Sklyanin Bracket Research Articles

Comments on all-loop constraints for scattering amplitudes and Feynman integrals

We comment on the status of “Steinmann-like” constraints, i.e. all-loop constraints on consecutive entries of the symbol of scattering amplitudes and Feynman integrals in planar mathcal{N} = 4 super-Yang-Mills, which have been crucial for the recent progress of the bootstrap program. Based on physical discontinuities and Steinmann relations, we first summarize all possible double discontinuities (or first-two-entries) for (the symbol of) amplitudes and integrals in terms of dilogarithms, generalizing well-known results for n = 6, 7 to all multiplicities. As our main result, we find that extended-Steinmann relations hold for all finite integrals that we have checked, including various ladder integrals, generic double-pentagon integrals, as well as finite components of two-loop NMHV amplitudes for any n; with suitable normalization such as minimal subtraction, they hold for n = 8 MHV amplitudes at three loops. We find interesting cancellation between contributions from rational and algebraic letters, and for the former we have also tested cluster-adjacency conditions using the so-called Sklyanin brackets. Finally, we propose a list of possible last-two-entries for MHV amplitudes up to 9 points derived from overline{Q} equations, which can be used to reduce the space of functions for higher-point MHV amplitudes.

Separation of variables for quadratic algebras: Algebras of Maillet and reflection-equation algebras

We consider the problem of variable separation for the classical integrable Hamiltonian systems possessing gl(n)-valued Lax matrices satisfying quadratic Poisson brackets of Freidel and Maillet [Phys. Lett. B 262(2-3), 278 (1991)] with spectral-parameter dependent a-b-c-d tensors. We formulate, in terms of the corresponding a-b-c-d tensors, a sufficient condition that guarantees that the separating functions A(u), B(u) of Sklyanin [Commun. Math. Phys. 150(1), 181 (1992)], Scott [e-print arXiv:hep-th 940303 (1994)], and Gekhtman [Commun. Math. Phys. 167, 593 (1995)] produce canonical coordinates. We consider an important subcase of a-b-c-d algebras, namely, the case of classical reflection equation algebras [E. Sklyanin, J. Phys. A: Math. Gen. 21(10), 2357 (1988)], and formulate, in terms of the corresponding r-s-matrices, the analogous sufficient condition that guarantees the canonicity of the constructed coordinates. For the case s = 0, we recover our previous results on the variable separation for quadratic Sklyanin brackets [B. Dubrovin and T. Skrypnyk, J. Math. Phys. 60, 093506 (2019)]. We consider an example of gl(n) ⊗ gl(n)-valued trigonometric a-b-c-d tensors that satisfy the considered condition and find a class of Lax matrices for them for which the obtained set of canonical coordinates is complete.

A Family of GL<sub>r</sub> Multiplicative Higgs Bundles on Rational Base

In this paper we study a restricted family of holomorphic symplectic leaves in the Poisson-Lie group ${\rm GL}_r(\mathcal{K}_{\mathbb{P}^1_x})$ with rational quadratic Sklyanin brackets induced by a one-form with a single quadratic pole at $\infty \in \mathbb{P}_{1}$. The restriction of the family is that the matrix elements in the defining representation are linear functions of $x$. We study how the symplectic leaves in this family are obtained by the fusion of certain fundamental symplectic leaves. These symplectic leaves arise as minimal examples of (i) moduli spaces of multiplicative Higgs bundles on $\mathbb{P}^{1}$ with prescribed singularities, (ii) moduli spaces of $U(r)$ monopoles on $\mathbb{R}^2 \times S^1$ with Dirac singularities, (iii) Coulomb branches of the moduli space of vacua of 4d $\mathcal{N}=2$ supersymmetric $A_{r-1}$ quiver gauge theories compactified on a circle. While degree 1 symplectic leaves regular at $\infty \in \mathbb{P}^1$ (Coulomb branches of the superconformal quiver gauge theories) are isomorphic to co-adjoint orbits in $\mathfrak{gl}_{r}$ and their Darboux parametrization and quantization is well known, the case irregular at infinity (asymptotically free quiver gauge theories) is novel. We also explicitly quantize the algebra of functions on these moduli spaces by presenting the corresponding solutions to the quantum Yang-Baxter equation valued in Heisenberg algebra (free field realization).

The Sklyanin bracket and cluster adjacency at all multiplicity

We argue that the Sklyanin Poisson bracket on Gr(4, n) can be used to efficiently test whether an amplitude in planar mathcal{N} = 4 supersymmetric Yang-Mills theory satisfies cluster adjacency. We use this test to show that cluster adjacency is satisfied by all one- and two-loop MHV amplitudes in this theory, once suitably regulated. Using this technique we also demonstrate that cluster adjacency implies the extended Steinmann relations at all particle multiplicities.

Classical r-matrices of real low-dimensional Jacobi–Lie bialgebras and their Jacobi–Lie groups

In this paper, we obtain the classical [Formula: see text]-matrices of real two- and three-dimensional Jacobi–Lie bialgebras. In this way, we classify all non-isomorphic real two- and three-dimensional coboundary Jacobi–Lie bialgebras and their types (triangular and quasi-triangular). Also, we obtain the generalized Sklyanin bracket formula by use of which, we calculate the Jacobi structures on the related Jacobi–Lie groups. Finally, we present a new method for constructing classical integrable systems using coboundary Jacobi–Lie bialgebras.

Poisson Yang-Baxter maps with binomial Lax matrices

A construction of multidimensional parametric Yang-Baxter maps is presented. The corresponding Lax matrices are the symplectic leaves of first-degree matrix polynomials equipped with the Sklyanin bracket. These maps are symplectic with respect to the reduced symplectic structure on these leaves and provide examples of integrable mappings. An interesting family of quadrirational symplectic YB maps on C4×C4 with 3 × 3 Lax matrices is also presented.

