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An Exact Prediction of N=4 SUSYM Theory for String Theory

Nadav Drukker, David J. Gross

TL;DR

The paper shows that the circular BPS-Wilson loop in N=4 SYM is governed by a conformal anomaly that localizes to a point, allowing an exact computation via a Gaussian matrix model to all orders in 1/N^2 and g^2N. It derives an explicit expression for ⟨W_circle⟩ in terms of Laguerre polynomials and Bessel functions, and proves a general relation ⟨W_C⟩ = F(λ,N)⟨W̃_{C̃}⟩ for any smooth loop, with F determined by the matrix model. The authors then compare these gauge-theory predictions with string theory in AdS5×S5, showing agreement at leading large-λ and all orders in g_s, and extend the analysis via S-duality and to multiply-wound loops. This work provides strong evidence for the AdS/CFT correspondence and suggests a powerful, exactly solvable sector in N=4 SYM described by a matrix model. The results point toward deep connections between anomalies, matrix models, and holographic duals, with potential extensions to α' corrections and broader observables.

Abstract

We propose that the expectation value of a circular BPS-Wilson loop in N=4 SUSYM can be calculated exactly, to all orders in a 1/N expansion and to all orders in g^2 N. Using the AdS/CFT duality, this result yields a prediction of the value of the string amplitude with a circular boundary to all orders in alpha' and to all orders in g_s. We then compare this result with string theory. We find that the gauge theory calculation, for large g^2 N and to all orders in the 1/N^2 expansion does agree with the leading string theory calculation, to all orders in g_s and to lowest order in alpha'. We also find a relation between the expectation value of any closed smooth Wilson loop and the loop related to it by an inversion that takes a point along the loop to infinity, and compare this result, again successfully, with string theory.

An Exact Prediction of N=4 SUSYM Theory for String Theory

TL;DR

The paper shows that the circular BPS-Wilson loop in N=4 SYM is governed by a conformal anomaly that localizes to a point, allowing an exact computation via a Gaussian matrix model to all orders in 1/N^2 and g^2N. It derives an explicit expression for ⟨W_circle⟩ in terms of Laguerre polynomials and Bessel functions, and proves a general relation ⟨W_C⟩ = F(λ,N)⟨W̃_{C̃}⟩ for any smooth loop, with F determined by the matrix model. The authors then compare these gauge-theory predictions with string theory in AdS5×S5, showing agreement at leading large-λ and all orders in g_s, and extend the analysis via S-duality and to multiply-wound loops. This work provides strong evidence for the AdS/CFT correspondence and suggests a powerful, exactly solvable sector in N=4 SYM described by a matrix model. The results point toward deep connections between anomalies, matrix models, and holographic duals, with potential extensions to α' corrections and broader observables.

Abstract

We propose that the expectation value of a circular BPS-Wilson loop in N=4 SUSYM can be calculated exactly, to all orders in a 1/N expansion and to all orders in g^2 N. Using the AdS/CFT duality, this result yields a prediction of the value of the string amplitude with a circular boundary to all orders in alpha' and to all orders in g_s. We then compare this result with string theory. We find that the gauge theory calculation, for large g^2 N and to all orders in the 1/N^2 expansion does agree with the leading string theory calculation, to all orders in g_s and to lowest order in alpha'. We also find a relation between the expectation value of any closed smooth Wilson loop and the loop related to it by an inversion that takes a point along the loop to infinity, and compare this result, again successfully, with string theory.

Paper Structure

This paper contains 13 sections, 84 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Gauge Theory Calculation
  3. Quadratic term
  4. The circle to all orders
  5. Arbitrary loops
  6. The Comparison with String Theory
  7. Circular loops
  8. S-duality
  9. Arbitrary loops
  10. Multiply wound loops
  11. Conclusions
  12. Matrix Model Calculation
  13. Explicit $1/N$ Expansion

Figures (3)

  • Figure 1: a. To go from a straight line to a circle one should include diagrams with some gluon propagators replaced by the total derivatives (dotted lines). Those give a boundary contribution only when all of them hit the point of inversion (marked by an x). b. Regardless of the rest of the diagram, the anomaly is dependent only on the vicinity of the inversion point and since it lives at one point, is given by the matrix model expression.
  • Figure 2: Two graphs contributing (a) to the open Wilson loop ${\left< {\tilde{W}_{\tilde{C}}} \right>}$ and (b) to the closed loop ${\left< {W_C} \right>}$. The curves are related by a conformal transformation, and the two diagrams differ by total derivatives
  • Figure 3: We show here some diagrams one gets by replacing gluon propagators by total derivatives (dotted lines). Diagrams (a) and (b) will not contribute, since not all the total derivatives hit the inversion point. (c) does contribute, since all the total derivatives can hit the origin. One gets diagram (d) by doing the same procedure to a sightly different graph. Summing (c), (d) and a few other such graphs gives the matrix model expression at order $\lambda^2$ times the rest of the diagram.