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BMN Correlators by Loop Equations

B. Eynard, C. Kristjansen

TL;DR

The paper develops a loop-equation framework to evaluate BMN sector correlators in N=4 SYM by mapping them to expectations of traces of words in a Gaussian complex matrix model. It introduces split and merge rules to build recursive relations among loop generating functionals, enabling analytic computation of the genus expansion to arbitrary order; explicit genus-one and genus-two results are recovered and extended. The authors provide detailed expressions for one- and multi-trace generating functions and demonstrate exact calculability of certain correlators via generating functionals, offering a systematic path to higher-impurity and higher-genus analyses. Overall, the work strengthens the bridge between BMN gauge theory calculations and matrix-model techniques, with implications for pp-wave string interactions and AdS/CFT correspondence.

Abstract

In the BMN approach to N=4 SYM a large class of correlators of interest are expressible in terms of expectation values of traces of words in a zero-dimensional Gaussian complex matrix model. We develop a loop-equation based, analytic strategy for evaluating such expectation values to any order in the genus expansion. We reproduce the expectation values which were needed for the calculation of the one-loop, genus one correction to the anomalous dimension of BMN-operators and which were earlier obtained by combinatorial means. Furthermore, we present the expectation values needed for the calculation of the one-loop, genus two correction.

BMN Correlators by Loop Equations

TL;DR

The paper develops a loop-equation framework to evaluate BMN sector correlators in N=4 SYM by mapping them to expectations of traces of words in a Gaussian complex matrix model. It introduces split and merge rules to build recursive relations among loop generating functionals, enabling analytic computation of the genus expansion to arbitrary order; explicit genus-one and genus-two results are recovered and extended. The authors provide detailed expressions for one- and multi-trace generating functions and demonstrate exact calculability of certain correlators via generating functionals, offering a systematic path to higher-impurity and higher-genus analyses. Overall, the work strengthens the bridge between BMN gauge theory calculations and matrix-model techniques, with implications for pp-wave string interactions and AdS/CFT correspondence.

Abstract

In the BMN approach to N=4 SYM a large class of correlators of interest are expressible in terms of expectation values of traces of words in a zero-dimensional Gaussian complex matrix model. We develop a loop-equation based, analytic strategy for evaluating such expectation values to any order in the genus expansion. We reproduce the expectation values which were needed for the calculation of the one-loop, genus one correction to the anomalous dimension of BMN-operators and which were earlier obtained by combinatorial means. Furthermore, we present the expectation values needed for the calculation of the one-loop, genus two correction.

Paper Structure

This paper contains 19 sections, 79 equations, 10 figures.

Table of Contents

  1. Introduction
  2. From ${\cal N}=4$ SYM to matrix model
  3. Definitions and Notation
  4. Loop Equations
  5. Split Rule
  6. Merge Rule
  7. Functionals needed for the one-loop, genus one computation
  8. $\omega(x)$ and $W_1(x,y)$ to leading order
  9. $W_2$, $U_1$ and $W_{1,1}$ to leading order
  10. $W_1(x,y)$ to next to leading order
  11. Functionals needed for the one-loop, genus two computation
  12. $W_3$, $U_2$, $W_{2,1}$ and $U_{1,1}$ to leading order
  13. $W_2$, $U_1$ and $W_{1,1}$ to next to leading order
  14. $W_1(x,y)$ to third order
  15. The general case
  16. ...and 4 more sections

Figures (10)

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  • ...and 5 more figures