gaudin models with irregular singularities abstract 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 b feigin et al 1994 17 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

homological and monodromy representations of framed braid groups in this paper we introduce two new classes of representations of the framed braid groups one is the homological representation constructed as the action of a mapping class group on a certain homology group the other is the monodromy representation of the confluent kz equation which is a generalization of the kz equation to have irregular singularities we also give a conjectural equivalence between these two classes of representations

improved parameterized and exact algorithms for cut problems on trees we study the multicut on trees and the generalized multiway cut on trees problems for the multicut on trees problem we present a parameterized algorithm that runs in time o k where 2 1 1 554 is the positive root of the polynomial x 4 2 x 2 1 this improves the current best algorithm of chen et al that runs in time o 1 619 k for the generalized multiway cut on trees problem we show that this problem is solvable in polynomial time if the number of terminal sets is fixed this answers an open question posed in a recent paper by liu and zhang by reducing the generalized multiway cut on trees problem to the multicut on trees problem our results give a parameterized algorithm that solves the generalized multiway cut on trees problem in time o k

stable bundles and integrable systems on considere la geometrie symplectique des fibres cotangents aux espaces de modules de fibres vectoriels stables sur une surface de riemann on montre que ce sont des systemes dynamiques hamiltoniens algebriquement completement integrables

quantization of the shift of argument subalgebras in type a abstract given a simple lie algebra g and an element g the corresponding shift of argument subalgebra of s g is poisson commutative in the case where is regular this subalgebra is known to admit a quantization that is it can be lifted to a commutative subalgebra of u g we show that if g is of type a then this property extends to arbitrary thus proving a conjecture of feigin frenkel and toledano laredo the proof relies on an explicit construction of generators of the center of the affine vertex algebra at the critical level

gaudin models for m n we establish the basics of the bethe ansatz for the gaudin model associated to the lie superalgebra x1d524 x1d529 m n in particular we prove the completeness of the bethe ansatz in the case of tensor products of fundamental representations

confluent kz equations for mathfrak sl _n with poincare rank 2 at infinity we construct confluent kz equations with poincar rank 2 at infinity for the case of and the integral representation for the solutions hamiltonians of these confluent kz equations are derived from suitable quantization of dlog constructed in the theory of monodromy preserving deformation mpd in jimbo et al 1981 physica d 2 306 52 our confluent kz equations can be viewed as a quantization of mpd with poincar rank 2 at infinity

multiple orthogonal polynomials and a counterexample to the gaudin bethe ansatz conjecture jacobi polynomials are polynomials whose zeros form the unique solution of the bethe ansatz equation associated with two irreducible modules we study sequences of polynomials whose zeros form the unique solution of the bethe ansatz equation associated with two highest weight irreducible modules with the restriction that the highest weight of one of the modules is a multiple of the first fundamental weight we describe the recursion which can be used to compute these polynomials moreover we show that the first polynomial in the sequence coincides with the jacobi pineiro multiple orthogonal polynomial and others are given by wronskian type determinants of jacobi pineiro polynomials as a byproduct we describe a counterexample to the bethe ansatz conjecture for the gaudin model

affine kac moody algebras and semi infinite flag manifolds we study representations of affine kac moody algebras from a geometric point of view it is shown that wakimoto modules introduced in 18 which are important in conformal field theory correspond to certain sheaves on a semi infinite flag manifold with support on its schhubert cells this manifold is equipped with a remarkable semi infinite structure which is discussed in particular the semi infinite homology of this manifold is computed the cousin grothendieck resolution of an invertible sheaf on a semi infinite flag manifold gives a two sided resolution of an irreducible representation of an affine algebras consisting of wakimoto modules this is just the brst complex as a byproduct we compute the homology of an algebra of currents on the real line with values in a nilpotent lie algebra

fock representations of the affine lie algebra a 1 1 the aim of this note is to show that the affine lie algebraa1 1 has a natural family v of fock representations on the spacec xi yj i andj parametrized by v c2 by corresponding the highest weight of to each the parameter spacec2 forms a double cover of the weight spacec 0 c 1 with singularities at linear forms of level 2 this number is 1 times the dual coxeter number our results contain explicit realizations of irreducible non integrable highest wieghta1 1 modules for generic v

