Letters in Mathematical Physics最新文献

We provide an analytic method for estimating the entanglement of the non-Gaussian energy eigenstates of disordered harmonic oscillator systems. We invoke the explicit formulas of the eigenstates of the oscillator systems to establish bounds for their (epsilon )-Rényi entanglement entropy (epsilon in (0,1)). Our methods result in a logarithmically corrected area law for the entanglement of eigenstates, corresponding to one excitation, of the disordered harmonic oscillator systems.

Conformal blocks of q, t-deformed Virasoro and ({mathcal {W}})-algebras are important special functions in representation theory with applications in geometry and physics. In the Nekrasov–Shatashvili limit (t rightarrow 1), whenever one of the representations is degenerate then conformal block satisfies a difference equation with respect to the coordinate associated with that degenerate representation. This is a stationary Schrodinger equation for an appropriate relativistic quantum integrable system. It is expected that generalization to generic (t ne 1) is a non-stationary Schrodinger equation where t parametrizes shift in time. In this paper we make the non-stationary equation explicit for the q, t-Virasoro block with one degenerate and four generic Verma modules and prove it when three modules out of five are degenerate, using occasional relation to Macdonald polynomials.

The Freudenthal–Tits magic square (mathfrak {m}(mathbb {A}_1,mathbb {A}_2)) for (mathbb {A}=mathbb {R},mathbb {C},mathbb {H},mathbb {O}) of semi-simple Lie algebras can be extended by including the sextonions (mathbb {S}). A series of non-reductive Lie algebras naturally appear in the new row associated with the sextonions, which we will call the intermediate exceptional series, with the largest one as the intermediate Lie algebra (E_{7+1/2}) constructed by Landsberg–Manivel. We study various aspects of the intermediate vertex operator (super)algebras associated with the intermediate exceptional series, including rationality, coset constructions, irreducible modules, (super)characters and modular linear differential equations. For all (mathfrak {g}_I) belonging to the intermediate exceptional series, the intermediate VOA (L_1(mathfrak {g}_I)) has characters of irreducible modules coinciding with those of the simple rational (C_2)-cofinite W-algebra (W_{-h^vee /6}(mathfrak {g},f_theta )) studied by Kawasetsu, with (mathfrak {g} ) belonging to the Cvitanović–Deligne exceptional series. We propose some new intermediate VOA (L_k(mathfrak {g}_I)) with integer level k and investigate their properties. For example, for the intermediate Lie algebra (D_{6+1/2}) between (D_6) and (E_7) in the subexceptional series and also in Vogel’s projective plane, we find that the intermediate VOA (L_2(D_{6+1/2})) has a simple current extension to a SVOA with four irreducible Neveu–Schwarz modules, and the supercharacters can be realized by a fermionic Hecke operator on the (N=1) Virasoro minimal model (L(c_{16,2},0)). We also provide some (super) coset constructions such as (L_2(E_7)/L_2(D_{6+1/2})) and (L_1(D_{6+1/2})^{otimes 2}!/L_2(D_{6+1/2})). In the end, we find that the theta blocks associated with the intermediate exceptional series produce some new holomorphic Jacobi forms of critical weight and lattice index.

We consider gauged linear sigma models with gauge group U(1) that exhibit a geometric as well as a Landau–Ginzburg phase. We construct defects that implement the transport of D-branes from the Landau–Ginzburg phase to the geometric phase. Through their fusion with boundary conditions these defects in particular provide functors between the respective D-brane categories. The latter map (equivariant) matrix factorizations to coherent sheaves and can be formulated explicitly in terms of complexes of matrix factorizations.

We revisit the construction of quantum Riemannian geometries on graphs starting from a hermitian metric compatible connection, which always exists. We use this method to find quantum Levi-Civita connections on the n-leg star graph for (n=2,3,4) and find the same phenomenon as recently found for the (A_n) Dynkin graph that the metric length for each inbound arrow has to exceed the length in the other direction by a multiple, here (sqrt{n}). We then study quantum geodesics on graphs and construct these on the 4-leg graph and on the integer lattice line (mathbb {Z}) with a general edge-symmetric metric.

The amplituhedron is a mathematical object which was introduced to provide a geometric origin of scattering amplitudes in (mathcal {N}=4) super Yang–Mills theory. It generalizes cyclic polytopes and the positive Grassmannian and has a very rich combinatorics with connections to cluster algebras. In this article, we provide a series of results about tiles and tilings of the (m=4) amplituhedron. Firstly, we provide a full characterization of facets of BCFW tiles in terms of cluster variables for (text{ Gr}_{4,n}). Secondly, we exhibit a tiling of the (m=4) amplituhedron which involves a tile which does not come from the BCFW recurrence—the spurion tile, which also satisfies all cluster properties. Finally, strengthening the connection with cluster algebras, we show that each standard BCFW tile is the positive part of a cluster variety, which allows us to compute the canonical form of each such tile explicitly in terms of cluster variables for (text{ Gr}_{4,n}). This paper is a companion to our previous paper “Cluster algebras and tilings for the (m=4) amplituhedron.”

We give an elementary proof of an inequality of Lin, Kim and Hsieh that implies strong subadditivity of the von Neumann entropy.

We give a general account of family algebras over a finitely presented linear operad. In a family algebra, each operation of arity n is replaced by a family of operations indexed by (Omega ^n), where (Omega ) is a set of parameters. We show that the operad, together with its presentation, naturally defines an algebraic structure on the set of parameters, which in turn is used in the description of the family version of the relations between operations. The examples of dendriform and duplicial family algebras (hence with two parameters) and operads are treated in detail, as well as the pre-Lie family case. Finally, free one-parameter duplicial family algebras are described, together with the extended duplicial semigroup structure on the set of parameters.

It is well-known that a formal deformation of a commutative algebra (mathcal {A}) leads to a Poisson bracket on (mathcal {A}) and that the classical limit of a derivation on the deformation leads to a derivation on (mathcal {A}), which is Hamiltonian with respect to the Poisson bracket. In this paper we present a generalization of it for formal deformations of an arbitrary noncommutative algebra (mathcal {A}). The deformation leads in this case to a Poisson algebra structure on (Pi (mathcal {A}){:}{=}Z(mathcal {A})times (mathcal {A}/Z(mathcal {A}))) and to the structure of a (Pi (mathcal {A}))-Poisson module on (mathcal {A}). The limiting derivations are then still derivations of (mathcal {A}), but with the Hamiltonian belong to (Pi (mathcal {A})), rather than to (mathcal {A}). We illustrate our construction with several cases of formal deformations, coming from known quantum algebras, such as the ones associated with the nonabelian Volterra chains, Kontsevich integrable map, the quantum plane and the quantized Grassmann algebra.

We study, via the eigenvalue theory of completely continuous operators, the existence and asymptotic behavior of nontrivial p-k-convex radial solutions for a p-k-Hessian equation. This is probably the first time that p-k-Hessian equations have been studied by employing this technique. Several new nonexistence conclusions are also derived in this paper.