Semiclassical Quantisation of Finite-Gap Strings

We perform a first principle semiclassical quantisation of the general finite-gap solution to the equations of a string moving on R x S^3. The derivation is only formal as we do not regularise divergent sums over stability angles. Moreover, with regards to the AdS/CFT correspondence the result is incomplete as the fluctuations orthogonal to this subspace in AdS_5 x S^5 are not taken into account. Nevertheless, the calculation serves the purpose of understanding how the moduli of the algebraic curve gets quantised semiclassically, purely from the point of view of finite-gap integration and with no input from the gauge theory side. Our result is expressed in a very compact and simple formula which encodes the infinite sum over stability angles in a succinct way and reproduces exactly what one expects from knowledge of the dual gauge theory. Namely, at tree level the filling fractions of the algebraic curve get quantised in large integer multiples of hbar = 1/lambda^{1/2}. At 1-loop order the filling fractions receive Maslov index corrections of hbar/2 and all the singular points of the spectral curve become filled with small half-integer multiples of hbar. For the subsector in question this is in agreement with the previously obtained results for the semiclassical energy spectrum of the string using the method proposed in hep-th/0703191. Along the way we derive the complete hierarchy of commuting flows for the string in the R x S^3 subsector. Moreover, we also derive a very general and simple formula for the stability angles around a generic finite-gap solution. We also stress the issue of quantum operator orderings since this problem already crops up at 1-loop in the form of the subprincipal symbol.Comment: 53 pages, 22 figures; some significant typos corrected, references adde

Large Winding Sector of AdS/CFT

We study a family of classical strings on R x S^3 subspace of the AdS_5 x S^5 background that interpolates between pulsating strings and single-spike strings. They are obtained from the helical strings of hep-th/0609026 by interchanging worldsheet time and space coordinates, which maps rotating/spinning string states with large spins to oscillating states with large winding numbers. From a finite-gap perspective, this transformation is realised as an interchange of quasi-momentum and quasi-energy defined for the algebraic curve. The gauge theory duals are also discussed, and are identified with operators in the non-holomorphic sector of N=4 super Yang-Mills. They can be viewed as excited states above the ``antiferromagnetic'' state, which is ``the farthest from BPS'' in the spin-chain spectrum. Furthermore, we investigate helical strings on AdS_3 x S^1 in an appendix.Comment: 1+52 pages, 10 figures, v2: references and comments added, v3: minor changes and a reference adde

Integrable degenerate E\mathcal E-models from 4d Chern-Simons theory

We present a general construction of integrable degenerate $\mathcal E$-models on a 2d manifold $\Sigma$ using the formalism of Costello and Yamazaki based on 4d Chern-Simons theory on $\Sigma \times \mathbb{C}P^1$. We begin with a physically motivated review of the mathematical results of [arXiv:2008.01829] where a unifying 2d action was obtained from 4d Chern-Simons theory which depends on a pair of 2d fields $h$ and $\mathcal L$ on $\Sigma$ subject to a constraint and with $\mathcal L$ depending rationally on the complex coordinate on $\mathbb{C}P^1$. When the meromorphic 1-form $\omega$ entering the action of 4d Chern-Simons theory is required to have a double pole at infinity, the constraint between $h$ and $\mathcal L$ was solved in [arXiv:2011.13809] to obtain integrable non-degenerate $\mathcal E$-models. We extend the latter approach to the most general setting of an arbitrary 1-form $\omega$ and obtain integrable degenerate $\mathcal E$-models. To illustrate the procedure we reproduce two well known examples of integrable degenerate $\mathcal E$-models: the pseudo dual of the principal chiral model and the bi-Yang-Baxter $\sigma$-model

