NSR Formalism Research Articles

Strings in Ramond-Ramond backgrounds from the Neveu-Schwarz-Ramond formalism

We treat RR flux backgrounds of type II string theory in the framework of closed superstring field theory based on the NSR formalism, focusing on two examples: (1) the pp-wave background supported by 5-form flux, and (2) AdS3 × S3 × M4 supported by mixed 3-form fluxes. In both cases, we analyze the classical string field solution perturbatively, and compute the correction to the dispersion relation of string states to quadratic order in the RR flux. In the first example, our result is in a delicate way consistent with that obtained from lightcone quantization of the Green-Schwarz string. In the second example, we will obtain numerically the mass corrections to pulsating type IIB strings in AdS3 × S3 × M4. Our results, valid at finite AdS radius, agree with previously known answers in the semiclassical limit and in the BMN limit respectively.

Some tree-level string amplitudes in the NSR formalism

We calculate tree level scattering amplitudes for open strings using the NSR formalism. We present a streamlined symmetry-based and pedagogical approach to the computations, which we first develop by checking two-, three-, and four-point functions involving bosons and fermions. We calculate the five-point amplitude for massless gluons and find agreement with an earlier result by Brandt, Machado and Medina. We then compute the five-point amplitudes involving two and four fermions respectively, the general form of which has not been previously obtained in the NSR formalism. The results nicely confirm expectations from the supersymmetric $F^4$ effective action. Finally we use the prescription of Kawai, Lewellen and Tye (KLT) to compute the amplitudes for the closed string sector.

THE WORLDSHEET PERSPECTIVE OF T-DUALITY SYMMETRY IN STRING THEORY

The purpose of this paper is to present a pedagogical review of T-duality in string theory. The evolution of the closed string is envisaged on the worldsheet in the presence of its massless excitations. The duality symmetry is studied when some of the spacial coordinates are compactified on d-dimensional torus, Td. The known results are reviewed to elucidate that equations of motion for the compact coordinates are O(d, d) covariant, d being the number of compact directions. Next, the vertex operators of excited massive levels are considered in a simple compactification scheme. It is shown that the vertex operators for each massive level can be cast in a T-duality invariant form in such a case. Subsequently, the duality properties of superstring is investigated in the NSR formulation for the massless backgrounds such as graviton and antisymmetric tensor. The worldsheet superfield formulation is found to be very suitable for our purpose. The Hassan–Sen compactification is adopted and it is shown that the worldsheet equations of motion for compact superfields are O(d, d) covariant when the backgrounds are independent of superfields along compact directions. The vertex operators for excited levels are presented in the NS–NS sector and it is shown that they can be cast in T-duality invariant form for the case of Hassan–Sen compactification scheme. An illustrative example is presented to realize our proposal.

Spacetime fermions in light-cone gauge superstring field theory and dimensional regularization

We consider the dimensional regularization of the light-cone gauge type II superstring field theories in the NSR formalism. In the previous work, we have calculated the tree-level amplitudes with external lines in the (NS,NS) sector using the regularization and shown that the desired results are obtained without introducing contact term interactions. In this work, we study the tree-level amplitudes with external lines in the Ramond sector. In order to deal with them, we propose a worldsheet theory to be used instead of that for the naive dimensional regularization. With the worldsheet theory, we regularize and define the tree-level amplitudes by analytic continuation. We show that the results coincide with those of the first quantized formulation.

Scattering of massive open strings in pure spinor

In Park (2008) [4], it was proposed that the D-brane geometry could be produced by open string quantum effects. In an effort to verify the proposal, we consider scattering amplitudes involving massive open superstrings. The main goal of this paper is to set the ground for two-loop “renormalization” of an oriented open superstring on a D-brane and to strengthen our skill in the pure spinor formulation of a superstring, an effective tool for multi-loop string diagrams. We start by reviewing scattering amplitudes of massless states in the 2D component method of the NSR formulation. A few examples of massive string scattering are worked out. The NSR results are then reproduced in the pure spinor formulation. We compute the amplitudes using the unintegrated form of the massive vertex operator constructed by Berkovits and Chandia (2002) [15]. We point out that it may be possible to discover new Riemann type identities involving Jacobi ϑ-functions by comparing a NSR computation and the corresponding pure spinor computation.

