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The present paper is devoted to studying 2-local superderivations on the super Virasoro algebra and the super W(2,2) algebra. It is proved that all 2-local superderivations on the super Virasoro al...

We show that direct limit completions of vertex tensor categories inherit vertex and braided tensor category structures, under conditions that hold for example for all known Virasoro and affine Lie algebra tensor categories. A consequence is that the theory of vertex operator (super)algebra extensions also applies to infinite-order extensions. As an application, we relate rigid and non-degenerate vertex tensor categories of certain modules for both the affine vertex superalgebra of $\mathfrak{osp}(1|2)$ and the $N=1$ super Virasoro algebra to categories of Virasoro algebra modules via certain cosets., Comment: 51 pages; in this version, Theorem 7.1 is improved by removing one of the conditions needed to apply it; corresponding improvements are made to the subsequent examples; final version to appear in Communications in Contemporary Mathematics

We define and study a class of $\mathcal{N}=2$ vertex operator algebras $\mathcal{W}_{\mathcal{\mathsf{G}}}$ labelled by complex reflection groups. They are extensions of the $\mathcal{N}=2$ super Virasoro algebra obtained by introducing additional generators, in correspondence with the invariants of the complex reflection group $\mathcal{\mathsf{G}}$. If $\mathcal{\mathsf{G}}$ is a Coxeter group, the $\mathcal{N}=2$ super Virasoro algebra enhances to the (small) $\mathcal{N}=4$ superconformal algebra. With the exception of $\mathcal{\mathsf{G}} = \mathbb{Z}_2$, which corresponds to just the $\mathcal{N}=4$ algebra, these are non-deformable VOAs that exist only for a specific negative value of the central charge. We describe a free-field realization of $\mathcal{W}_{\mathcal{\mathsf{G}}}$ in terms of rank$(\mathcal{\mathsf{G}})$ $\beta \gamma bc$ ghost systems, generalizing a construction of Adamovic for the $\mathcal{N}=4$ algebra at $c = -9$. If $\mathcal{\mathsf{G}}$ is a Weyl group, $\mathcal{W}_{\mathcal{\mathsf{G}}}$ is believed to coincide with the $\mathcal{N}=4$ VOA that arises from the four-dimensional super Yang-Mills theory whose gauge algebra has Weyl group $\mathcal{\mathsf{G}}$. More generally, if $\mathcal{\mathsf{G}}$ is a crystallographic complex reflection group, $\mathcal{W}_{\mathcal{\mathsf{G}}}$ is conjecturally associated to an $\mathcal{N}=3$ $4d$ superconformal field theory. The free-field realization allows to determine the elusive `$R$-filtration' of $\mathcal{W}_{\mathcal{\mathsf{G}}}$, and thus to recover the full Macdonald index of the parent $4d$ theory, Comment: 70 pages

The appearance of L$_\infty$ structures for supersymmetric symmetry algebras in two-dimensional conformal field theories is investigated. Looking at the simplest concrete example of the ${\cal N}=1$ super-Virasoro algebra in detail, we investigate whether an extension to a super-L$_\infty$ algebra is sufficient to capture all appearing signs., Comment: 16 pages, v2: final version to be published

We study global subalgebras of superconformal algebras in two dimensions and their unitary representations. Global superconformal multiplets are decomposed into conformal multiplets using Racah-Speiser algorithm, revealing many essential aspects of superconformal theories such as stress-energy tensor, conserved current, supersymmetric deformation and supersymmetry enhancement. Character formulae for the representations are presented. We further find a collection of conserved charges that are $k$-forms under the R-symmetry, which must be part of the super Virasoro algebra with $\mathcal{N} \geq 3$ supersymmetries., Comment: 61 pages, 15 figures; v2: minor edits, references added

We study 2d N=4 superconformal field theories, focusing on its application on numerical bootstrap study. We derive the superconformal block by utilizing the global part of the super Virasoro algebra and set up the crossing equations for the non-BPS long-multiplet 4-point function. Along the way, we build global N=4 superconformal short and long multiplets, and compute all possible 2,3-point functions of long-multiplets that are needed to construct the superconformal blocks and the crossing equations. Since we consider a long-multiplet 4-point function, the number of crossing equations is huge and we expect it to give a strong constraint than the usual superconformal bootstrap analysis, which relies on BPS 4-point functions. In addition, we present an alternative way to derive crossing equations using N=4 superspace and comment on a puzzle., Comment: 40 pages. Mathematica files attached; v3: a final version to appear in JHEP, with minor corrections

