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Integrability and the AdS/CFT correspondence

Didina Serban

Abstract

The description of gauge theories at strong coupling is one of the long-standing problems in theoretical physics. The idea of a relation between strongly coupled gauge theories and string theory was pioneered by 't Hooft, Wilson and Polyakov. A decade ago, Maldacena made this relation explicit by conjecturing the exact equivalence of a conformally invariant theory in four dimensions, the maximally supersymmetric Yang-Mills theory, with string theory in the AdS5 x S5 background. Other examples of correspondence between a conformally invariant theory and string theory in an AdS background were discovered recently. The comparison of the two sides of the correspondence requires the use of non-perturbative methods. The discovery of integrable structures in gauge theory and string theory led to the conjecture that the two theories are integrable for any value of the coupling constant and that they share the same integrable structure defined non-perturbatively. The last eight years brought remarkable progress in identifying this solvable model and in explicitly solving the problem of computing the spectrum of conformal dimensions of the theory. The progress came from the identification of the dilatation operator with an integrable spin chain and from the study of the string sigma model. In this thesis, I present the evolution of the concept of integrability in the framework of the AdS/CFT correspondence and the the main results obtained using this approach.

Integrability and the AdS/CFT correspondence

Abstract

The description of gauge theories at strong coupling is one of the long-standing problems in theoretical physics. The idea of a relation between strongly coupled gauge theories and string theory was pioneered by 't Hooft, Wilson and Polyakov. A decade ago, Maldacena made this relation explicit by conjecturing the exact equivalence of a conformally invariant theory in four dimensions, the maximally supersymmetric Yang-Mills theory, with string theory in the AdS5 x S5 background. Other examples of correspondence between a conformally invariant theory and string theory in an AdS background were discovered recently. The comparison of the two sides of the correspondence requires the use of non-perturbative methods. The discovery of integrable structures in gauge theory and string theory led to the conjecture that the two theories are integrable for any value of the coupling constant and that they share the same integrable structure defined non-perturbatively. The last eight years brought remarkable progress in identifying this solvable model and in explicitly solving the problem of computing the spectrum of conformal dimensions of the theory. The progress came from the identification of the dilatation operator with an integrable spin chain and from the study of the string sigma model. In this thesis, I present the evolution of the concept of integrability in the framework of the AdS/CFT correspondence and the the main results obtained using this approach.

Paper Structure

This paper contains 47 sections, 462 equations, 9 figures.

Table of Contents

  1. Introduction: the gauge-string correspondence
  2. The ${\cal N}=4$ super Yang-Mills theory
  3. The action
  4. Symmetries
  5. Oscillator representation of $PSU(2,2|4)$
  6. The dilatation operator as the Hamiltonian of an integrable spin chain
  7. Bethe ansatz solution for the Heisenberg model
  8. The $so(6)$ sector at one loop
  9. The full $psu(2,2|4)$ Bethe ansatz at one loop
  10. Perturbative integrability in ${\cal N}=4$ SYM
  11. The notion of perturbative integrability and the asymptotic Bethe ansatz.
  12. The Inozemtsev model
  13. The three loop dilatation operator and Inozemtsev model
  14. Strings on $AdS_5\times S^5$
  15. The Lagrangian
  16. ...and 32 more sections

Figures (9)

  • Figure 1: Two Dynkin diagrams for $psu(2,2|4)$.
  • Figure 2: The Dynkin diagram and the Dynkin labels ${\alpha}_r\cdot w$ corresponding to the $so(6)$ Bethe ansatz (\ref{['NBA']}).
  • Figure 3: Duality transformation on a piece of Dynkin diagram .
  • Figure 4: Connectivity of the nodes of the Dynkin diagram for the all-loop Bethe ansatz. The dotted lines represent the nodes which are connected only at $g\neq 0$.
  • Figure 5: Structure of the representation (\ref{['su22twisted']}) of the centrally extended $su(2|2)$ algebra. The links in red (green) represent action of two $su(2|1)$ subalgebras. The dotted links disappear at $g=0$, leaving $su(2|2)$ four dimensional representations at constant length.
  • ...and 4 more figures