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Wilson loops in N=4 SYM theory: rotation in S5

A. A. Tseytlin, K. Zarembo

TL;DR

This work analyzes Wilson loops in N=4 SYM with non-constant scalar coupling, implemented as rotation in S^5, across Minkowski and Euclidean signatures. It reveals a nontrivial Minkowski–Euclidean mismatch: Minkowski rotating loops lack a straightforward open-string dual, whereas Euclidean results map coherently to AdS5 × S5 minimal surfaces at strong coupling. The study provides explicit one-loop perturbative results and classical-string solutions, highlighting a delicate analytic continuation between signatures and a second-order phase transition in the Euclidean antiparallel-rotation case. The findings underscore the nuanced role of S^5 rotation in AdS/CFT and point to Euclidean formulations as the more robust setting for connecting Wilson loops to string-theoretic duals, with potential links to BMN-like sectors via large R-charge OPE data.

Abstract

We study Wilson loops in N=4 SYM theory which are non-constant in the scalar (S5) directions and open string solutions associated with them in the context of AdS/CFT correspondence. An interplay between Minkowskian and Euclidean pictures turns out to be non-trivial for time-dependent Wilson loops. We find that in the S5-rotating case there appears to be no direct open-string duals for the Minkowskian Wilson loops, and their expectation values should be obtained by analytic continuation from the Euclidean-space results. In the Euclidean case, we determine the dependence of the ``quark - anti-quark'' potential on the rotation parameter, both at weak coupling (i.e. in the 1-loop perturbative SYM theory) and at strong coupling (i.e. in the classical string theory in AdS5 x S5).

Wilson loops in N=4 SYM theory: rotation in S5

TL;DR

This work analyzes Wilson loops in N=4 SYM with non-constant scalar coupling, implemented as rotation in S^5, across Minkowski and Euclidean signatures. It reveals a nontrivial Minkowski–Euclidean mismatch: Minkowski rotating loops lack a straightforward open-string dual, whereas Euclidean results map coherently to AdS5 × S5 minimal surfaces at strong coupling. The study provides explicit one-loop perturbative results and classical-string solutions, highlighting a delicate analytic continuation between signatures and a second-order phase transition in the Euclidean antiparallel-rotation case. The findings underscore the nuanced role of S^5 rotation in AdS/CFT and point to Euclidean formulations as the more robust setting for connecting Wilson loops to string-theoretic duals, with potential links to BMN-like sectors via large R-charge OPE data.

Abstract

We study Wilson loops in N=4 SYM theory which are non-constant in the scalar (S5) directions and open string solutions associated with them in the context of AdS/CFT correspondence. An interplay between Minkowskian and Euclidean pictures turns out to be non-trivial for time-dependent Wilson loops. We find that in the S5-rotating case there appears to be no direct open-string duals for the Minkowskian Wilson loops, and their expectation values should be obtained by analytic continuation from the Euclidean-space results. In the Euclidean case, we determine the dependence of the ``quark - anti-quark'' potential on the rotation parameter, both at weak coupling (i.e. in the 1-loop perturbative SYM theory) and at strong coupling (i.e. in the classical string theory in AdS5 x S5).

Paper Structure

This paper contains 17 sections, 64 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Minkowski Wilson loops with $S^5$-rotation
  3. Perturbative SYM results
  4. Single line
  5. Anti-parallel lines
  6. Open strings in $AdS_5$ with Minkowski signature
  7. Single line
  8. Anti-parallel lines
  9. Euclidean Wilson loops with rotation in $S^5$
  10. Perturbative SYM results
  11. Single line
  12. Antiparallel lines
  13. Open string minimal surface
  14. Single line
  15. Anti-parallel lines
  16. ...and 2 more sections

Figures (2)

  • Figure 1: The energy ($U(l)$, bold line) and the angular momentum ($F(l)$) as functions of $l=\nu L$. Both curves go from bottom to top as $v$ goes from 0 to 1.
  • Figure 2: $W(l)$ as a function of $l$ (bold curve). The area of the unstable solution A (thin curve). The dashed line corresponds to pure Coulomb potential at $\nu=0$.