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Usual approaches to quantisation of a 3-Toda lead to numerical calculations requiring many steps that can be time consuming to insure their reliability. In order to reduce as much as possible the numerical part of the EKB quantisation procedure, and then to ease numerical calculations, we propose a reformulation of the mathematical framework with more adapted variables. The resulting equations and procedure might be easily implemented in a short Mathematica code. This more explicit framework will be useful for studying quantum Toda-Bianchi IX models in quantum cosmology where the true Bianchi IX anisotropy Hamiltonian can be approximated by a 3-particle Toda system., submitted Eur. Phys. J. C

We perform a systematic study of integrated four-point functions of half-BPS operators in four-dimensional $\mathcal{N}=4$ super Yang-Mills theory with gauge group $SU(N)$. These observables, defined by a certain spacetime integral of $\langle\mathcal{O}_2\mathcal{O}_2\mathcal{O}_p\mathcal{O}_p\rangle$ where $\mathcal{O}_p$ is a superconformal primary of charge $p$, are known to be computable by supersymmetric localization, yet are non-trivial functions of the complexified gauge coupling $\tau$. We find explicit and remarkably simple results for several classes of these observables, exactly as a function of $N$ and $\tau$. Their physical and formal properties are greatly illuminated upon employing the $SL(2,\mathbb{Z})$ spectral decomposition: in this S-duality-invariant eigenbasis, the integrated correlators are fixed simply by polynomials in the spectral parameter. These polynomials are determined recursively by linear algebraic equations relating different $N$ and $p$, such that all integrated correlators are ultimately fixed in terms of the integrated stress tensor multiplets in the $SU(2)$ theory. Our computations include the full matrix of integrated correlators at low values of $p$, and a certain infinite class involving operators of arbitrary $p$. The latter satisfy an open lattice chain equation for all $N$, reminiscent of the Toda equation obeyed by extremal correlators in $\mathcal{N}=2$ superconformal theories. We compute ensemble averages of these observables and analyze our solutions at large $N$, confirming and predicting features of semiclassical AdS$_5\, \times$ S$^5$ supergravity amplitudes., Comment: 42+24 pages

We derive a large-scale hydrodynamic equation, including diffusive and dissipative effects, for systems with generic static position-dependent driving forces coupling to local conserved quantities. We show that this equation predicts entropy increase and thermal states as the only stationary states. The equation applies to any hydrodynamic system with any number of local, parity and time-symmetric conserved quantities, in arbitrary dimension. It is fully expressed in terms of elements of an extended Onsager matrix. In integrable systems, this matrix admits an expansion in the density of excitations. We evaluate exactly its two-particle–hole contribution, which dominates at low density, in terms of the scattering phase and dispersion of the quasiparticles, giving a lower bound for the extended Onsager matrix and entropy production. We conclude with a molecular dynamics simulation, demonstrating thermalisation over diffusive time scales in the Toda interacting particle model with an inhomogeneous energy field.

International audience; We discuss exactly solvable systems involving integrals of motion with higher powers of momenta. If one of these integrals is chosen for the Hamiltonian, we obtain a higher-derivative system involving ghosts, i.e. a system whose Hamiltonian is not bounded neither from below, nor from above. However, these ghosts are benign: there is no collapse and unitarity is not violated. As an example, we consider the 3-particle Toda periodic chain, with the cubic invariant I chosen for the Hamiltonian. The classical trajectories exhibit regular oscillations, and the spectrum of the quantum Hamiltonian is discrete running from minus to plus infinity. We also discuss the classical dynamics of a perturbed system with the Hamiltonian H = I + v, where v is an oscillator potential. Such a system is not exactly solvable, but its classical trajectories exhibit not regular, but still benign behaviour without collapse. This means that also the corresponding quantum problem is well defined. The same observation can be made for exactly solvable (1+1)-dimensional field theories involving an infinite number of conservation laws: any of them can be chosen for the Hamiltonian. We illustrate this for the Sine-Gordon and KdV models. In the latter case, the Lagrangian and standard integrals of motion involve higher spatial rather than temporal derivatives. But one can always interchange x and t, after which we obtain a system with benign ghosts.

Communications in mathematical physics -, - (2019). doi:10.1007/s00220-019-03617-y, This paper investigates the relations between the Toda conformal field theories,quantum group theory and the quantisation of moduli spaces of flat connections.We use the free field representation of theW-algebras to define natural bases for spacesof conformal blocks of the Toda conformal field theory associated to the Lie algebrasl3 on the three-punctured sphere with representations of generic type associated to thethree punctures. The operator-valued monodromies of degenerate fields can be used todescribe the quantisation of the moduli spaces of flat SL(3)-connections. It is shown thatthe matrix elements of the monodromies can be expressed as Laurent polynomials ofmore elementary operatorswhich have a simple definition in the free field representation.These operators are identified as quantised counterparts of natural higher rank analogsof the Fenchel–Nielsen coordinates from Teichmüller theory. Possible applications tothe study of the non-Lagrangian SUSY field theories are briefly outlined., Published by Springer, Heidelberg

