In this paper we define a quantum version of the ``fusion'' tensor product of two representations... more In this paper we define a quantum version of the ``fusion'' tensor product of two representations of an affine Kac-Moody algebra.It is replaced by what we call fusion action of the category of finite-dimensional representations of quantum affine algebra on its highest weight representations. We construct a quantum version of the associativity constraint. We give categorical treatment of the subject and related questions ( like quantum Knizhnik-Zamolodchikov equations).
We study refined and motivic wall-crossing formulas in N=2 supersymmetric gauge theories with SU(... more We study refined and motivic wall-crossing formulas in N=2 supersymmetric gauge theories with SU(2) gauge group and N_f < 4 matter hypermultiplets in the fundamental representation. Such gauge theories provide an excellent testing ground for the conjecture that "refined = motivic."
This allows us to give a more precise mathematical formulation of the (conjectural) alternative d... more This allows us to give a more precise mathematical formulation of the (conjectural) alternative description of the Fukaya-Seidel category of a Kahler manifold endowed with a holomorphic Morse function.
The quantum groups gl ∞ and A ∞ are constructed. The representation theory of these algebras is d... more The quantum groups gl ∞ and A ∞ are constructed. The representation theory of these algebras is developed and the universal R-matrix is presented.
In present paper we develop the deformation theory of operads and algebras over operads. Free res... more In present paper we develop the deformation theory of operads and algebras over operads. Free resolutions (constructed via Boardman-Vogt approach) are used in order to describe formal moduli spaces of deformations. We apply the general theory to the proof of Deligne's conjecture. The latter says that the Hochschild complex of an associative algebra carries a canonical structure of a dg-algebra
We study refined and motivic wall-crossing formulas in {{mathcal N}=2} supersymmetric gauge theor... more We study refined and motivic wall-crossing formulas in {{mathcal N}=2} supersymmetric gauge theories with SU(2) gauge group and N f < 4 matter hypermultiplets in the fundamental representation. Such gauge theories provide an excellent testing ground for the conjecture that "refined = motivic."
Advances in Theoretical and Mathematical Physics, Oct 1, 2014
ABSTRACT BPS quivers for N=2 SU(N) gauge theories are derived via geometric engineering from deri... more ABSTRACT BPS quivers for N=2 SU(N) gauge theories are derived via geometric engineering from derived categories of toric Calabi-Yau threefolds. While the outcome is in agreement of previous low energy constructions, the geometric approach leads to several new results. An absence of walls conjecture is formulated for all values of N, relating the field theory BPS spectrum to large radius D-brane bound states. Supporting evidence is presented as explicit computations of BPS degeneracies in some examples. These computations also prove the existence of BPS states of arbitrarily high spin and infinitely many marginal stability walls at weak coupling. Moreover, framed quiver models for framed BPS states are naturally derived from this formalism, as well as a mathematical formulation of framed and unframed BPS degeneracies in terms of motivic and cohomological Donaldson-Thomas invariants. We verify the conjectured absence of BPS states with &quot;exotic&quot; SU(2)_R quantum numbers using motivic DT invariants. This application is based in particular on a complete recursive algorithm which determine the unframed BPS spectrum at any point on the Coulomb branch in terms of noncommutative Donaldson-Thomas invariants for framed quiver representations.
ABSTRACT The universal quantumR-matrix is obtained in the case of the affine Kac-Moody Lie algebr... more ABSTRACT The universal quantumR-matrix is obtained in the case of the affine Kac-Moody Lie algebra sl(2).
The paper is devoted to the comparison of the Fukaya category (it is responcible for the A-side o... more The paper is devoted to the comparison of the Fukaya category (it is responcible for the A-side of mirror symmetry) with the category of holonomic modules over the quantized algebra of functions on the same symplectic manifold. We conjecture that these categories become $A_{\infty}$-equivalent after a twist by a kind of integral transformation.
... of two-dimensional surfaces on the Hochschild chain complex of an A-infinity algebra with ...... more ... of two-dimensional surfaces on the Hochschild chain complex of an A-infinity algebra with ... projective variety X is A∞-equivalent to the category of perfect modules over a ... Then our conjecture becomes the classical theorem which claims degeneration of the spectral sequence ...