M theory through the looking glass: Tachyon condensation in theE8heterotic string

We study the spacetime decay to nothing in string theory and M-theory. First we recall a nonsupersymmetric version of heterotic M-theory, in which bubbles of nothing -- connecting the two E_8 boundaries by a throat -- are expected to be nucleated. We argue that the fate of this system should be addressed at weak string coupling, where the nonperturbative instanton instability is expected to turn into a perturbative tachyonic one. We identify the unique string theory that could describe this process: The heterotic model with one E_8 gauge group and a singlet tachyon. We then use worldsheet methods to study the tachyon condensation in the NSR formulation of this model, and show that it induces a worldsheet super-Higgs effect. The main theme of our analysis is the possibility of making meaningful alternative gauge choices for worldsheet supersymmetry, in place of the conventional superconformal gauge. We show in a version of unitary gauge how the worldsheet gravitino assimilates the goldstino and becomes dynamical. This picture clarifies recent results of Hellerman and Swanson. We also present analogs of R_\xi gauges, and note the importance of logarithmic CFT in the context of tachyon condensation.

Non(anti)commutative Script N = 2 supersymmetric gauge theory from superstrings in graviphoton background

We study open string amplitudes with the D3-branes in type IIB superstring theory compactified on C^2/Z_2. We introduce constant graviphoton background along the branes and calculate disk amplitudes using the NSR formalism. We take the zero slope limit and investigate the effective Lagrangian on the D3-branes deformed by the graviphoton background. We find that the deformed Lagrangian agrees with that of N=2 supersymmetric U(N) gauge theory defined in non(anti)commutative N=1 superspace by choosing appropriate graviphoton background. It is also shown that abelian gauge theory defined in N=2 harmonic superspace with specific non-singlet deformation is consistent with the deformed theory.

Hybrid superstrings on singular Calabi–Yau fourfolds

Two-dimensional hybrid superstring on singular Calabi–Yau manifolds is studied by the field redefinition of the NSR formalism. The compactification on singular Calabi–Yau fourfold is described by N=2 super-Liouville theory and N=2 Landau–Ginzburg models. We examine the world-sheet topological N=4 superconformal algebra, which is useful to identify physical states of the theory. It is shown that the space–time superconformal symmetry is not compatible with this original topological algebra. A new model is proposed by modifying the topological superconformal generators, which is consistent with space–time N=2 superconformal symmetry.

WORLD SHEET LOGARITHMIC CFT IN AdS STRINGS, GHOST-MATTER MIXING AND M THEORY

We discuss several closely related concepts in the NSR formulation of superstring theory. We demonstrated that recently proposed NSR model for superstrings on AdS 5× S5 is described by the world sheet logarithmic conformal field theory (LCFT). The origin of LCFT on a world sheet is closely connected to the matter-ghost mixing in the structure of a brane-like vortex operators. We suggest a dynamical origin of M theory as a string theory with an extra dimension given by bosonized superconformal ghosts.

Quantization of superstrings in Ramond-Ramond backgrounds

We present a perturbative study of Ramond-Ramond backgrounds in the NSR formalism. We show how to perform sigma-model computations and discuss in detail the structure of the BRST charge and picture-changing operators. Contact terms play a vital role in the analysis. We also give evidence for a two loop non-renormalization theorem for the background beta functions.

On the NSR formulation of string theory on AdS5× S5

We discuss the NSR formulation of the superstring action on AdS 5× S 5 proposed recently by Kallosh and Tseytlin in the Green-Schwarz formalism. We show that the stress-energy tensor corresponding to the NSR action for AdS superstring contains the branelike terms, corresponding to exotic massless vertex operators (referred to as the branelike vertices)- the 5-form e φ ψ m1 ⋯ ψ m5 and the 3-form ∂( e φ ψ m 1 ⋯ ψ m 3 ), multiplied by ∂X m . The corresponding sigma-model action has the manifest SO(1,3)× SO(6) invariance of superstring theory on AdS 5× S 5. We argue that adding the branelike terms is equivalent to curving the space-time to obtain the AdS 5× S 5 background. We commence the study of the proposed NSR sigma-model by analyzing the scattering amplitudes involving the branelike vertex operators. The analysis shows quite an unusual momentum dependence of these scattering amplitudes.

