Letters in Mathematical Physics最新文献

We initiate the mathematical study of the boundary Carrollian conformal algebra (BCCA), an infinite-dimensional Lie algebra recently discovered in the context of Carrollian physics. The BCCA is an intriguing object from both physical and mathematical perspectives, since it is a filtered but not graded Lie algebra. In this paper, we first construct some modules for the BCCA and one of its subalgebras, which we call (mathcal {O}), by restriction of well-known modules of the BMS₃ and Witt algebras, respectively. Along the way, we prove the irreducibility criteria for the so-called induced modules of the BMS₃ algebra (which we prefer to call massive modules to avoid ambiguity) and show that this is the same criteria for the irreducibility of the Verma modules of the BMS₃ algebra. Interestingly, the modules generated by the action of the BCCA on the generating vector of the massive modules are also irreducible under the same criteria. When these criteria hold, every massive module decomposes into a direct sum of two BCCA-submodules, each of which we conjecture to be indecomposable. Meanwhile, restricting Verma modules to the BCCA and (mathcal {O}) leads to free or “almost-free” modules, which are not particularly interesting from a representation-theoretic viewpoint. This motivates the construction of BCCA modules intrinsically. To do this, we go through some structure theory on the BCCA to define a new basis and a decreasing filtration on the algebra, using which we construct Whittaker modules over the BCCA and the subalgebra (mathcal {O}) and prove criteria for their irreducibility.

This work focuses on the Cauchy problem for the nonlocal modified Korteweg–de Vries equation

with the oscillating step-like boundary conditions: (u(x,t)rightarrow 0) as (xrightarrow -infty ) and (u(x,t)backsimeq Acos (2Bx+8B^3t)) as (xrightarrow infty ), where (A,B>0) are arbitrary constants. The main goal is to develop the Riemann–Hilbert formalism for this problem, paying a particular attention to the case of the “pure oscillating step” initial data, that is, (u(x,0)=0) for (x<0) and (u(x,0)=Acos (2Bx)) for (xge 0). Also, we derive three new families of two-soliton solutions, which correspond to the values of A and B satisfying (B<frac{A}{4}), (B>frac{A}{4}) and (B=frac{A}{4}).

We verify a conjecture of [1] stating that certain physically motivated BRST reductions of beta-gamma systems and free fermions are isomorphic to (L_1(mathfrak {psl}_{n|n})), the simple quotients of the affine vertex superalgebras associated to (mathfrak {psl}_{n|n}) at level 1, and that the respective associated varieties are isomorphic as Poisson varieties to the minimal nilpotent orbit closures (overline{mathbb {O}_{textrm{min}}(mathfrak {sl}_n)}). This shows in particular that (L_1(mathfrak {psl}_{n|n})) is quasi-lisse. Combining this with other results in the literature (in particular [2]), this paper provides a concrete and interesting example of how one can extract two symplectic dual varieties from a vertex superalgebra.

We consider a class of two-dimensional tight binding models displaying conical intersections of the Bloch bands at the Fermi level. The setting includes the case of generic transitions between quantum Hall phases. We consider the longitudinal conductivity, as given by Kubo formula, describing the variation of the current after introducing a space-homogeneous electric field, in an adiabatic way. We obtain an explicit expression for the longitudinal conductivity, completely determined by the number of conical intersections and by the shape of the cones. In particular, the formula reproduces the known quantized values found for graphene and for the critical Haldane model. Furthermore, we discuss the validity of Kubo formula in presence of conical intersections in the spectrum, starting from the time-dependent Schrödinger equation. For electric fields which are weak and slowly varying in space and in time, we prove the validity of linear response from quantum dynamics.

We present a formula for the connected (n)-point functions of a tau-function of the BKP hierarchy by embedding the BKP hierarchy into the KP hierarchy. This formula differs from that given by Wang and Yang [20]. We prove that these two formulae are equivalent.

In this paper, we establish equivariant mirror symmetry for footballs (mathcal {F}(m,r)). This extends the results by Fang et al. [13], where the projective line was considered and the results by Tang of weighted projective lines, on arXiv:1712.04836. More precisely, we prove the equivalence of the R-matrices for A-model and B-model at large radius limit and establish isomorphism for R-matrices for general radius. We further demonstrate that the graph sum of higher genus cases is the same for both models, hence establishing equivariant mirror symmetry for footballs. In the last two sections the large radius limit and equivariant limit are considered, resulting in a generalized Bouchard–Mariño conjecture and Norbury–Scott conjecture, respectively.

Motivated by the work of Kontsevich–Soibelman on the comparison of isomorphisms conjecture for closed algebraic 1-forms, we establish a Riemann–Hilbert correspondence of Deligne–Malgrange type. As an application, we prove a variant of the comparison of isomorphisms theorem for a simple class of algebraic 1-forms on complex curves.

Tom Leinster gave a bijective correspondence between the set of operators on a finite-dimensional vector space V and the set of pairs consisting of a nilpotent operator and a vector in V. Over a finite field this bijection implies that the probability that an operator be nilpotent is the reciprocal of the number of vectors in V. We generalize this correspondence to pairs of operators in opposite directions between pairs of vector spaces and determine the probability that a random pair of operators be nilpotent. We also determine the set-theoretical counterpart of this construction and compute the number of eventually constant pairs of maps between two finite sets, closely related to the number of spanning trees in a complete bipartite graph.