The event ID is the same for both carriers

MAJOR TOPICS

Luminy segment:

(a) Affine and double affine Hecke algebras and their relations to integrable models,
(b) Geometric Langlands program and the theory of affine flag varieties,
(c) Finite characteristic and the p-adic Langlands program.

Cargese segment:

(a) Super Yang-Mills theory, including applications to the Langlands program,
(b) AdS-CFT correspondence and its relation to integrable models,
(c) Conformal and superconformal field theory in diverse dimensions.

Affine and Double affine Hecke algebras are now indispensable in modern representation theory, combinatorics, geometry of various homogeneous spaces, harmonic analysis and the theory of hypergeometric, spherical and Whittaker functions. The applications in mathematical physics include quantum many-body problems, Knizhnik-Zamolodchikov equations, Verlinde algebras, Gromov-Witten invariants of flag varieties and more. The focus will be on the geometric aspects of the theory, however, quite a few related topics will be presented, for instance, various aspects of the Khovanov-Lauda-Rouquier categorization construction.

Geometric Langlands program and homogeneous spaces of loop groups will be among the key topics of the mathematical part of the conference. The ideas of Drinfeld's proof of Langlands conjectures for GL(2) in the functional field case have led Beilinson and Drinfeld to a geometric counterpart of Langlands conjectures, which can be viewed as a categorification of classical Langlands program. While the classical theory is concerned with the spaces of automorphic forms, the geometric one deals with categories of l-adic sheaves or D-modules, which suggests a new approach to representations of Lie algebras and generalizations.

The p-adic Langlands Program includes p-adic representations of reductive groups, especially those that occur in the p-adic cohomology of Shimura varieties and similar spaces, p-adic deformations of classical automorphic Hecke eigenforms with p-adic Hecke eigenvalues and deformations of p-adic Galois representations. This theory has developed into a promising field, where especially in recent years, the power of the theory automorphic forms has been enhanced by considering the structure of eigenvarieties. Such considerations have been directly involved in many (perhaps most) of the important advances in number theory in the past few decades.

One of the main topics of the 2006 meeting was the relation between the geometric Langlands program and four-dimensional gauge theory due to Witten, Kapustin and others. This continues to develop. Gukov and Witten generalized the gauge theory approach to the geometric Langlands program to the tamely ramified case and later to the wildly ramified case (Witten), the relations with Fukaya categories for Higgs bundles were found (Kapustin) and more. It is expected that the derived Satake correspondence established by Bezrukavnikov and Finkelberg can be interpreted as the equivalence of categories of loop operators in a pair of 4D topological gauge theories related by S-duality.

Conformal field theories, the AdS/CFT correspondence, and integrability are some of the most active areas in modern theoretical physics. They significantly shape the ideas and the activity of scientists in high energy physics, condensed matter theory, particle physics, cosmology and have a strong influence on mathematics. In recent years there have been many breakthroughs  in our understanding of various important systems and models, such as super-Yang-Mills theories, strings in curved backgrounds, Liouville theory, topological field theory and string theory, and in many related areas. The physics part of the program of the meeting is especially aimed at exchanges between physicists and mathematicians, which may give a new momentum to the whole area.

OUTLINE OF MAJOR DIRECTIONS

The focus of the following scientific description of PhyMSI is on the major topics and connections between the programs. It is not a systematic overview, and many important developments and contributors are omitted for lack of space. The bibliography consists of very few relevant general and introductory books and articles;
they do not always reflect the latest developments.

The areas of LP, Affine Flag Varieties, AHA-DAHA, SYM, CFT are among the most active areas of current research in mathematics and physics. They are so deeply interconnected that experts in one of these fields need to constantly monitor progress in the others; the program we suggest will provide an opportunity to do so. The focus will be on the connections between the programs.