opers with irregular singularity and spectra of the shift of argument subalgebra joint work with boris feigin and edward frenkel math qa 0712 1183 let g be a semisimple complex lie algebra and u g its universal enveloping algebra the algebra u g bears a natural filtration by the degree with respect to the generators the associated graded algebra is the symmetric algebra s g c g by the poincare birkhoff witt theorem the commutator on u g defines the poisson bracket on s g let zs g s g be the center of s g with respect to the poisson bracket and let g g be a regular semisimple element due to the result of mischenko and fomenko 1978 the algebra a s g generated by the elements where zs g is commutative with respect to the poisson bracket and has maximal possible transcendence degree equal to 12 dim g rk g if is a regular element contained in the cartan subalgebra h g then h a and if f is the principal nilpotent element then the subalgebra af contains the principal nilpotent subalgebra zg f it was recently shown that the shift of argument subalgebras can be quantized

confluent primary fields in the conformal field theory for any complex simple lie algebra we generalize the primary fields in the wess zumino novikov witten conformal field theory for a case with irregular singularities we refer to these generalized primary fields as confluent primary fields we present the screening currents ward identity a recursion rule for computing the expectation values of the products of confluent primary fields in the case of the expectation values of the products of confluent primary fields are integral formulas of solutions to confluent knizhnik zamolodchikov kz equations given in jimbo et al 2008 j phys a math theor 41 175205 by computing the operator product expansion of the energy momentum tensor t z and the confluent primary fields we obtain new differential operators moreover in the case of these differential operators are the same as those of the confluent kz equations jimbo et al 2008

extended trigonometric cherednik algebras and nonstationary schrodinger equations with delta potentials we realize an extended version of the trigonometric cherednik algebra as affine dunkl operators involving heaviside functions we use the quadratic casimir element of the extended trigonometric cherednik algebra to define an explicit nonstationary schrodinger equation with delta potential we use coordinate bethe ansatz methods to construct solutions of the nonstationary schrodinger equation in terms of generalized bethe wave functions it is shown that the generalized bethe wave functions satisfy affine difference knizhnik zamolodchikov equations as functions of the momenta the relation to the vector valued root system analogs of the quantum bose gas on the circle with delta function interactions is indicated

on the consistent separation of scale and variance for gaussian random fields we present fixed domain asymptotic results that establish consistent estimates of the variance and scale parameters for a gaussian random field with a geometric anisotropic matern autocovariance in dimension d 4 when d 4 we show that one can estimate the coefficient on the principle irregular term accurately enough to get a consistent estimate of the coefficient on the second irregular term these two coefficients can then be used to separate the scale and variance we extend our results to the general problem of estimating a variance and geometric anisotropy for more general autocovariance functions our results illustrate the interaction between the accuracy of estimation the smoothness of the random field the dimension of the observation space and the number of increments used for estimation as a corollary our results establish the orthogonality of matern gaussian random fields with different parameters when d 4 the case d 4 is still open

a rigid irregular connection on the projective line in this paper we construct a connection on the trivial g bundle on p 1 for any simple complex algebraic group g which is regular outside of the points 0 and has a regular singularity at the point 0 with principal unipotent monodromy and has an irregular singularity at the point with slope 1 h the reciprocal of the coxeter number of g the connection which admits the structure of an oper in the sense of beilinson and drinfeld appears to be the characteristic 0 counterpart of a hypothetical family of l adic representations which should parametrize a specific automorphic representation under the global langlands correspondence these l adic representations and their characteristic 0 counterparts have been constructed in some cases by deligne and katz our connection is constructed uniformly for any simple algebraic group and characterized using the formalism of opers it provides an example of the geometric langlands correspondence with wild ramification we compute the de rham cohomology of our connection with values in a representation v of g and describe the differential galois group of as a subgroup of g

compatibility of the feigin frenkel isomorphism and the harish chandra isomorphism for jet algebras let g be a simple finite dimensional complex lie algebra with a cartan subalgebra h and weyl group w let g denote the lie algebra of n jets on g a theorem of rais and tauvel and geoffriau identifies the centre of the category of g modules with the algebra of functions on the variety of n jets on the affine space h w on the other hand a theorem of feigin and frenkel identifies the centre of the category of critical level smooth modules of the corresponding affine kac moody algebra with the algebra of functions on the ind scheme of opers for the langlands dual group we prove that these two isomorphisms are compatible by defining the higher residue of opers with irregular singularities we also define generalized verma and wakimoto modules and relate them by a nontrivial morphism