Exact field theory dualities can be implemented by duality domain walls such that passing any operator through the interface maps it to the dual operator. This paper describes the S-duality wall of four-dimensional ${\cal N}=2$ SU(N) SQCD with 2N hypermultiplets in terms of fields on the defect, namely three-dimensional ${\cal N}=2$ SQCD with gauge group U(N-1) and 2N flavours, with a monopole superpotential. The theory is self-dual under a duality found by Benini, Benvenuti and Pasquetti, in the same way that T[SU(N)] (the S-duality wall of ${\cal N}=4$ super Yang-Mills) is self-mirror. The domain-wall theory can also be realized as a limit of a USp(2N-2) gauge theory; it reduces to known results for N=2. The theory is found through the AGT correspondence by determining the braiding kernel of two semi-degenerate vertex operators in Toda CFT., v2: correct superpotential to fix symmetries, better introduction and references, 42 pages

In this note, we establish several interesting connections between the supergroup gauge theories and the super integrable systems, i.e. gauge theories with supergroups as their gauge groups and integrable systems defined on superalgebras. In particular, we construct the super-characteristic polynomials of super-Toda lattice and elliptic double Calogero-Moser system by considering certain orbifolded instanton partition functions of their corresponding supergroup gauge theories. We also derive an exotic generalization of sl(2) XXX spin chain arising from the instanton partition function of SQCD with supergauge group, and study its Bethe ansatz equation., Comment: 39 pages, 2 tables, fixed typo

We consider general Darboux maps arising from intertwining relations on second order, linear partial differential operators, as deformations of the classical, Laplace case. We present Lax pairs for the corresponding relations on invariants and discuss the conditions for a lattice structure analogous to 2D Toda theory.

We investigate Fuchsian equations arising in the context of 2-dimensional conformal field theory (CFT) and we apply the Katz theory of Fucshian rigid systems to solve some of these equations. We show that the Katz theory provides a precise mathematical framework to answer the question whether the fusion rules of degenerate primary fields are enough for determining the differential equations satisfied by their correlation functions. We focus on the case of W3 Toda CFT: we argue that the differential equations arising for four-point conformal blocks with one n-th level semi-degenerate field and a fully-degenerate one in the fundamental sl3 representation are associated to Fuchsian rigid systems. We show how to apply Katz theory to determine the explicit form of the differential equations, the integral expression of solutions and the monodromy group representation. The theory of twisted homology is also used in the analysis of the integral expression. This approach allows to construct the corresponding fusion matrices and to perform the whole bootstrap program: new explicit factorization of W3 correlation functions as well shift relations between structure constants are also provided., Comment: 49 pages

We give a microscopic two dimensional ${\cal N}=(2,2)$ gauge theory description of arbitrary M2-branes ending on $N_f$ M5-branes wrapping a punctured Riemann surface. These realize surface operators in four dimensional ${\cal N}=2$ field theories. We show that the expectation value of these surface operators on the sphere is captured by a Toda CFT correlation function in the presence of an additional degenerate vertex operator labelled by a representation ${\cal R}$ of $SU(N_f)$, which also labels M2-branes ending on M5-branes. We prove that symmetries of Toda CFT correlators provide a geometric realization of dualities between two dimensional gauge theories, including ${\cal N}=(2,2)$ analogues of Seiberg and Kutasov--Schwimmer dualities. As a bonus, we find new explicit conformal blocks, braiding matrices, and fusion rules in Toda CFT., 70 pages + appendices. v2: references added. v3: correct theta angle shifts to match the version published in 2016

The Toda flow is a Hamiltonian system which evolves on the dual of the Borel subalgebra of a complex Lie algebra g. The dual of the Borel subalgebra can be identified with an affine subspace of its negative plus the element given by the sum of the simple root vectors in g. The system has been proven completely integrable in the Liouville sense on a generic coadjoint orbit for the Borel subgroup. This paper gives a verification of integrability of the Toda flow on classical simple Lie algebras and describes a method for the construction of a complete collection of integrals of motion for each. After this description, an implementation of the outlined procedures is given in the Maple programming environment, together with explicit examples, demonstrating both the accuracy of the procedure and the efficacy of the Maple programming code.

We present a new matrix-valued isospectral ordinary differential equation that asymptotically block-diagonalizes $n\times n$ zero-diagonal Jacobi matrices employed as its initial condition. This o.d.e.\ features a right-hand side with a nested commutator of matrices, and structurally resembles the double-bracket o.d.e.\ studied by R.W.\ Brockett in 1991. We prove that its solutions converge asymptotically, that the limit is block-diagonal, and above all, that the limit matrix is defined uniquely as follows: For $n$ even, a block-diagonal matrix containing $2\times 2$ blocks, such that the super-diagonal entries are sorted by strictly increasing absolute value. Furthermore, the off-diagonal entries in these $2\times 2$ blocks have the same sign as the respective entries in the matrix employed as initial condition. For $n$ odd, there is one additional $1\times 1$ block containing a zero that is the top left entry of the limit matrix. The results presented here extend some early work of Kac and van Moerbeke., Comment: 19 pages, 3 figures, conjecture from previous version is added as assertion (iv) of the main theorem including a proof; other major changes