DOUBLE AFFINE HECKE ALGEBRAS

Affine Hecke algebras appeared as a tool in the theory of automorphic functions. Now they are indispensable in modern representation theory, combinatorics, geometry, harmonic analysis, mathematical physics and the theory of special functions. The Langlands program, the theory quantum groups, invariants of knots, the modern theory of the symmetric group and Weyl groups, Lusztig's modular conjectures and applications to the theory of spherical and hypergeometric functions are well known examples.

Double Affine Hecke algebras (DAHA) are a fundamental development. They unify Affine Hecke algebras and Heisenberg-Weyl algebras. After they were employed to prove Macdonald's conjectures including the celebrated constant term conjecture, they have attracted a lot of attention. DAHA are expected to play a major role in representation theory and applications.

Concerning the theory of DAHA (applications and generalizations), there is significant progress in the following directions: harmonic analysis and the theory of the hypergeometric function (Cherednik, Delorme, Opdam, Stokman and quite a few others), DAHA methods in the global quantization of certain algebraic varieties (in the range from Hilbert schemes to del Pezzo surfaces), the theory of Calogero-Moser varieties, symplectic reflection algebras and the wreath products, the theory of unitary representations of DAHA (it is in the beginning but there are already interesting results due to Etingof et.al.). Let us also mention recent related progress in the theory of Affine Hecke algebras.

The localization functor by Ginzburg, Guay, Opdam and Rouquier, including its applications to the q-Schur algebras, is an important development based on the Knizhnik-Zamolodchikov equations. It is related to the parabolic induction due to Etingof and Bezrukavnikov, a new powerful tool in the DAHA theory.

Quite a few specialists work now on the geometric methods in the DAHA theory, including applications to the classification of DAHA modules (Vasserot, Varagnolo and others). One of the major directions here concerns applications to the Hilbert schemes and similar objects (Ginzburg, Gordon, Griffeth, Grojnowski, Martino, Peng Shan, Rouquier, Schiffmann, Suzuki, Varagnolo, Vasserot and other specialists). It includes the latest results on the "categorization" in relation to the Khovanov-Lauda-Rouquier algebras.

DAHA are directly connected with the K-theory (and homology) of affine flag varieties and provide an approach to the so-called double arithmetic (Braverman, Gaitsgory, Kapranov, Kazhdan and others). There are important relations to the quantum toroidal algebras and the theory of the affine Satake isomorphism.

Applications to the quantum Langlands program are expected. For instance, the affine exponents, which govern the irreducibility of the polynomial representation of DAHA, satisfy the quantum Langlands duality of new type, with an extra parameter t.

The algebraic and analytic theory of "global" spherical and Whittaker functions is an important recent development, including known and expected connections of these functions with the Gromov-Witten invariants, the Givental-Lee formula in the quantum K-theory of Grassmannians and the IC-theory of affine flag varieties.

These developments, the related progress in the theory of Affine Hecke algebras and connections with physics will be in the focus of the DAHA part of the conference.

THE LANGLANDS PROGRAM

This will be the major theme of the conference. The Langlands Program has emerged in the late 1960s in the form of a series of far-reaching conjectures tying together seemingly unrelated objects in number theory, algebraic geometry, and the theory of automorphic forms (such as Galois representations, motives, and automorphic forms). In recent years it was realized that the Langlands conjectures (in the function field case) may be formulated geometrically, thereby allowing one to state them over an arbitrary field (e.g., the field of complex numbers). This approach has led to Drinfeld's proof of the Langlands conjecture for GL(2) in the function field case.

L. Lafforgue has proved the Langlands conjecture for GL(n) in the case of the field of functions on a curve over a finite field. Geometry of bundles on curves with additional structures (shtukas) plays an important role in his proof. Beilinson and Drinfeld suggested a variant of the geometric Langlands correspondence over complex field. It relates Hecke eigensheaves on the moduli stack of G-bundles over a complex curve X and local systems for the Langlands dual group of G. They construct this correspondence via quantization of an integrable system on the cotangent bundle to the moduli space of G-bundles defined by Hitchin. Their work uses in an essential way methods from physics, especially, conformal field theory. Recently, there was a significant development in this direction in the GL(n)-case by Gaitsgory and Frenkel, Gaitsgory, Vilonen.

