Andrei Mikhailov

Sine-Gordon-like action for the superstring in AdS(5) x S-5

We propose an action for a sine-Gordon-like theory, which reproduces the classical equations of motion of the Green-Schwarz-Metsaev-Tseytlin superstring on AdS5 × S^5. The action is relativistically invariant. It is a mass-deformed gauged WZW model for SO(4, 1) × SO(5)/SO(4) × SO(4) interacting with fermions.

An action variable of the sine–Gordon model

Abstract It was conjectured that the classical bosonic string in AdS times a sphere has a special action variable which corresponds to the length of the operator on the field theory side. We discuss the analogous action variable in the sine–Gordon model. We explain the relation between this action variable and the Backlund transformations and show that the corresponding hidden symmetry acts on breathers by shifting their phase. It can be considered a nonlinear analogue of splitting the solution of the free field equations into positive- and negative-frequency parts.

A nonlocal Poisson bracket of the sine-Gordon model

It is well known that the classical string on a two-sphere is more or less equivalent to the sine-Gordon model. We consider the non-abelian dual of the classical string on a two-sphere. We show that there is a projection map from the phase space of this model to the phase space of the sine-Gordon model. The corresponding Poisson structure of the sine-Gordon model is nonlocal with one integration.

Slow evolution of nearly-degenerate extremal surfaces

Abstract It was conjectured recently that the string worldsheet theory for the fast moving string in AdS times a sphere becomes effectively first order in the time derivative and describes the continuous limit of an integrable spin chain. In this paper we will try to make this statement more precise. We interpret the first order theory as describing the long term evolution of the tensionless string perturbed by a small tension. The long term evolution is a Hamiltonian flow on the moduli space of periodic trajectories. It should correspond to the renormgroup flow on the field theory side.

Supersymmetric null-surfaces

Single trace operators with the large R-charge in supersymmetric Yang-Mills theory correspond to the null-surfaces in AdS(5) x S-5. We argue that the moduli space of the null-surfaces is the space of contours in the super-grassmanian parametrizing the complex (2|2)-dimensional subspaces of the complex (4|4)-dimensional space. The odd coordinates on this super-grassmanian correspond to the fermionic degrees of freedom of the superstring.

Algebra of transfer-matrices and Yang-Baxter equations on the string worldsheet in AdS(5) x S(5)

Integrability of the string worldsheet theory in AdS_5 × S^5 is related to the existence of a flat connection depending on the spectral parameter. The transfer matrix is the open-ended Wilson line of this flat connection. We study the product of transfer matrices in the near-flat space expansion of the AdS_5 × S^5 string theory in the pure spinor formalism. The natural operations on Wilson lines with insertions are described in terms of r- and s-matrices satisfying a generalized classical Yang–Baxter equation. The formalism is especially transparent for infinite or closed Wilson lines with simple gauge invariant insertions.

Notes on Fast Moving Strings

We review the recent work on the mechanics of fast moving strings in anti-de Sitter space times a sphere and discuss the role of conserved charges. An interesting relation between the local conserved charges of rigid solutions was found in the earlier work. We propose a generalization of this relation for arbitrary solutions, not necessarily rigid. We conjecture that an infinite combination of local conserved charges is an action variable generating periodic trajectories in the classical string phase space. It corresponds to the length of the operator on the field theory side.

Symmetries of massless vertex operators in AdS_5 × S^5

The worldsheet sigma model of the superstring in AdS_5×S^5 has a one-parameter family of flat connections parametrized by the spectral parameter. The corresponding Wilson line is not BRST invariant for an open contour, because the BRST transformation leads to boundary terms. These boundary terms define a cohomological complex associated with the endpoint of the contour. We study the cohomology of this complex for Wilson lines in some infinitedimensional representations. We find that for these representations the cohomology is nontrivial at the ghost number 2. This implies that it is possible to define a BRST invariant open Wilson line. The central point in the construction is the existence of massless vertex operators transforming exactly covariantly under the action of the global symmetry group. In flat space, massless vertices transform covariantly only up to adding BRST-exact terms. But in AdS we show that it is possible to define vertices so that they transform exactly covariantly.

Plane wave limit of local conserved charges

We study the plane wave limit of the Backlund transformations for the classical string in AdS space times a sphere and obtain an explicit expression for the local conserved charges. We show that the Pohlmeyer charges become in the plane wave limit the local integrals of motion of the free massive field. This fixes the coefficients in the expansion of the anomalous dimension as the sum of the Pohlmeyer charges.

Anomalous dimension and local charges

AdS space is the universal covering of a hyperboloid. We consider the action of the deck transformations on a classical string worldsheet in AdS5 × S5. We argue that these transformations are generated by an infinite linear combination of the local conserved charges. We conjecture that a similar relation holds for the corresponding operators on the field theory side. This would be a generalization of the recent field theory results showing that the one loop anomalous dimension is proportional to the Casimir operator in the representation of the Yangian algebra.