Entwining Yang-Baxter maps and integrable lattices

Yang–Baxter (YB) map systems (or set-theoretic analoga of entwining YB structures) are presented. They admit zero curvature representations with spectral parameter depended Lax triples L1, L2, L3 derived from symplectic leaves of 2 � 2 binomial matrices equipped with the Sklyanin bracket. A unique factorization condition of the Lax triple implies a 3-dimensional compatibility property of these maps. In case L1 = L2 = L3 this property yields the set-theoretic quantum Yang-Baxter equation, i.e. the YB map property. By considering periodic ‘staircase’ initial value problems on quadrilateral lattices, these maps give rise to multidimensional integrable mappings which preserve the spectrum of the corresponding monodromy matrix.

Yang–Baxter maps with first-degree-polynomial 2 × 2 Lax matrices

A family of non-parametric Yang–Baxter (YB) maps is constructed by re-factorization of 2 × 2 matrix polynomials of first degree. These maps are Poisson with respect to the Sklyanin bracket. For each Casimir function a parametric Poisson YB map is generated by reduction on the corresponding level set. By considering a complete set of Casimir functions symplectic multi-parametric YB maps are derived. These maps are quadrirational with explicit formulae in terms of matrix operations. Their Lax matrices are, by construction, 2 × 2 first-degree-polynomial in the spectral parameter and are classified by THE Jordan normal form of the leading term. Degenerate parametric YB maps constructed as limits of the non-degenerate ones are connected to known integrable systems on quadgraphs.

10.1007/s11819-008-1005-1

The bi-hamiltonian structures for the Goryachev-Chaplygin top are constructed by using the Chaplygin variables and the Sklyanin bracket.

A family of the Poisson brackets compatible with the Sklyanin bracket

We introduce a family of compatible Poisson brackets on the space of 2 × 2 polynomial matrices, which contains the Sklyanin bracket, and use it to derive a multi-Hamiltonian structure for a set of integrable systems that includes the XXX Heisenberg magnet, the open and periodic Toda lattices, the discrete self-trapping model and the Goryachev–Chaplygin gyrostat.

Classification of real three-dimensional Lie bialgebras and their Poisson–Lie groups

Classical r-matrices of the three-dimensional real Lie bialgebras are obtained. In this way all three-dimensional real coboundary Lie bialgebras and their types (triangular, quasitriangular or factorizable) are classified. Then, by using the Sklyanin bracket, the Poisson structures on the related Poisson-Lie groups are obtained.

Commutative Poisson Subalgebras for Sklyanin Brackets and Deformations of Some Known Integrable Models

We construct hierarchies of commutative Poisson subalgebras for Sklyanin brackets. Each of the subalgebras is generated by a complete set of integrals in involution. Some new integrable systems and schemes for separation of variables for them are elaborated using various well-known representations of the brackets. The constructed models include deformations for the Goryachev–Chaplygin top, the Toda chain, and the Heisenberg model.

Becchi–Rouet–Stora–Tyutin formalism and zero locus reduction

In the Becchi–Rouet–Stora–Tyutin (BRST) quantization of gauge theories, the zero locus ZQ of the BRST differential Q carries an (anti)bracket whose parity is opposite to that of the fundamental bracket. Observables of the BRST theory are in a 1:1 correspondence with Casimir functions of the bracket on ZQ. For any constrained dynamical system with the phase space N0 and the constraint surface Σ, we prove its equivalence to the constrained system on the BFV-extended phase space with the constraint surface given by ZQ. Reduction to the zero locus of the differential gives rise to relations between bracket operations and differentials arising in different complexes (the Gerstenhaber, Schouten, Berezin–Kirillov, and Sklyanin brackets); the equation ensuring the existence of a nilpotent vector field on the reduced manifold can be the classical Yang–Baxter equation. We also generalize our constructions to the bi-QP manifolds which from the BRST theory viewpoint correspond to the BRST–anti-BRST-symmetric quantization.

(1+1) Schrödinger Lie bialgebras and their Poisson-Lie groups

All Lie bialgebra structures for the (1+1)-dimensional centrally extended Schrodinger algebra are explicitly derived and proved to be of the coboundary type. Therefore, since all of them come from a classical r-matrix, the complete family of Schrodinger Poisson-Lie groups can be deduced by means of the Sklyanin bracket. All possible embeddings of the harmonic oscillator, extended Galilei and gl(2) Lie bialgebras within the Schrodinger classification are studied. As an application, new quantum (Hopf algebra) deformations of the Schrodinger algebra, including their corresponding quantum universal R-matrices, are constructed.

Double quantization of CPn type orbits by generalized Verma modules

It is known that symmetric orbits in g ∗ for any simple Lie algebra g are equipped with a Poisson pencil generated by the Kirillov-Kostant-Souriau bracket and the reduced Sklyanin bracket associated to the “canonical” R-matrix. We realize quantization of the Poisson pencil CP n type orbits (i.e. orbits in sl(n + 1) ∗ whose real compact form is CP n ) by means of q-deformed Verma modules.

Bicovariant 3D calculus for SLq(2) from Poisson–Lie structures

All possible graded Poisson–Lie structures on the external algebra of SL(2) are described. We prove that differential Poisson–Lie structures prolonging the Sklyanin brackets do not exist on SL(2). There are two and only two graded Poisson–Lie structures on SL(2) and neither of them can be obtained by a reduction of graded Poisson–Lie structures on the external algebra of GL(2). Both of them can be quantized and as a result we get new graded algebras of quantum right-invariant forms on SLq(2) with three generators.