On integrable field theories as dihedral affine Gaudin models

We introduce the notion of a classical dihedral affine Gaudin model, associated with an untwisted affine Kac-Moody algebra $\widetilde{\mathfrak{g}}$ equipped with an action of the dihedral group $D_{2T}$, $T \geq 1$ through (anti-)linear automorphisms. We show that a very broad family of classical integrable field theories can be recast as examples of such classical dihedral affine Gaudin models. Among these are the principal chiral model on an arbitrary real Lie group $G_0$ and the $\mathbb{Z}_T$-graded coset $\sigma$-model on any coset of $G_0$ defined in terms of an order $T$ automorphism of its complexification. Most of the multi-parameter integrable deformations of these $\sigma$-models recently constructed in the literature provide further examples. The common feature shared by all these integrable field theories, which makes it possible to reformulate them as classical dihedral affine Gaudin models, is the fact that they are non-ultralocal. In particular, we also obtain affine Toda field theory in its lesser-known non-ultralocal formulation as another example of this construction. We propose that the interpretation of a given classical non-ultralocal integrable field theory as a classical dihedral affine Gaudin model provides a natural setting within which to address its quantisation. At the same time, it may also furnish a general framework for understanding the massive ODE/IM correspondence since the known examples of integrable field theories for which such a correspondence has been formulated can all be viewed as dihedral affine Gaudin models.Comment: 103 pages; v2: minor changes and new section 5.3.4 added. Accepted for publication in Int. Math. Res. No

Local charges in involution and hierarchies in integrable sigma-models

Integrable σ-models, such as the principal chiral model, ℤT-coset models for T∈ℤ≥2 and their various integrable deformations, are examples of non-ultralocal integrable field theories described by (cyclotomic) r/s-systems with twist function. In this general setting, and when the Lie algebra 픤 underlying the r/s-system is of classical type, we construct an infinite algebra of local conserved charges in involution, extending the approach of Evans, Hassan, MacKay and Mountain developed for the principal chiral model and symmetric space σ-model. In the present context, the local charges are attached to certain `regular' zeros of the twist function and have increasing degrees related to the exponents of the untwisted affine Kac-Moody algebra 픤ˆ associated with 픤. The Hamiltonian flows of these charges are shown to generate an infinite hierarchy of compatible integrable equations

Affine Gaudin models and hypergeometric functions on affine opers

We conjecture that quantum Gaudin models in affine types admit families of higher Hamiltonians, labelled by the (countably infinite set of) exponents, whose eigenvalues are given by functions on a space of meromorphic opers associated with the Langlands dual Lie algebra. This is in direct analogy with the situation in finite types. However, in stark contrast to finite types, we prove that in affine types such functions take the form of hypergeometric integrals, over cycles of a twisted homology defined by the levels of the modules at the marked points. That result prompts the further conjecture that the Hamiltonians themselves are naturally expressed as such integrals. We go on to describe the space of meromorphic affine opers on an arbitrary Riemann surface. We prove that it fibres over the space of meromorphic connections on the canonical line bundle Ω. Each fibre is isomorphic to the direct product of the space of sections of the square of Ω with the direct product, over the exponents j not equal to 1, of the twisted cohomology of the jth tensor power of Ω

The Magic Renormalisability of Affine Gaudin Models

We study the renormalisation of a large class of integrable $\sigma$-models obtained in the framework of affine Gaudin models. They are characterised by a simple Lie algebra $\mathfrak{g}$ and a rational twist function $\varphi(z)$ with simple zeros, a double pole at infinity but otherwise no further restrictions on the pole structure. The crucial tool used in our analysis is the interpretation of these integrable theories as $\mathcal{E}$-models, which are $\sigma$-models studied in the context of Poisson-Lie T-duality and which are known to be at least one- and two-loop renormalisable. The moduli space of $\mathcal{E}$-models still contains many non-integrable theories. We identify the submanifold formed by affine Gaudin models and relate its tangent space to curious matrices and semi-magic squares. In particular, these results provide a criteria for the stability of these integrable models under the RG-flow. At one loop, we show that this criteria is satisfied and derive a very simple expression for the RG-flow of the twist function, proving a conjecture made earlier in the literature.Comment: 29 pages, 1 figur