Communications in Mathematical Physics最新文献

We study the algebraic structures which govern the deformation of supersymmetric intersections of M2 and M5 branes. The universal algebras on M2 and M5 branes are deformed double current algebra of (mathfrak {gl}_K) and (mathfrak {gl}_K)-extended (mathcal {W}_{infty })-algebra respectively. We give a new presentation of the deformed double current algebra of (mathfrak {gl}_K), and we give a rigorous mathematical construction of the (mathfrak {gl}_K)-extended (mathcal {W}_{infty })-algebra. A new presentation of the affine Yangian of (mathfrak {gl}_K) is also obtained. We construct various coproducts of these algebras, which are expected to encode the fusions of defects in twisted M-theory. The matrix extended Miura operators are identified as intertwiners in certain bimodules of these algebras.

For a real distribution (mathcal {D}) on the interval [0, L] with (widetilde{mathcal { D}}) the associated even distribution on the interval ([-L, L]), we prove that if the associated quadratic form with Schwartz kernel (widetilde{mathcal {D}}(x - y)) defines a lower-bounded selfadjoint operator on (L^2([-frac{L}{2}, frac{L}{2}])), whose lowest spectral value (lambda ) is a simple, isolated eigenvalue with even eigenfunction (xi ), then all the zeros of the entire function (widehat{xi }(z)), the Fourier transform of (xi ), lie on the real line. The proof proceeds in five steps. (1) We give a (C^*)-algebraic proof of a corollary of Carathéodory–Fejér’s 1911 structure theorem for Toeplitz matrices: if (T in M_n(mathbb {C})) is a Hermitian, positive semidefinite Toeplitz matrix of rank (n - 1), and (xi in ker T), then the polynomial (P(z) = sum xi _j z^j) has all its zeros on the unit circle. (2) We formulate and prove a continuous analogue of this result, replacing the Toeplitz matrix with a convolution operator with continuous kernel (h(x - y)), and the polynomial P(z) with the Fourier transform of the eigenfunction corresponding to the largest eigenvalue. (3) We analyze finite-dimensional truncations of the quadratic forms defined by real, even distributions (mathcal {D}) on ([-L, L]), and observe that the resulting matrices exhibit a structure previously encountered in perturbative expansions of the spectral action. (4) We establish an analogue of Carathéodory–Fejér’s corollary for matrices of this specific structure, thereby extending the zero localization result beyond the classical Toeplitz setting. (5) Finally, we apply a classical theorem of Hurwitz concerning the zeros of uniform limits of holomorphic functions to deduce the general result stated above.

For the one-dimensional Facilitated Exclusion Process with initial state a product measure of density (rho =1/2-delta ), (delta ge 0), there exists an infinite-time limiting state (nu _rho ) in which all particles are isolated and hence cannot move. We study the variance V(L), under (nu _rho ), of the number of particles in an interval of L sites. Under (nu _{1/2}) either all odd or all even sites are occupied, so that (V(L)=0) for L even and (V(L)=1/4) for L odd: the state is hyperuniform (Torquato in Phys Rep 745:1–95, 2018), since V(L) grows more slowly than L. We prove that for densities approaching 1/2 from below there exist three regimes in L, in which the variance grows at different rates: for (Lgg delta ^{-2}), (V(L)simeq rho (1-rho )L), just as in the initial state; for (A(delta )ll Lll delta ^{-2}), with (A(delta )=delta ^{-2/3}) for L odd and (A(delta )=1) for L even, (V(L)simeq CL^{3/2}) with (C=2sqrt{2/pi }/3); and for (Lll delta ^{-2/3}) with L odd, (V(L)simeq 1/4). The analysis is based on a careful study of a renewal process with a long tail. Our study is motivated by simulation results showing similar behavior in higher dimensions; we discuss this background briefly.

The classical Chern correspondence states that a choice of Hermitian metric on a holomorphic vector bundle determines uniquely a unitary ‘Chern connection’. We generalize the Chern correspondence to the context of higher gauge theory, where the structure group of the bundle is categorified. For this, we define connective structures on multiplicative gerbes and propose a natural notion of complexification for an important class of 2-groups. Using this, we put forward a new notion of connection which is well-suited for describing holomorphic principal 2-bundles for these 2-groups, and apply it to establish a Chern correspondence. This relates holomorphic principal 2-bundles with holomorphic connective structure to supersymmetric configurations in string theory.

