Ramond Sector Research Articles

Whittaker modules over the N = 2 Neveu–Schwarz algebra

In this paper, we study Whittaker modules over the [Formula: see text] Neveu–Schwarz algebra [Formula: see text]. We first classify all simple finite-dimensional modules over the positive part of [Formula: see text]. Motivated by this classification, we define Whittaker modules and investigate their simplicity. Finally, we also study the simplicity of the quotient modules of the universal Whittaker module if it is not simple.

S-move matrix in the NS sector of N = 1 super Liouville field theory

In this paper we calculate matrix of modular transformations of the one-point toric conformal blocks in the Neveu-Schwarz sector of N = 1 super Liouville field theory. For this purpose we use explicit expression for this matrix as integral of product of certain elements of fusion matrix. This integral is computed using the chain of integral identities for supersymmetric hyperbolic gamma functions derived by the degeneration of the integrals of parafermionic elliptic gamma functions.

Higher-point gauge-theory couplings of massive spin-2 states in four-dimensional string theories

We explicitly compute the Neveu-Schwarz sector conventional type-I superstring tree-level amplitudes at five points after compactifying to 4D, express the QFT building block in the helicity basis, and give several attempts toward arbitrary n points. More specifically, we consider the interaction of one first excited level and otherwise massless states of conventional type-I superstrings, where the four-dimensional states can, for instance, be realized via D3 branes. We construct the amplitude by using the Berends-Giele currents. From the recursion of Berends-Giele currents, we can generate the higher point amplitude. We also apply the BCFW recursion with massive external legs shifted and get the amplitude for arbitrary n points. Published by the American Physical Society 2024

Non-Weight Modules over the N = 1 Heisenberg–Virasoro Superalgebra

In this paper, we mainly study free U(h)-modules over the N=1 Heisenberg–Virasoro superalgebra. We construct a family of non-weight modules that are free of rank 2 when regarded as modules over the Cartan subalgebra. Moreover, we classify the free U(h)-modules of rank 2 over the N = 1 Heisenberg–Virasoro superalgebra and provide the necessary and sufficient conditions for such g-modules to be isomorphic.

Local derivations and local automorphisms on the super Virasoro algebras

This paper aims to study the local derivations, 2-local automorphisms and local automorphisms on the super-Virasoro algebras. The primary focus is to establish that every local derivation of the super-Virasoro algebras is indeed a derivation, and to demonstrate that every local or 2-local automorphism of the super-Virasoro algebras is an automorphism.

Conformal super Virasoro algebra: Matrix model and quantum deformed algebra

In this paper, we construct the super Virasoro algebra with an arbitrary conformal dimension [Formula: see text] from the generalized [Formula: see text]-deformed quantum algebra and investigate the [Formula: see text]-deformed super Virasoro algebra with the particular conformal dimension [Formula: see text] Furthermore, we perform the [Formula: see text]-conformal Virasoro [Formula: see text]-algebra, the [Formula: see text]-conformal super Virasoro [Formula: see text]-algebra ([Formula: see text]-even) and discuss a toy model for the [Formula: see text]-conformal Virasoro constraints and [Formula: see text]-conformal super Virasoro constraints. Besides, we generalized the notion of the [Formula: see text]-elliptic Hermitian matrix model with an arbitrary conformal dimension [Formula: see text]. Finally, we deduce relevant particular cases generated by quantum algebras known in the literature.

2-Local superderivations on infinite-dimensional Lie superalgebras

The present paper is devoted to studying 2-local superderivations on the twisted N = 2 superconformal algebra, the Ramond N = 2 super-Virasoro algebra and the super W -algebra S W ( 3 2 , 3 2 ) . It is proved that all 2-local superderivations on these Lie superalgebras are superderivations.

Hecke relations among 2d fermionic RCFTs

Recently, Harvey and Wu proposed a suitable Hecke operator for vector-valued SL(2, ℤ) modular forms to connect the characters of different 2d rational conformal field theories (RCFTs). We generalize such an operator to the 2d fermionic RCFTs and call it fermionic Hecke operator. The new Hecke operator naturally maps the Neveu-Schwarz (NS) characters of a fermionic theory to the NS characters of another fermionic theory. Mathematically, it is the natural Hecke operator on vector-valued Γθ modular forms of weight zero. We find it can also be extended to NS~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overset{\\sim }{\ extrm{NS}} $$\\end{document} and Ramond (R) sectors by combining the characters of the two sectors together. We systematically study the fermionic Hecke relations among 2d fermionic RCFTs with up to five NS characters and find that almost all known supersymmetric RCFTs can be realized as fermionic Hecke images of some simple theories such as supersymmetric minimal models. We also study the coset relations between fermionic Hecke images with respect to c = 12k holomorphic SCFTs.