In 2012 Gamayun, Iorgov, Lisovyy conjectured an explicit expression for the Painlev\'e VI $\tau$~function in terms of the Liouville conformal blocks with central charge $c=1$. We prove that proposed expression satisfies Painlev\'e VI $\tau$~function bilinear equations (and therefore prove the conjecture). The proof reduces to the proof of bilinear relations on conformal blocks. These relations were studied using the embedding of a direct sum of two Virasoro algebras into a sum of Majorana fermion and Super Virasoro algebra. In the framework of the AGT correspondence the bilinear equations on the conformal blocks can be interpreted in terms of instanton counting on the minimal resolution of $\mathbb{C}^2/\mathbb{Z}_2$ (similarly to Nakajima-Yoshioka blow-up equations)., Comment: 35 pages, v2 misprints corrected, v4 references added, misprints corrected, v5 small improvements

There are two extensions of Virasoro algebra with particular importance in superstring theory: the Ramond algebra and the Neveu-Schwarz algebra, which are ℤ2-graded extensions of the Virasoro algebra. In this paper, we show that the support of a simple weight module over the Ramond algebra with an infinite-dimensional weight space coincides with the weight lattice and that all intersections of non-trivial weight spaces and odd part or even part of the module are infinite-dimensional. This result together with the one that we have obtained over the Neveu-Schwarz algebra generalizes the result for the Virasoro algebra to the super-Virasoro algebras.

The asymptotic symmetries described in Chap. 9 suggest that the quantization of asymptotically flat gravitational fields in three dimensions provides unitary representations of BMS\(_3\).

We construct infinite dimensional spectral triples associated with representations of the super-Virasoro algebra. In particular the irreducible, unitary positive energy representation of the Ramond algebra with central charge c and minimal lowest weight h=c/24 is graded and gives rise to a net of even theta-summable spectral triples with non-zero Fredholm index. The irreducible unitary positive energy representations of the Neveu-Schwarz algebra give rise to nets of even theta-summable generalised spectral triples where there is no Dirac operator but only a superderivation., 27 pages; v2: a comment concerning the difficulty in defining cyclic cocycles in the NS case have been added

In this paper, Lie super-bialgebra structures on Ramond N = 2 super-Virasoro algebras L are considered. It is proved that all such Lie super-bialgebras are coboundary triangular Lie super-bialgebras.

In this paper we obtain that every super-Virasoro algebra admits only triangular coboundary Lie super-bialgebra structures and this is proved mainly based on the computation of derivations from the super-Virasoro algebra to the tensor product of its adjoint module.

We define the worldline harmonic SU(2|1) superspace and its analytic subspace as a deformation of the flat N=4, d=1 harmonic superspace. The harmonic superfield description of the two mutually mirror off-shell (4,4,0) SU(2|1) supermultiplets is developed and the corresponding invariant actions are presented, as well as the relevant classical and quantum supercharges. Whereas the \sigma-model actions exist for both types of the (4,4,0) multiplet, the invariant Wess-Zumino term can be defined only for one of them, thus demonstrating non-equivalence of these multiplets in the SU(2|1) case as opposed to the flat N=4, d=1 supersymmetry. A superconformal subclass of general SU(2|1) actions invariant under the trigonometric-type realizations of the supergroup D(2,1;\alpha) is singled out. The superconformal Wess-Zumino actions possess an infinite-dimensional supersymmetry forming the centerless N=4 super Virasoro algebra. We solve a few simple instructive examples of the SU(2|1) supersymmetric quantum mechanics of the (4,4,0) multiplets and reveal the SU(2|1) representation contents of the corresponding sets of the quantum states., Comment: 0 + 37 pages, typos corrected, a reference updated

Similarly to the ordinary bosonic Liouville field theory, in its N=1 supersymmetric version an infinite set of operator valued relations, the ``higher equations of motions'', holds. Equations are in one to one correspondence with the singular representations of the super Virasoro algebra and enumerated by a couple of natural numbers $(m,n)$. We demonstrate explicitly these equations in the classical case, where the equations of type $(1,n)$ survive and can be interpreted directly as relations for classical fields. General form of the higher equations of motion is established in the quantum case, both for the Neveu-Schwarz and Ramond series., Comment: Two references added

In this note, we determine the structure of the pre-Verma modules \(N(z,\frac{1}{24}z)\) over the N = 1 Ramond algebra. The results of this note together with the results obtained in Iohara and Koga [Adv Math 178:1–65, 2003] completely determine the structure of Verma modules over the N = 1 super Virasoro algebra both in Neveu Schwarz and Ramond sectors.

