Spectral Flow Operator Research Articles

Mirror symmetry and new approach to constructing orbifolds of Gepner models

Motivated by the principles of the conformal bootstrap, primarily the principle of Locality, simultaneously with the requirement of space-time supersymmetry, we reconsider constructions of compactified superstring models. Starting from requirements of space-time supersymmetry and mutual locality, we construct a complete set of physical fields of orbifolds of Gepner models. To technically implement this, we use spectral flow generators to construct all physical fields from the chiral primary fields. The set of these spectral flow operators forms a so-called admissible group Gadm, which defines a given orbifold. The action of these operators produces a collection of physical fields consistent with the action of supersymmetry generators. The selection of mutually local fields from this collection is carried out using the mirror group Gadm⁎. The permutation of Gadm and Gadm⁎ replaces the original orbifold with a mirror one that satisfies the same conditions as the original one. This also implies that the resulting model is modular invariant.

On spectrally flowed local vertex operators in AdS$_3$

We provide a novel local definition for spectrally flowed vertex operators in the SL(2,\mathbb{R})SL(2,ℝ)-WZW model, generalising the proposal of Maldacena and Ooguri in [arXiv:hep-th/0111180] for the singly-flowed case to all \omega>1ω>1. This allows us to establish the precise connection between the computation of correlators using the so-called spectral flow operator, and the methods introduced recently by Dei and Eberhardt in [arXiv:2105.12130] based on local Ward identities. We show that the auxiliary variable yy used in the latter paper arises naturally from a point-splitting procedure in the space-time coordinate. The recursion relations satisfied by spectrally flowed correlators, which take the form of partial differential equations in yy-space, then correspond to null-state conditions for generalised spectral flowed operators. We highlight the role of certain SL(2,\mathbb{R})SL(2,ℝ) discrete module isomorphisms in this context, and prove the validity of the conjecture put forward in [arXiv:2105.12130] for yy-space structure constants of three-point functions with arbitrary spectral flow charges.

Note on the spectral flow operator

The insertion of the spectral flow operator in a string scattering amplitude on AdS$_3\times \mathcal{N}$ produces a change in the winding number of one of the incoming (or outgoing) states, making it possible to compute amplitudes of processes in which winding number in AdS$_3$ is not conserved. The insertion of such operator, however, might seem artificial from the worldsheet theory perspective, as it appears as an unintegrated vertex operator of conformal dimension zero that does not represent any normalizable state. Here, we show that the spectral flow operator naturally emerges in the Liouville field theory description of the WZW correlation functions once it is combined with a series of duality relations among conformal integrals. By considering multiple insertions of spectral flow operators, we study the dependence on the moduli for arbitrary number of them, and we show explicitly that the amplitude does not depend on the specific locations of the accessory insertions in the worldsheet, as required by consistency. This generalizes previous computations in which particular cases were considered. This can also be thought of as an alternative proof of the WZW-Liouville correspondence in the case of maximally winding violating correlators.

Discrete symmetries in the heterotic-string landscape

We describe a new type of discrete symmetry that relates heterotic-string models. It is based on the spectral flow operator which normally acts within a general N = (2, 2) model and we use this operator to construct a map between N = (2, 0) models. The landscape of N = (2, 0) models is of particular interest among all heterotic-string models for two important reasons: Firstly, N =1 spacetime SUSY requires (2, 0) superconformal invariance and secondly, models with the well motivated by the Standard Model SO(10) unification structure are of this type. This idea was inspired by a new discrete symmetry in the space of fermionic ℤ2 × ℤ2 heterotic-string models that exchanges the spinors and vectors of the SO(10) GUT group, dubbed spinor-vector duality. We will describe how to generalize this to arbitrary internal rational Conformal Field Theories.

Spectral flow as a map between [formula omitted]-models

The space of (2,0) models is of particular interest among all heterotic-string models because it includes the models with the minimal SO(10) unification structure, which is well motivated by the Standard Model of particle physics data. The fermionic Z2×Z2 heterotic-string models revealed the existence of a new symmetry in the space of string configurations under the exchange of spinors and vectors of the SO(10) GUT group, dubbed spinor–vector duality. In this paper we generalize this idea to arbitrary internal rational conformal field theories (RCFTs). We explain how the spectral flow operator normally acting within a general (2,2) theory can be used as a map between (2,0) models. We describe the details, give an example and propose more simple currents that can be used in a similar way.

