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

Nonlinear Realization and Hidden Local Symmetries

Recent studies of QCD Green's functions and their applications in hadronic physics are reviewed. We briefly discuss the issues of gauge fixing, BRS invariance and positivity. Evidence for the violation of positivity by quarks and transverse gluons in the covariant gauge is collected, and it is argued that this is one manifestation of confinement. 
We summarise the derivation of the Dyson-Schwinger equations (DSEs) of QED and QCD. The influence of instantons on DSEs in a 2-dimensional model is mentioned. Solutions for the Green's functions in QED in 2+1 and 3+1 dimensions provide tests of various schemes to truncate DSEs. We discuss possible extensions to QCD and their limitations. Truncation schemes for DSEs of QCD are discussed in the axial gauge and in the Landau gauge. We review the available results from a systematic non-perturbative expansion scheme established for Landau gauge QCD. Comparisons to related lattice results, where available, are presented. 
The applications of QCD Green's functions to hadron physics are summarized. Properties of ground state mesons are discussed on the basis of the Bethe-Salpeter equation for quarks and antiquarks. The Goldstone nature of pseudoscalar mesons and mechanisms of diquark confinement are reviewed. We discuss some properties of ground state baryons based on their description as Bethe-Salpeter/Faddeev bound states of quark-diquark correlations in the quantum field theory of confined quarks and gluons.

The Infrared Behavior of QCD Green's Functions - Confinement, Dynamical Symmetry Breaking, and Hadrons as Relativistic Bound States

The infrared behaviour of QCD Green's functions ☆: Confinement, dynamical symmetry breaking, and hadrons as relativistic bound states

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

We consider aspects of dynamical baryons in a holographic dual of QCD that is formulated on the basis of a D4/D8-brane configuration. We construct a soliton solution carrying a unit baryon number and show that it is obtained as an instanton solution of four-dimensional Yang-Mills theory with fixed size. The Chern-Simons term on the flavor D8-branes plays a crucial role of protecting the instanton from collapsing to zero size. By quantizing the collective coordinates of the soliton, we derive the baryon spectra. Negative-parity baryons as well as baryons with higher spins and isospins can be obtained in a simple manner.

/pdf/baryons-from-instantons-in-holographic-qcd-4kwok5dgos.pdf

Baryons from instantons in holographic QCD

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.

Covariant string field theory.

Manifestly covariant field theory of interacting string I

Skew-symmetric tensor gauge field theory dynamically realized in the QCD U(1) channel

We consider aspects of dynamical baryons in a holographic dual of QCD that is proposed on the basis of a D4/D8-brane configuration. We construct a soliton solution carrying a unit baryon number and show that it is given by an instanton solution of four-dimensional Yang-Mills theory with fixed size. The Chern-Simons term on the flavor D8-branes plays a crucial role of protecting the instanton from collapsing to zero size. By quantizing the collective coordinates of the soliton, we work out the baryon spectra. Negative-parity baryons as well as baryons with higher spins and isospins can be obtained in a simple manner.

Hiroyuki Hata

Papers

Baryons from instantons in holographic QCD

Covariant string field theory.

Manifestly covariant field theory of interacting string I

Skew-symmetric tensor gauge field theory dynamically realized in the QCD U(1) channel

Baryons from instantons in holographic QCD