In this article, we study the structure of Verma modules and Fock modules over the twisted sector of the N =2 super Virasoro algebras.

The quantum super-algebra structure on the deformed super Virasoro algebra is investigated. More specifically we established the possibility of defining a nontrivial Hopf super-algebra on both one and two-parameters deformed super Virasoro algebras.

We demonstrate that all rational models of the N=2 super Virasoro algebra are unitary. Our arguments are based on three different methods: we determine Zhu's algebra (for which we give a physically motivated derivation) explicitly for certain theories, we analyse the modular properties of some of the vacuum characters, and we use the coset realisation of the algebra in terms of su_2 and two free fermions. Some of our arguments generalise to the Kazama-Suzuki models indicating that all rational N=2 supersymmetric models might be unitary., LaTeX (+amssym.def), 28 pages; minor changes in content, some references added, final version

We present a mechanism to construct four-dimensional charged massless Ramond states using the discrete states of a fivebrane Liouville internal conformal field theory. This conformal field theory has background charge, and admits an inner product which allows positive norm states. A connection among supergravity soliton solutions, Liouville conformal field theory, non-critical string theory and their gauge symmetry properties is given. A generalized construction of the SU(2) super Kac-Moody algebra mixing with the N=1 super Virasoro algebra is analyzed. How these Ramond states evade the DKV no-go theorem is explained., Comment: 20 pages, plain Tex, no figures, published version

Applying the paraquantization of order Q to an open bosonic and spining strings, modified Virasoro and super-Virasoro algebra are obtained. It is shown that the anomaly c-number term is a linear function of Q.

We continue our study of the AGT correspondence between instanton counting on C^2/Z_p and Conformal field theories with the symmetry algebra A(r,p). In the cases r=1, p=2 and r=2, p=2 this algebra specialized to: A(1,2)=H+sl(2)_1 and A(2,2)=H+sl(2)_2+NSR. As the main tool we use a new construction of the algebra A(r,2) as the limit of the toroidal gl(1) algebra for q,t tend to -1. We claim that the basis of the representation of the algebra A(r,2) (or equivalently, of the space of the local fields of the corresponding CFT) can be expressed through Macdonald polynomials with the parameters q,t go to -1. The vertex operator which naturally arises in this construction has factorized matrix elements in this basis. We also argue that the singular vectors of the $\mathcal{N}=1$ Super Virasoro algebra can be realized in terms of Macdonald polynomials for a rectangular Young diagram and parameters q,t tend to -1., 34 pages, v2: references added, misprints corrected; v3: exposition improved, new section inserted; v4: misprints corrected, Propositions 3.1,3.2,4.1 are called Conjectures, new subsection 5.3 were included, reference added. Version for JHEP

We construct higher-spin N=1 superalgebras as extensions of the super-Virasoro algebra containing generators for all spins s≥3/2. We find two distinct classical (Poisson) algebras on the phase superspace. Our results indicate that only one of them can be consistently quantized.

We investigate AdS3/CFT2 correspondence in three dimensional supergravity. We construct a current for general coordinate invariance and that for local supersymmetry via covariant approach. Hamiltonian and supercharge are well defined in terms of vielbein and spin connection. After discussing asymptotic supersymmetry group at the boundary of AdS3 geometry, we show that there exist a direct product of Virasoro algebras at the boundary. We also show that one of them is extended to super Virasoro algebra., 19 pages. references added, typos corrected and discussions on the Dirac bracket of the Hamiltonian and the supercharge are added in section 6

We study the supersymmetric Gelfand-Dickey algebras associated with the superpseudodifferential operators of positive as well as negative leading order. We show that, upon the usual constraint, these algebras contain the N=2 super Virasoro algebra as a subalgebra as long as the leading order is odd. The decompositions of the coefficient functions into N=1 primary fields are then obtained by covariantizing the superpseudodifferential operators. We discuss the problem of identifying N=2 supermultiplets and work out a couple of supermultiplets by explicit computations., Comment: 19 pages (Plain TeX), NHCU-HEP-94-13

We obtained a quantum deformation of super Virasoro algebra by using the Arik-Coon’sq algebra.

