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$Q \bar{Q}$ Potential from Strings in Curved Spacetime - Classical Results

Y. Kinar, E. Schreiber, J. Sonnenschein

TL;DR

The paper develops a unified classical framework in which Wilson loops are realized as string world sheets in curved spacetimes with an extra dimension, parameterized by metric functions $f(s)$ and $g(s)$. It proves a sharp confinement criterion: confinement occurs if and only if $f(0)>0$, in which case the string tension is $f(0)$, and it derives the large-$l$ behavior of the quark–antiquark potential $E(l)$ including subleading corrections. The corrections are either exponential (in a critical case) or negative-power in $l$ (in noncritical cases), with explicit constants determined by local expansions of $f$ and $g$ near the minimum or divergence of the extra dimension. The general results are then applied to a broad set of backgrounds—AdS5×S5, non-conformal D$p$-brane setups, finite-temperature YM, MQCD, rotating branes, and Polyakov’s non-critical string—recovering known results (such as Maldacena’s $E(l) o l$ behavior) and clarifying the structure and magnitude of the corrections. The work also discusses invariance under reparameterizations of the extra dimension and the distinction between even and odd string configurations, laying a foundation for future inclusion of quantum fluctuations.

Abstract

We compute the leading behaviour of the quark anti-quark potential from a generalized Nambu-Goto action associated with a curved space-time having an "extra dimension". The extra dimension can be the radial coordinate in the AdS/CFT correspondence, the Liouville field in Polyakov's approach, or an internal dimension in MQCD. In particular, we derive the condition for confinement, and in the case it occurs we find the string tension and the correction to the linear potential.

$Q \bar{Q}$ Potential from Strings in Curved Spacetime - Classical Results

TL;DR

The paper develops a unified classical framework in which Wilson loops are realized as string world sheets in curved spacetimes with an extra dimension, parameterized by metric functions $f(s)$ and $g(s)$. It proves a sharp confinement criterion: confinement occurs if and only if $f(0)>0$, in which case the string tension is $f(0)$, and it derives the large-$l$ behavior of the quark–antiquark potential $E(l)$ including subleading corrections. The corrections are either exponential (in a critical case) or negative-power in $l$ (in noncritical cases), with explicit constants determined by local expansions of $f$ and $g$ near the minimum or divergence of the extra dimension. The general results are then applied to a broad set of backgrounds—AdS5×S5, non-conformal D$p$-brane setups, finite-temperature YM, MQCD, rotating branes, and Polyakov’s non-critical string—recovering known results (such as Maldacena’s $E(l) o l$ behavior) and clarifying the structure and magnitude of the corrections. The work also discusses invariance under reparameterizations of the extra dimension and the distinction between even and odd string configurations, laying a foundation for future inclusion of quantum fluctuations.

Abstract

We compute the leading behaviour of the quark anti-quark potential from a generalized Nambu-Goto action associated with a curved space-time having an "extra dimension". The extra dimension can be the radial coordinate in the AdS/CFT correspondence, the Liouville field in Polyakov's approach, or an internal dimension in MQCD. In particular, we derive the condition for confinement, and in the case it occurs we find the string tension and the correction to the linear potential.

Paper Structure

This paper contains 18 sections, 7 theorems, 61 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. The quark anti--quark potential
  3. The proof
  4. Monotonicity of $l(s_0)$
  5. Asymptotics of $l(s_0)$ and $K(s_0)$
  6. Asymptotics of $E(l)$
  7. Applications and verifications of the general result
  8. invariance under reparameterizations of $s$
  9. General considerations
  10. The $AdS_5\times S^5$ dual of the ${\cal N}=4$ SYM in four dimensions
  11. Non-conformal cases with sixteen supersymmetries
  12. Dual models of pure YM theory in three and four dimensions
  13. Dual models of pure YM theory at finite temperature
  14. Pure YM from rotating branes
  15. MQCD
  16. ...and 3 more sections

Key Result

Theorem 1

Let $f(s),g(s) > 0$ for $s > 0$, and let $f(s)$ be monotone increasing. Assume that $s(x)$ described by (sofx) has the global minimum value of $E$. Then $s_0$ is monotone decreasing as a function of $l$.

Figures (3)

  • Figure 1: (a) the geodesic lines $G^{(1)},G^{(2)}$. (b) $G^{(2)}$ and the new line having smaller energy.
  • Figure 2: (a) The minimal energy odd string. (b) The derived even string
  • Figure 3: (a) The minimal energy even string. (b) The derived odd string

Theorems & Definitions (9)

  • Theorem 1
  • Definition 1
  • Lemma 1
  • Definition 2
  • Lemma 2
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5