A very important development is the proof by Ngo of the Langlands-Shelstad Fundamental Lemma over a local field of characteristic p>0. The latter has resisted all attempts to prove it for more than twenty years and is directly connected with the geometric Langlands program. There are interesting new developments connecting the Fundamental Lemma with the theory of the global Springer fiber, defined in terms of the Hitchin system, and DAHA (Yun). Taylor and Harris recent breakthrough results on the Sato-Tate conjecture must be of course mentioned too.

Geometric Langlands program and homogeneous spaces of loop groups

As was mentioned above, the ideas of Drinfeld's proof of Langlands conjectures for GL(2) in the functional field case have led Beilinson and Drinfeld to a geometric counterpart of Langlands conjectures. This can be viewed as a categorification of classical Langlands conjectures - while the classical theory is concerned with the spaces of automorphic forms, the geometric one deals with categories of l-adic sheaves or D-modules. (The idea that an l-adic sheaf can be thought of as an enrichment of a function on the set of points of an algebraic variety over a finite field goes back to Grothendieck).

In the recent years it has been realized that the categories arising as categorification of representation of p-adic groups oftentimes admit an alternative interpretation as categories of representations, which are of significant independent interest. Examples (some proved, some conjectured) include categories of representations of Kac-Moody Lie algebras, of semi-simple Lie algebras in positive characteristic and quantum groups at roots of unity.
The compatibility between local and global Langlands conjectures should then provide some rich structure on the above representation category, such as a braided monoidal or modular structure. Thus geometric Langlands program suggests a new approach to representations of Lie algebras and generalizations. The first result in this direction has been obtained by Kazhdan and Lusztig in the 90's. Recent works by several authors (Bezrukavnikov-Mirkovic, Gaitsgory-Frenkel et. al.) provide a wealth of open problems in the area.

On the other hand, categorical counterparts of known constructions from harmonic analysis of p-adic groups are expected to provide more satisfying results in the geometric case. So far the local geometric theory has only been developed in the unramified (geometric Satake of Lusztig, Drinfeld and Mirkovic-Vilonen) or tamely ramified (Arkhipov-Bezrukavnikov) case. The available information suggests that the theory of Whittaker model works in the ideal way in this setting - this is an essential ingredient in Gaitsgory-Lurie's current approach to geometric Langlands as an equivalence of 3 dimensional topological field theories.
Tamely ramified geometric Langlands is closely related to the work of Gukov and Witten (see below), while subsequent work by Witten on wild ramification suggests an approach to further steps of local geometric Langlands duality. This is expected to have significant applications to classical theory, namely to the study of L-packets and stability phenomenon in representation theory of p-adic groups.

P-adic Langlands program

The p-adic Langlands Program is to be taken in its broadest sense. Thus it includes the recent and on-going work regarding:

A) p-adic (infinite dimensional) representations of (a large family of) reductive groups, and especially those representations that occur in the p-adic cohomology of (p-adic towers of) Shimura varieties and similar spaces,

B) p-adic deformations of classical automorphic Hecke eigenforms with p-adic Hecke eigenvalues, and C) deformations of p-adic Galois representations.

The intimate connection between these is given, conjecturally, by a version of the Langlands philosophy that, it is to be hoped, will work very well with the current approach, i.e., viewing these items as parameterized by certain p-adic analytic spaces (called eigenvarieties). Just as Langlands' philosophy might be considered a modern non-abelian counterpart to Class Field Theory, the deformational approach alluded to is a modern "non-abelian" extension of the classical Iwasawa theory. This p-adic deformational view of things is an attempt to give the fullest expression to the striking fact that classical modular forms (technically: eigenforms for the Hecke operators and of finite slope) are intimately tied one to another by a very important and remarkably intensive web of congruences.

