Communications in Mathematical Physics最新文献

In this work, we consider critical planar site percolation on the triangular lattice and derive sharp estimates on the asymptotics of the probability of half-plane j-arm events for (jge 1) and planar (polychromatic) j-arm events for (j>1), building upon a recent, not yet peer-reviewed result of Binder and Richards (Convergence rates of random discrete model curves approaching sle curves in the scaling limit. Preprint, 2020). These estimates greatly improve previous results and in particular answer (a large part of) a question of Schramm (ICM Proc., 2006).

We consider the log-gamma polymer in the half-space with bulk weights distributed as ({text {Gamma}}^{-1}(2theta )) and diagonal weights as ({text {Gamma}}^{-1}(alpha +theta )) for (theta >0) and (alpha >-theta ). We show that in the bound phase, i.e., when (alpha in (-theta ,0)), the endpoint of the polymer lies within an O(1) stochastic window of the diagonal. This result gives the first rigorous proof of the pinned phenomena for the half-space polymers in the bound phase conjectured by Kardar Kardar (Phys Rev Lett 55:2235, 1985). We also show that the limiting quenched endpoint distribution of the polymer around the diagonal is given by a random probability mass function proportional to the exponential of a random walk with log-gamma type increments.

Connectedness and bipartiteness are basic properties of classical graphs, and the purpose of this paper is to investigate the case of quantum graphs. We introduce the notion of connectedness and bipartiteness of quantum graphs in terms of graph homomorphisms. This paper shows that regular tracial quantum graphs have the same algebraic characterization of connectedness and bipartiteness as classical graphs. We also prove the equivalence between bipartiteness and two-colorability of quantum graphs by comparing two notions of graph homomorphisms: one respects adjacency matrices and the other respects edge spaces. In particular, all kinds of quantum two-colorability are mutually equivalent for regular connected tracial quantum graphs.

We consider the quantum difference equation of the Hilbert scheme of points in ({{mathbb {C}}}^2). This equation is the K-theoretic generalization of the quantum differential equation discovered by A. Okounkov and R. Pandharipande in [27]. We obtain two explicit descriptions for the monodromy of these equations - representation-theoretic and algebro-geometric. In the representation-theoretic description, the monodromy acts via certain explicit elements in the quantum toroidal algebra . In the algebro-geometric description, the monodromy features as transition matrices between the stable envelope bases in equivariant K-theory and elliptic cohomology. Using the second approach we identify the monodromy matrices for the differential equation with the K-theoretic R-matrices of cyclic quiver varieties, which appear as subvarieties in the 3D-mirror Hilbert scheme. Most of the results in the paper are illustrated by explicit examples for cases (n=2) and (n=3) in the Appendix.

Topological T-duality is a relationship between pairs (E, P) over a fixed space X, where (E rightarrow X) is a principal torus bundle and (P rightarrow E) is a twist, such as a gerbe for a principal (PU({mathcal {H}}))-bundle. This is of interest to topologists because of the T-duality transformation: a T-duality relation between pairs (E, P) and (({hat{E}}, {hat{P}})) comes with an isomorphism (with degree shift) between the twisted K-theory of E and the twisted K-theory of ({hat{E}}). We formulate topological T-duality for circle bundles in the equivariant setting, following the definition of Bunke, Rumpf, and Schick. We define the T-duality transformation in equivariant K-theory and show that it is an isomorphism for all compact Lie groups, equal to its own inverse and uniquely characterized by naturality and a normalization for trivial situations.

We construct examples of deformed Hermitian Yang–Mills connections and deformed (textrm{Spin}(7))-instantons (also called (textrm{Spin}(7)) deformed Donaldson–Thomas connections) on the cotangent bundle of (mathbb {C}mathbb {P}^2) endowed with the Calabi hyperKähler structure. Deformed (textrm{Spin}(7))-instantons on cones over 3-Sasakian 7-manifolds are also constructed. We show that these can be used to distinguish between isometric structures and also between (textrm{Sp}(2)) and (textrm{Spin}(7)) holonomy cones. To the best of our knowledge, these are the first non-trivial examples of deformed (textrm{Spin}(7))-instantons.

