Communications in Mathematical Physics最新文献

We specialise the construction of orbifold graph TQFTs introduced in Carqueville et al. (Orbifold graph TQFTs) to Reshetikhin–Turaev defect TQFTs. We explain that the modular fusion category (mathcal {C}_mathcal {A}) constructed in Mulevičius and Runkel (Quant Topol 13(3):459–523, 2023. https://doi.org/10.4171/QT/170) from an orbifold datum (mathcal {A}) in a given modular fusion category (mathcal {C}) is a special case of the Wilson line ribbon categories introduced as part of the general theory of orbifold graph TQFTs. Using this, we prove that the Reshetikhin–Turaev TQFT obtained from (mathcal {C}_mathcal {A}) is equivalent to the orbifold of the TQFT for (mathcal {C}) with respect to the orbifold datum (mathcal {A}).

In this paper we characterize all distributional limits of the quadratic chaos (T_n =sum _{1le u< vle n} a_{u, v} X_u X_v), where (((a_{u, v}))_{1le u,vle n}) is a ({0, 1})-valued symmetric matrix with zeros on the diagonal and (X_1, X_2, ldots , X_n) are i.i.d.  mean 0 variance 1 random variables with common distribution function F. In particular, we show that any distributional limit of (S_n:=T_n/sqrt{{text {Var}}[T_n]}) can be expressed as the sum of three independent components: a Gaussian, a (possibly) infinite weighted sum of independent centered chi-squares, and a Gaussian mixture with a random variance. As a consequence, we prove a fourth moment theorem for the asymptotic normality of (S_n), which applies even when F does not have finite fourth moment. More formally, we show that (S_n) converges to N(0, 1) if and only if the fourth moment of (S_n) (appropriately truncated when F does not have finite fourth moment) converges to 3 (the fourth moment of the standard normal distribution). The proofs combine a Lindeberg-type replacement argument and combinatorial moment calculations using results of Erdős and Alon on extremal subgraph counts.

We prove that in the 2D Ising model with a weak bidimensional quasi-periodic disorder in the interaction, the critical behavior is the same as in the non-disordered case; that is, the critical exponents for the specific heat and energy-energy correlations are identical, and no logarithmic corrections are present. The disorder produces a quasi-periodic modulation of the amplitude of the correlations and a renormalization of the velocities, that is, the coefficients of the rescaling of positions, and of the critical temperature. The result establishes the validity of the prediction based on the Harris–Luck criterion, and it provides the first rigorous proof of universality in the Ising model in the presence of quasi-periodic disorder in both directions and for any angle. Small divisors are controlled assuming a Diophantine condition on the frequencies, and the convergence of the series is proved by Renormalization Group analysis.

We propose a novel way to define imaginary root subgroups associated with (timelike) imaginary roots of hyperbolic Kac–Moody algebras. Using in an essential way the theory of unitary irreducible representation of covers of the group SO(2, 1), these imaginary root subgroups act on the complex Kac–Moody algebra viewed as a Hilbert space. We illustrate our new view on Kac–Moody groups by considering the example of a rank-two hyperbolic algebra that is related to the Fibonacci numbers. We also point out some open issues and new avenues for further research, and briefly discuss the potential relevance of the present results for physics and current attempts at unification.

Let M be an n-dimensional Frobenius manifold. Fix (kappa in {1,dots ,n}). Assuming certain invertibility, Dubrovin introduced the Legendre-type transformation (S_kappa ), which transforms M to an n-dimensional Frobenius manifold (S_kappa (M)). In this paper, we show that these (S_kappa (M)) share the same monodromy data at the Fuchsian singular point of the Dubrovin connection, and that for the case when M is semisimple they also share the same Stokes matrix and the same central connection matrix. A straightforward application of the monodromy identification is the following: if we know the monodromy data of some semisimple Frobenius manifold M, we immediately obtain those of its Legendre-type transformations. Another application gives the identification between the (kappa )th partition function of a semisimple Frobenius manifold M and the topological partition function of (S_{kappa }(M)).

We prove the convergence of Araki–Haag detectors in any Haag–Kastler quantum field theory with an upper and lower mass gap. We cover the case of a single Araki–Haag detector on states of bounded energy, which are selected from the absolutely continuous part of the energy-momentum spectrum sufficiently close to the lower boundary of the multi-particle spectrum. These states essentially encompass those states in the multi-particle spectrum lying below the three-particle threshold. In our proof, we draw on insights from proofs of asymptotic completeness in quantum mechanics. Notably, we apply Mourre’s conjugate operator method for the first time within the framework of Haag–Kastler quantum field theory. Furthermore, we discuss applications of our findings for the problem of asymptotic completeness in local relativistic quantum field theory.

We construct a cocycle that, for a given n-manifold, maps a pair of asymptotically locally hyperbolic (ALH) metrics to a tractor-valued ((n-1))-form field on the conformal infinity. This requires the metrics to be asymptotically related to a given order that depends on the dimension. It then provides a local geometric quantity on the boundary that is naturally associated to the pair and can be interpreted as a relative energy-momentum density. It is distinguished as a geometric object by its property of being invariant under suitable diffeomorphisms fixing the boundary, and that act on (either) one of the argument metrics. Specialising to the case of an ALH metric h that is suitably asymptotically related to a locally hyperbolic conformally compact metric, we show that the cocycle determines an absolute invariant c(h), which still is local in nature. This tractor-valued ((n-1))-form field on the conformal infinity is canonically associated to h (i.e. is not dependent on other choices) and is equivariant under the appropriate diffeomorphisms. Finally specialising further to the case that the boundary is a sphere and that a metric h is asymptotically related to a hyperbolic metric on the interior, we show that the invariant c(h) can be integrated over the boundary. The result pairs with solutions of the KID (Killing initial data) equation to recover the known description of hyperbolic mass integrals of Wang, and Chruściel–Herzlich.

We compute the precise logarithmic corrections to Alexander–Orbach behaviour for various quantities describing the geometric and spectral properties of the four-dimensional uniform spanning tree. In particular, we prove that the volume of an intrinsic n-ball in the tree is (n^2 (log n)^{-1/3+o(1)}), that the typical intrinsic displacement of an n-step random walk is (n^{1/3} (log n)^{1/9-o(1)}), and that the n-step return probability of the walk decays as (n^{-2/3}(log n)^{1/9-o(1)}).

We prove a q-refined tropical correspondence theorem for higher genus descendant logarithmic Gromov–Witten invariants with a (lambda _g) class in toric surfaces. Specifically, a generating series of such logarithmic Gromov–Witten invariants agrees with a q-refined count of rational tropical curves satisfying higher valency conditions. As a corollary, we obtain a geometric proof of the deformation invariance of this tropical count. In particular, our results give an algebro-geometric meaning to the tropical count defined by Blechman and Shustin. Our strategy is to use the logarithmic degeneration formula, and the key new technique is to reduce to computing integrals against double ramification cycles and connect these integrals to the non-commutative KdV hierarchy.

The random XXZ quantum spin chain manifests localization (in the form of quasi-locality) in any fixed energy interval, as previously proved by the authors. In this article it is shown that this property implies slow propagation of information, one of the putative signatures of many-body localization (MBL), in the same energy interval.