In this talk, I will introduce the chiral coordinate Bethe ansatz for anisotropic spin chains with periodic boundaries, including the XYZ, XY, and XX models. First, we construct a set of factorized chiral vectors with a fixed number of kinks, which form an invariant subspace of the Hilbert space. Next, we propose a modified coordinate Bethe ansatz method to solve the XYZ model, based on the action of the Hamiltonian on the chiral vectors. For the XY and XX models, we demonstrate that our Bethe ansatz yields all normalized eigenstates and the complete spectrum of the Hamiltonian. The differences between our approach and other methods are also discussed.

Brickwork circuits composed of the Yang-Baxter gates are integrable. It becomes an important tool to study the quantum many-body system out of equilibrium. I will talk about the properties of Yang-Baxter gates via the quantum information theory. We find that only certain two-qubit gates can be converted to the Yang-Baxter gates via the single-qubit gate operations. I will also talk about some possible extensions of the integrable circuits. Numerical analysis suggests that there is a broad class of circuits that are integrable, which are beyond the standard algebraic Bethe ansatz method.

Reference: [1] K. Zhang, K. Hao, K. Yu, V. Korepin, and W.-L. Yang, Geometric representations of braid and Yang-Baxter gates, J. Phys. A: Math. Theor. 57 445303, arXiv:2406.08320 (2024).

[2] K. Zhang, K. Yu, K. Hao, and V. Korepin, Optimal realization of Yang-Baxter gate on quantum computers, Adv. Quantum Technol. 2024, 2300345, arXiv:2307.16781 (2024).

Quantum non-integrability, or the absence of local conserved quantity, is a necessary condition for various empirical laws observed in macroscopic systems. Examples are thermal equilibration, the Kubo formula in linear response theory, and the Fourier law in heat conduction, all of which require non-integrability. From these facts, it is widely believed that integrable systems are highly exceptional, and non-integrability is ubiquitous in generic quantum many-body systems. Many numerical simulations also support this expectation. Nevertheless, conventional approaches in mathematical physics cannot address this belief, and establishing non-integrability of vast majority of generic quantum many-body systems is left as an open problem.

 In this study, we address this problem and provide an affirmative result for a wide class of quantum many-body systems. Precisely, we rigorously classify the integrability and non-integrability of all spin-1/2 chains with symmetric nearest-neighbor interactions [1]. Our classification demonstrates that except for the known integrable models, all systems are indeed non-integrable. This result provides a rigorous proof of the ubiquitousness of non-integrability, as well as the absence of undiscovered integrable systems which remains to be discovered. Moreover, it is proved that there is no partially integrable systems with finite number of local conserved quantities.

 In addition, recent extensions of non-integrability proofs to spin-1 systems [2] and others will be presented.

 [1] M. Yamaguchi, Y. Chiba, N. Shiraishi, ``Complete Classification of Integrability and Non-integrability for Spin-1/2 Chain with Symmetric Nearest-Neighbor Interaction,'' arXiv:2411.02162

 [2] A. Hokkyo, M. Yamaguchi, Y. Chiba, ``Proof of the absence of local conserved quantities in the spin-1 bilinear-biquadratic chain and its anisotropic extensions,'' arXiv:2411.04945 

Weight systems are functions on chord diagrams satisfying the so-called Vassiliev 4-term relations. They are closely related to finite-type knot invariants. Certain weight systems can be derived from graph invariants and Lie algebra.  

In a recent paper by M. Kazarian and the speaker, a recurrence for the Lie algebras $\mathfrak{so}(N)$ weight systems has been suggested; the recurrence allows one to construct the universal $\mathfrak{so}$ weight system. The construction is based on an extension of the $\mathfrak{so}$ weight systems to permutations.

Another recent paper, by M. Kazarian, N. Kodaneva, and the S. Lando, shows that under the specific substitution for the Casimir elements, the leading term in $N$ of the universal $\mathfrak{gl}$ weight system becomes the chromatic polynomial of the intersection graph of the chord diagram.

In this talk, we establish a similar result for the universal $\mathfrak{so}$ weight system. that is the leading term of the universal $\mathfrak{so}$ weight system also becomes the chromatic polynomial under a specific substitution.

The talk is based on a joint work arxiv: 2411.01128 with Sergei Lando.

We present unconventional constructions of Schur/Grothendieck polynomials from the viewpoint of quantum integrability. First, we present a construction of Schur/Grassmannian Grothendieck polynomials using a degeneration of higher rank rational/quantum R-matrices, which is different from the Bethe vector or Fomin-Kirillov type constructions.