We investigate extensions of the N=2 super Virasoro algebra by one additional super primary field and its charge conjugate. Using a supersymmetric covariant formalism we construct all N=2 super W-algebras up to spin 5/2 of the additional generator. Led by these first examples we close with some conjectures on the classification of N=2 ${\cal SW}(1,\Dt)$ algebras., 33 pages, plain tex, BONN-HE-92-23

In the first part of this paper we present the general structure of Osp(1|2) invariant superconformal algebras. We show how Osp(1|2) invariance puts severe constraints on the operator-product expansion of two superconformal fields. Then we introduce Osp(1|2)-invariant normal-ordered products and apply the whole formalism to the construction of extensions of the N = 1 super virasoro algebra, so-called super W-algebras. We compute explicitly the first super W-algebra with three generators, the SW ( 3 2 , 3 2 , 2) algebra. This algebra exists for generic values of the central charge and one additional free parameter.

We undertake a systematic study of the possible extensions of the N = 1 super Virasoro algebra by a superprimary field of spin [Formula: see text]. Besides new extensions, which exist only for specific values of the central charge, we find a new nonlinear algebra (super W2) generated by a spin 2 superprimary which is associative for all values of the central charge. Furthermore, the spin 3 extension is argued to be the symmetry algebra of the m = 6 super Virasoro unitary minimal model, by exhibiting the (A7, D4)-type modular invariant as diagonal in terms of extended characters.

The N = 2 (modified) super KdV and super sine-Gordon equations based on the affine Lie super algebra A(1,1)(1) = SL(2|2)(1)/〈1〉 of rank 2 are constructed. The associated super Gel'fand-Dikii algebra is shown to give the classical N = 2 super Virasoro algebra. The connection of generalized N = 2 super KdV hierarchies to the super KP hierarchy is discussed.

We use a fermionic realization of the SU(2)2×SU(2)2 algebra, and extend the generalized Sugawara like construction to the supersymmetric case. In this framework, a two parameter set of solutions for the N= 1 superconformal algebra at c=32 is obtained.

Using the previous construction of the geometrical representation (GR) of the centerless 1D, N = 4 extended Super Virasoro algebra, we construct the corresponding Short Distance Operation Product Expansions for the deformed version of the algebra. This algebra differs from the regular algebra by the addition of terms containing the Levi-Civita tensor. How this addition changes the super-commutation relations and affects the Short Distance Operation Product Expansions (OPEs) of the associated fields is investigated. The Method of Coadjoint Orbits, which removes the need first to find Lagrangians invariant under the action of the symmetries, is used to calculate the expansions. Finally, an alternative method involving Clifford algebras is investigated for comparison., Comment: 20 pages, corrected typos

The method of coadjoint orbits produces for any infinite dimensional Lie (super) algebra A with nontrivial central charge an action for scalar (super) fields which has at least the symmetry A. In this article, we try to make this method accessible to a larger audience by analyzing several examples in more detail than in the literature. After working through the Kac-Moody and Virasoro cases, we apply the method to the super Virasoro algebra and reobtain the supersymmetric extension of Polyakov's local nonpolynomial action for two-dimensional quantum gravity. As in the Virasoro case this action corresponds to the coadjoint orbit of a pure central extension. We further consider the actions corresponding to the other orbits of the super Virasoro algebra. As a new result we construct the actions for the N = 2 super Virasoro algebra.

We show that the deformations of Virasoro and super Virasoro algebra, constructed earlier on an abstract mathematical background, emerge after Wick rotation, within an exact treatment of discrete-time free field models on a circle. The deformation parameter is $e^\lambda$, where $\lambda=\tau/\rho$ is the ratio of the discrete-time scale $\tau$ and the radius $\rho$ of the compact space., Comment: 3 pages, latex, no figures

We continue the development of the actions, S_{AFF}, by examining the cases where there are N fermionic degrees of freedom associated with a 0-brane. These actions correspond to the interaction of the N-extended super Virasoro algebra with the supergraviton and the associated SO(N) gauge field that accompanies the supermultiplet. The superfield formalism is used throughout so that supersymmetry is explicit., Comment: PACS: 04.65.+e, 11.15.-q, 11.25.-w, 12.60.J

