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## Painleve Field Theory

Cornell University
,
2013.

We propose multidimensional versions of the Painleve VI equation and its degenerations. These field theories are related to the isomonodromy problems on flat holomorphic infinite rank bundles over elliptic curves and take the form of non-autonomous Hamiltonian equations. The modular parameter of curves plays the role of "time". Reduction of the field equations to the zero modes leads to SL(N,C) monodromy preserving equations. In particular, the latter coincide with the Painleve VI equation for N=2. We consider two types of the bundles. In the first one the group of automorphisms is the centrally and cocentrally extended loop group L(SL(N,C)) or some multiloop group. In the case of the Painleve VI field theory in D=1+1 the four constants of the Painleve VI equation become dynamical fields. The bundles of the second type are defined by the group of automorphisms of the noncommutative torus. They lead to the equations in dimension 2+1. In both cases we consider trigonometric, rational and scaling limits of the theories. Generically (except some degenerate cases) the derived equations are nonlocal. We consider Whitham quasiclassical limit to integrable systems. In this way we derive two and three dimensional integrable nonlocal versions of the integrable Euler-Arnold tops.

A. Levin, Olshanetsky M., Zotov A., / Cornell University. Series math "arxiv.org". 2013.

We consider the isomonodromy problems for flat $G$-bundles over punctured
elliptic curves $\Sigma_\tau$ with regular singularities of connections at
marked points. The bundles are classified by their characteristic classes.
These classes are elements of the second cohomology group
$H^2(\Sigma_\tau,{\mathcal Z}(G))$, where ${\mathcal Z}(G)$ is the center of
$G$. For any complex simple Lie group $G$ and arbitrary class we define ...

Added: December 27, 2013

Levin A., Olshanetsky M., Zotov A., / Cornell University. Series math "arxiv.org". 2014.

We construct special rational ${\rm gl}_N$ Knizhnik-Zamolodchikov-Bernard
(KZB) equations with $\tilde N$ punctures by deformation of the corresponding
quantum ${\rm gl}_N$ rational $R$-matrix. They have two parameters. The limit
of the first one brings the model to the ordinary rational KZ equation. Another
one is $\tau$. At the level of classical mechanics the deformation parameter
$\tau$ allows to extend the ...

Added: January 23, 2015

A. Levin, Olshanetsky M., Zotov A., Nuclear Physics B 2014 Vol. 887 P. 400-422

In our recent paper we suggested a natural construction of the classical relativistic integrable tops in terms of the quantum R -matrices. Here we study the simplest case – the 11-vertex R -matrix and related gl2 rational models. The corresponding top is equivalent to the 2-body Ruijsenaars–Schneider (RS) or the 2-body Calogero–Moser (CM) model depending ...

Added: January 22, 2015

Covolo T., Ovsienko V., Poncin N., Journal of Geometry and Physics 2012 Vol. 62 P. 2294-2319

We define the notions of trace, determinant and, more generally, Berezinian of matrices over a (Z_2)^n graded commutative associative algebra. The applications include a new approach to the classical theory of matrices with coefficients in a Clifford algebra, in particular of quaternionic matrices. In a special case, we recover the classical Dieudonn\'e determinant of quaternionic ...

Added: September 28, 2015

A. Levin, Olshanetsky M., Zotov A., Journal of High Energy Physics 2014 Vol. 2014 No. 7:12 P. 1-39

We describe classical top-like integrable systems arising from the quantum exchange relations and corresponding Sklyanin algebras. The Lax operator is expressed in terms of the quantum non-dynamical R-matrix even at the classical level, where the Planck constant plays the role of the relativistic deformation parameter in the sense of Ruijsenaars and Schneider (RS). The integrable ...

Added: January 23, 2015

Levin A., Olshanetsky M., Zotov A., / Cornell University. Series math "arxiv.org". 2014.

e describe classical top-like integrable systems arising from the quantum
exchange relations and corresponding Sklyanin algebras. The Lax operator is
expressed in terms of the quantum non-dynamical $R$-matrix even at the
classical level, where the Planck constant plays the role of the relativistic
deformation parameter in the sense of Ruijsenaars and Schneider (RS). The
integrable systems (relativistic tops) are described ...

Added: January 23, 2015

Levin A., Olshanetsky M., Zotov A., / Cornell University. Series math "arxiv.org". 2014.

In our recent paper we suggested a natural construction of the classical relativistic integrable tops in terms of the quantum R-matrices. Here we study the simplest case -- the 11-vertex R-matrix and related gl_2 rational models. The corresponding top is equivalent to the 2-body Ruijsenaars-Schneider (RS) or the 2-body Calogero-Moser (CM) model depending on its ...

Added: January 23, 2015

Braverman A., Michael Finkelberg, Nakajima H., / Cornell University. Series math "arxiv.org". 2014.

We describe the (equivariant) intersection cohomology of certain moduli spaces ("framed Uhlenbeck spaces") together with some structures on them (such as e.g.\ the Poincar\'e pairing) in terms of representation theory of some vertex operator algebras ("W-algebras"). ...

Added: January 30, 2015

Braverman A., Dobrovolska G., Michael Finkelberg, / Cornell University. Series math "arxiv.org". 2014.

Let G be an almost simple simply connected group over complex numbers. For a positive element α of the coroot lattice of G let Z^α denote the space of based maps from the projective line to the flag variety of G of degree α. This space is known to be isomorphic to the space of ...

