The moduli space *M*(*r*,*n*) of framed torsion free sheaves on the projective plane with rank *r* and second Chern class equal to *n* has the natural action of the (*r*+2)-dimensional torus. In this paper, we look at the fixed point set of different one-dimensional subtori in this torus. We prove that in the homogeneous case the generating series of the numbers of the irreducible components has a beautiful decomposition into an infinite product. In the case of odd *r*, these infinite products coincide with certain Virasoro characters. We also propose a conjecture in a general quasihomogeneous case.

Let (M,I,J,K) be a hyperkahler manifold, and Z⊂(M,I) a complex subvariety in (M,I). We say that Z is trianalytic if it is complex analytic with respect to J and K, and absolutely trianalytic if it is trianalytic with respect to any hyperk\"ahler triple of complex structures (M,I,J′,K′) containing I. For a generic complex structure I on M, all complex subvarieties of (M,I) are absolutely trianalytic. It is known that a normalization Z′ of a trianalytic subvariety is smooth; we prove that b2(Z′) is no smaller than b2(M) when M has maximal holonomy (that is, M is IHS). To study absolutely trianalytic subvarieties further, we define a new geometric structure, called k-symplectic structure; this structure is a generalization of the hypersymplectic structure. A k-symplectic structure on a 2d-dimensional manifold X is a k-dimensional space R of closed 2-forms on X which all have rank 2d or d. It is called non-degenerate if the set of all degenerate forms in R is a smooth, non-degenerate quadric hypersurface in R. We consider absolutely trianalytic tori in a hyperkahler manifold M of maximal holonomy. We prove that any such torus is equipped with a non-degenerate k-symplectic structure, where k=b2(M). We show that the tangent bundle TX of a k-symplectic manifold is a Clifford module over a Clifford algebra Cl(k−1). Then an absolutely trianalytic torus in a hyperkahler manifold M with b2(M)≥2r+1 is at least 2r−1-dimensional.

An LCK manifold with potential is a compact quotient of a Kähler manifold X equipped with a positive Kähler potential f, such that the monodromy group acts on X by holomorphic homotheties and multiplies f by a character. The LCK rank is the rank of the image of this character, considered as a function from the monodromy group to real numbers. We prove that an LCK manifold with potential can have any rank between 1 and b1(M). Moreover, LCK manifolds with proper potential (ones with rank 1) are dense. Two errata to our previous work are given in the last Section.

We discuss the Poisson structures on Lie groups and propose an explicit construction of the integrable models on their appropriate Poisson submanifolds. The integrals of motion for the SL(N)-series are computed in cluster variables via the Lax map. This construction, when generalised to the co-extended loop groups, gives rise not only to alternative descriptions of relativistic Toda systems, but allows to formulate in general terms some new class of the integrable models. We discuss the subtleties of this Lax map related to the ambiguity in projection to the trivial co-extension and propose a way to write the spectral curve equation, which fixes this ambiguity, both for the Toda chains and their generalisations.

We present a summary of current knowledge about the AGT relations for conformal blocks with additional insertion of the simplest degenerate operator, and a special choice of the corresponding intermediate dimension, when the conformal blocks satisfy hypergeometric-type differential equations in position of the degenerate operator. A special attention is devoted to representation of conformal block through the beta-ensemble resolvents and to its asymptotics in the limit of large dimensions (both external and intermediate) taken asymmetrically in terms of the deformation epsilon-parameters. The next-to-leading term in the asymptotics defines the generating differential in the Bohr-Sommerfeld representation of the one-parameter deformed Seiberg-Witten prepotentials (whose full two-parameter deformation leads to Nekrasov functions). This generating differential is also shown to be the one-parameter version of the single-point resolvent for the corresponding beta-ensemble, and its periods in the perturbative limit of the gauge theory are expressed through the ratios of the Harish-Chandra function. The Shroedinger/Baxter equations, considered earlier in this context, directly follow from the differential equations for the degenerate conformal block. This provides a powerful method for evaluation of the single-deformed prepotentials, and even for the Seiberg-Witten prepotentials themselves. We mostly concentrate on the representative case of the insertion into the four-point block on sphere and one-point block on torus.