Advances in Theoretical and Mathematical Physics, 2014
ABSTRACT BPS quivers for N=2 SU(N) gauge theories are derived via geometric engineering from deri... more ABSTRACT BPS quivers for N=2 SU(N) gauge theories are derived via geometric engineering from derived categories of toric Calabi-Yau threefolds. While the outcome is in agreement of previous low energy constructions, the geometric approach leads to several new results. An absence of walls conjecture is formulated for all values of N, relating the field theory BPS spectrum to large radius D-brane bound states. Supporting evidence is presented as explicit computations of BPS degeneracies in some examples. These computations also prove the existence of BPS states of arbitrarily high spin and infinitely many marginal stability walls at weak coupling. Moreover, framed quiver models for framed BPS states are naturally derived from this formalism, as well as a mathematical formulation of framed and unframed BPS degeneracies in terms of motivic and cohomological Donaldson-Thomas invariants. We verify the conjectured absence of BPS states with &quot;exotic&quot; SU(2)_R quantum numbers using motivic DT invariants. This application is based in particular on a complete recursive algorithm which determine the unframed BPS spectrum at any point on the Coulomb branch in terms of noncommutative Donaldson-Thomas invariants for framed quiver representations.
The structures of Poisson Lie groups on a simple compact group are parametrized by pairs (α,w), w... more The structures of Poisson Lie groups on a simple compact group are parametrized by pairs (α,w), where aeR, ueA 2 \) R , and ί) R is a real Cartan subalgebra of complexifϊcation of Lie algebra of the group in question. In the present article the description of the symplectic leaves for all pairs (α, u) is given. Also, the corresponding quantized algebras of functions are constructed and their irreducible representations are described. In the course of investigation Schubert cells and quantum tori appear. At the end of the article the quantum analog of the Weyl group is constructed and some of its applications, among them the formula for the universal ^-matrix, are given.
We study refined and motivic wall-crossing formulas in N = 2 supersymmetric gauge theories with S... more We study refined and motivic wall-crossing formulas in N = 2 supersymmetric gauge theories with SU (2) gauge group and N f < 4 matter hypermultiplets in the fundamental representation. Such gauge theories provide an excellent testing ground for the conjecture that "refined = motivic." . 81T30, 14N35, 14E15, 05A18.
In this paper we define a quantum version of the ``fusion'' tensor product of two representations... more In this paper we define a quantum version of the ``fusion'' tensor product of two representations of an affine Kac-Moody algebra.It is replaced by what we call fusion action of the category of finite-dimensional representations of quantum affine algebra on its highest weight representations. We construct a quantum version of the associativity constraint. We give categorical treatment of the subject and related questions ( like quantum Knizhnik-Zamolodchikov equations).
We study refined and motivic wall-crossing formulas in N=2 supersymmetric gauge theories with SU(... more We study refined and motivic wall-crossing formulas in N=2 supersymmetric gauge theories with SU(2) gauge group and N_f < 4 matter hypermultiplets in the fundamental representation. Such gauge theories provide an excellent testing ground for the conjecture that "refined = motivic."
This allows us to give a more precise mathematical formulation of the (conjectural) alternative d... more This allows us to give a more precise mathematical formulation of the (conjectural) alternative description of the Fukaya-Seidel category of a Kahler manifold endowed with a holomorphic Morse function.
The quantum groups gl ∞ and A ∞ are constructed. The representation theory of these algebras is d... more The quantum groups gl ∞ and A ∞ are constructed. The representation theory of these algebras is developed and the universal R-matrix is presented.
In present paper we develop the deformation theory of operads and algebras over operads. Free res... more In present paper we develop the deformation theory of operads and algebras over operads. Free resolutions (constructed via Boardman-Vogt approach) are used in order to describe formal moduli spaces of deformations. We apply the general theory to the proof of Deligne's conjecture. The latter says that the Hochschild complex of an associative algebra carries a canonical structure of a dg-algebra
We study refined and motivic wall-crossing formulas in {{mathcal N}=2} supersymmetric gauge theor... more We study refined and motivic wall-crossing formulas in {{mathcal N}=2} supersymmetric gauge theories with SU(2) gauge group and N f < 4 matter hypermultiplets in the fundamental representation. Such gauge theories provide an excellent testing ground for the conjecture that "refined = motivic."
