https://www.physics.rutgers.edu/~friedan/papers/Nucl_Phys_B271_93_1986.pdf

Conformal invariance, supersymmetry and string theory

The zero modes of closed strings on a torus — the torus coordinates plus dual coordinates conjugate to winding number — parameterize a doubled torus. In closed string field theory, the string field depends on all zero-modes and so can be expanded to give an infinite set of fields on the doubled torus. We use string field theory to construct a theory of massless fields on the doubled torus. Key to the consistency is a constraint on fields and gauge parameters that arises from the L0−0 = 0 condition in closed string theory. The symmetry of this double field theory includes usual and `dual diffeomorphisms', together with a T-duality acting on fields that have explicit dependence on the torus coordinates and the dual coordinates. We find that, along with gravity, a Kalb-Ramond field and a dilaton must be added to support both usual and dual diffeomorphisms. We construct a fully consistent and gauge invariant action on the doubled torus to cubic order in the fields. We discuss the challenges involved in the construction of the full nonlinear theory. We emphasize that the doubled geometry is physical and the dual dimensions should not be viewed as an auxiliary structure or a gauge artifact.

Double Field Theory

UNC-MATH-92/2originally April 27, 1990, revised September 24, 1992INTRODUCTION TO SH LIE ALGEBRAS FOR PHYSICISTSTom LadaJim StasheffMuch of point particle physics can be described in terms of Lie algebras andtheir representations. Closed string ﬁeld theory, on the other hand, leads to ageneralization of Lie algebra which arose naturally within mathematics in the studyof deformations of algebraic structures [SS]. It also appeared in work on higherspin particles [BBvD]. Representation theoretic analogs arose in the mathematicalanalysis of the Batalin-Fradkin-Vilkovisky approach to constrained Hamiltonians[S6].The sh Lie algebra of closed string ﬁeld theory [SZ], [KKS], [K], [Wies], [WZ],[Z] is deﬁned on the full Fock complex of the theory, with the BRST diﬀerential Q.Following Zwiebach [Z], we stipulate that the string ﬁelds B

Introduction to SH Lie algebras for physicists

This paper is devoted to recent progress made towards the understanding of closed bosonic and fermionic string perturbation theory, formulated in a Lorentz-covariant way on Euclidean space-time. Special emphasis is put on the fundamental role of Riemann surfaces and supersurfaces. The differential and complex geometry of their moduli space is developed as needed. New results for the superstring presented here include the supergeometric construction of amplitudes, their chiral and superholomorphic splitting and a global formulation of supermoduli space and amplitudes.

The Geometry of String Perturbation Theory

Hamiltonian form of the path integral for theories with a gauge freedom

Covariant quantization of string based on BRS invariance

The covariant string field theory is presented in its full detail for the closed-bosonic-string case. The properties of the interaction vertex of three closed strings are fully clarified. We thus establish the nilpotency of the full nonlinear Becchi-Rouet-Stora (BRS) transformation as well as the BRS invariance of our gauged-fixed action. This, on the other hand, also establishes the gauge invariance of our gauge-unfixed action and the group law of the gauge transformations, which were reported previously. The general N-point amplitudes for arbitrary external closed-string states are computed explicitly at the tree level. In particular, when the external states are on shell and physical, we prove that the amplitudes become independent of the choice of the external string-length parameters and correctly reproduce the usual dual amplitudes.

Kaku Ogawa

Papers

Covariant quantization of string based on BRS invariance

Covariant string field theory.

Manifestly covariant field theory of interacting string I

Pregeometrical String Field Theory: Creation of Space-time and Motion

The gauge invariant action of the interacting string field