Superstring perturbation theory and Ramond-Ramond backgrounds

We consider perturbative Type II superstring theory in the covariant NSR formalism in the presence of NSNS and RR backgrounds. A concrete example that we have in mind is the geometry of D3-branes which in the near-horizon region is AdS_5 x S_5, although our methods may be applied to other backgrounds as well. We show how conformal invariance of the string path integral is maintained order by order in the number of holes. This procedure makes uses of the Fischler-Susskind mechanism to build up the background geometry. A simple formal expression is given for a \sigma-model Lagrangian. This suggests a perturbative expansion in 1/g^2N and 1/N. As applications, we consider at leading order the mixing of RR and NSNS states, and the realization of the spacetime supersymmetry algebra.

Picture-changing operators and space-time supersymmetry

We explore the geometrical properties of fermionic vertex operators for a NSR superstring in order to establish the connection between world-sheet and target space supersymmetries. The mechanism of picture changing is obtained as a result of imposing certain constraints on a world-sheet gauge group of the NSR superstring theory. We find that picture-changing operators of different integer ghost numbers form a polynomial ring. By using properties of the picture-changing formalism, we establish a relation between the NSR and GS string theories. We show that, up to picture-changing transformations, the stress-energy tensor of the N = 1 NSR superstring theory can be obtained from the stress-energy tensor of the N = 1 GS superstring theory in a flat background by a simple field redefinition. The equations of motion of a GS superstring are shown to be fulfilled in the NSR operator formalism; they are also shown to be invariant under κ-symmetry, in terms of operator products in the NSR theory. This allows us to derive the space-time supersymmetry transformation laws for the NSR string theory. Then, we explore the properties of the κ-symmetry in the NSR formalism and find that it leads to some new relations between bosonic and fermionic correlation functions.

Open-closed transition for the NSR spinning string

Based on the overlaps of string wave functions, we construct the Lorentz covariant and BRST-invariant vertex operators for the transitions between one open string and one closed string in the NSR formalism. Ghost contributions are determined by imposing BRST invariance on the complete vertex. Especially, an extended supercurrent insertion is required at the interaction point in order to retain BRST invariance. By comparing the transition vertices constructed from different superghost vacuua, this insertion is shown to play the role of a picture-changing operator.

Covariant two-loop fermion emission amplitude in closed superstring theories

We calculate the two-fermion-two-boson scattering amplitude at two loops in the NSR formalism of the heterotic and type II superstring theories by using the hyperelliptic representation of the genus 2 Riemann surface and the covariant fermion emission vertex. This computation entails the use of a modular invariant combination of a Θ-function and prime forms which we prove.

Construction of vertex operators using operator formalism techniques

We derive vertex operators in oscillator form as an application of the conserved charges method developed by Vafa for the operator formalism in higher genus Riemann surfaces. This construction proves to be clear, direct and valid for the bosonic and fermionic strings as well as for twisted strings on orbifolds. We discuss the method and construct vertex operators for the bosonic string moving on Z N orbifolds and for the fermionic string in the NSR formulation.

Point-wise vanishing of two-loop contributions to 1,2,3-point functions in the NSR formalism

A point-wise vanishing on the module space of contributions to 1,2,3-point functions in superstring theory is demonstrated, provided that both odd moduli are located at one of the ramification points. An analysis of generalized Riemann identities, which are important for development of NSR formalism on hyperelliptic surfaces, is presented.

FOUR DIMENSIONAL SUPERSTRING THEORIES IN ORBIFOLD-LIKE CONSTRUCTION

We present a construction of string theories based on the orbifold-like boundary condition on the world sheet fields in the NSR formalism. A simple model is given.

Covariant one-loop fermion emission amplitudes in closed string theories

We calculate the 2 fermion-2 boson and the 4 fermion scattering amplitudes at one loop in the NSR formulation of the heterotic and type II superstring theories using the covariant fermion emission vertex. The results are shown to agree with well known amplitudes calculated in the light-cone gauge in the Green-Schwarz formulation. The agreement of the 4-fermion amplitude entails the use of a new ϑ-function identity which we also prove.