This has developed into a promising field, where especially in recent years, the power of the theory automorphic forms (for a large collection of reductive groups) has been enhanced by considering the structure of eigenvarieties. Such considerations have been directly involved in many (perhaps most) of the important advances in number theory in the past few decades. For example, congruences played its role in the dramatic proof of modularity of elliptic curves over Q, a few years ago, as it has done more recently in the resolution of Serre's Conjecture and in the advances regarding the Sato-Tate Conjecture.

Galois representations require special comment. The classical (arithmetic) Langlands program posits a strong relation between p-adic representations of absolute Galois groups of number fields and automorphic forms that are simultaneous eigenforms for all the local Hecke algebras. Now p-adic representations of Galois groups naturally live in families (analogous to the representation varieties of fundamental groups that appear in topology), known (for technical and historical reasons) as deformation spaces of Galois representations, while it has long been known that classical modular forms (and, more generally, automorphic forms) are intimately connected by a very important and remarkably intensive web of congruences.

These congruences reflect the fact that automorphic forms themselves can be placed in p-adic families, which are a reflection under the global Langlands correspondence of the families of Galois representations parameterized by deformation spaces. In other words, the Langlands correspondence admits a p-adic interpolation, and the study of this interpolation, in all its aspects, is the subject of the p-adic Langlands Program.

Because a priori quite different automorphic forms and Galois representations can be linked together (and hence influence one another) by the p-adic families that contain them, the study of these families is thus a powerful tool for studying questions in algebraic number theory, and it has been directly involved in many of the important advances in number theory in the past few decades. For example, it played a key role in the dramatic proof of modularity of elliptic curves over Q, a few years ago, as well as in the proof of the Sato-Tate conjecture.

In recent years, led by Breuil, Colmez, Schneider, Teitelbaum, and others, it has been realized that the representation-theoretic methods of the classical Langlands program can themselves be "p-adically interpolated'', leading to a fruitful theory of "p-adic representations of p-adic groups''. These methods have found serious application to the global problems of the p-adic Langlands program, leading, for an example, to an almost complete resolution of the Fontaine-Mazur conjecture for 2-dimensional Galois representations, by Kisin and Emerton. At the same time, some of the phenomena that arise in this new p-adic representation theory seem to be similar to phenomena arising in some of the local aspects of the geometric Langlands program, and perhaps also in the theory of DAHA.

The p-adic Langlands program part of the conference will focus on all this current work and its relationship to the other facets of the theory. It will include more classical aspects too such as the Iwasawa theory.

SUPER YANG-MILLS THEORIES AND CONFORMAL FIELD THEORIES

These continue to be major topics in theoretical particle physics, with surprising new developments every year.

Significant progress in four-dimensional supersymmetric gauge theory is being made based on the exact instanton calculus of Nekrasov, built on the technique developed by Losev, Moore, Nekrasov and Shatashvili in 1994-1999. In 2003, he and Okounkov proved the Seiberg-Witten conjecture for the low energy description of N=2 super Yang-Mills theory and observed the relation of the instanton partition functions to two dimensional conformal theories. Over the subsequent years, these techniques were used to compute effective superpotentials in N=1 theories. More recently, a very detailed connection between the vacuum structure of gauge theories and integrable spin systems has been developed by Gerasimov, Nekrasov and Shatashvili. These connections involve a great deal of the mathematical structure of quantum integrability, which opens a wide range of possibilities for cooperation with experts in representation theory, including Kac-Moody algebras, super Lie algebras, AHA-DAHA, and with mathematicians from other fields.

A very interesting recent development is a unified understanding of a large class of four-dimensional N=2 superconformal field theories, due to Gaiotto and collaborators. Starting from the somewhat mysterious "(2,0)'' supersymmetric theories in six dimensions, one can compactify on a genus g Riemann surface with n punctures, to obtain a family of theories parameterized by the moduli space of complex structures of these surfaces, M_{g,n}. All of the old and many new results for S-dualities arise as relations between degeneration limits in M_{g,n}.

