Vogel's universality implies a unified description of the adjoint sector of representation theory for simple Lie algebras in terms of three parameters <math altimg='si1.svg'><mi>&#945;</mi><mo>,</mo><mi>&#946;</mi><mo>,</mo><mi>&#947;</mi></math>, which are homogeneous coordinates of Vogel's plane. Actually this is true (if at all) only for a piece of representation theory captured by knot/Chern-Simons theory, where some irreducible representations are often undistinguishable and combined into new &#8220;universally-irreducible&#8221; entities (uirreps). We consider from this point of view the recently discovered Macdonald deformation of quantum dimensions, for which a kind of universality holds for the ADE series. The claim is that universal are not Macdonald dimensions themselves, but their products with Littlewood-Richardson coefficients, which themselves are functions of <italic>q</italic> and <italic>t</italic> in Macdonald theory. These products are precisely what arises in knot/refined Chern-Simons theory. Actually, we consider the simplest decomposition of adjoint square into six uirreps and obtain the universal formulas for hyperpolynomials of the Hopf link and, more generally, of the torus links <math altimg='si2.svg'><mi>T</mi><mo stretchy='false'>[</mo><mn>2</mn><mo>,</mo><mn>2</mn><mi>n</mi><mo stretchy='false'>]</mo></math>.

Russia

Liudmila Bishler

Andrei Mironov

Alexei Morozov

Macdonald deformation of Vogel's universality and link hyperpolynomials

We demonstrate the consistency of character expansion for the Itzykson-Zuber (IZ) model in terms of Schur polynomials with the old formulas for pair correlators with the IZ measure. An essential new feature of the correlators is that they are not symmetric in eigenvalues – and thus can not be expressed through Schur polynomials only. Instead, we demonstrate that an expression is possible in terms of Schur derivatives. This opens a new way to study arbitrary IZ correlators of any order in character expansion.

Hasib Sifat

Itzykson-Zuber correlators from character expansion

In the tight binding model with multiple degenerate vacua we might treat wave function overlaps as instanton tunnelings between different wells (vacua). An amplitude for such a tunneling process might be constructed as $$ {\textrm{T}}_{i\to j}\sim {e}^{-{S}_{\textrm{inst}}}{\textbf{v}}_j^{+}{\textbf{v}}_i^{-} $$ , where there is canonical instanton action suppression, and $$ {\textbf{v}}_i^{-} $$ annihilates a particle in the $\textit{i}$ vacuum, whereas $$ {\textbf{v}}_j^{+} $$ creates a particle in the $\textit{j}$ vacuum. Adiabatic change of the wells leads to a Berry-phase evolution of the couplings, which is described by the zero-curvature Gauss-Manin connection, i.e. by a quantum $\textit{R}$-matrix. Zero-curvature is actually a consequence of level repulsion or topological protection, and its implication is the Yang-Baxter relation for the $\textit{R}$-matrices. In the simplest case the story is pure Abelian and not very exciting. But when the model becomes more involved, incorporates supersymmetry, gauge and other symmetries, such amplitudes obtain more intricate structures. Operators $$ {\textbf{v}}_i^{-} $$ , $$ {\textbf{v}}_j^{+} $$ might also evolve from ordinary Heisenberg operators into a more sophisticated algebraic object — a “tunneling algebra”. The result for the tunneling algebra would depend strongly on geometry of the QFT we started with, and, unfortunately, at the moment we are unable to solve the reverse engineering problem. In this note we revise few successful cases of the aforementioned correspondence: quantum algebras $\textit{U}$ ( $$ \mathfrak{g} $$ ) and affine Yangians $\textit{Y}$( $$ \hat{\mathfrak{g}} $$ ). For affine Yangians we demonstrate explicitly how instantons “perform” equivariant integrals over associated quiver moduli spaces appearing in the alternative geometric construction.

Dmitry Galakhov

Tunnels under geometries (or instantons know their algebras)

The Macdonald finite-difference Hamiltonian is lifted to a super-generalization. In addition to canonical bosonic time variables <math altimg='si1.svg'><msub><mrow><mi>p</mi></mrow><mrow><mi>k</mi></mrow></msub></math> new Grassmann time variables <math altimg='si2.svg'><msub><mrow><mi>&#952;</mi></mrow><mrow><mi>k</mi></mrow></msub></math> are introduced, and the Hamiltonian is represented as a differential operator acting on a space of functions of both types of variables <math altimg='si1.svg'><msub><mrow><mi>p</mi></mrow><mrow><mi>k</mi></mrow></msub></math> and <math altimg='si2.svg'><msub><mrow><mi>&#952;</mi></mrow><mrow><mi>k</mi></mrow></msub></math>. Eigenfunctions for this Hamiltonian are a suitable generalization of Macdonald polynomials to super-Macdonald polynomials discussed earlier in the literature. Peculiarities of the construction in comparison to the canonical bosonic case are discussed.

