Letters in Mathematical Physics最新文献

In this work, we provide a classification scheme for topological phases of certain systems whose observable algebra is described by a trivial (C^*)-bundles. The classification is based on the study of the homotopy classes of configurations, which are maps from a quantum parameter space to the space of pure states of a reference fiber (C^*)-algebra. Both the quantum parameter space and the fiber algebra are naturally associated with the observable algebra. A list of various examples described in the last section shows that the common classification scheme of non-interacting topological insulators of type A is recovered inside this new formalism.

We consider the Schrödinger operators which are constructed from the (lambda )-opers corresponding to solutions of the (widehat{mathfrak {sl}}_2) Gaudin Bethe Ansatz equations. We define and study the connection coefficients called the Q-functions. We conjecture that the Q-functions obtained from the (lambda )-opers coincide with the Q-functions of the Bazhanov–Lukyanov–Zamolodchikov opers with the monster potential related to the quantum KdV flows. We give supporting evidence for this conjecture. In particular, we give a rigorous proof that the Q-functions of (lambda )-opers satisfy the QQ and TQ relations.

We construct defects describing the transition between different phases of gauged linear sigma models with higher rank abelian gauge groups, as well as defects embedding these phases into the GLSMs. Our construction refers entirely to the sector protected by B-type supersymmetry, decoupling the gauge sector. It relies on an abstract characterization of such transition defects and does not involve an actual perturbative analysis. It turns out that the choices that are required to characterize consistent transition defects match with the homotopy classes of paths between different phases. Our method applies to non-anomalous as well as anomalous GLSMs, and we illustrate both cases with examples. This includes the GLSM associated to the resolution of the (A_N) singularity and one describing the entire parameter space of (N=2) minimal models, in particular, the relevant flows between them. Via fusion with boundary conditions, the defects we construct yield functors describing the transport of D-branes on parameter space. We find that our results match with known results on D-brane transport.

The Kontsevich–Soibelman wall-crossing formula is known to control the jumping behaviour of BPS state-counting indices in four-dimensional theories with (mathcal {N}=2) supersymmetry. The formula can take two equivalent forms: a “fermionic” form with nice positivity properties and a “bosonic” form with a clear physical interpretation. In an important class of examples, the fermionic form of the formula has a mathematical categorification involving PBW bases for a Cohomological Hall Algebra. The bosonic form lacks an analogous categorification. We construct an equivalence of chain complexes, which categorifies the simplest example of the bosonic wall-crossing formula: the bosonic pentagon identity for the quantum dilogarithm. The chain complexes can be promoted to differential-graded algebras which we relate to the PBW bases of the relevant CoHA by a certain quadratic duality. The equivalence of complexes then follows from the relation between quadratic duality and Koszul duality. We argue that this is a special case of a general phenomenon: the bosonic wall-crossing formulae are categorified to equivalences of (A_infty ) algebras which are quadratic dual to PBW presentations of algebras which underlie the fermionic wall-crossing formulae. We give a partial interpretation of our differential-graded algebras in terms of a holomorphic-topological version of BPS webs.

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.