In a recent work, Alday, Gaiotto and Tachikawa have related the partition functions for the N=2 gauge theories, proposed by Gaiotto, to conformal blocks of two dimensional Liouville theory. In a particular case, the full partition function of the theories compactified on a four-sphere (computed by Pestun) becomes the full correlation functions of the Liouville theory. There is a much more detailed relation between the conformal field theories in 2 and 4 dimensions, which reflects the existence of the six dimensional superconformal theory.

SYM and geometric LP

One of the main topics of the 2006 meeting was the relation between the geometric Langlands program and four-dimensional gauge theory due to Witten, Kapustin and others. This continues to develop. In 2006 Gukov and Witten generalized the gauge theory approach to the geometric Langlands program to the tamely ramified case. This was accomplished by considering surface operators in N=4 super Yang-Mills theory. In 2007, Witten further generalized it to the wildly ramified case. In 2008, Kapustin studied the quantum geometric Langlands duality from the gauge theory viewpoint and showed that it can be reformulated in terms of equivalence of categories of A-branes (i.e. Fukaya categories) for Higgs bundle moduli spaces. In 2009, Kapustin systematically studied the category of Wilson loop operators and showed that it is equivalent to the equivariant derived category of coherent sheaves on the Lie algebra of the complexified gauge group.

On the side of mathematics, the latter development indicates that the derived Satake correspondence established by Bezrukavnikov and Finkelberg can be interpreted as the equivalence of categories of loop operators in a pair of 4D topological gauge theories related by S-duality. One expects that 2-categories of surface operators and 3-categories of domain wall operators in 4D topological gauge theories are similarly equivalent; a better understanding of these could shed light on the quantum geometric Langlands duality.

Many interesting examples of domain walls and boundary conditions in N=4 super Yang-Mills theory have been constructed by Gaiotto and Witten in 2008; they also proposed a general prescription for mapping observables between S-dual theories. There was also some progress in extending the methods of Kapustin and Witten to Yang-Mills theories with N=2 supersymmetry. In particular, in 2008 Kapustin and Saulina studied the algebra of Wilson-'t Hooft operators using these methods. More generally, the study of loop and surface operators in topologically twisted N=2 super Yang-Mills theories appears to be an arena for fruitful interaction between physicists and representation theorists.

AdS-CFT correspondence and integrability

This was another major topic of the 2006 meeting, and the discussions there led to major advances in this area. Perhaps the most famous of these was a 2006 work of Beisert, Eden and Staudacher which conjectured an exact formula for the cusp anomalous dimension of Wilson loops in N=4 super Yang-Mills at arbitrary finite coupling. This built on years of effort by dozens of physicists to give the first exact results in four-dimensional gauge theory at finite coupling, and is considered a milestone in the field. Rapid progress in this area continues, and there is now a conjecture for how to obtain exact results for the FULL spectrum of anomalous dimensions of ALL operators in N=4 SYM. One must solve a so-called Y-system of equations, obtained from the thermodynamic Bethe ansatz applied to the N=4 integrable spin chain. This conjecture was formulated in a 2009 paper of Gromov-Kazakov-Vieira, who followed up with the numerical solution of Y-system for all values of the coupling for the most interesting short operator, the Konishi operator.

Integrability is also a key concept in understanding scattering amplitude calculations and their symmetries, as demonstrated in recent works of Alday-Maldacena, Drummond- Korchemsky- Sokatchev, and Beisert and collaborators. In a 2009 work of Drummond, Henn and Plefka, it was shown that by applying superconformal invariance to the tree level n-point scattering amplitudes, one obtains a Yangian symmetry. This is very surprising as previous constructions of the Yangian applied only to infinite spin chains, but here it arises in a finite system. This opens a new field of interesting activity.

