Communications in Mathematical Physics最新文献

We introduce a purely Lie algebraic formalization of the Feigin–Tipunin’s geometric construction of logarithmic CFTs/VOAs. After reformulating the geometric representation theory of FT construction under this new setting, within this framework, we uniformly construct the (multiplet) principal W-algebras at positive integer level associated with any simple Lie algebra ({mathfrak {g}}) and Lie superalgebra (mathfrak {osp}(1|2r)), thereby establishing Weyl-type character formulas and simplicity theorems that extend the first author’s previous results.

We consider the Hurwitz spaces of ramified coverings of ({mathbb {P}}^1) with prescribed ramification profile over the point at infinity. By means of a particular symmetric bidifferential on a compact Riemann surface, we introduce quasi-homogeneous differentials. By following Dubrovin, we construct on Hurwitz spaces a family of Frobenius manifold structures associated with the quasi-homogeneous differentials. We explicitly derive new generating formulas for the corresponding prepotentials. This produces quasi-homogeneous solutions to the following generalized WDVV associativity equations: (F_ieta ^{-1}F_j=F_jeta ^{-1}F_i), where the invertible constant matrix (eta ) is a linear combination of the matrices (F_j). In particular, our approach provides another look at Dubrovin’s construction of semisimple Hurwitz-Frobenius manifolds and establishes an alternative practical method to calculate their primary free energy functions . As applications, we use our formalism to obtain various explicit quasi-homogeneous solutions to the WDVV equations in genus one and give a new proof of Ramanujan’s differential equations for Eisenstein series.

We study the perturbative path integral of Chern–Simons theory (the effective BV action on zero-modes) in Lorenz gauge, expanded around a (possibly non-acyclic) flat connection, as a family over the smooth irreducible stratum ({mathcal {M}}'subset {mathcal {M}}) of the moduli space of flat connections. We prove that it is horizontal with respect to the Grothendieck connection up to a BV-exact term. From it, we construct a volume form on ({mathcal {M}}')—the “global partition function”—whose cohomology class is independent of the metric, and so is a 3-manifold invariant. As an element of the construction, we construct an extension of the perturbative partition function to a nonhomogeneous form on the space of triples ((A,A',g)) consisting of (1) a “kinetic” flat connection A around which Chern–Simons action is expanded, (2) a “gauge-fixing” flat connection (A'), (3) a metric g. This extension is horizontal with respect to an appropriate Gauss-Manin superconnection (which involves the BV operator as a degree zero component).

Let (X, g) be a closed, connected surface, with variable negative curvature. We consider the distribution of eigenvalues of the Laplacian on random covers (X_nrightarrow X) of degree n. We focus on the ensemble variance of the smoothed number of eigenvalues of the square root of the positive Laplacian (sqrt{Delta }) in windows ([lambda -frac{1}{L},lambda +frac{1}{L}]), over the set of n-sheeted covers of X. We first take the limit of large degree (nrightarrow +infty ), then we let the energy (lambda ) go to (+infty ) while the window size (frac{1}{L}) goes to 0. In this ad hoc limit, local energy averages of the variance converge to an expression corresponding to the variance of the same statistic when considering instead spectra of large random matrices of the Gaussian Orthogonal Ensemble (GOE). By twisting the Laplacian with unitary representations, we are able to observe different statistics, corresponding to the Gaussian Unitary Ensemble (GUE) when time reversal symmetry is broken. These results were shown by F. Naud for the model of random covers of a hyperbolic surface. For an individual cover (X_nrightarrow X), we consider spectral fluctuations of the counting function on (X_n) around the ensemble average. In the high energy regime, for a typical cover (X_nrightarrow X) of large degree, these fluctuations are shown to approach the GOE result, a phenomenon called ergodicity in Random Matrix Theory. An analogous result for random covers of hyperbolic surfaces was obtained by Y. Maoz.

A braid representation is a monoidal functor from the braid category ({mathsf B}), for example given by a solution to the constant Yang-Baxter equation. Given a monoidal category (textsf{C}) with (ob(textsf{C})={mathbb N}), a rank-N charge-conserving representation, or spin-chain representation, is a strict monoidal functor F from (textsf{C}) to the category ({textsf{Match}}^N) of rank-N charge-conserving matrices (see below for definition), that is natural in the sense that (F(1)=1). In this work we construct all spin-chain braid representations, and classify up to suitable notions of isomorphism.

The recently introduced tile codes are a promising alternative to surface codes, combining two-dimensional locality with higher encoding efficiency. While surface codes are well understood in terms of their logical operators and boundary behavior, much less is known about tile codes. In this work, we establish a natural and precise description of their logical operator space. We prove that, under mild assumptions, any tile code admits a canonical symplectic basis of logical operators supported along lattice boundaries, which can be generated efficiently by a simple cellular automaton with the number of update rules only depending on the non-locality of the tile code. Further, we develop algebraic and algebro-geometric frameworks for tile codes, by resolving them by translationally invariant Pauli stabilizer models and showing that they arise as derived sections of a Koszul complex on (mathbb {P}^1 times mathbb {P}^1). Finally, we introduce the concept of derived automorphisms for quantum codes. These are automorphism-like operations that can exist even for codes that do not have symmetries. We explain how derived automorphisms can be implemented for tile codes in a low-overhead and fault-tolerant manner by extending the lattice on one side and shrinking it on the other. While this operation is trivial for the surface code, it induces a product of logical CNOT gates on the encoded information. Our results provide new structural insights into tile codes and lay the groundwork for tile codes as building blocks for fault-tolerant quantum computation.