A particular result towards a positive solution of a problem by V.I. Arnol’d about a higher analog of the ergodic asymptotic invariant of magnetic fields is presented.

This is a note on Beauville's problem (solved by Greb, Lehn, and Rollenske in the non-algebraic case, and by Hwang and Weiss in general) whether a lagrangian torus on an irreducible holomorphic symplectic manifold is a fiber of a lagrangian fibration. We provide a different very short solution in the non-algebraic case, and make some observations suggesting a different approach in the algebraic case. © 2013 Elsevier B.V.

A mathematically correct description is presented on the interrelations between the dynamics of divergence free vector fields on an oriented 3-dimensional manifold M and the dynamics of Hamiltonian systems. It is shown that for a given divergence free vector field Xwith a global cross-section there exist some 4-dimensional symplectic manifold M̃⊃M and a smooth Hamilton function H:M̃→R such that for some c∈R one gets M={H=c}and the Hamiltonian vector field XH restricted on this level coincides with X. For divergence free vector fields with singular points such an extension is impossible but the existence of local cross-section allows one to reduce the dynamics to the study of symplectic diffeomorphisms in some sub-domains of M. We also consider the case of a divergence free vector field X with a smooth integral having only finite number of critical levels. It is shown that such a noncritical level is always a 2-torus and restriction of X on it possesses a smooth invariant 2-form. The linearization of the flow on such a torus (i.e. the reduction to the constant vector field) is not always possible in contrast to the case of an integrable Hamiltonian system but in the analytic case (M and X are real analytic), due to the Kolmogorov theorem, such a linearization is possible for tori with Diophantine rotation numbers.

We study Frobenius manifolds of rank three and dimension one that are related to submanifolds of certain Frobenius manifolds arising in mirror symmetry of elliptic orbifolds. We classify such Frobenius manifolds that are defined over an arbitrary field K⊂C via the theory of modular forms. By an arithmetic property of an elliptic curve Eτ defined over K associated to such a Frobenius manifold, it is proved that there are only two such Frobenius manifolds defined over C satisfying a certain symmetry assumption and thirteen Frobenius manifolds defined over Q satisfying a weak symmetry assumption on the potential.

We study the relation between the Frobenius manifolds of GW theory and the Hurwitz– Frobenius manifold.Weprove that the Frobenius manifold given by the orbifold GW theory of P^1 (2, 2, 2, 2) is isomorphic to the submanifold in the Hurwitz–Frobenius manifold of ramified coverings of the sphere by the genus 1 curve with the ramification profile (2, 2, 2, 2) over infinity.

We provide a construction of Saito primitive forms for Gepner singularity by studying the relation between Saito primitive forms for Gepner singularities and primitive forms for singularities of the form F_{k,n} = ∑^n_{i=1} x^k_i invariant under the natural S_n-action.

In our previous publications we introduced differential calculus on the enveloping algebras U(gl(m)) similar to the usual calculus on the commutative algebra . The main ingredients of our calculus are quantum partial derivatives which turn into the usual partial derivatives in the classical limit. In the particular case m=2 we prolonged this calculus on a central extension A of the algebra U(gl(2)). In the present paper we consider the problem of a further extension of the quantum partial derivatives on the skew-field of the algebra A and define the corresponding de Rham complex. As an application of the differential calculus we suggest a method of transferring dynamical models defined on an extended to an extended algebra U(u(2)). We call this procedure the quantization with noncommutative configuration space. In this sense we quantize the Dirac monopole and find a solution of this model.

A detailed construction of the universal integrability objects related to the integrable systems associated with the quantum loop algebra Uq(L(sl2)) is given. The full proof of the functional relations in the form independent of the representation of the quantum loop algebra on the quantum space is presented. The case of the general gradation and general twisting is treated. The specialization of the universal functional relations to the case when the quantum space is the state space of a discrete spin chain is described. This is a digression of the corresponding consideration for the case of the quantum loop algebra Uq(L(sl3)) with an extension to the higher spin case.