invariants of the vacuum module associated with the lie superalgebra mathfrak gl 1 1 we describe the algebra of invariants of the vacuum module associated with an affinization of the lie superalgebra we give a formula for its hilbert poincare series in a fermionic cancellation free form which turns out to coincide with the generating function of the plane partitions over the hook our arguments are based on a super version of the beilinson drinfeld rais tauvel theorem which we prove by producing an explicit basis of invariants of the symmetric algebra of polynomial currents associated with we identify the invariants with affine supersymmetric polynomials via a version of the chevalley theorem

wakimoto modules opers and the center at the critical level abstract wakimoto modules are representations of affine kac moody algebras in fock modules over infinite dimensional heisenberg algebras in this paper we present the construction of the wakimoto modules from the point of view of the vertex algebra theory we then use wakimoto modules to identify the center of the completed universal enveloping algebra of an affine kac moody algebra at the critical level with the algebra of functions on the space of opers for the langlands dual group on the punctured disc giving another proof of the theorem of b feigin and the author

estimating structured correlation matrices in smooth gaussian random field models this article considers the estimation of structured correlation matrices in infinitely differentiable gaussian random field models the problem is essentially motivated by the stochastic modeling of smooth deterministic responses in computer experiments in particular the log likelihood function is determined explicitly in closed form and the sieve maximum likelihood estimators are shown to be strongly consistent under mild conditions

feigin frenkel center in types b c and d for each simple lie algebra mathfrak g consider the corresponding affine vertex algebra v mathrm crit mathfrak g at the critical level the center of this vertex algebra is a commutative associative algebra whose structure was described by a remarkable theorem of feigin and frenkel about two decades ago however only recently simple formulas for the generators of the center were found for the lie algebras of type a following talalaev s discovery of explicit higher gaudin hamiltonians we give explicit formulas for generators of the centers of the affine vertex algebras v mathrm crit mathfrak g associated with the simple lie algebras mathfrak g of types b c and d the construction relies on the schur weyl duality involving the brauer algebra and the generators are expressed as weighted traces over tensor spaces and equivalently as traces over the spaces of singular vectors for the action of the lie algebra mathfrak sl 2 in the context of howe duality this leads to explicit constructions of commutative subalgebras of the universal enveloping algebras mathrm u mathfrak g t and mathrm u mathfrak g and to higher order hamiltonians in the gaudin model associated with each lie algebra mathfrak g we also introduce analogues of the bethe subalgebras of the yangians mathrm y mathfrak g and show that their graded images coincide with the respective commutative subalgebras of mathrm u mathfrak g t

bethe subalgebras in twisted yangians we study analogues of the yangian of the lie algebra mathfrak g mathfrak l n for the other classical lie algebras mathfrak s mathfrak o n and mathfrak s mathfrak p n we call them twisted yangians they are coideal subalgebras in the yangian of mathfrak g mathfrak l n and admit homomorphisms onto the universal enveloping algebras u mathfrak s mathfrak o n and u mathfrak s mathfrak p n respectively in every twisted yangian we construct a family of maximal commutative subalgebras parametrized by the regular semisimple elements of the corresponding classical lie algebra the images in u mathfrak s mathfrak o n and u mathfrak s mathfrak p n of these subalgebras are also maximal commutative

center of the quantum affine vertex algebra in type a abstract we consider the quantum vertex algebra associated with the double yangian in type a as defined by etingof and kazhdan we show that its center is a commutative associative algebra and construct algebraically independent families of topological generators of the center at the critical level

remarks on the confluent kz equation for mathfrak sl _2 and quantum painleve equations for the lie algebra we give an explicit form of an irregular singular version of the kz equation in the simplest non trivial cases these kz equations reproduce the quantum painlev equations possessing affine weyl group symmetry found previously by one of the authors

segal sugawara vectors for the lie algebra of type g2 explicit formulas for segal sugawara vectors associated with the simple lie algebra mathfrak g of type g 2 are found by using computer assisted calculations this leads to a direct proof of the feigin frenkel theorem describing the center of the corresponding affine vertex algebra at the critical level as an application we give an explicit solution of vinberg s quantization problem by providing formulas for generators of maximal commutative subalgebras of u mathfrak g we also calculate the eigenvalues of the hamiltonians on the bethe vectors in the gaudin model associated with mathfrak g