Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The su(2|3) Dynamic Spin Chain

Niklas Beisert

TL;DR

The paper investigates whether the integrable structure observed at one loop in N=4 SYM extends to higher orders within the su(2|3) subsector. By enforcing the su(2|3) symmetry algebra and BMN scaling, Beisert derives the dilatation (Hamiltonian) operator up to three loops, revealing a dynamic spin chain where the number of sites can fluctuate yet integrability persists. The work provides strong evidence for all-loop integrability in planar N=4 SYM by showing that higher-loop charges Q_n(g) can be consistently defined and that δH(g) commutes with these charges, while matching known BMN results. It also demonstrates that length fluctuations, parity degeneracies, and multiplet shortening play crucial roles in constraining the spectrum and preserving integrability in a nontrivial, dynamically evolving spin chain. The findings suggest that symmetry and scaling considerations alone can strongly fix higher-loop corrections and offer a path toward understanding the full all-order integrable structure of the theory.

Abstract

The complete one-loop, planar dilatation operator of the N=4 superconformal gauge theory was recently derived and shown to be integrable. Here, we present further compelling evidence for a generalisation of this integrable structure to higher orders of the coupling constant. For that we consider the su(2|3) subsector and investigate the restrictions imposed on the spin chain Hamiltonian by the symmetry algebra. This allows us to uniquely fix the energy shifts up to the three-loop level and thus prove the correctness of a conjecture in hep-th/0303060. A novel aspect of this spin chain model is that the higher-loop Hamiltonian, as for N=4 SYM in general, does not preserve the number of spin sites. Yet this dynamic spin chain appears to be integrable.

The su(2|3) Dynamic Spin Chain

TL;DR

The paper investigates whether the integrable structure observed at one loop in N=4 SYM extends to higher orders within the su(2|3) subsector. By enforcing the su(2|3) symmetry algebra and BMN scaling, Beisert derives the dilatation (Hamiltonian) operator up to three loops, revealing a dynamic spin chain where the number of sites can fluctuate yet integrability persists. The work provides strong evidence for all-loop integrability in planar N=4 SYM by showing that higher-loop charges Q_n(g) can be consistently defined and that δH(g) commutes with these charges, while matching known BMN results. It also demonstrates that length fluctuations, parity degeneracies, and multiplet shortening play crucial roles in constraining the spectrum and preserving integrability in a nontrivial, dynamically evolving spin chain. The findings suggest that symmetry and scaling considerations alone can strongly fix higher-loop corrections and offer a path toward understanding the full all-order integrable structure of the theory.

Abstract

The complete one-loop, planar dilatation operator of the N=4 superconformal gauge theory was recently derived and shown to be integrable. Here, we present further compelling evidence for a generalisation of this integrable structure to higher orders of the coupling constant. For that we consider the su(2|3) subsector and investigate the restrictions imposed on the spin chain Hamiltonian by the symmetry algebra. This allows us to uniquely fix the energy shifts up to the three-loop level and thus prove the correctness of a conjecture in hep-th/0303060. A novel aspect of this spin chain model is that the higher-loop Hamiltonian, as for N=4 SYM in general, does not preserve the number of spin sites. Yet this dynamic spin chain appears to be integrable.

Paper Structure

This paper contains 41 sections, 137 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction and conclusions
  2. The Model
  3. A Subsector of $\mathcal{N}=4$ SYM
  4. Fields.
  5. States.
  6. Parity.
  7. Adjoint.
  8. Spin Chain.
  9. The Algebra
  10. Representations
  11. Constituents.
  12. Shortenings.
  13. Fluctuations in Length.
  14. From $\mathcal{N}=4$ SYM to $\mathfrak{su}(2|3)$
  15. One-Loop
  16. ...and 26 more sections

Figures (5)

  • Figure 1: The structures for the construction of $H_2$. Straight and zigzag lines correspond to bosons and fermions, respectively.
  • Figure 2: The structures of $H_2$ which are compatible with $\mathfrak{su}(2|3)$ symmetry at leading order. A straight+zigzag line correspond to a supermultiplet.
  • Figure 3: The structures for the construction of $H_3,Q_1,S_1$. The number of spin sites is not conserved here.
  • Figure 4: Closure of the algebra at $\mathcal{O}(g^3)$ fixes the relative coefficients within $H_2$.
  • Figure 5: A generally non-planar interaction (left) can become planar when acting on short states (right).