of a topological supercharge we construct an algebra over the operad of little disks. We also explain the role of the twisting homomorphism allowing us to construct an algebra over the operad of framed little disks. Finally, we give a complete classification of topological supercharges and twisting homomorphisms in dimensions $1$ through $10$., 56 pages

In this chapter we introduce a number of notational conventions, describe our primary object of study—the (0,2) supersymmetry algebra, and give a Lagrangian field realization of this structure.