Another interesting development in the AdS/CFT integrability is the discovery of a three-dimensional maximally supersymmetric Chern-Simons theory, which also appears to be integrable. This AdS/CFT integrability is the first example of the "exactly solvable'' 3D and 4D gauge theories with a nontrivial strong coupling regime, outside of the topological BPS sector. This revives the fading hopes of the existence of a true analytical approach to the realistic strongly coupled 4D gauge theories, such as QCD or the standard model.

Many of these areas have already attracted mathematicians. Links between the geometric Langlands correspondence and conformal field theory are a recognized and promising direction of joint work of mathematicians and physicists. Almost all topics of the SYM part of the project heavily involve integrable systems in a fundamental way.

MORE ON MAJOR LINES OF INTERACTION

There is an important conceptual connection between LP and CFT. The main object of the geometric Langlands program is the quotient G(O)\ G(A)/ G(K), where G is a reductive algebraic group, K the field of rational functions k(X) on a smooth projective curve X over the ground field k, A is the ring of adeles of K, and O the ring of integers in K. If k is the field of complex numbers, then G(A) is essentially a product of affine Kac-Moody groups, and the Langlands program in this case is closely related to the representation theory of affine Kac-Moody algebras, vertex algebras, and the Wess-Zumino-Witten model. The double coset space G(O)\ G(A)/ G(K) is nothing but the moduli stack of principal G-bundles on X in this case, which is the main geometric object in conformal field theories with G-symmetry.

On physics side, the Super Yang-Mills theory already proved to be very important to understand the geometric Langlands program and related matters (Witten, Kapustin, Gukov, Donagi and others). Technically, the Langlands correspondence appears at level of D=2, N=4 (D stays for the dimension of the theory, N for the "number" of super-symmetries). There are recent relations to the Satake correspondence by Bezrukavnikov and Finkelberg and other aspects of the theory of affine flag varieties.

The quantum geometric (local) Langlands program is a breakthrough development, which opens a road for alternative mathematical and physical approaches. For instance, the theory of the DAHA-Fourier transform and the so-called q,t-localization functor seem of real importance for the quantum Langlands program. Concerning DAHA, they also provide an important link between the real/complex and p-adic harmonic analysis and are hopefully connected with the theory of eigenvarieties.

Let us discuss some concrete aspects and approaches.

Q-Whittaker functions

One of the simplest examples of the local Langlands correspondence is the Shintani and Casselman-Shalika formula, which establishes a connection of the values of the p-adic Whittaker function for the Lie group G with the characters of its Langlands dual G'. Recently, the theory of these formulas was significantly developed by Gerasimov et.al. and Cherednik. Namely, certain special values of the q-Whittaker function appeared connected with the Demazure characters for the affine dual Lie group; the connection goes via the q,t-theory. When t is added to the theory, the Shintani-type special value formulas give the Macdonald polynomials.

The q-Whittaker functions are solutions of the q-Toda eigenvalue problem. They are directly related to the Givental-Lee theory and are expected to have other important applications in mathematics and physics. Almost certainly the quantum Langlands program is among such applications, which will be discussed during the conference.
The theory with t is, currently, beyond the existing setting of the quantum Langlands correspondence. There is growing evidence that DAHA and their degenerations may play an important role in the geometric Langlands program and related directions of the theory of affine flag varieties.

Affine Satake isomorphism

The Shintani-type formulas are closely connected with the geometric Satake isomorphism}, which is essentially the equivalence of tensor categories of the category of G'[[t]]-equivariant D-modules on the affine flag variety G'((t))/G'[[t]] and the category Rep (G) due to Drinfeld, Lusztig and others. Here G' is the Langlands dual of a semisimple complex Lie group G, the affine flag variety is defined as the space of all meromorphic loops divided by the holomorphic ones; Rep (G) is the tensor category of finite-dimensional representations of the Lie group G.

