Explicit Computation of a Galois Representation Attached to an Eigenform Over $${\text {SL}}_3$$ from the $${{\text {H}}}_{\acute{\mathrm{e}}\mathrm{t}}^2$$ of a Surface

We describe a method to compute mod $$\ell $$ ℓ Galois representations contained in the $${{\text {H}}}_{\acute{\mathrm{e}}\mathrm{t}}^2$$ H e ´ t 2 of surfaces. We apply this method to the case of a representation with values in $${\text {GL}}_3(\mathbb {F}_9)$$ GL 3 ( F 9 ) attached to an eigenform over a congruence subgroup of $${\text {SL}}_3$$ SL 3 . We obtain, in particular, a polynomial with Galois group isomorphic to the simple group $${\text {PSU}}_3(\mathbb {F}_9)$$ PSU 3 ( F 9 ) and ramified at 2 and 3 only.

Counting D1-D5-P microstates in supergravity

We quantize the D1-D5-P microstate geometries known as superstrata directly in supergravity. We use Rychkov's consistency condition [hep-th/0512053] which was derived for the D1-D5 system; for superstrata, this condition turns out to be strong enough to fix the symplectic form uniquely. For the (1,0,n) superstrata, we further confirm this quantization by a bona-fide explicit computation of the symplectic form using the semi-classical covariant quantization method in supergravity. We use the resulting quantizations to count the known supergravity superstrata states, finding agreement with previous countings that the number of these states grows parametrically smaller than those of the corresponding black hole.

Real-time warm pions from the lattice using an effective theory

Lattice measurements provide adequate information to fix the parameters of long-distance effective field theories in Euclidean time. Using such a theory, we examine the analytic continuation of long-distance correlation functions of composite operators at finite temperature from Euclidean to Minkowski space–time. We show through an explicit computation that the analytic continuation of the pion correlation function is possible and gives rise to nontrivial effects. Among them is the possibility, supported by lattice computations of Euclidean correlators, that long distance excitations can be understood in terms of (very massive) pions even at temperatures higher than the QCD crossover temperature.

Six-point conformal blocks in the snowflake channel

We compute d-dimensional scalar six-point conformal blocks in the two possible topologies allowed by the operator product expansion. Our computation is a simple application of the embedding space operator product expansion formalism developed recently. Scalar six-point conformal blocks in the comb channel have been determined not long ago, and we present here the first explicit computation of the scalar six-point conformal blocks in the remaining inequivalent topology. For obvious reason, we dub the other topology the snowflake channel. The scalar conformal blocks, with scalar external and exchange operators, are presented as a power series expansion in the conformal cross-ratios, where the coefficients of the power series are given as a double sum of the hypergeometric type. In the comb channel, the double sum is expressible as a product of two 3F2-hypergeometric functions. In the snowflake channel, the double sum is expressible as a Kampé de Fériet function where both sums are intertwined and cannot be factorized. We check our results by verifying their consistency under symmetries and by taking several limits reducing to known results, mostly to scalar five-point conformal blocks in arbitrary spacetime dimensions.

Explicit computation of some families of Hurwitz numbers, II

We continue our computation, using a combinatorial method based on Gronthendieck’s dessins d’enfant, of the number of (weak) equivalence classes of surface branched covers matching certain specific branch data. In this note we concentrate on data with the surface of genus g as source surface, the sphere as target surface, 3 branching points, degree 2k, and local degrees over the branching points of the form [2, …, 2], [2h + 1, 3, 2, …, 2], $\begin{array}{} \displaystyle \pi=[d_i]_{i=1}^\ell. \end{array}$ We compute the corresponding (weak) Hurwitz numbers for several values of g and h, getting explicit arithmetic formulae in terms of the di’s.

Proposed measurement of simultaneous particle and wave properties of electric current in a superconductor

In a microscopic quantum system one cannot perform a simultaneous measurement of particle and wave properties. This, however, may not be true for macroscopic quantum systems. As a demonstration, we propose to measure the local macroscopic current passed through two slits in a superconductor. According to the theory based on the linearized Ginzburg–Landau equation for the macroscopic pseudo wave function, the streamlines of the measured current should have the same form as particle trajectories in the Bohmian interpretation of quantum mechanics. By an explicit computation we find that the streamlines should show a characteristic wiggling, which is a consequence of quantum interference.

A sparsity augmented probabilistic collaborative representation based classification method

In order to enhance the performance of image recognition, a sparsity augmented probabilistic collaborative representation based classification (SA-ProCRC) method is presented. The proposed method obtains the dense coefficient through ProCRC, then augments the dense coefficient with a sparse one, and the sparse coefficient is attained by the orthogonal matching pursuit (OMP) algorithm. In contrast to conventional methods which require explicit computation of the reconstruction residuals for each class, the proposed method employs the augmented coefficient and the label matrix of the training samples to classify the test sample. Experimental results indicate that the proposed method can achieve promising results for face and scene images. The source code of our proposed SA-ProCRC is accessible at https://github.com/yinhefeng/SAProCRC

Computation of the unipotent Albanese map on elliptic and hyperelliptic curves

We study the unipotent Albanese map appearing in the non-abelian Chabauty method of Minhyong Kim. In particular we explore the explicit computation of the p-adic de Rham period map $$j^{dr}_n$$ j n dr on elliptic and hyperelliptic curves over number fields via their universal unipotent connections $${\mathscr {U}}$$ U . Several algorithms forming part of the computation of finite level versions $$j^{dr}_n$$ j n dr of the unipotent Albanese maps are presented. The computation of the logarithmic extension of $${\mathscr {U}}$$ U in general requires a description in terms of an open covering, and can be regarded as a simple example of computational descent theory. We also demonstrate a constructive version of a lemma of Hadian used in the computation of the Hodge filtration on $${\mathscr {U}}$$ U over affine elliptic and odd hyperelliptic curves. We use these algorithms to present some new examples describing the co-ordinates of some of these period maps. This description will be given in terms iterated p-adic Coleman integrals. We also consider the computation of the co-ordinates if we replace the rational basepoint with a tangential basepoint, and present some new examples here as well.