Supersymmetric partition function hierarchies and character expansions

We construct the supersymmetric beta and (q, t)-deformed Hurwitz–Kontsevich partition functions through W-representations and present the corresponding character expansions with respect to the Jack and Macdonald superpolynomials, respectively. Based on the constructed beta and (q, t)-deformed superoperators, we further give the supersymmetric beta and (q, t)-deformed partition function hierarchies through W-representations. We also present the generalized super Virasoro constraints, where the constraint operators obey the generalized super Virasoro algebra and null super 3-algebra. Moreover, the superintegrability for these (non-deformed) supersymmetric hierarchies is shown by their character expansions, i.e., <character>sim character.

Irreducible modules for super-Virasoro algebras from algebraic D-modules

In this paper, we introduce a new family of functors from the category of modules for the Weyl algebra to the category of modules for the super-Virasoro algebras. The properties of these functors are investigated, with an emphasis on irreducibility preservation and natural isomorphisms. By utilizing these functors, we recover some old irreducible super-Virasoro modules, including those from the irreducible intermediate series as well as irreducible U(h)-free modules. Additionally, we provide several families of new irreducible super-Virasoro modules via our constructed functors.

Non-weight modules over N = 1 Lie superalgebras of Block type

Abstract In this paper, a class of non-weight modules over N = 1 {N=1} Lie superalgebras of Block type (without center) is studied. These modules over Ramond algebras of Block type are free of rank 1 when restricted to the universal enveloping algebra of the Cartan subalgebra, while such modules are free of rank 2 over the Neveu–Schwarz algebras of Block type. We present the classification of these modules. Furthermore, the simplicity of these modules is determined.

Fermionic CFTs from classical codes over finite fields

We construct a class of chiral fermionic CFTs from classical codes over finite fields whose order is a prime number. We exploit the relationship between classical codes and Euclidean lattices to provide the Neveu–Schwarz sector of fermionic CFTs. On the other hand, we construct the Ramond sector using the shadow theory of classical codes and Euclidean lattices. We give various examples of chiral fermionic CFTs through our construction. We also explore supersymmetric CFTs in terms of classical codes by requiring the resulting fermionic CFTs to satisfy some necessary conditions for supersymmetry.

Closed string theory without level-matching at the free level

In its traditional form, the string field in closed string field theory is constrained by the level-matching condition, which is imposed beside the action. By analogy with the similar problem for the Ramond sector, it was understood by Okawa and Sakaguchi how to lift this condition and work with unconstrained field by introducing spurious free fields. These authors also pointed out that new backgrounds may exist thanks to a new gauge field which is trivial on flat space, but can generate fluxes on a toroidal background. In this paper, we perform a complete study of the free theory at the tachyonic and massless levels with the aim of setting the stage for studying backgrounds without level-matching.

On classification of fermionic rational conformal field theories

We systematically study how the integrality of the conformal characters shapes the space of fermionic rational conformal field theories in two dimensions. The integrality suggests that conformal characters on torus with a given choice of spin structures should be invariant under a principal congruence subgroup of PSL(2, ℤ). The invariance strongly constrains the possible values of the central charge as well as the conformal weights in both Neveu-Schwarz and Ramond sectors, which improves the conventional holomorphic modular bootstrap method in a significant manner. This allows us to make much progress on the classification of fermionic rational conformal field theories with the number of independent characters less than five.

Simple restricted modules over the N = 1 Ramond algebra as weak modules for vertex operator superalgebras

In the present paper, a class of new simple modules over the N=1 Ramond algebra are constructed, which are induced from simple modules over some finite dimensional solvable Lie superalgebras. These new modules are simple restricted modules over the N=1 Ramond algebra. Combined with the result in [22], a classification of simple weak ψ-twisted W¯(0,c)-modules under certain conditions is also given. At last, some examples of simple restricted N=1 Ramond modules as various versions of Whittaker modules are presented (classical Whittaker modules were studied in [17]).

