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DOI: 10.1007/s00220-019-03299-6
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In the continuity of our previous paper (Crampe et al. in Commun Math Phys 349:271, 2017, arXiv:1509.05516 ), we define three new algebras, ${\mathcal{A}_{\mathfrak{n}}(a,b,c)}$ , ${\mathcal{B}_{\mathfrak{n}}}$ and ${\mathcal{C}_{\mathfrak{n}}}$ , that are close to the braid algebra. They allow to build solutions to the Yang-Baxter equation with spectral parameters. The construction is based on a baxterisation procedure, similar to the one used in the context of Hecke or BMW algebras. The ${\mathcal{A}_{\mathfrak{n}}(a,b,c)}$ algebra depends on three arbitrary parameters, and when the parameter a is set to zero, we recover the algebra ${\mathcal{M}_{\mathfrak{n}}(b,c)}$ already introduced elsewhere for purpose of baxterisation. The Hecke algebra (and its baxterisation) can be recovered from a coset of the ${\mathcal{A}_{\mathfrak{n}}(0,0,c)}$ algebra. The algebra ${\mathcal{A}_{\mathfrak{n}}(0,b,-b^2)}$ is a coset of the braid algebra. The two other algebras ${\mathcal{B}_{\mathfrak{n}}}$ and ${\mathcal{C}_{\mathfrak{n}}}$ do not possess any parameter, and can be also viewed as a coset of the braid algebra.

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Ref Arxiv: 1903.08179
DOI: 10.1016/j.nuclphysb.2019.114720
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3 Citations
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Using Sklyanin's classical theory of integrable boundary conditions, we use the Hamiltonian approach to derive new integrable boundary conditions for the Ablowitz-Ladik model on the finite and half infinite lattice. In the case of half infinite lattice, the special and new emphasis of this paper is to connect directly the Hamiltonian approach, based on the classical $r$-matrix, with the zero curvature representation and B\"acklund transformation approach that allows one to implement a nonlinear mirror image method and construct explicit solutions. It is shown that for our boundary conditions, which generalise (discrete) Robin boundary conditions, a nontrivial extension of the known mirror image method to what we call {\it time-dependent boundary conditions} is needed. A careful discussion of this extension is given and is facilitated by introducing the notion of intrinsic and extrinsic picture for describing boundary conditions. This gives the specific link between Sklyanin's reflection matrices and B\"acklund transformations combined with folding, {\it in the case of non-diagonal reflection matrices}. All our results reproduce the known Robin boundary conditions setup as a special case: the diagonal case. Explicit formulas for constructing multisoliton solutions on the half-lattice with our time-dependent boundary conditions are given and some examples are plotted.

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Ref Arxiv: 1811.02763
DOI: 10.1063/1.5111292
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The $sl_N$-Onsager algebra has been introduced by Uglov and Ivanov in 1995. In this letter, a FRT presentation of the $sl_N$-Onsager algebra is given, its current algebra and commutative subalgebra are constructed. Certain quotients of the $sl_N$-Onsager algebra are then considered, which produce `classical' analogs of higher rank extensions of the Askey-Wilson algebra. As examples, the cases $N=3$ and $N=4$ are described in details.

Commentaires: 13 pages

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Ref Arxiv: 1810.10547
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DOI: 10.1103/PhysRevD.99.023503
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4 Citations
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The growth of large scale structure is a battle between gravitational attraction and cosmic acceleration. We investigate the future behavior of cosmic growth under both general relativity (GR) and modified gravity during prolonged acceleration, deriving analytic asymptotic behaviors and showing that gravity generally loses and growth ends. We also note that the “why now” problem is equally striking when viewed in terms of the shutdown of growth. For many models inside GR the gravitational growth index γ also shows today as a unique time between constant behavior in the past and a higher asymptotic value in the future. Interestingly, while f(R) models depart in this respect dramatically from GR today and in the recent past, their growth indices are identical in the asymptotic future and past.

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Ref Arxiv: 1806.07232
DOI: 10.1007/s11005-019-01182-y
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4 Citations
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Automorphisms of the infinite dimensional Onsager algebra are introduced. Certain quotients of the Onsager algebra are formulated using a polynomial in these automorphisms. In the simplest case, the quotient coincides with the classical analog of the Askey-Wilson algebra. In the general case, generalizations of the classical Askey-Wilson algebra are obtained. The corresponding class of solutions of the non-standard classical Yang-Baxter algebra are constructed, from which a generating function of elements in the commutative subalgebra is derived. We provide also another presentation of the Onsager algebra and of the classical Askey-Wilson algebras.

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Ref Arxiv: 1804.06928
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DOI: 10.4310/ATMP.2019.v23.n3.a2
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Inspired by the split attractor flow conjecture for multi-centered black hole solutions in $N = 2$ supergravity, we propose a formula expressing the BPS index $\Omega (\gamma , z)$ in terms of ‘attractor indices’ $\Omega_{\ast} (\gamma_i)$. The latter count BPS states in their respective attractor chamber. This formula expresses the index as a sum over stable flow trees weighted by products of attractor indices. We show how to compute the contribution of each tree directly in terms of asymptotic data, without having to integrate the attractor flow explicitly. Furthermore, we derive new representations for the index which make it manifest that discontinuities associated to distinct trees cancel in the sum, leaving only the discontinuities consistent with wall-crossing. We apply these results in the context of quiver quantum mechanics, providing a new way of computing the Betti numbers of quiver moduli spaces, and compare them with the Coulomb branch formula, clarifying the relation between attractor and single-centered indices.

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Ref Arxiv: 1611.10327
DOI: 10.1142/S0217751X19500040
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General forms of the K ̈ahler and superpotenials that lead toconsistent low energy broken Super-symmetry originating fromN= 1 Supergravity have been classified and used for model buildingsince more than three decades. We point out the incompleteness of this classification when hiddensector vacuum expectation values are of the order of the Planck mass. Focusing in this paper mainly on the case of minimal Kahler potential, we adopt a rigorous approach that retrieves on the one hand the known forms, and demonstrate on the other hand the existence of a whole set of new forms for the superpotential of which we give a complete classification. The latter forms involve a new type of chiral superfields having the unusual property of belonging neither to the hidden sector nor to the conventional observable sector. Comparing the obtained forms with the conventional ones, we argue how new possibilities for model building can arise, and discuss the gravity mediation of soft as well as additional hard (but parametrically small) Supersymmetry breaking, in the presence of the new type of chiral superfields. In the simplest case, we study the vacuum structure, characterize the masses and couplings of the scalar components to the hidden and observable sectors and discuss briefly the physical role they could play. In the generic case, we estimate the magnitude and possible consequences of the hard breaking of Supersymmetry in terms of the interplay between hidden and visible sectors mass scales.

Commentaires: 63 pages