On a class of conformal E\mathcal{E}-models and their chiral Poisson algebras

In this paper, we study conformal points among the class of $\mathcal{E}$-models. The latter are $\sigma$-models formulated in terms of a current Poisson algebra, whose Lie-theoretic definition allows for a purely algebraic description of their dynamics and their 1-loop RG-flow. We use these results to formulate a simple algebraic condition on the defining data of such a model which ensures its 1-loop conformal invariance and the decoupling of its observables into two chiral Poisson algebras, describing the classical left- and right-moving fields of the theory. In the case of so-called non-degenerate $\mathcal{E}$-models, these chiral sectors form two current algebras and the model takes the form of a WZW theory once realised as a $\sigma$-model. The case of degenerate $\mathcal{E}$-models, in which a subalgebra of the current algebra is gauged, is more involved: the conformal condition yields a wider class of theories, which includes gauged WZW models but also other examples, seemingly different, which however sometimes turn out to be related to gauged WZW models based on other Lie algebras. For this class, we build non-local chiral fields of parafermionic-type as well as higher-spin local ones, forming classical $\mathcal{W}$-algebras. In particular, we find an explicit and efficient algorithm to build these local chiral fields. These results (and their potential generalisations discussed at the end of the paper) open the way for the quantisation of a large class of conformal $\mathcal{E}$-models using the standard operator formalism of two-dimensional CFT.Comment: 67+10 page

Contribution of the RSCM Geothermometry to Detect the Thermal Anomalies and Peak Temperatures Induced by Fluid Circulation in Metasediments

International audienceThe occurrence of deposits hosted by carbonaceous materials-rich metasediments is widespread. For this reason, we propose in this study to investigate the potential of the Raman Spectroscopy of Carbonaceous Material (RSCM) geothermometry to detect thermal anomalies in hydrothermal ore deposits environment. The chosen geological context is the Lucia subterrane in the Franciscan Complex (California, USA), which includes gold-bearing quartz veins (Underwood et al., 1995). Estimated Raman temperatures 1) confirmed the increase in the metamorphic grade towards the north already shown by Underwood et al. (1995), using classical methods like mineralogy and vitrinite reflectance and 2) exhibit anomalous values. These anomalies are probably due to the later hydrothermal event. This result suggests that RSCM could be used as a reliable tool to determine thermal anomalies caused by hot fluid-flow

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

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

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

A unifying 2d action for integrable σ-models from 4d Chern-Simons theory

In the approach recently proposed by K. Costello and M. Yamazaki, which is based on a four-dimensional variant of Chern-Simons theory, we derive a simple and unifying two-dimensional form for the action of many integrable $\sigma$-models which are known to admit descriptions as affine Gaudin models. This includes both the Yang-Baxter deformation and the $\lambda$-deformation of the principal chiral model. We also give an interpretation of Poisson-Lie $T$-duality in this setting and derive the action of the $\mathsf{E}$-model.Comment: 37 page

Cubic hypergeometric integrals of motion in affine Gaudin models

© 2020 International Press of Boston, Inc. This is the accepted manuscript version of an article which has been published in final form at https://dx.doi.org/10.4310/ATMP.2020.v24.n1.a5.We construct cubic Hamiltonians for quantum Gaudin models of affine types $\hat{\mathfrak{sl}}_M$. They are given by hypergeometric integrals of a form we recently conjectured in arXiv:1804.01480. We prove that they commute amongst themselves and with the quadratic Hamiltonians. We prove that their vacuum eigenvalues, and their eigenvalues for one Bethe root, are given by certain hypergeometric functions on a space of affine opers.Peer reviewedFinal Accepted Versio