In the affine theory, the parameter q=exp{π i c} associated with the center charge c is added. Accordingly, the equivalence becomes between the categories of the c-twisted N'((t))-equivariant D-modules on the affine flag variety for the standard unipotent subgroup N' in G', and the category Rep_q (G) of finite-dimensional representations of the corresponding quantum group, the q-deformation of the universal enveloping algebra for G. It is due to Lurie and Gaitsgory; q is generic here. The most important part of the local Langlands correspondence is a connection of this claim and the one for 1/c. It requires using derived and triangular categories (and the language of 2-categories). Importantly, the quantum conjecture is much more symmetric than the non-quantum one. The original idea goes back to B. Feigin and Stoyanovski; there are important recent results by B. Feigin, E. Frenkel and others.

The geometric theory of the affine Satake isomorphism and related theory of affine Hall functions is closely connected with their algebraic theory based on DAHA. Namely, the algebraic Satake an isomorphism establishes an isopmophism between the spaces of DAHA-coinvariants

P-adic LP and Hecke algebras

The representation theory that arises in the p-adic LP involves the representations of p-adic reductive groups (e.g. GL_n(Q_p)) on p-adic topological vectors spaces. This representation theory has a hybrid flavor, involving aspects of both the classical smooth representation theory of p-adic reductive groups, with its combinatorial, Hecke-algebra-theoretic flavor, and more continuous aspects. For example, the composition series for the so-called locally analytic principal series seem to be governed by a combination of two different classical theories: the theory of composition series for smooth principal series (or equivalently, for modules over the affine Hecke algebra), and the theory of compositions series for Verma modules.

This suggests that there may be links between this p-adic representation theory and other types of representation theory that appear in the theory of DAHA, Kac-Moody algebras, and other aspects of representation theory related to geometric Langlands program. Investigating these possible links may suggest new techniques and points of view for studying all these various kinds of representation theory.

Verlinde algebras

The Conformal Field Theory is the "closest" to DAHA in the modern physics. The connection mainly goes through the Knizhnik-Zamolodchikov equations (and closely related Quantum Many-Body Problems). For this particular program, the following seems important.

The so-called logarithmic CFT is a certain variant of the usual CFT. It dramatically increases the number of applications. For instance, it is known or conjectured that the correlation functions of quite a few models (say, various types of dimer models) are described by the logarithmic CFT. Many of this models are connected with affine Hecke algebras and their degenerations. The following could be a perfect explanation.

Conjecturally, the Verlinde-type algebras corresponding to the logarithmic conformal field theory (they are non-semisimple commutative algebras, generally, with the action of the S,T-operators (Verlinde) can be described as quotients of the polynomial representation at roots of unity. Here DAHA serves both, the Kac-Moody (massive) and the Virasoro (massless) cases. The minimal models in the limit c → 1 are supposed to lead to an infinite dimensional Verlinde-type algebra. It may be the polynomial representation itself.

Bethe ansatz

According to Nekrasov and Shatashvili, the description of vacuum states for the theories a) D=2, b) D=3, c) D=4 are related to the following 3 variants of the coordinate Bethe Ansatz: a) rational (classical), b) trigonometric (or hyperbolic), and c) the one in terms of the elliptic functions. Correspondingly, these cases are connected to the following integrable chains: a) the rational Quantum Many-Body Problem (Calogero, Koornwinder and others), b) its trigonometric counterpart (Sutherland, Heckman, Opdam) and c) the elliptic one (Olshanetsky- Perelomov and others). All these integrable chains have difference counterparts (Macdonald, Ruijsenaars, Cherednik).

DAHA provide the best known tools for managing these six types of QMBP. The trigonometric BA is associated with the classification of the representations of unitary type at trivial center charge and at roots of unity.
There are some recent similar results on the interpretation of classical (rational) Bethe ansatz for the Yang particles (with the delta-function interaction) due to Emsiz, Opdam and Stokman. The correspondence between BA and QMBP, including the elliptic case, is an important line of recent research, both, in mathematics and physics.

