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Supersymmetric null-surfaces

Andrei Mikhailov

TL;DR

The paper proposes that large-R-charge single-trace operators in N=4 SYM map to parametrized null-surfaces in AdS5×S5, with the renormalization-group flow acting as a slow evolution on contours in the super-Grassmannian Gr(2|2,4|4). It derives the one-loop anomalous dimension as a Hamiltonian functional on these null-surfaces, showing consistency with the slow evolution picture. Supersymmetry is incorporated by promoting the moduli space to the super-Grassmannian, where fermionic degrees of freedom correspond to odd directions of the supermanifold and yield supersymmetric null-surfaces. Together, the results provide a Hamiltonian framework linking gauge-theory operator dynamics to the classical string worldsheet in AdS5×S5, including a fermionic extension via the supercoset structure.

Abstract

Single trace operators with the large R-charge in supersymmetric Yang-Mills theory correspond to the null-surfaces in $AdS_5\times S^5$. 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.

Supersymmetric null-surfaces

TL;DR

The paper proposes that large-R-charge single-trace operators in N=4 SYM map to parametrized null-surfaces in AdS5×S5, with the renormalization-group flow acting as a slow evolution on contours in the super-Grassmannian Gr(2|2,4|4). It derives the one-loop anomalous dimension as a Hamiltonian functional on these null-surfaces, showing consistency with the slow evolution picture. Supersymmetry is incorporated by promoting the moduli space to the super-Grassmannian, where fermionic degrees of freedom correspond to odd directions of the supermanifold and yield supersymmetric null-surfaces. Together, the results provide a Hamiltonian framework linking gauge-theory operator dynamics to the classical string worldsheet in AdS5×S5, including a fermionic extension via the supercoset structure.

Abstract

Single trace operators with the large R-charge in supersymmetric Yang-Mills theory correspond to the null-surfaces in . We argue that the moduli space of the null-surfaces is the space of contours in the super-Grassmanian parametrizing the complex -dimensional subspaces of the complex -dimensional space. The odd coordinates on this super-Grassmanian correspond to the fermionic degrees of freedom of the superstring.

Paper Structure

This paper contains 8 sections, 61 equations, 1 figure.

Table of Contents

  1. Introduction.
  2. Single trace operators with large spin.
  3. Coherent states.
  4. Anomalous dimension.
  5. Fast moving superstrings.
  6. Anomalous dimension as a deck transformation.
  7. Supersymmetric null surfaces.
  8. The Hamiltonian system.

Figures (1)

  • Figure 1: Eq. (\ref{['GaugeChoiceForPsi']}) shows that $\Psi_{++}\in \rho_{SU(2,2)}\otimes \rho_{SU(4)}$ defines a linear map $\Psi_{++}:\;\rho_{SU(4)}^*\to \rho_{SU(2,2)}$ such that $L_S^*$ goes into $L_A^{\perp}$ and $L_S^{*\perp}$ goes into $L_A$. Therefore $(\Psi_{++},\Psi_{++}^*)$ defines a linear automorphism of $\rho_{SU(2,2)}\oplus \rho_{SU(4)}^*$ and theorefore an infinitesimal variation of the plane $L_A\oplus L_S^*\subset \rho_{SU(2,2)}\oplus \rho_{SU(4)}^*$.