We describe actions that correspond to the interaction of the Super Virasoro algebra with supergravitons. These new field theories introduce a superfield that corresponds to dual elements of the super Virasoro algebra. We are also able to extend the definition of these field theories to higher dimensions. We explicitly exhibit the 2, 3 and 4 dimensional cases. Remarkably, the fundamental prepotentials describing these dual elements of the super Virasoro algebra, in each model, agrees with the known prepotentials of the corresponding supergravity theory. These theories might be important in the quantization of the super Virasoro group, supergravity and in AdS and super AdS gravity., 12 pages

We consider N=1,2 superconformal mechanics in 0+1 dimensions and show that if the Hamiltonian is invertible the superconformal generators can be used to construct half of the super Virasoro algebra. The full algebra can be derived if the special conformal generator is also invertible. The generators are quantized and a general prescription is given for the construction of the N=1 algebra independently of the specific details of the superconformal mechanics provided that in addition its quadratic Casimir operator vanishes., Latex, 12 pages, references added, a minor correction

The q-deformed super Virasoro algebra proposed by Chaichian and Presnajder is examined. Presented is the realizations by the FFZ algebra (the magnetic translation algebra) defined on a two-dimensional lattice with a supersymmetric Hamiltonian., Comment: Latex 9 pages, a couple of short comments are added

Studied is the deformation of super Virasoro algebra proposed by Belov and Chaltikhian. Starting from abstract realizations in terms of the FFZ type generators, various connections of them to other realizations are shown, especially to deformed field representations, whose bosonic part generator is recently reported as a deformed string theory on a noncommutative world-sheet. The deformed Virasoro generators can also be expressed in terms of ordinary free fields in a highly nontrivial way., Comment: neutral fields are replaced by complex fields almost everywhere in Sects. 6 and 7

Utilizing sets of super-vector fields (derivations), explicit expressions are obtained for; (a.) the 1D, N-extended superconformal algebra, (b.) the 1D, N-extended super Virasoro algebra for N = 1, 2 and 4 and (c.) a geometrical realization (GR) covering algebra that contains the super Virasoro algebra for arbitrary values of N. By use of such vector fileds, the super Virasoro algebra is embedded as a geometrical and model-independent structure in 1D and 2D Aleph-null-extended superspace., Comment: 13 pages, Latex

A new Lax operator is proposed from the viewpoint of constructing the integrable hierarchies related with N=2 super $W_n$ algebra. It is shown that the Poisson algebra associated to the second Hamiltonian structure for the resulted hierarchy contains the N=2 super Virasoro algebra as a proper subalgebra. The simplest cases are discussed in detail. In particular, it is proved that the supersymmetric two-boson hierarchy is one of N=2 supersymmetric KdV hierarchies. Also, a Lax operator is supplied for one of N=2 supersymmetric Boussinesq hierarchies., 11 pages, AMS-LaTex, to appear in Phys. Lett. A

We address the question whether so-called m-invariants of the N=2 super Virasoro algebra can be used for the construction of reasonable four-dimensional string models. It turns out that an infinite subset of those are pathological in the sense that - although N=2 supersymmetric - the Ramond sector is not isomorphic to the Neveu-Schwarz sector. Consequently, these two properties are independent and only requiring both guarantees an N=1 space-time supersymmetric string spectrum. However, the remaining 529 consistent spectra - 210 of them are mirrors of Gepner models and 76 real orbifolds - show exact mirror symmetry and are contained in a recent classification of orbifolds of Gepner models., 11 pages, plain TeX, no postscript figures

We study the nonunitary representations of N = 2 Super Virasoro algebra for the rational central charges ĉ < 1. The resolutions for the irreducible representations of N = 2 SVir in terms of the “2-d gravity modules” are obtained and their characters are computed. The correspondence between N = 2 nonunitary minimal models and the Virasoro (q,p) minimal models coupled to 2-d gravity is shown at the level of states. We also define the Hamiltonian reduction of the BRST complex of \(\widehat{sl}\left( N \right)/\widehat{sl}\left( N \right)\) coset to the BRST complex of the W-gravity coupled to the W matter. The case \( \widehat{{sl}}\left( 2 \right) \) is considered explicitly. Finally, we trace down the mechanism of the correspondence between \( \widehat{{sl}}\left( 2 \right)/\widehat{{sl}}\left( 2 \right) \) coset and 2-d gravity models.