A Symplectic Structure for String Theory on Integrable Backgrounds

TL;DR

The paper develops a consistent Poisson-bracket framework for classical strings on $\mathbb{R}\times S^3$ by applying Maillet's regularisation to non-ultralocal brackets, obtaining an infinite tower of commuting charges. Using these brackets, it derives the correct symplectic form on the finite-gap moduli and constructs action-angle variables, with the actions identified as the filling fractions of the spectral curve cuts. The algebro-geometric data consist of the spectral curve $\Sigma$, the differential $dp$, and a divisor $\hat{\gamma}(0,0)$, whose Poisson brackets become canonical after a suitable change of variables. These results underpin the leading-order semiclassical quantisation of strings on $\mathrm{AdS}_{5}\times S^{5}$ and predict integer-valued filling fractions in this sector.

Abstract

We define regularised Poisson brackets for the monodromy matrix of classical string theory on R x S^3. The ambiguities associated with Non-Ultra Locality are resolved using the symmetrisation prescription of Maillet. The resulting brackets lead to an infinite tower of Poisson-commuting conserved charges as expected in an integrable system. The brackets are also used to obtain the correct symplectic structure on the moduli space of finite-gap solutions and to define the corresponding action-angle variables. The canonically-normalised action variables are the filling fractions associated with each cut in the finite-gap construction. Our results are relevant for the leading-order semiclassical quantisation of string theory on AdS_5 x S^5 and lead to integer-valued filling fractions in this context.

Figures (2)

• Figure 1: The cycle $\mathcal{A}_I$ and path $\mathcal{B}_I$ for the cut $\mathcal{C}_I$.
• Figure 2: Example of a path lifting required in computing Poisson brackets of transition matrices on $S^1$ of the form $\{ T(\gamma,x) \, \overset{\otimes}, \, T(\gamma',x') \}$.