Consider a bipartite quantum system, where Alice and Bob jointly possess a pure state (|psi rangle ). Using local quantum operations on their respective subsystems, and unlimited classical communication, Alice and Bob may be able to transform (|psi rangle ) into another state (|phi rangle ). Famously, Nielsen’s theorem [28] provides a necessary and sufficient algebraic criterion for such a transformation to be possible (namely, the entanglement spectrum of (|phi rangle ) should majorise the entanglement spectrum of (|psi rangle )). In the paper where Nielsen proved this theorem, he conjectured that in the limit of large dimensionality, for almost all pairs of states (|psi rangle , |phi rangle ) (according to the natural unitary invariant measure) such a transformation is not possible. That is to say, typical pairs of quantum states (|psi rangle , |phi rangle ) are entangled in fundamentally different ways, that cannot be converted to each other via local operations and classical communication. Via Nielsen’s theorem, this conjecture can be equivalently stated as a conjecture about majorisation of spectra of random matrices from the so-called trace-normalised complex Wishart–Laguerre ensemble. Concretely, let X and Y be independent (n times m) random matrices whose entries are i.i.d. standard complex Gaussians; then Nielsen’s conjecture says that the probability that the spectrum of (X X^dagger / operatorname {tr}(X X^dagger )) majorises the spectrum of (Y Y^dagger / operatorname {tr}(Y Y^dagger )) tends to zero as both n and m grow large. We prove this conjecture, and we also confirm some related predictions of Cunden et al. [12].

We propose a symmetrization relation between BPS quivers encoding 4d (mathcal {N}=2) theories and symmetric quivers associated to 3d (mathcal {N}=2) theories. We analyse in detail the symmetrization of BPS quivers for a series of (A_m) Argyres-Douglas theories by engineering 3d-4d systems in geometric backgrounds involving appropriate 3-manifolds and Riemann surfaces. We discuss properties of these geometric backgrounds and derive the corresponding quiver partition functions from the perspective of skein modules, which forms the foundation of the symmetrization map for the minimal chamber. We also prove that the structure of wall-crossing in 4d (A_m) Argyres-Douglas theories is isomorphic to the structure of unlinking of symmetric quivers encoding their partner 3d theories, which allows for a proper definition of the symmetrization map outside the minimal chamber. Finally, we show that the Schur indices of 4d theories are captured by symmetric quivers that include symmetrization of 4d BPS quivers.

In this work, we consider decoupling a bipartite quantum state via a general quantum channel. We propose a joint state-channel decoupling approach to obtain a one-shot error exponent bound without smoothing, in which the trace distance is used to measure how good the decoupling is. The established exponent is expressed in terms of a sum of two sandwiched Rényi entropies, one quantifying the amount of initial correlation between the state and environment, while the other characterizing the effectiveness of the quantum channel. This gives an explicit exponential decay of the decoupling error for the entire achievable region, which was missing in the previous results [Commun. Math. Phys. 328, 2014]. Moreover, it strengthens the error exponent bound obtained in a recent work [IEEE Trans. Inf. Theory, 69(12), 2023], for exponent from the channel part. As an application, we establish a one-shot error exponent bound for quantum channel coding given by a sandwiched Rényi coherent information.

The asymptotic equipartition property (AEP) states that in the limit of a large number of independent and identically distributed (i.i.d.) random experiments, the output sequence is virtually certain to come from the typical set, each member of which is almost equally likely. This property is a form of the law of large numbers and lies at the heart of information theory. In this work, we prove a generalized quantum AEP beyond the i.i.d. framework where the random samples are drawn from two sets of quantum states. In particular, under suitable assumptions on the sets, we prove that all operationally relevant divergences converge to the quantum relative entropy between the sets. More specifically, both the quantum hypothesis testing relative entropy (a smoothed form of the min-relative entropy) and the smoothed max-relative entropy approach the regularized relative entropy between the sets. Notably, the asymptotic limit has explicit convergence guarantees and can be efficiently estimated through convex optimization programs, despite the regularization, provided that the sets have efficient descriptions. The generalized AEP directly implies a new quantum Stein’s lemma for conducting quantum hypothesis testing between two sets of quantum states. Combined with the accompanying work [IEEE TIT 72(4):2330-2342 (2026)], this addresses open questions raised by Brandão et al. [IEEE TIT 66(8):5037–5054 (2020)] and Mosonyi et al. [IEEE TIT 68(2):1032-1067 (2022)], which seek a Stein’s lemma with computational efficiency. Moreover, we propose a new framework for quantum resource theory in which state transformations are performed without requiring precise characterization of the states being manipulated, making it more robust to imperfections. We demonstrate the reversibility (also referred to as the second law) of such a theory and identify the regularized relative entropy as the unique measure of the resource in this new framework.