The effective string spectrum in the orthogonal gauge

TL;DR

The paper tests the Polchinski–Strominger framework for effective strings in the orthogonal gauge by computing the leading correction to the Nambu–Goto spectrum at order $1/R^{5}$. Starting from the PS action with the next-to-leading term, the authors perform a careful higher-order expansion, implement field redefinitions to maintain a tractable canonical structure, and derive corrected Virasoro generators. They show that the energy shifts for excited states arise from diagonal pieces of the corrected $L_0$ and compute explicit representations- dependent corrections, finding agreement with the static gauge results when the leading coefficient is fixed to $c_4=rac{D-26}{192\pi}=-rac{eta}{16\pi}$. This concordance provides a nontrivial consistency check for the PS framework and strengthens the link between orthogonal-gauge and static-gauge formulations of the effective string action, with implications for universal predictions of the leading deviation from Nambu–Goto energies. The work also highlights how the only surviving corrections at this order are tied to the transverse structure via the operator $oldsymbol{ extSigma}^{ij} ilde{oldsymbol{ extSigma}}_{ij}$, underscoring a potential universality of the leading long-string corrections.

Abstract

The low-energy effective action on long string-like objects in quantum field theory, such as confining strings, includes the Nambu-Goto action and then higher-derivative corrections. This action is diffeomorphism-invariant, and can be analyzed in various gauges. Polchinski and Strominger suggested a specific way to analyze this effective action in the orthogonal gauge, in which the induced metric on the worldsheet is conformally equivalent to a flat metric. Their suggestion leads to a specific term at the next order beyond the Nambu-Goto action. We compute the leading correction to the Nambu-Goto spectrum using the action that includes this term, and we show that it agrees with the leading correction previously computed in the static gauge. This gives a consistency check for the framework of Polchinski and Strominger, and helps to understand its relation to the theory in the static gauge.