Ramond Sector Research Articles

Notes on superconformal representations in two dimensions

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 N≥3 supersymmetries.

Cohomology of q-Deformed Witt–Virasoro Superalgebras of the Hom Type

UDC 512.5 We study Virasoro-type extensions of the $q$-deformed Witt Hom – Lie superalgebras. Moreover, we provide the cohomology of the $q$-deformed Witt – Virasoro superalgebras of the Hom type.

The Conway Moonshine Module is a reflected K3 theory

Recently, Duncan and Mack-Crane established an isomorphism, as Virasoro modules at central charges c = 12, between the space of states of the Conway Moonshine Module and the space of states of a special K3 theory that was extensively studied some time ago by Gaberdiel, Volpato and the two authors. In the present work, we lift this result to the level of modules of the extensions of these Virasoro algebras to N = 4 super Virasoro algebras. Moreover, we relate the super vertex operator algebra and module structure of the Conway Moonshine Module to the operator product expansion of this special K3 theory by a procedure we call reflection. This procedure can be applied to certain superconformal field theories, transforming all fields to holomorphic ones. It also allows to describe certain superconformal field theories within the language of super vertex operator algebras. We discuss reflection and its limitations in general, and we argue that through reflection, the Conway Moons hine Module inherits from the K3 theory a richer structure than anticipated so far. The comparison between the Conway Moonshine Module and the K3 theory is considerably facilitated by exploiting the free fermion description as well as the lattice vertex operator algebra descript ion of both theories. We include an explicit construction of cocycles for the relevant charge lattice s, which are half integral. The transition from the K3 theory to the Conway Moonshine Module via reflection promotes the latter to the role of a medium that collects the symmetries of K3 theories from distinct points of the moduli space, thus uncovering a version of symmetry surfing in this context.

On the supersymmetric extension of asymptotic symmetries in three spacetime dimensions

In this work we obtain known and new supersymmetric extensions of diverse asymptotic symmetries defined in three spacetime dimensions by considering the semigroup expansion method. The super-BMS_3, the superconformal algebra and new infinite-dimensional superalgebras are obtained by expanding the super-Virasoro algebra. The new superalgebras obtained are supersymmetric extensions of the asymptotic algebras of the Maxwell and the mathfrak {so}(2,2)oplus mathfrak {so}(2,1) gravity theories. We extend our results to the mathcal {N}=2 and mathcal {N}=4 cases and find that R-symmetry generators are required. We also show that the new infinite-dimensional structures are related through a flat limit ell rightarrow infty .

A family of non-weight modules over the super-Virasoro algebras

In this paper, a family of non-weight modules over the super-Virasoro algebras are constructed. These modules over the Ramond algebra are free of rank 1 when restricted as modules over the Cartan subalgebra h, while these modules over the Neveu-Schwarz algebra are free of rank 2 when restricted as modules over the Cartan subalgebra H. We obtain a sufficient and necessary condition for such modules to be simple. Moreover, we determine the isomorphism classes of these modules. Finally, we show that these modules constitute a complete classification of free U(h)-modules of rank 1 over the super-Virasoro algebra of Ramond type, and also constitute a complete classification of free U(H)-modules of rank 2 over the super-Virasoro algebra of Neveu-Schwarz type.

Supersymmetric Euler equations associated to the N ≤ 3 Neveu-Schwarz algebra

We give a systematic study about supersymmetric Euler equations on the smooth dual NSreg*(N) of the Neveu-Schwarz algebra NS(N) for N ≤ 3. Let A be the inertia operator and c1,c2∈R, we will show that the N = 2 supersymmetric Euler equation with A=c1+c2D2 is local bi-super-Hamiltonian with the freezing point (c1,c2)∈NSreg*(2), which is similar to that of the Virasoro algebra NS(0), and the N = 3 supersymmetric Euler equation with A=c2D3∂−1 is local bi-super-Hamiltonian with the freezing point (0,c2)∈NSreg*(3), which is similar to that of the N = 1 Neveu-Schwarz algebra NS(1).

Two-loop scattering amplitudes: double-forward limit and colour-kinematics duality

We propose new formulae for the two-loop n-point D-dimensional integrands of scattering amplitudes in Yang-Mills theory and gravity. The loop integrands are written as a double-forward limit of tree-level trivalent diagrams, and are inferred from the formalism of the two-loop scattering equations. We discuss the relationship between the formulae for non-supersymmetric theories and the Neveu-Schwarz sector of the formulae for maximally supersymmetric theories, which can be derived from ambitwistor strings. An important property of the loop integrands is that they are expressed in a representation that includes linear-type propagators. This representation exhibits a loop-level version of the colour- kinematics duality, which follows directly from tree level via the double-forward limit.

Simple restricted modules for Neveu-Schwarz algebra

In this paper, we give a construction of simple modules generalizing both highest weight and Whittaker modules for the Neveu-Schwarz algebra, in the spirit of the work of Mazorchuk and Zhao on simple Virasoro modules. We establish a 1-1 correspondence between simple restricted Neveu-Schwarz modules and simple modules of a family of finite dimensional solvable Lie superalgebras associated to the Neveu-Schwarz algebra. Moreover, for two of these superalgebras all simple modules are classified.

