Letters in Mathematical Physics最新文献

We consider quantum mechanical systems of spin chain type, with finite-dimensional Hilbert spaces and (mathcal {N}=2) or (mathcal {N}=4) supersymmetry, described in (mathcal {N}=2) superspace in terms of nonlinear chiral multiplets. We prove that they are natural truncations of 1D sigma models, whose target spaces are (textsf {SU}(n)) (co)adjoint orbits. As a first application, we compute the Witten indices of these finite-dimensional models showing that they reproduce the Dolbeault and de Rham indices of the target space. The problem of finding the exact spectra of generalized Laplace operators on such orbits is shown to be equivalent to the diagonalization of spin chain Hamiltonians.

In this paper, we have developed Cauchy matrix approach to construct the q-difference two-dimensional Toda lattice (q-2DTL) and q-difference sine-Gordon (q-sG) equation, and explore their integrability such as Lax pair and explicit solutions. By leveraging specific dispersion relations pertaining to r and s of the Sylvester equation (KM + ML = rs^top ), we establish the q-2DTL and derive its Lax pair. We also clarify the connection of the (tau ) function of the q-2DTL with Cauchy matrix approach. Besides, explicit solutions of the q-2DTL are formulated and classified by comprehensively investigating its underlying systems of linear q-difference equations. As typical examples, the dynamical behaviors of both soliton solutions and a double-pole solution are simulated numerically. Under the assumption (K = L), we demonstrate how to reduce the q-sG equation from the q-2DTL both by Cauchy matrix approach and by 2-periodic reductions. Besides, the bilinear representation for the q-sG equation is reported for the first time. Furthermore, rich solutions such as kink solutions and breathers are explicitly presented and graphically illustrated for the q-sG equation.

In previous work (Sati and Voronov in Commun Math Phys 400:1915–1960, 2023. https://doi.org/10.1007/s00220-023-04643-7, in Adv Theor Math Phys 28(8):2491–2601, 2024. https://doi.org/10.4310/atmp.241119034750), we introduced Mysterious Triality, extending the Mysterious Duality (Iqbal et al. in Adv Theor Math Phys 5:769–808, 2002. https://doi.org/10.4310/ATMP.2001.v5.n4.a5) between physics and algebraic geometry to include algebraic topology in the form of rational homotopy theory. Starting with the rational Sullivan minimal model of the 4-sphere (S^4), capturing the dynamics of M-theory via Hypothesis H, this progresses to the dimensional reduction of M-theory on torus (T^k), (k ge 1), with its dynamics described via the iterated cyclic loop space ({mathcal {L}}_c^k S^4) of the 4-sphere. From this, we also extracted data corresponding to the maximal torus/Cartan subalgebra and the Weyl group of the exceptional Lie group/algebra of type (E_k). In this paper, we discover much richer symmetry by extending the action of the Cartan subalgebra by symmetries of the equations of motion of ((11-k))d supergravity to a maximal parabolic subalgebra (mathfrak {p}_k^{k(k)}) of the Lie algebra (mathfrak {e}_{k(k)}) of the U-duality group. We do this by constructing the action on the rational homotopy model of the slightly more symmetric than ({mathcal {L}}_c^k S^4) toroidification ({mathcal {T}}^k S^4), which is another bookkeeping device for the equations of motion. To justify these results, we identify the minimal model of the toroidification ({mathcal {T}}^k S^4), generalizing the results of Vigué-Poirrier, Sullivan, and Burghelea, and establish an algebraic toroidification/totalization adjunction.

The long-time asymptotic behaviors of the rarefaction problem for the focusing nonlinear Schrödinger equation with discrete spectrum are analyzed via the Riemann–Hilbert formulation. It is shown that for the rarefaction problem with pure step initial condition there are three asymptotic sectors in time–space: the plane wave sector, the 1-phase elliptic wave sector and the vacuum sector, while for the rarefaction problem with general initial data there are five asymptotic sectors in time–space: the plane wave sector, the sector of plane wave with soliton transmission, the sector of plane wave with phase shift, the sector of 1-phase elliptic wave with phase shift and the vacuum sector with phase shift. The leading-order term of each sector along with the corresponding error estimate is given by adopting the Deift–Zhou nonlinear steepest-descent method for Riemann–Hilbert problems. The asymptotic solutions match very well with the results from Whitham modulation theory and the direct numerical simulations.

Let (mu ) be an arbitrary composition of (M+N) and let (mathfrak {s}) be an arbitrary (0^{M}1^{N})-sequence. The present paper is devoted to extending parabolic presentations, depending on (mu ) and (mathfrak {s}), of the super Yangian (Y_{M|N}) associated with the general linear Lie superalgebra ({mathfrak gmathfrak l}_{M|N}), to a field of positive characteristic.

We construct four edge contractions for the affine super Yangian of type A. As an application, by using these edge contractions, we give a homomorphism from the affine super Yangian of type A to the universal enveloping algebra of the non-rectangular W-superalgebra of type A.

We revisit the perturbative theory of infinite dimensional integrable systems developed by P. Deift and X. Zhou [8], aiming to provide new and simpler proofs of some key (L^infty ) bounds and (L^p) a priori estimates. Our proofs emphasizes a further step towards understanding focussing problems and extends the applicability to other integrable models. As a concrete application, we examine the perturbation of the one-dimensional defocussing cubic nonlinear Schrödinger equation by a localized higher-order term. We introduce improved estimates to control the power of the perturbative term and demonstrate that the perturbed equation exhibits the same long-time behavior as the completely integrable nonlinear Schrödinger equation.

We consider real tensors of order D, that is D-dimensional arrays of real numbers (T_{a^1a^2 dots a^D}), where each index (a^c) can take N values. The tensor entries (T_{a^1a^2 dots a^D}) have no symmetry properties under permutations of the indices. The invariant polynomials built out of the tensor entries are called trace invariants. We prove that for a Gaussian random tensor with (Dge 3) indices (that is such that the entries (T_{a^1a^2 dots a^D}) are independent identically distributed Gaussian random variables) the cumulant, or connected expectation, of a product of trace invariants is not always suppressed in scaling in N with respect to the product of the expectations of the individual invariants. Said otherwise, not all the multi-trace expectations factor at large N in terms of the single-trace ones and the Gaussian scaling is not subadditive on the connected components. This is in stark contrast to the (D=2) case of random matrices in which the multi-trace expectations always factor at large N. The best one can do for (Dge 3) is to identify restricted families of invariants for which the large N factorization holds and we check that this indeed happens when restricting to the family of melonic observables, the dominant family in the large N limit.

This is a review of the relationship between Fay identities and Hirota equations in integrable systems, reformulated in a geometric language compatible with recent Topological Recursion formalism. We write Hirota equations as trans-series and Fay identities as spinor functional relations. We also recall several constructions of how some solutions to Fay/Hirota equations can be built from Riemann surface geometry.

We consider trapped Bose gases in three dimensions in the Gross–Pitaevskii regime whose low energy states are well known to exhibit Bose–Einstein condensation. That is, the majority of the particles occupies the same condensate state. We prove exponential control of the number of particles orthogonal to the condensate state, generalizing recent results from Nam and Rademacher (Trans Am Math Soc, 2023, arXiv:2307.10622) for translation invariant systems.