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Structure Constants from Q-Systems and Separation of Variables

Till Bargheer, Carlos Bercini, Gabriel Lefundes, Paul Ryan

Abstract

We introduce a novel method to compute structure constants from Q-functions in the scalar sector of planar N=4 super Yang-Mills (SYM) and related theories. The method derives from operatorial as well as functional separation of variables, and the structure constants are expressed as determinants of matrices whose entries are integrals over products of Q-functions. In this framework, each operator is twisted by an external angle, mirroring the cusped Maldacena-Wilson loop. The structure constants of local single-trace operators in N=4 SYM are recovered in the untwisting limit, where we obtain a one-to-one correspondence between our key building blocks and those of the Hexagon formalism. Retaining appropriate twists, our structure constants also perfectly match those of the orbifold points of N=4 SYM. Our results thus far are valid at leading order in the weak-coupling expansion, but their formulation in terms of Q-functions provides a natural starting point for including loop corrections. Many of the methods we develop in this work apply more generally to the computation of correlation functions in integrable models.

Structure Constants from Q-Systems and Separation of Variables

Abstract

We introduce a novel method to compute structure constants from Q-functions in the scalar sector of planar N=4 super Yang-Mills (SYM) and related theories. The method derives from operatorial as well as functional separation of variables, and the structure constants are expressed as determinants of matrices whose entries are integrals over products of Q-functions. In this framework, each operator is twisted by an external angle, mirroring the cusped Maldacena-Wilson loop. The structure constants of local single-trace operators in N=4 SYM are recovered in the untwisting limit, where we obtain a one-to-one correspondence between our key building blocks and those of the Hexagon formalism. Retaining appropriate twists, our structure constants also perfectly match those of the orbifold points of N=4 SYM. Our results thus far are valid at leading order in the weak-coupling expansion, but their formulation in terms of Q-functions provides a natural starting point for including loop corrections. Many of the methods we develop in this work apply more generally to the computation of correlation functions in integrable models.

Paper Structure

This paper contains 60 sections, 191 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Setup
  3. Result
  4. Left-Right Quantum Symmetry
  5. Derivation
  6. SoV Bases.
  7. Overlaps and Norm.
  8. Structure Constant.
  9. Comparison with Hexagons
  10. Untwisting Limits and Orbifolds
  11. Twisted theory.
  12. Taking the untwisting limit.
  13. Counting untwisted structure constants.
  14. Conclusion
  15. Review of the su(4) Sector
  16. ...and 45 more sections

Figures (1)

  • Figure 1: Twisted three-point function of one excited operator of size $L_1$ with $N$ excitations and two protected operators of sizes $L_2$ and $L_3$. The red lines represent the propagators of the excitations $\Phi_i \in \{X,Y,\bar{X},\bar{Y},\bar{Z}\}$ coming from the non-BPS operator. The bridge lengths $\ell_{ij} = (L_i+L_j-L_k)/2$ denote the number of propagators between operators $\mathcal{O}_i$ and $\mathcal{O}_j$. In particular, we use $L_1\equiv L$, $\ell_{12} = \ell$ and $\ell_{13}=L-\ell$ to denote the length and bridges of the excited operator.