Coulomb integrals for theSL(2,R)Wess-Zumino-Novikov-Witten model

We review the Coulomb gas computation of three-point functions in the $SL(2,\mathbb{R})$ Wess-Zumino-Novikov-Witten model and obtain explicit expressions for generic states. These amplitudes have been computed in the past by this and other methods but the analytic continuation in the number of screening charges required by the Coulomb gas formalism had only been performed in particular cases. After showing that ghost contributions to the correlators can be generally expressed in terms of Schur polynomials, we solve Aomoto integrals in the complex plane, a new set of multiple integrals of Dotsenko-Fateev type. We then make use of monodromy invariance to analytically continue the number of screening operators and prove that this procedure gives results in complete agreement with the amplitudes obtained from the bootstrap approach. We also compute a four-point function involving a spectral flow operator and we verify that it leads to the one unit spectral flow three-point function according to a prescription previously proposed in the literature. In addition, we present an alternative method to obtain spectral flow nonconserving $n$-point functions through well defined operators and we prove that it reproduces the exact correlators for $n=3$. Independence of the result on the insertion points of these operators suggests that it is possible to violate winding number conservation modifying the background charge.

Note on Z2 symmetries of the Knizhnik-Zamolodchikov equation

We continue the study of hidden Z2 symmetries of the four-point sl(2)̂k Knizhnik-Zamolodchikov equation initiated by Giribet [Phys. Lett. B 628, 148 (2005)]. Here, we focus our attention on the four-point correlation function in those cases where one spectral flowed state of the sector ω=1 is involved. We give a formula that shows how this observable can be expressed in terms of the four-point function of non spectral flowed states. This means that the formula holding for the winding violating four-string scattering processes in AdS3 has a simple expression in terms of the one for the conservative case, generalizing what is known for the case of three-point functions, where the violating and the nonviolating structure constants turn out to be connected one to each other in a similar way. What makes this connection particularly simple is the fact that, unlike what one would naively expect, it is not necessary to explicitly solve the five-point function containing a single spectral flow operator to this end. Instead, nondiagonal functional relations between different solutions of the Knizhnik-Zamolodchikov equation turn out to be the key point for this short path to exist. Considering such functional relation is necessary but it is not sufficient; besides, the formula also follows from the relation existing between correlators in both Wess-Zumino-Novikov-Witten (WZNW) and Liouville conformal theories.

Three-point function in perturbed Liouville gravity

Abstract Three-point correlation function in perturbed conformal field theory coupled to two-dimensional quantum gravity (perturbed Liouville gravity) is explicitly computed by using the free field approach. The representation considered here is the one recently proposed in [G. Giribet, Nucl. Phys. B 737 (2006) 209] to describe the string theory in AdS 3 space. Consequently, this computation extends previous results which presented free field calculations of particular cases of string amplitudes, and confirms that the free field approach leads to the exact result. Remarkably, this representation allows to compute winding violating three-point functions without making use of the spectral flow operator.

CLOSED N=2 STRINGS: PICTURE-CHANGING, HIDDEN SYMMETRIES AND SDG HIERARCHY

We study the action of picture-changing and spectral flow operators on a ground ring of ghost number zero operators in the chiral BRST cohomology of the closed N=2 string and describe an infinite set of symmetry charges acting on physical states. The transformations of physical string states are compared with symmetries of self-dual gravity which is the effective field theory of the closed N=2 string. We derive all infinitesimal symmetries of the self-dual gravity equations in (2+2)-dimensional space–time and introduce an infinite hierarchy of commuting flows on the moduli space of self-dual metrics. The dependence on moduli parameters can be recovered by solving the equations of the SDG hierarchy associated with an infinite set of Abelian symmetries generated recursively from translations. These nonlocal Abelian symmetries are shown to coincide with the hidden Abelian string symmetries responsible for the vanishing of most scattering amplitudes. Therefore, N=2 string theory "predicts" not only self-dual gravity but also the SDG hierarchy.

Integration measure and spectral flow in the critical N=2 String

I present the moduli space of the (2+2)-dimensional critical closed fermionic string with two world-sheet supersymmetries. The integration of fermionic and Maxwell moduli in the presence of punctures yields the string measure for n-point amplitudes at arbitrary genus and instanton number. Generalized picture-changing and spectral-flow operators emerge, connecting different instanton sectors. Tree and loop amplitudes are computed.

The string measure and spectral flow of critical N = 2 strings

The general structure of N = 2 moduli space at arbitrary genus and instanton number is investigated. The N = 2 NSR string measure is calculated, yielding picture- and U(1) ghost number-changing operator insertions. An explicit formula for the spectral flow operator acting on vertex operators is given, and its effect on N = 2 string amplitudes is discussed.