We study K-theory classes of Hamiltonian loop group spaces represented by admissible Fredholm complexes. We prove various equivariant index formulae in this context. In a sequel to this article we show that, when specialized to a family of non-local elliptic boundary value problems over the moduli space of framed flat connections on a surface, one obtains a gauge theory analogue of the Teleman–Woodward index formula.

In this paper, we establish the Anderson localization, strong dynamical localization and the ((frac{1}{2}-))-Hölder continuity of the integrated density of states (IDS) for some multi-dimensional discrete quasi-periodic (QP) Schrödinger operators with asymmetric (C^2)-cosine type potentials. We extend both the iteration scheme of Cao-Shi-Zhang (Commun Math Phys 404(1):495–561, 2023) and the interlacing method of Forman and VandenBoom (Localization and Cantor spectrum for quasiperiodic discrete Schrödinger operators with asymmetric, smooth, cosine-like sampling functions. arXiv:2107.05461, 2021) to handle asymmetric Rellich functions with collapsed gaps.

Despite the plausible necessity of topological protection for realizing scalable quantum computers, the conceptual underpinnings of topological quantum logic gates had arguably remained shaky, both regarding their physical realization as well as their information-theoretic nature. Building on recent results on defect branes in string/M-theory (Sati and Schreiber in Rev Math Phys, 2023. https://doi.org/10.1142/S0129055X23500095. [arXiv:2203.11838]) and on their holographically dual anyonic defects in condensed matter theory (Sati and Schreiber in Rev Math Phys 35(03):2350001, 2023. https://doi.org/10.1142/S0129055X23500010. [arXiv:2206.13563]), here we explain [as announced in Sati and Schreiber (PlanQC 2022:33, 2022, [arXiv:2209.08331], [ncatlab.org/schreiber/show/Topological+Quantum+Programming+in+TED-K])] how the specification of realistic topological quantum gates, operating by anyon defect braiding in topologically ordered quantum materials, has a surprisingly slick formulation in parameterized point-set topology, which is so fundamental that it lends itself to certification in modern homotopically typed programming languages, such as cubical Agda. We propose that this remarkable confluence of concepts may jointly kickstart the development of topological quantum programming proper as well as of real-world application of homotopy type theory, both of which have arguably been falling behind their high expectations; in any case, it provides a powerful paradigm for simulating and verifying topological quantum computing architectures with high-level certification languages aware of the actual physical principles of realistic topological quantum hardware. In companion articles (Sati and Schreiber in The Quantum Monadology, [arXiv:2310.15735], Sati and Schreiber in Entanglement of Sections: The pushout of entangled and parameterized quantum information [arXiv:2309.07245]) [announced in Schreiber (Quantum types via Linear Homotopy Type Theory, talk at Workshop on Quantum Software @ QTML2022, Naples, 2022, [ncatlab.org/schreiber/files/QuantumDataInLHoTT-221117.pdf])], we explain how further passage to “dependent linear” homotopy types naturally extends this scheme to a full-blown quantum programming/certification language in which our topological quantum gates may be compiled to verified quantum circuits, complete with quantum measurement gates and classical control.

We study the asymptotic growth of the degeneracy of the BPS local operators with scaling dimension n/2 in the three-dimensional superconformal field theories describing N M2-branes. From the large N supersymmetric indices we obtain the asymptotic formulas for degeneracies of the M2-brane SCFTs according to the Meinardus theorem. We observe an intriguing universal (n^{2/3}) growth of the degeneracies in various theories of M2-brane SCFTs. We also determine the coefficients of (n^{2/3}) growth as well as further corrections in these theories explicitly.