The free field representation for the GL(1∣1) WZW model revisited

The GL(1∣1) WZW model in the free field realization that uses the bc system is revisited. By bosonizing the bc system we describe the Neveu-Schwarz and Ramond sector modules and in terms of the subspaces of a given fermion number l. We show that there are two sectors of mutually local operators, each consists of all Neveu-Schwarz operators and of Ramond operators with either integer or half-integer spins. Conformal blocks and structure constants are found for operators that correspond the highest weight vectors of the spaces . The crossing and braiding matrices are considered and the hexagon and pentagon equations are shown to be satisfied for typical modules. The degenerate case of conformal blocks with atypical (logarithmic) modules as intermediate states is considered. The known conformal block decomposition of correlation functions in the degenerate case is shown to be related to the degeneration splitting in the crossing and braiding relations. The scalar product in atypical modules is discussed. The decomposition of unity in the full correlation functions in the degenerate case in terms of this scalar product is explained.

Graded extension of Thomas-Whitehead gravity

Thomas-Whitehead (TW) gravity was recently introduced as a projective gauge theory of gravity over a d-dimensional manifold that embeds reparametrization invariance into the action functional for gravitation through the use of the Thomas-Whitehead connection. The projective invariance in this d-dimensional theory enjoys an intimate relationship with the Virasoro coadjoint elements found in string theory as one of the components of the connection, ${\mathcal{D}}_{ab}$, is directly related to the coadjoint elements of the Virasoro algebra. TW gravity exploits projective Gauss-Bonnet terms in the action functional which allows the theory to collapse to Einstein's theory of general relativity in the limit that ${\mathcal{D}}_{ab}$ vanishes. In this paper we develop the graded extension of TW gravity, super-TW gravity, in the framework of a DeWitt supermanifold. We construct the Lagrangian for super-TW gravity, give a detailed derivation of the classical field equations, and discuss the graded extension of the projective connection as a prelude to a future understanding of TW-supergravity (which has manifest supersymmetry) and its relationship to the super-Virasoro algebra.

Wilson lines construction of [formula omitted] conformal blocks

We study N=1 superconformal theory in the context of AdS/CFT correspondence in the large central charge limit using Chern-Simons formulation of 3d gravity. In this limit conformal dimensions of a subclass of so-called light primary superfields remain finite and are governed by osp(1|2) subalgebra of N=1 super-Virasoro algebra. We describe the construction of osp(1|2) conformal blocks in terms of Wilson lines of the Chern-Simons 3d gravity. We consider examples of two and three-point blocks on the sphere and one-point torus blocks of light superprimary fields, which belong to finite-dimensional representations of osp(1|2). We study the correlation function for lower and upper components of the primary osp(1|2) doublets and show that the associated conformal blocks are obtained via Wilson line construction in Chern-Simons theory.

ℛ(p,q)-deformed super Virasoro n-algebras

In this paper, we construct the super Witt algebra and super Virasoro algebra in the framework of the [Formula: see text]-deformed quantum algebras. Moreover, we perform the super [Formula: see text]-deformed Witt [Formula: see text]-algebra, the [Formula: see text]-deformed Virasoro [Formula: see text]-algebra and discuss the super [Formula: see text]-Virasoro [Formula: see text]-algebra ([Formula: see text] even). Besides, we define and construct another super [Formula: see text]-deformed Witt [Formula: see text]-algebra and study a toy model for the super [Formula: see text]-Virasoro constraints. Relevant particular cases induced from the quantum algebras known in the literature are deduced from the formalism developed.

Neveu-Schwarz-Ramond open superstring in the proper-time gauge: Free field theory

We study the Neveu-Schwarz-Ramond (NSR) open super-string theory in the proper-time gauge. The string field action is obtained by evaluating the Polyakov string path integral. In this study, we focus on the open-string free-field action, which corresponds to the string path integral on a strip. Depending on the periodicity of the fermion fields, the open super-string has two sectors: The Neveu-Schwarz (NS) and Ramond (R) sectors. We can impose the gauge conditions to fix the (super-) reparametrization invariance on the two-dimensional metric and its super-partner on the string world sheet to secure the covariance, in contrast to the light-cone gauge condition. Accordingly, the proper-time emerges in the NS sector and both proper-time and its super-partner appear in the R-sector. Integration leads to free-string field actions in both sectors.