[Paper Review] Spectrum to all orders of Polchinski-Strominger Effective String Theories of the Drummond Type

N. D. Hari Dass, Yashas Bharadwaj|ArXiv.org|Oct 29, 2009

Quantum chaos and dynamical systems4 references10 citations

TL;DR

This paper analyzes the spectral impact of Drummond-type effective string actions at order $ R^{-6} $ within the Polchinski-Strominger framework. Using a covariant formalism and on-shell stress tensor analysis, it demonstrates that these actions—though non-trivial—do not modify the spectrum of the Nambu-Goto theory, confirming universality of the spectrum to all orders in $ R^{-1} $, consistent with earlier results for conformally invariant actions.

ABSTRACT

Drummond had proposed four actions for Polchinski-Strominger effective string theories at order $R^{-6}$, where $2πR$ is the length of the (closed) string >. In \cite{covariant} it had been shown, based on covariance arguments, that only two of them are independent. We analyse the spectral content of effective string theories with these two actions. We show that the inclusion of these actions does not yield corrections to the spectrum of Nambu-Goto theory \cite{Arv}.

Motivation & Objective

• To determine whether Drummond's $ R^{-6} $ effective string actions alter the spectrum of the Nambu-Goto theory.
• To identify the independent actions among Drummond's four proposed terms using covariance and conformal invariance.
• To extend the all-order spectral universality result—previously shown for conformally invariant actions—to the Drummond-type actions.
• To clarify the physical role of higher-order curvature corrections in effective string theories, particularly in the context of QCD-strings.

Proposed method

• Constructing manifestly covariant actions using scalar densities built from worldsheet Levi-Civita tensors, target-space metric $ \eta_{\mu\nu} $, and covariant derivatives of the embedding fields $ X^\mu $.
• Implementing a conformal gauge where $ g_{++} = g_{--} = 0 $, and using $ L = g_{+-} = \partial_+X \cdot \partial_-X $ as a $ (1,1) $-tensor to ensure conformal invariance.
• Deriving the on-shell stress tensor $ T_{--} $ from the action and decomposing it into holomorphic and non-holomorphic parts using $ X^\mu = X^\mu_{\text{cl}} + F^\mu(\tau^+) + G^\mu(\tau^-) + H^\mu(\tau^+, \tau^-) $.
• Applying the equation $ D_+ T_{--} = \partial_+ T_{--} = 2\pi E \cdot \partial_-X $ to show that only holomorphic terms contribute to the spectrum, and proving that all terms in the Drummond actions are non-holomorphic.
• Using the fact that $ \partial_+ L $ is non-holomorphic due to $ L $'s dependence on $ F_+, G_-, H_+, H_- $, which implies $ D_{++}X^\mu $ and its derivatives are non-holomorphic, so no contribution to $ T_{--}^{\text{holo}} $.
• Concluding that since the on-shell stress tensor remains unchanged, the spectrum is identical to that of the Nambu-Goto theory.

Experimental results

Research questions

• RQ1 Do the $ R^{-6} $ actions proposed by Drummond modify the spectrum of the Nambu-Goto effective string theory?
• RQ2 Which of Drummond’s four $ R^{-6} $ actions are physically independent when conformal invariance is imposed?
• RQ3 Can the spectral universality of effective string theories—previously shown for conformally invariant actions—be extended to Drummond-type actions?
• RQ4 Why do non-trivial higher-order curvature corrections fail to alter the on-shell stress tensor and thus the spectrum?

Key findings

• Only two of Drummond’s four $ R^{-6} $ actions are independent when covariance and conformal invariance are enforced.
• The covariantized Drummond actions do not contribute to the holomorphic part of the on-shell stress tensor $ T_{--}^{\text{holo}} $, as all terms are non-holomorphic.
• The on-shell stress tensor remains identical to that of the Nambu-Goto theory, implying no correction to the spectrum.
• The spectrum of effective string theories containing these actions is therefore identical to that of the free bosonic string theory to all orders in $ R^{-1} $.
• This confirms spectral universality beyond $ R^{-3} $, extending the result of [12] to a second class of all-order conformally invariant actions.
• The physical content of these higher-order actions remains hidden in the non-holomorphic sector, suggesting that spectral observables are insensitive to such corrections.