Given a countable discrete amenable group, we study conditions under which a set map into a Banach space (or more generally, a complete semi-normed space) can be realized as the ergodic sum of a vector under a group representation, such that the realization is asymptotically indistinguishable from the original map. We show that for uniformly bounded group representations, this property is characterized by the class of bounded asymptotically additive set maps, extending previous work for sequences in Banach spaces and on the case of a single non-expansive linear map. Additionally, we develop a relative version of this characterization, identifying when the additive realization can be chosen within a prescribed target set. As an application, our results generalize central aspects of thermodynamic formalism, bridging the additive and asymptotically additive frameworks.

We consider the Cauchy problem for the equations of pressureless gases in two space dimensions. For a generic set of smooth initial data (density and velocity), it is known that the solution loses regularity at a finite time (t_0), where both the density and the velocity gradient become unbounded. Aim of this paper is to provide an asymptotic description of the solution beyond the time of singularity formation. For (t>t_0) we show that a singular curve is formed, where the mass has positive density w.r.t. 1-dimensional Hausdorff measure. The system of equations describing the behavior of the singular curve is not hyperbolic. Working within a class of analytic data, local solutions can be constructed using a version of the Cauchy–Kovalevskaya theorem. For this purpose, by a suitable change of variables we rewrite the evolution equations as a first order system of Briot–Bouquet type, to which a general existence-uniqueness theorem can then be applied.

Although wave kinetic equations have been rigorously derived in dimension (d ge 2), both the physical and mathematical theory of wave turbulence in dimension (d = 1) is less understood. Here, we look at the one-dimensional MMT (Majda, McLaughlin, and Tabak) model on a large interval of length L with nonlinearity of size (alpha ), restricting to the case where there are no derivatives in the nonlinearity. The dispersion relation here is (|k|^sigma ) for (0 < sigma le 2) and (sigma ne 1), and when (sigma = 2), the MMT model specializes to the cubic nonlinear Schrödinger (NLS) equation. In the range of (1 < sigma le 2), the proposed collision kernel in the kinetic equation is trivial, begging the question of what is the appropriate kinetic theory in that setting. In this paper we study the kinetic limit (L rightarrow infty ) and (alpha rightarrow 0) under various scaling laws (alpha sim L^{-gamma }) and exhibit the wave kinetic equation up to timescales (T sim L^{-epsilon }alpha ^{-frac{5}{4}}) (or (T sim L^{-epsilon } T_{textrm{kin}}^{frac{5}{8}})). In the case of a trivial collision kernel, our result implies there can be no nontrivial dynamics of the second moment up to timescales (T_{textrm{kin}}).

A rational normal scroll structure on an ((n+1))-dimensional manifold M is defined as a field of rational normal scrolls of degree (n-1) in the projectivised cotangent bundle ({mathbb {P}} T^*M). We show that geometry of this kind naturally arises on solutions of various 4D dispersionless integrable hierarchies of heavenly type equations. In this context, rational normal scrolls coincide with the characteristic varieties (principal symbols) of the hierarchy. Furthermore, such structures automatically satisfy an additional property of involutivity. Our main result states that involutive scroll structures are themselves governed by a dispersionless integrable hierarchy, namely, the hierarchy of conformal self-duality equations.

The Moonshine module is a (c=24) conformal field theory (CFT) whose automorphism group is the Monster group. It was argued by Dixon, Ginsparg, and Harvey in Dixon et al. (Commun Math Phys 119:221–241, 1988. https://doi.org/10.1007/BF01217740) that there exists a spin lift of the Moonshine CFT with superconformal symmetry. Reference Dixon et al. (1988) did not provide an explicit construction of a superconformal current. The present paper provides an explicit construction of a supercurrent. In fact, we will construct several superconformal currents in a spin lift of the Moonshine CFT using techniques developed in Harvey and Moore (JHEP 05:146, 2020. https://doi.org/10.1007/JHEP05(2020)146. arXiv:2003.13700 [hep-th]). In particular, our construction relies on error correcting codes.

This paper consists of two parts. In the first part, we prove that when ({mathfrak {g}}) is a simple basic Lie superalgebra with a principal odd nilpotent element f, the W-algebra (W^k({mathfrak {g}}, F)) for (F=-frac{1}{2}{[}f,f{]}) is isomorphic to the SUSY W-algebra (W^k(bar{{mathfrak {g}}},f)) via screening operators, which implies the supersymmetry of (W^k({mathfrak {g}}, F)). In the second part, we show that a finite SUSY W-algebra, which is a Hamiltonian reduction of (U(widetilde{{mathfrak {g}}})) for the SUSY Takiff algebra (widetilde{{mathfrak {g}}}={mathfrak {g}}otimes wedge (theta )) is isomorphic to the Zhu algebra of a SUSY W-algebra. As a corollary, we show that a finite SUSY principal W-algebra is isomorphic to a finite principal W-algebra.