Second, using the q=0 version of the three-dimensional $R$-matrix satisfying the tetrahedron equation introduced by Bazhanov-Sergeev and further studied by Kuniba-Maruyama-Okado, we show that a class of three-dimensional partition functions can be expressed as Schur polynomials. Keys of our derivation in both constructions are the multiple commutation relations between quantum algebras. 

Partly based on joint work with Shinsuke Iwao and Ryo Ohkawa.

Higher Bessel functions are the solutions to the quantum differential equations for $\mathbb{P}^{N-1}$. These functions are connected to the periods of the Dwork families via the Laplace transform, and the functions themselves are exponential integrals. In my talk, I will show how product formulas for these irregular special functions lead to other geometric differential equations associated with higher-dimensional families of algebraic varieties. I will discuss the geometric and algebraic properties of the periods for these families and later provide further perspectives on these correspondences.

Work in collaboration with V.Rubtsov and D. van Straten.

Isomonodromic tau functions have explicit expressions as sums of conformal blocks (or Nekrasov functions), so-called Kyiv formulas, found by Gamayun, Iorgov, Lisovyy. Zeros of these tau functions correspond to the situation when 2*2 isomonodromic problem becomes the quantum mechanical problem, e.g., with potential $\cosh x$. This way we get exact quantization conditions for the latter. Expansion around zero of the tau function is also worth studying, since its modular properties are well-defined and imply the so-called holomorphic anomaly equation for $E_2$ dependence of conformal block. 

The talk will be partially based on the papers https://arxiv.org/abs/2410.17868 and https://arxiv.org/abs/2105.00985.

The Deligne's category is a formal way to define an interpolation of the category of finite-dimensional representations of the Lie group $GL(n)$ to any complex number $n$. It is used in various constructions, which all together can be named as representation theory in complex rank. In the talk, I will present one of such constructions, namely, the interpolation of the algebra of higher Gaudin Hamiltonians (the Bethe algebra) associated with the Lie algebra $gl(n)$.

One can also interpolate monodromy-free differential operators of order $n$ desribing eigenvectors of Gaudin Hamiltonians, obtaining "monodromy-free" pseudo-differential operators. In joint work with L. Rybnikov and B. Feigin arXiv:2304.04501, we prove that the Bethe algebra in Deligne's category is isomorphic to the algebra of functions on certain pseudo-differential operators. Our work is motivated by the Bethe ansatz conjecture for the case of Lie superalgebra $gl(m|n)$. The conjecture says that eigenvectors in this case are described by ratios of differential operators of orders $m$ and $n$. We prove that such ratios are "monodromy-free" pseudo-differential operators.

The percolation problem provides one of the basic examples of phase transition and critical behavior manifested in the statistics of percolation clusters. The critical bond percolation model on a square lattice is closely related to the $O(1)$ dense loop model, which, in turn, can be mapped on the exactly solvable six-vertex model at special values of the Boltzmann weights, known as the Razumov-Stroganov combinatorial point. This point is known for providing the possibility  to obtain exact results in finite-size systems. I will review the latest results on calculating the exact densities of percolation clusters in critical percolation, as well as loops in the $O(1$) dense loop model on an infinite  cylinder of a finite circumference.

We prove that the supersymmetric deformed $\mathbb{CP}^1$ sigma model (the generalization of the Fateev-Onofri-Zamolodchikov model) admits an equivalent description as a generalized Gross-Neveu model. This formalism is useful for the study of renormalization properties and particularly for calculation of the one- and two-loop $\beta$-function. Remarkably we find new Nahm-type conditions, which guarantee renormalizability and supersymmetric invariance. We show that in the UV the superdeformed model flows to the super-Thirring CFT, for which we also develop a superspace approach. It is then demonstrated that the super-Thirring model is equivalent to a sigma model with the cylinder $\mathbb{R}\times S^1$ target space by an explicit computation of the correlation functions on both sides. Apart from that, we observe that the original model has another interesting conformal limit, given by the supercigar model, for which we also find a chiral dual and explicitly demonstrate agreement of the four-point functions on both sides. In addition, we investigate novel relations of our construction through mirror symmetry and dimensional reduction, which in the framework of $\sigma$-models on toric varieties maps to a class of $\cl{N}=2$ Liouville (Landau-Ginzburg class), as well as topological theories in higher $D$.

Recent success in the study of Baxter Q operators in Ruijsenaars hyperbolic system led to the establishment, besides of bispectral duality, of the duality concerning reflection of the coupling constant. It also gives a way to prove the orthogonality and completeness of the wave functions. The corresponding integral transform, defined for complex-valued parameters, can be regarded as a generalization of the Laplace transform. We prove an analog of the classical inversion formula and apply it for establishing L2 isomorphisms of Ruijsenaars spectral transform in 4 regimes of unitarity of the system. 