Nikita Tselousov

Super-Hamiltonians for super-Macdonald polynomials

A simple geometric way is suggested to derive the Ward identities in the Chern-Simons theory, also known as quantum A- and C-polynomials for knots. In quasi-classical limit it is closely related to the well publicized augmentation theory and contact geometry. Quantization allows to present it in much simpler terms, what could make these techniques available to a broader audience. To avoid overloading of the presentation, only the case of the colored Jones polynomial for the trefoil knot is considered, though various generalizations are straightforward. Restriction to solely Jones polynomials (rather than full HOMFLY-PT) is related to a serious simplification, provided by the use of Kauffman calculus. Going beyond looks realistic, however it remains a problem, both challenging and promising.

On geometric bases for A-polynomials of knots

In this note, we aim to review algorithms for constructing crystal representations of quiver Yangians in detail. Quiver Yangians are believed to describe an action of the BPS algebra on BPS states in systems of D-branes wrapping toric Calabi-Yau three-folds. Crystal modules of these algebras originate from molten crystal models for Donaldson-Thomas invariants of respective three-folds. Despite the fact that this subject was originally at the crossroads of algebraic geometry with effective supersymmetric field theories, equivariant toric action simplifies applied calculations drastically. So the sole pre-requisite for this algorithm’s implementation is linear algebra. It can be easily taught to a machine with the help of any symbolic calculation system. Moreover, these algorithms may be generalized to toroidal and elliptic algebras and exploited in various numerical experiments with those algebras. We illustrate the work of the algorithms in applications to simple cases of   <math> <mi>Y</mi> <mfenced> <msub> <mi>sl</mi> <mn>2</mn> </msub> </mfenced> </math>  $$ \textrm{Y}\left({\mathfrak{sl}}_2\right) $$ ,   <math> <mi>Y</mi> <mfenced> <msub> <mover> <mi>gl</mi> <mo>̂</mo> </mover> <mn>1</mn> </msub> </mfenced> </math>  $$ \textrm{Y}\left({\hat{\mathfrak{gl}}}_1\right) $$  and   <math> <mi>Y</mi> <mfenced> <msub> <mover> <mi>gl</mi> <mo>̂</mo> </mover> <mrow> <mfenced> <mn>1</mn> </mfenced> <mn>1</mn> </mrow> </msub> </mfenced> </math>  $$ \textrm{Y}\left({\hat{\mathfrak{gl}}}_{\left.1\right|1}\right) $$ .

Alexei Gavshin

Algorithms for representations of quiver Yangian algebras

We discuss generalization of famous Macdonald polynomials for the case of super-Young diagrams that contain half-boxes on the equal footing with full boxes. These super-Macdonald polynomials are polynomials of extended set of variables: usual <math><msub><mrow><mi>p</mi></mrow><mrow><mi>k</mi></mrow></msub></math> variables are accompanied by anti-commuting Grassmann variables <math><msub><mrow><mi>θ</mi></mrow><mrow><mi>k</mi></mrow></msub></math>. Starting from recently defined super-Schur polynomials and exploiting orthogonality relations with triangular decompositions we are able to fully determine super-Macdonald polynomials. These polynomials have similar properties to canonical Macdonald polynomials – they respect two different orderings in the set of (super)-Young diagrams simultaneously.