Added: February 3, 2015

A. Levin, Olshanetsky M., Zotov A., / Cornell University. Series math "arxiv.org". 2015.

It was shown in our previous paper that quantum ${\rm gl}_N$ $R$-matrices
satisfy noncommutative analogues of the Fay identities in ${\rm gl}_N^{\otimes
3}$. In this paper we extend the list of $R$-matrix valued elliptic function
identities. We propose counterparts of the Fay identities in ${\rm
gl}_N^{\otimes 2}$, the symmetry between the Planck constant and the spectral
parameter, quasi-periodicities with respect ...

Added: February 3, 2015

Kurnosov N., / Cornell University. Series math "arxiv.org". 2015.

We prove that a generic complex deformation of a generalized Kummer variety contains no complex analytic tori. ...

Added: October 16, 2015

Entov M., Verbitsky M., / Cornell University. Series math "arxiv.org". 2014.

Let M be a closed symplectic manifold of volume V. We say that M admits a full symplectic packing by balls if any collection of symplectic balls of total volume less than V admits a symplectic embedding to M. In 1994 McDuff and Polterovich proved that symplectic packings of Kahler manifolds can be characterized in ...

Added: February 5, 2015

Verbitsky M., / Cornell University. Series math "arxiv.org". 2013.

Let M be a hyperkaehler manifold, and η a closed, positive (1,1)-form which is degenerate everywhere on M. We associate to η a family of complex structures on M, called a degenerate twistor family, and parametrized by a complex line. When η is a pullback of a Kaehler form under a Lagrangian fibration L, all ...

Added: December 27, 2013

Verbitsky M., / Cornell University. Series math "arxiv.org". 2013.

Let M be a compact complex manifold. The corresponding Teichmuller space $\Teich$ is a space of all complex structures on M up to the action of the group of isotopies. The group Γ of connected components of the diffeomorphism group (known as the mapping class group) acts on $\Teich$ in a natural way. An ergodic ...

Added: December 27, 2013

Andrey Soldatenkov, Misha Verbitsky, / Cornell University. Series math "arxiv.org". 2014.

Let $(M,I,J,K)$ be a hyperkahler manifold, and $Z\subset (M,I)$ 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 ...

Added: September 5, 2014

Déev R. N., / Cornell University. Series arXiv "math". 2016.

Essential dimension of a family of complex manifolds is the dimension of the image of its base in the Kuranishi space of the fiber. We prove that any family of hyperk\"ahler manifolds over a compact simply connected base has essential dimension not greater than 1. A similar result about families of complex tori is also ...

Added: September 23, 2016

Ivan Cheltsov, Martinez-Garcia J., / Cornell University. Series math "arxiv.org". 2014.

For every smooth del Pezzo surface $S$, smooth curve $C\in|-K_{S}|$ and $\beta\in(0,1]$, we compute the $\alpha$-invariant of Tian $\alpha(S,(1-\beta)C)$ and prove the existence of K\"ahler--Einstein metrics on $S$ with edge singularities along $C$ of angle $2\pi\beta$ for $\beta$ in certain interval. In particular we give lower bounds for the invariant $R(S,C)$, introduced by Donaldson as ...

Added: February 5, 2015

Providence : American Mathematical Society, 2014

Added: September 15, 2016

Verbitsky M., Grantcharov G., Lejmi M., / Cornell University. Series math "arxiv.org". 2014.

A hypercomplex manifold M is a manifold equipped with three complex structures satisfying quaternionic relations. Such a manifold admits a canonical torsion-free connection preserving the quaternion action, called Obata connection. A quaternionic Hermitian metric is a Riemannian metric on which is invariant with respect to unitary quaternions. Such a metric is called HKT if it ...

Added: September 19, 2014

Khoroshkin S. M., Shapiro A., Journal of Geometry and Physics 2010 Vol. 60 No. 11 P. 1833-1851

In this article, we give an explicit formula for the universal weight function of the quantum twisted affine algebra Uq(A(2)2 ). The calculations use the technique of projecting products of Drinfeld currents onto the intersection of Borel subalgebras of different types. ...

Added: September 26, 2012

Gavrylenko P., Iorgov N., Lisovyy O., Journal of Physics A: Mathematical and Theoretical 2012 Vol. 45 No. 2 P. 025402

We study U(1) twist fields in a two-dimensional lattice theory of massive Dirac fermions. Factorized formulas for finite-lattice form factors of these fields are derived using elliptic parametrization of the spectral curve of the model, elliptic determinant identities and theta functional interpolation. We also investigate the thermodynamic and infinite-volume scaling limit, where the corresponding expressions ...

Added: October 20, 2014

Kamenova L., Lu S., Verbitsky M., / Cornell University. Series math "arxiv.org". 2013.

The Kobayashi pseudometric on a complex manifold $M$ is the maximal pseudometric such that any holomorphic map from the Poincare disk to $M$ is distance-decreasing. Kobayashi has conjectured that this pseudometric vanishes on Calabi-Yau manifolds. Using ergodicity of complex structures, we prove this result for any hyperkaehler manifold if it admits a deformation with a ...

Added: August 28, 2013

Kurnosov N., / Cornell University. Series math "arxiv.org". 2014.

Let M be a compact irreducible hyperkahler manifold, from Bogomolov inequality [V1] we obtain forbidden values of the second Betti number b_2 in arbitrary dimension. ...

Added: February 21, 2014

Verbitsky M., Communications in Mathematical Physics 2013 Vol. 324 No. 1 P. 173-177

Let M be an almost complex manifold equipped with a Hermitian form such that its de Rham differential has Hodge type (3,0)+(0,3), for example a nearly Kahler manifold. We prove that any connected component of the moduli space of pseudoholomorphic curves on M is compact. This can be used to study pseudoholomorphic curves on a ...

Added: February 16, 2013