We review our results on bi-Hamiltonian structures of trigonometric spin Sutherland models built on collective spin variables. Our basic observation was that the cotangent bundle $T^*U(n)$ and its holomorphic analogue $T^*GL(n, \mathbb{C})$, as well as $T^* GL(n, \mathbb{C})_{\mathbb{R}}$, carry a natural quadratic Poisson bracket, which is compatible with the canonical linear one. The quadratic bracket arises by change of variables and analytic continuation from an associated Heisenberg double. Then, the reductions of $T^*U(n)$ and $T^*GL(n, \mathbb{C})$ by the conjugation actions of the corresponding groups lead to the real and holomorphic spin Sutherland models, respectively, equipped with a bi-Hamiltonian structure. The reduction of $T^* GL(n, \mathbb{C})_{\mathbb{R}}$ by the group $U(n) \times U(n)$ gives a generalized Sutherland model coupled to two $\mathfrak{u}(n)^*$-valued spins. We also show that a bi-Hamiltonian structure on the associative algebra $\mathfrak{gl}(n, \mathbb{R})$ that appeared in the context of Toda models can be interpreted as the quotient of compatible Poisson brackets on $T^*GL(n, \mathbb{R})$. Before our work, all these reductions were studied using the canonical Poisson structures of the cotangent bundles, without realizing the bi-Hamiltonian aspect.

We use the Hamiltonian reduction method to construct the Ruijsenaars dual systems to generalized Toda chains associated with the classical Lie algebras of types B, C, D. The dual systems turn out to be the B, C and D analogues of the rational Goldfish model, which is, as in the type A case, the strong coupling limit of rational Ruijsenaars systems. We explain how both types of systems emerge in the reduction of the cotangent bundle of a Lie group and provide the formulae for dual Hamiltonians. We compute explicitly the higher Hamiltonians of Goldfish models using the Cauchy-Binet theorem.

The subject of the talk is the delay version of the Painleve-I equation obtained as a delay periodic reduction of Shabat's dressing chain. We study the formal entire solutions to this equation and introduce a new family of interesting polynomials (called Bernoulli-Catalan polynomials). Using the recent results by Di Yang, Don Zagier and Youjin Zhang, we apply the theory of these polynomials to the problem of calculation of the Masur-Veech volumes of the moduli spaces of meromorphic quadratic differentials.   The talk is based on a joint work with John Gibbons and Alex Stokes.

Anti-self-dual Yang-Mills (ASDYM) equations have played important roles in quantum field theory (QFT), geometry and integrable systems for more than 50 years. In particular, instantons, global solutions of them, have revealed nonperturbative aspects of QFT ['t Hooft,...] and have given a new insight into the study of the four-dimensional geometry [Donaldson]. Furthermore, it is well known as the Ward conjecture that the ASDYM equations can be reduced to many integrable systems, such as the KdV eq. and Toda eq. Integrability aspects of them can be understood from the viewpoint of the twistor theory [Mason-Woodhouse,...]. The ASDYM equation is realized as the equation of motion of the four-dimensional Wess-Zumino-Witten (4dWZW) model in Yang's form. The 4dWZW model is analogous to the two dimensional WZW model and possesses aspects of conformal field theory and twistor theory [Losev-Moore-Nekrasov-Shatashvili,...].

On the other hand, 4d Chern-Simons (CS) theory has connections to many solvable models such as spin chains and principal chiral models [Costello-Witten-Yamazaki, ...]. These two theories (4dCS and 4dWZW) have been derived from a 6dCS theory like a ``double fibration'' [Costello, Bittleston-Skinner].

This suggests a nontrivial duality correspondence between the 4dWZW model and the 4dCS theory. We note that the Ward conjecture holds mostly in the split signature (+,+,−,−) and then the 4dWZW model describes the open N=2 string theory in the four-dimensional space-time. Hence a unified theory of integrable systems (6dCS-->4dCS/4dWZW) can be proposed in this context with the split signature.

In this talk, I would like to discuss integrability aspects of the ASDYM equation and construct soliton/instanton solutions of it by the Darboux/ADHM procedures, respectively. We calculate the 4dWZW action density of the solutions and found that the soliton solutions behaves as the KP-type solitons, that is, the one-soliton solution has localized action (energy) density on a 3d hyperplane in 4-dimensions (soliton wall) and the N-soliton solution describes N intersecting soliton walls with phase shifts. Our soliton solutions would describe a new-type of physical objects (3-brane) in the N=2 string theory. If time permits, I would mention reduction to lower-dimensions and extension to noncommutative spaces.