WZWH-model

Recently the partition function of the Wess-Zumino-Witten model extended by the Higgs fields (i.e., extended by the Hitchin system) was considered thoroughly by Gerasimov and Shatashvili. This theory is connected with solutions of the Bethe-type equations depending on an extra parameter t, which fits well the DAHA theory. There are important recent developments concerning the "superpotententials" and related BA. Relations with the Langlands program were recently established by Nekrasov and Witten.

The Luminy-2006 conference stimulated the geometric Langlands program and related theory of Hecke algebras a great deal. The quantum LP is one of important mathematical recent developments. The recent progress in our understanding of (related) 2-categories and affine Grassmannians may be useful for the surface operators and boundary conditions in 4D gauge theory (Gaiotto, Gukov, Witten). The "elliptic" 6D theory attracts now a lot of attention.

This track is connected with other instances of Bethe Ansatz and Nested Bethe Ansatz (discussed above) and seems an important direction. The curves of arbitrary genus are included in the WZWH model, similar to the way they appear in the WZW model. It may give a valuable link from the local quantum geometric Langlands correspondence to the global one (for arbitrary curves).

The AdS-CFT correspondence

The N=4 SYM-theory is dual to the superstring theory in AdS₅ x S⁵ (Maldacena and many others). This is expected to lead to its asymptotic "integrability". Quite a few features of this theory are now associated with some structures of the integrable models (like chiral fields). One of the main mathematical tools here is the SU(2|2) dynamic S-matrix found by Beisert. Recently, he connected it with Shastry's S-matrix from the theory of the Hubbard model. The latter provides a fundamental example of the so-called Nested Bethe Ansatz. There are important recent developments by Kazakov et.al. toward integrability of the general theory (in the planar limit).

It is not impossible that (extended) SU(2|2) is connected with DAHA in small ranks (or their variants). For instance, the DAHA for A₁ is directly connected with the super Lie algebra osp(2|1). However, it is more likely that the relation to DAHA can be as follows. The Nested Bethe Ansatz is expected to be associated with reducible but non-decomposable representations of DAHA. The Bethe ansatz for the Hubbard model, a challenge for mathematicians, is expected to be of this type.

Some other related topics

Among other related developments, let us mention the Razumov-Stroganov conjecture (recently proved by Cantini and Sportiello) and related tiling problems, enumeration problems for plane partitions, and the theory of alternating sign matrices. There are results by Di Francesco, Zinn-Justin, Zuber and others. Here special solutions of the quantum Knizhnik-Zamolodchikov equation plays an important role; QKZ is one of the major topics in the theory of CFT and DAHA. The DAHA at cubic root of unity has quite a remarkable structure; which is certainly related to the generalizations of the Razumov-Stroganov conjecture.

The connections with the theory of cluster algebras with potentials with DAHA attaract a lot of attention, for instance, the recent proof by B. Keller and others the Zamolodchikov conjecture. Among other topics that may be represented let us mention Khovanov-Lauda-Rouquier theory (the categorization using quantum groups, Hecke algebras and similar objects) including recent developments due to Brundan and Kleshchev and the Fock-Goncharov results on the quantum theory of the Techmüller space.

Recent results by Braverman, Maulik and Okounkov demonstrate the conceptual relation of the KZ-type connections coming from AHA and DAHA with the standard connections on the quantum cohomology (Dubrovin and others) of the flag varieties. This is a part a general program, aimed at Nakajima varieties and other similar varieties, which allows one to associate a quantum R-matrix (with rational dependence of the parameter) and a shift operator with the multiplication in quantum cohomology.

SOME INTRODUCTORY TEXTS

[1] L. Baulieu, P. Di Francesco, M. Douglas, V. Kazakov, M. Picco, P. Windey (Eds.),
Strings, Branes and Dualities,
Proceedings of the NATO Advanced Study Institute, Nato Science Series C, V. 520
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