Macdonald polynomials for super-partitions

BPS algebras are the symmetries of a wide class of brane-inspired models. They are closely related to Yangians – the peculiar and somewhat sophisticated limit of DIM algebras. Still they possess some simple and explicit representations. We explain here that for  $$Y(\widehat{\mathfrak {gl}}_r)$$  <math> <mrow> <mi>Y</mi> <mo>(</mo> <msub> <mover> <mi>gl</mi> <mo>^</mo> </mover> <mi>r</mi> </msub> <mo>)</mo> </mrow> </math>   these representations are related to Uglov polynomials, whose families are also labeled by natural r. They arise in the limit  $$\hbar {\longrightarrow } 0$$  <math> <mrow> <mi>ħ</mi> <mo>⟶</mo> <mn>0</mn> </mrow> </math>   from Macdonald polynomials, and generalize the well-known Jack polynomials ( $$\beta $$  <math> <mi>β</mi> </math>  -deformation of Schur functions), associated with  $$r=1.$$  <math> <mrow> <mi>r</mi> <mo>=</mo> <mn>1</mn> <mo>.</mo> </mrow> </math>   For  $$r=2$$  <math> <mrow> <mi>r</mi> <mo>=</mo> <mn>2</mn> </mrow> </math>   they approximate Macdonald polynomials with the accuracy  $$O(\hbar ^2),$$  <math> <mrow> <mi>O</mi> <mo>(</mo> <msup> <mi>ħ</mi> <mn>2</mn> </msup> <mo>)</mo> <mo>,</mo> </mrow> </math>   so that they are eigenfunctions of two immediately available commuting operators, arising from the  $$\hbar $$  <math> <mi>ħ</mi> </math>  -expansion of the first Macdonald Hamiltonian. These operators have a clear structure, which is easily generalizable, – what provides a technically simple way to build an explicit representation of Yangian  $$Y(\widehat{\mathfrak {gl}}_2),$$  <math> <mrow> <mi>Y</mi> <mo>(</mo> <msub> <mover> <mi>gl</mi> <mo>^</mo> </mover> <mn>2</mn> </msub> <mo>)</mo> <mo>,</mo> </mrow> </math>   where  $$U^{(2)}$$  <math> <msup> <mi>U</mi> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </msup> </math>   are associated with the states  $$|\lambda {\rangle },$$  <math> <mrow> <mo>|</mo> <mi>λ</mi> <mo>⟩</mo> <mo>,</mo> </mrow> </math>   parametrized by chess-colored Young diagrams. An interesting feature of this representation is that the odd time-variables  $$p_{2n+1}$$  <math> <msub> <mi>p</mi> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </math>   can be expressed through mutually commuting operators from Yangian, however even time-variables  $$p_{2n}$$  <math> <msub> <mi>p</mi> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msub> </math>   are inexpressible. Implications to higher r become now straightforward, yet we describe them only in a sketchy way.

Simple representations of BPS algebras: the case of  $$Y(\widehat{\mathfrak {gl}}_2)$$  <math> <mrow> <mi>Y</mi> <mo>(</mo> <msub> <mover> <mi>gl</mi> <mo>^</mo> </mover> <mn>2</mn> </msub> <mo>)</mo> </mrow> </math>

BPS states in supersymmetric theories can admit additional algebro-geometric structures in their spectra, described as quiver Yangian algebras. Equivariant fixed points on the quiver variety are interpreted as vectors populating a representation module, and matrix elements for the generators are then defined as Duistermaat-Heckman integrals in the vicinity of these points. The well-known wall-crossing phenomena are that the fixed point spectrum establishes a dependence on the stability (Fayet-Illiopolous) parameters ζ, jumping abruptly across the walls of marginal stability, which divide the ζ-space into a collection of stability chambers — “phases” of the theory. The standard construction of the quiver Yangian algebra relies heavily on the molten crystal model, valid in a sole cyclic chamber where all the ζ-parameters have the same sign. We propose to lift this restriction and investigate the effects of the wall-crossing phenomena on the quiver Yangian algebra and its representations — starting with the example of affine super-Yangian  $${\text{Y}}\left({\widehat{\mathfrak{g}\mathfrak{l}}}_{1\left|1\right.}\right)$$ . In addition to the molten crystal construction more general atomic structures appear, in other non-cyclic phases (chambers of the ζ-space). We call them glasses and also divide in a few different classes. For some of the new phases we manage to associate an algebraic structure again as a representation of the same affine Yangian  $${\text{Y}}\left({\widehat{\mathfrak{g}\mathfrak{l}}}_{1\left|1\right.}\right)$$ . This observation supports an earlier conjecture that the BPS algebraic structures can be considered as new wall-crossing invariants.

Wall-crossing effects on quiver BPS algebras

We review the main ideas underlying the emerging theory of Yangians—the new type of hidden symmetry in string-inspired models. Their classification by quivers is a far-going generalization of simple Lie algebras classification by Dynkin diagrams. However, this is still a kind of project, while a more constructive approach goes through toric Calabi-Yau spaces, related supersymmetric systems, and the Duistermaat-Heckman or equivariant integrals between the fixed points in the Atiyah-Drinfeld-Hitchin-Manin (ADHM)-like moduli spaces. These fixed points are classified by crystals (Young-type diagrams), and Yangian generators describe “instanton” transitions between them. Detailed examples will be presented elsewhere.