This talk is based on our works: [arXiv:2212.11800, 2106.01353, 2004.09248, 2004.01718] and forthcoming papers.

Applying the recently developed method-the off-diagonal Bethe ansatz method, we construct the exact solutions of the Heisenberg spin chain with various boundary conditions. The results allow us to calculate the boundary energy of the system in the thermodynamic limit. The method used here can be generalized to study the thermodynamic properties and boundary energy of other high rank models with non-diagonal boundary fields.

Many special functions appearing in the study of integrable systems have their finite field counterparts with extensive connections with number theory and algebraic geometry. For instance, it is well known that Gauss sums are finite field analogues of Gamma-functions and Kloosterman sums are finite field analogues of Bessel functions. In this talk I will present a new approach of studying certain special functions over finite fields using representation theory of finite Chevalley groups. Namely, I will first define finite field analogues of Gamma-functions and Whittaker functions and then identify them as matrix elements of representations of (subgroups of) general linear groups over a finite field and compare them with their counterparts defined over real groups.

For any abelian variety X with an action of a finite complex reflection group W, Etingof, Felder, Ma and Veselov constructed a family of integrable systems on T^*X. When X is a product of n copies of an elliptic curve E and W=S_n, this reproduces the usual elliptic Calogero­­-Moser system. Recently, together with Philip Argyres (Cincinnati) and Yongchao Lü (KIAS), we proposed that many of these integrable systems at the classical level can be interpreted as Seiberg­­-Witten integrable systems of certain super­symmetric quantum field theories. I will describe our progress in understanding this connection for the case X=E^n where E is an elliptic curve with the symmetry group Z_m (of order m=2,3,4,6), and W is the wreath product of Z_m and S_n. I will mostly talk about n=1 case, which is already rather interesting. Based on: arXiv 2309.12760.

The tunneling effect predicted by B.Josephson (Nobel Prize, 1973) concerns the Josephson junction: two superconductors separated by a arrow dielectric. It states existence of a supercurrent through it and equations governing it. The overdamped Josephson junction is modeled by a family of differential equations on 2-torus depending on 3 parameters:  $B$ (abscissa), $A$ (ordinate), $\omega$ (frequency). We study its rotation number $\rho(B, A; \omega)$ as a function of  $(B, A)$ with fixed $\omega$. The phase-lock areas are those level sets $L_r:=\{\rho=r\}$ that have non-empty interiors. They exist only for integer rotation number values $r$: this is the rotation number quantization effect discovered by Buchstaber, Karpov and Tertychnyi. They are analogues of the famous Arnold tongues. Each $L_r$ is an infinite chain  of  domains going vertically to infinity and separated by points called constrictions (expect for those with $A=0$). See the phase-lock area portraits for $\omega=2$, 1,  0.3 at the presentation.

We show that: 1)  all constrictions in $L_r$ lie in the vertical line $\{ B=\omega r\}$;  2)  each constriction is positive, that is, some its punctured neighborhood in the vertical line lies in $\operatorname{Int}(L_r)$. These results, obtained in collaboration with Yulia Bibilo, confirm experiences of physicists (pictures from physical books of 1970-th) and two mathematical conjectures.

The proof  uses an equivalent description of model by linear systems of differential equations on $\oc$ (found by Buchstaber, Karpov and Tertychnyi), their isomonodromic deformations described by Painleve 3 equations and methods of the theory of slow-fast systems. If the time allows we will discuss new results and open questions.

In this talk, we mainly introduce some background of the structure, the representation and the quantization of the generalized intersection matrix algebras. Then we introduce a result on finite dimensional modules over indefinite type Kac-Moody Lie algebras. It is given in a joint work with Hongmei Hu and Yilan Tan.

The prominent role of matrix models in physics and mathematics is well known. It is especially interesting that some of those models are exactly solvable, meaning the one can find explicit formulas for correlation functions. This phenomenon has also been called superintegrability of matrix models. I will present some recent attempt to study it systematically and search for its algebraic origins. It leads to an interesting connection with the rapidly developing field of BPS algebras and their representations.

We discuss several results on work in collaboration (with V.Mishnyakov, A.Popolitov, F.Liu, R.Wang and with B. Kang, K.Wu, W.Z. Zhao). We construct W-representations for multi-character expansions, which involve a generic number of sets of time variables. We propose integral representations for such kind of partition functions which are given by tensor models and multi-matrix models with multi-trace couplings. In addition, we present the W-representation for a two-tensor model with order-3. We derive the compact expressions of correlators from the W-representation, and analyze the free energy in the large N limit. By establishing the correspondence between the two-color Dyck order in Fredkin spin chain and the tree operator on the ring, we prove that the entanglement scaling of Fredkin spin chain beyond the logarithmic scaling in ordinary critical systems.