Peroxysome proliferator activated receptor coactivator-1 gene (PPARGC1A) is a positional and functional candidate gene for milk fat yield. It has key role in energy, fat and glucose metabolism. Single nucleotide polymorphisms (SNPs) in Exon-8 of PPARGC1A are reported to be associated with milk fat yield in dairy cattle. In the present investigation PPARGC1A was partially amplified (around 767bp) by designing gene specific primer and confirm by sequencing the amplicon and its comparison with the PPARGC1A gene of bovine. Comparative study of PPARGC1A among different breeds of buffaloes reveals different level of mutations with respect to its gene sequence 0.013-1.69% and protein sequence 0.42% to 2.99%, Similarly the protein structures modeled from their sequences were compared by structural superposition that shows variations (RMSD) from 0.736 to 1.507. Furthermore, the sequences were used to generate a dendrogram. It reveals that Murrah and reference are very close to each other, similarly Toda, Bhadawari and Surti are closely related, whereas Pandharpuri is separated from both the cluster. Especially the variations are more at the binding site of this protein that may be the cause that different breeds have different percentage of milk fat. Further study is underway to detect polymorphism and associate them with milk fat related traits in buffalo.

The Evolution of the Toda Habitation in the Nilgiri Hills (Tamil Nadu) : Back to its Sources. The Toda are a small community of traditional pastoralists (about 1 200), who have turned progressively to farming for the last 25 years. Their hamlets, known under the name of mund or mod are widely scattered on the heights of the Nilgiri Hills (between 2 000 and 2 400 m) in Southern India. Hamlets are the center of tribal life. They depict human activity, relationships and the environment. The traditional Toda habitation played an important role - and still does - in regrouping, expressing and teaching, in a metaphorical way, the values and principles concerned with planning and the relation to space., Les Toda sont une petite communauté (environ 1 200), de pasteurs traditionnels qui se sont tournés progressivement depuis 25 ans vers l’agriculture. Leurs hameaux, connus sous le nom de mund ou mod sont disséminés largement sur les prairies tempérées des hauteurs des Nilgiri (entre 2 000 et 2 400 m) en Inde du Sud. Le hameau est le lieu de vie de la tribu. Il reflète l’activité, les relations humaines et le milieu naturel. La maison traditionnelle toda avait et a encore, dans une dimension transformée, mission de concentrer, d’exprimer, d’enseigner les principes ou les valeurs liés à l’aménagement et à l’articulation à l’espace., Chapoullié Véronique. L’évolution de la maison toda dans le massif des Nilgiris (Tamil Nadu) ou retour à ses origines ?. In: Les montagnes tropicales : identités, mutations, développement. Table-Ronde, Bordeaux-Pessac, 27 et 28 novembre 1998. Bordeaux : Presses Universitaires de Bordeaux, 2001. pp. 479-488. (Espaces tropicaux, 16)

The KP and 2D Toda tau-functions of hypergeometric type that serve as generating functions for weighted single and double Hurwitz numbers are related to the topological recursion programme. A graphical representation of such weighted Hurwitz numbers is given in terms of weighted constellations. The associated classical and quantum spectral spectral curves are derived, and these are interpreted combinatorially in terms of the graphical model. The pair correlators are given a finite Christoffel-Darboux representation and determinantal expressions are obtained for the multipair correlators. The genus expansion of the multicurrent correlators is shown to provide generating series for weighted Hurwitz numbers of fixed ramification profile lengths. The WKB series for the Baker function is derived and used to deduce the loop equations and the topological recursion relations in the case of polynomial weight functions., 77 pages, references added, several typos corrected, clarification added on nonpolynomial weight generating function and examples

Multiparametric families of hypergeometric $\tau$-functions of KP or Toda type serve as generating functions for weighted Hurwitz numbers, providing weighted enumerations of branched covers of the Riemann sphere. A graphical interpretation of the weighting is given in terms of constellations mapped onto the covering surface. The theory is placed within the framework of topological recursion, with the Baker function at ${\bf t} ={\bf 0}$ shown to satisfy the quantum spectral curve equation, whose classical limit is rational. A basis for the space of formal power series in the spectral variable is generated that is adapted to the Grassmannian element associated to the $\tau$-function. Multicurrent correlators are defined in terms of the $\tau$-function and shown to provide an alternative generating function for weighted Hurwitz numbers. Fermionic VEV representations are provided for the adapted bases, pair correlators and multicurrent correlators. Choosing the weight generating function as a polynomial, and restricting the number of nonzero "second" KP flow parameters in the Toda $\tau$-function to be finite implies a finite rank covariant derivative equation with rational coefficients satisfied by a finite "window" of adapted basis elements. The pair correlator is shown to provide a Christoffel-Darboux type finite rank integrable kernel, and the WKB series coefficients of the associated adjoint system are computed recursively, leading to topological recursion relations for the generators of the weighted Hurwitz numbers., Comment: 30 pages, 2 figures. References updated. Weighting of constellations revised. Weighting of Hurwitz numbers corrected