Advances in Theoretical and Mathematical Physics, Oct 1, 2014
ABSTRACT BPS quivers for N=2 SU(N) gauge theories are derived via geometric engineering from deri... more ABSTRACT BPS quivers for N=2 SU(N) gauge theories are derived via geometric engineering from derived categories of toric Calabi-Yau threefolds. While the outcome is in agreement of previous low energy constructions, the geometric approach leads to several new results. An absence of walls conjecture is formulated for all values of N, relating the field theory BPS spectrum to large radius D-brane bound states. Supporting evidence is presented as explicit computations of BPS degeneracies in some examples. These computations also prove the existence of BPS states of arbitrarily high spin and infinitely many marginal stability walls at weak coupling. Moreover, framed quiver models for framed BPS states are naturally derived from this formalism, as well as a mathematical formulation of framed and unframed BPS degeneracies in terms of motivic and cohomological Donaldson-Thomas invariants. We verify the conjectured absence of BPS states with &quot;exotic&quot; SU(2)_R quantum numbers using motivic DT invariants. This application is based in particular on a complete recursive algorithm which determine the unframed BPS spectrum at any point on the Coulomb branch in terms of noncommutative Donaldson-Thomas invariants for framed quiver representations.
ABSTRACT The universal quantumR-matrix is obtained in the case of the affine Kac-Moody Lie algebr... more ABSTRACT The universal quantumR-matrix is obtained in the case of the affine Kac-Moody Lie algebra sl(2).
The paper is devoted to the comparison of the Fukaya category (it is responcible for the A-side o... more The paper is devoted to the comparison of the Fukaya category (it is responcible for the A-side of mirror symmetry) with the category of holonomic modules over the quantized algebra of functions on the same symplectic manifold. We conjecture that these categories become $A_{\infty}$-equivalent after a twist by a kind of integral transformation.
... of two-dimensional surfaces on the Hochschild chain complex of an A-infinity algebra with ...... more ... of two-dimensional surfaces on the Hochschild chain complex of an A-infinity algebra with ... projective variety X is A∞-equivalent to the category of perfect modules over a ... Then our conjecture becomes the classical theorem which claims degeneration of the spectral sequence ...
Advances in Theoretical and Mathematical Physics, 2014
ABSTRACT BPS quivers for N=2 SU(N) gauge theories are derived via geometric engineering from deri... more ABSTRACT BPS quivers for N=2 SU(N) gauge theories are derived via geometric engineering from derived categories of toric Calabi-Yau threefolds. While the outcome is in agreement of previous low energy constructions, the geometric approach leads to several new results. An absence of walls conjecture is formulated for all values of N, relating the field theory BPS spectrum to large radius D-brane bound states. Supporting evidence is presented as explicit computations of BPS degeneracies in some examples. These computations also prove the existence of BPS states of arbitrarily high spin and infinitely many marginal stability walls at weak coupling. Moreover, framed quiver models for framed BPS states are naturally derived from this formalism, as well as a mathematical formulation of framed and unframed BPS degeneracies in terms of motivic and cohomological Donaldson-Thomas invariants. We verify the conjectured absence of BPS states with &quot;exotic&quot; SU(2)_R quantum numbers using motivic DT invariants. This application is based in particular on a complete recursive algorithm which determine the unframed BPS spectrum at any point on the Coulomb branch in terms of noncommutative Donaldson-Thomas invariants for framed quiver representations.
The structures of Poisson Lie groups on a simple compact group are parametrized by pairs (α,w), w... more The structures of Poisson Lie groups on a simple compact group are parametrized by pairs (α,w), where aeR, ueA 2 \) R , and ί) R is a real Cartan subalgebra of complexifϊcation of Lie algebra of the group in question. In the present article the description of the symplectic leaves for all pairs (α, u) is given. Also, the corresponding quantized algebras of functions are constructed and their irreducible representations are described. In the course of investigation Schubert cells and quantum tori appear. At the end of the article the quantum analog of the Weyl group is constructed and some of its applications, among them the formula for the universal ^-matrix, are given.
We study refined and motivic wall-crossing formulas in N = 2 supersymmetric gauge theories with S... more We study refined and motivic wall-crossing formulas in N = 2 supersymmetric gauge theories with SU (2) gauge group and N f < 4 matter hypermultiplets in the fundamental representation. Such gauge theories provide an excellent testing ground for the conjecture that "refined = motivic." . 81T30, 14N35, 14E15, 05A18.
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Papers by Yan Soibelman