Young supertableaux and the large superconformal algebra

In this paper we consider unitary highest weight irreducible representations of the ‘Large’ superconformal algebra in the Ramond sector as infinite-dimensional graded modules of its zero mode subalgebra, . We describe how representations of may be classified using Young supertableaux, and use the decomposition of as an module to discuss the states which contribute to the supersymmetric index I1, previously proposed in the literature by Gukov, Martinec, Moore and Strominger.

Topological recursion in the Ramond sector

We investigate supereigenvalue models in the Ramond sector and their recursive structure. We prove that the free energy truncates at quadratic order in Grassmann coupling constants, and consider super loop equations of the models with the assumption that the 1/N expansion makes sense. Subject to this assumption, we obtain the associated genus-zero algebraic curve with two ramification points (one regular and the other irregular) and also the supersymmetric partner polynomial equation. Starting with these polynomial equations, we present a recursive formalism that computes all the correlation functions of these models. Somewhat surprisingly, correlation functions obtained from the new recursion formalism have no poles at the irregular ramification point due to a supersymmetric correction — the new recursion may lead us to a further development of supersymmetric generalizations of the Eynard-Orantin topological recursion.

Whittaker modules for the super-Virasoro algebras

In this paper, we define and study Whittaker modules for the super-Viraoro algebras, including the Neveu-Schwarz algebra and the Ramond algebra. We classify the simple Whittaker modules and obtain necessary and sufficient conditions for irreducibility of these modules.

On the spectrum and spacetime supersymmetry of heterotic ambitwistor string

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.

Branching Functions for Admissible Representations of Affine Lie Algebras and Super-Virasoro Algebras

We explicitly calculate the branching functions arising from the tensor product decompositions between level 2 and principal admissible representations over sl ^ 2 . In addition, investigating the characters of the minimal series representations of super-Virasoro algebras, we present the tensor product decompositions in terms of the minimal series representations of super-Virasoro algebras for the case of principal admissible weights.

Heterotic string field theory with cyclic $L_\infty$ structure

We construct a complete heterotic string field theory that includes both the Neveu-Schwarz and Ramond sectors. We give a construction of general string products, which realizes a cyclic L-infinity structure and thus provides with a gauge-invariant action in the homotopy algebraic formulation. Through a map of the string fields, we also give the Wess-Zumino-Witten-like action in the large Hilbert space, and verify its gauge invariance independently.

VOAs labelled by complex reflection groups and 4d SCFTs

We define and study a class of mathcal{N} = 2 vertex operator algebras {mathcal{W}}_{mathrm{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 G. If 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 G = ℤ2, 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}}_{mathrm{G}} in terms of rank(G) βγbc ghost systems, generalizing a construction of Adamovic for the mathcal{N} = 4 algebra at c = −9. If G is a Weyl group, {mathcal{W}}_{mathrm{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 G. More generally, if G is a crystallographic complex reflection group, {mathcal{W}}_{mathrm{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}}_{mathrm{G}} , and thus to recover the full Macdonald index of the parent 4d theory.

The first cohomology of N = 1 super Virasoro algebras

In this paper, we determine the first cohomology groups of centerless N = 1 super Virasoro algebras with coefficients in their tensor density modules. As applications, we classify all central extensions and derivations of the complete spectrum-generating superalgebra.

Three-dimensional Poincaré supergravity and [formula omitted]-extended supersymmetric BMS3 algebra

A new approach for obtaining the three-dimensional Chern-Simons supergravity for the Poincaré algebra is presented. The N-extended Poincaré supergravity is obtained by expanding the super Lorentz theory. We extend our procedure to their respective asymptotic symmetries and show that the N=(1,2,4) super-BMS3 appear as expansions of one Virasoro superalgebra. Interestingly, the N-extended super-BMS3 obtained here are not only centrally extended but also endowed with internal symmetry. We also show that the N-extended super Poincaré algebras with both central and automorphism generators are finite subalgebras.

String-Theory Realization of Modular Forms for Elliptic Curves with Complex Multiplication

It is known that the L-function of an elliptic curve defined over Q is given by the Mellin transform of a modular form of weight 2. Does that modular form have anything to do with string theory? In this article, we address a question along this line for elliptic curves that have complex multiplication defined over number fields. So long as we use diagonal rational N=(2,2) superconformal field theories for the string-theory realizations of the elliptic curves, the weight-2 modular form turns out to be the Boltzmann-weighted (q^{L_0-c/24}-weighted) sum of U(1) charges with F e^{ \pi i F} insertion computed in the Ramond sector.

2d small N=4 Long-multiplet superconformal block

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.

On the existence of an L\u221e structure for the super-Virasoro algebra

The appearance of L∞ structures for supersymmetric symmetry algebras in two-dimensional conformal field theories is investigated. Looking at the simplest concrete example of the mathcal{N}=1 super-Virasoro algebra in detail, we investigate whether an extension to a super-L∞ algebra is sufficient to capture all appearing signs.