Gaudin models with irregular singularities

We introduce a class of quantum integrable systems generalizing the Gaudin model. The corresponding algebras of quantum Hamiltonians are obtained as quotients of the center of the enveloping algebra of an affine Kac-Moody algebra at the critical level, extending the construction of higher Gaudin Hamiltonians from hep-th/9402022 to the case of non-highest weight representations of affine algebras. We show that these algebras are isomorphic to algebras of functions on the spaces of opers on P^1 with regular as well as irregular singularities at finitely many points. We construct eigenvectors of these Hamiltonians, using Wakimoto modules of critical level, and show that their spectra on finite-dimensional representations are given by opers with trivial monodromy. We also comment on the connection between the generalized Gaudin models and the geometric Langlands correspondence with ramification.Comment: Latex, 72 pages. Final version to appear in Advances in Mathematic

Thiopurines in inflammatory bowel disease. How to optimize thiopurines in the biologic era?

Thiopurines have been a cornerstone in the treatment of inflammatory bowel disease (IBD). Although they have been used for more than 50 years, there are still some unsolved issues about their efficacy and, also, some safety concerns, mainly the risk of myelosuppression and life-threatening lymphoproliferative disorders. Furthermore, the development of biological therapy raises the question whether there is still a role for thiopurines in the IBD treatment algorithm. On the other hand, limited cost and wide availability make thiopurines a reasonable option in settings of limited resources and increasing prevalence of IBD. In fact, there is a growing interest in optimizing thiopurine therapy, since pharmacogenomic findings suggest that a personalized approach based on the genotyping of some molecules involved in its metabolism could be useful to prevent side effects. Polymorphisms of thiopurine methyltransferase enzyme (TPMT) that result in low enzymatic activity have been associated with an increased risk of myelotoxicity, especially in Caucasians; however, in Asians it is assumed that the variants of nudix hydrolase 15 (NUDT15) are more relevant in the development of toxicity. Age is also important, since in elderly patients the risk of complications seems to be increased. Moreover, the primo-infection of Epstein Barr virus and cytomegalovirus under thiopurine treatment has been associated with severe lymphoproliferative disorders. In addition to assessing individual characteristics that may influence thiopurines treatment outcomes, this review also discusses other strategies to optimize the therapy. Low-dose thiopurines combined with allopurinol can be used in hypermethylators and in thiopurine-related hepatotoxicity. The measurement of metabolites could be useful to assess compliance, identify patients at risk of adverse events and also facilitating the management of refractory patients. Thioguanine is also a rescue therapy in patients with toxicity related to conventional thiopurine therapy. Finally, the current indications for thiopurines in monotherapy or in combination with biologics, as well as the optimal duration of treatment, are also reviewed. © Copyright © 2021 Gargallo-Puyuelo, Laredo and Gomollón

Dynamical differential equations compatible with rational qKZ equations

For the Lie algebra $gl_N$ we introduce a system of differential operators called the dynamical operators. We prove that the dynamical differential operators commute with the $gl_N$ rational quantized Knizhnik-Zamolodchikov difference operators. We describe the transformations of the dynamical operators under the natural action of the $gl_N$ Weyl group.Comment: 7 pages, AmsLaTe

Irregular Wakimoto modules and the Casimir connection

We study some non-highest weight modules over an affine Kac-Moody algebra at non-critical level. Roughly speaking, these modules are non-commutative localizations of some non-highest weight "vacuum" modules. Using free field realization, we embed some rings of differential operators in endomorphism rings of our modules. These rings of differential operators act on a localization of the space of coinvariants of any module over the Kac-Moody algebra with respect to a certain level subalgebra. In a particular case this action is identified with the Casimir connection.Comment: Final version, available at Springerlink.co