The topological vertex, developed by Aganagic, Klemm, Marino and Vafa, provides an explicit algorithm to compute the open Gromov-Witten invariants of smooth toric Calabi-Yau threefolds in mathematics, as well as the A-model topological string amplitudes in physics. In this talk, I will introduce our recent work on the connection between the topological vertex and multi-component KP hierarchy.

This talk is based on a joint work with Zhiyuan Wang and Jian Zhou.

BCOV’s Feynman rule is a conjectural algorithm used to compute the higher genus Gromov-Witten invariants of Calabi-Yau threefolds. The Feynman graph that appears in BCOV’s rule can be interpreted as a form of geometric quantization. In this presentation, I will attempt to extract it from the A-model perspective and realize it as Givental’s R-matrix quantization action. Finally, I will explain how mixed field theory applies to this quantization formalism of the Feynman rule. This talk is based on a series of joint works with H.-L. Chang, J. Li, W.-P. Li, and Y. Zhou.

In the talk, I shall present our joint paper with A.Gerasimov and D.Lebedev. In this paper, we develop a representation theory approach to the study of generalized hypergeometric functions of Gelfand, Kapranov and Zelevisnky (GKZ). We show that the GKZ hypergeometric functions may be identified with matrix elements of non-reductive Lie algebras L(N) of oscillator type. The Whittaker functions associated with principal series representations of gl(n,R) being special cases of GKZ hypergeometric functions, thus admit along with a standard matrix element representations associated with reductive Lie algebra gl(n,R), another matrix element representation in terms of L(n(n-1)).

Weight systems, which are functions on chord diagrams satisfying certain 4-term relations, appear naturally in Vassiliev's theory of nite type knot invariants. In particular, a weight system can be constructed from any nite dimensional Lie algebra endowed with a nondegenerate invariant bilinear form. Recently, M. Kazarian suggested to extend the gl(N)-weight system from chord diagrams (treated as involutions without fixed point) to arbitrary permutations, which led to a recurrence formula allowing for an effective computation of its values, elaborated by Zhuoke Yang. In turn, the recurrence helped to unify the gl(N) weight systems, for N = 1, 2, 3, . . . , into a universal gl-weight system. The latter takes values in the ring of polynomials C[N][C1, C2, . . . ] in finitely many variables C1, C2, . . . (Casimir elements), whose coefficients are polynomials in N. The universal gl-weight system carries a lot of information about chord diagrams and intersection graphs. The talk will address the question which graph invariants can be extracted from it. We will discuss the interlace polynomial, the enhanced skew-characteristic polynomial, and the chromatic polynomial. In particular, we show that the interlace polynomial of the intersection graphs can be obtained by a specific substitution for the variables N, C1, C2, . . . . This allows one to extend it from chord diagrams to arbitrary permutations. Questions concerning other graph and delta-matroid invariants and their presumable extensions will be formulated. The talk is based on a work of the speaker and a PhD student Nadezhda Kodaneva.

Björner and Ekedahl [Ann. of Math. (2), 170(2): 799-817, 2009] pioneered the study of length-enumerating sequences associated with parabolic lower Bruhat intervals in crystallographic Coxeter groups. In this talk, we study the asymptotic behavior of these sequences in affine Weyl groups. We prove that the length-enumerating sequences associated with the dominant intervals corresponding to a dominant coroot lattice element are ``asymptotically'' log-concave. More precisely, we prove that a certain sequence of discrete measures naturally constructed from the length-enumerating sequences converges weakly to a continuous measure constructed from a certain polytope. Moreover, a certain sequence of step functions naturally constructed from the length-enumerating sequences uniformly converges to the density function of that continuous measure, which implies the weak convergence and that the sequences of numbers of elements in each layer of the dilated dominant interval converge to a sequence of volumes of hyperplane sections of the polytope. By the Brunn–Minkovski inequality, the density function is log-concave. Our approach relies on the ``dominant lattice formula'', which yields a new bridge between the discrete nature of Betti numbers of parabolic affine Schubert varieties and the continuous nature of the geometry of convex polytopes. Our technique can be seen as a refinement in our context of the classical Ehrhart's theory relating the volume of a polytope and the number of lattice points the polytope contains, by replacing the volume by volumes of transversal sections and the number the total lattice points by the number of lattice points of a given length. Joint with Gaston Burrull and Hongsheng Hu.