Jintai DingCPS Lab, Chongqing University, China; Department of Mathematical Sciences, University of Cincinnati, USACrystal CloughDepartment of Mathematical Sciences, University of Cincinnati, USARoberto AraujoFaculdade de Computação, Universidade Federal do Pará, Brazil
Information TheoryAlgebraic Geometrymathscidoc:2207.19001
Finite Fields and Their Applications, 26, 32-48, 2014.3
In this paper, we prove that the degree of regularity of square systems, a subfamily of the HFE systems, over a prime finite field of odd characteristic q is exactly q and, therefore, prove that inverting square systems algebraically using Gröbner basis algorithm is exponential, when q = \Omega(n), where n is the number of variables of the system.
Sibasish BanerjeeWeyertal 86-90, Department of Mathematics, University of Cologne, 50679, Cologne, Germany; Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, GermanyPietro LonghiInstitute for Theoretical Physics, ETH Zurich, 8093, Zurich, SwitzerlandMauricio Andrés Romo JorqueraYau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China
Symplectic GeometryAlgebraic GeometryarXiv subject: High Energy Physics - Theory (hep-th)mathscidoc:2207.34001
This paper studies a notion of enumerative invariants for stable A-branes, and discusses its relation to invariants defined by spectral and exponential networks. A natural definition of stable A-branes and their counts is provided by the string theoretic origin of the topological A-model. This is the Witten index of the supersymmetric quantum mechanics of a single D3 brane supported on a special Lagrangian in a Calabi-Yau threefold. Geometrically, this is closely related to the Euler characteristic of the A-brane moduli space. Using the natural torus action on this moduli space, we reduce the computation of its Euler characteristic to a count of fixed points via equivariant localization. Studying the A-branes that correspond to fixed points, we make contact with definitions of spectral and exponential networks. We find agreement between the counts defined via the Witten index, and the BPS invariants defined by networks. By extension, our definition also matches with Donaldson-Thomas invariants of B-branes related by homological mirror symmetry.
We study hybrid models arising as homological projective duals (HPD) of certain projective embeddings f:X→P(V) of Fano manifolds X. More precisely, the category of B-branes of such hybrid models corresponds to the HPD category of the embedding f. B-branes on these hybrid models can be seen as global matrix factorizations over some compact space B or, equivalently, as the derived category of the sheaf of A-modules on B, where A is an A_∞ algebra. This latter interpretation corresponds to a noncommutative resolution of B. We compute explicitly the algebra A by several methods, for some specific class of hybrid models, and find that in general it takes the form of a smash product of an A_∞ algebra with a cyclic group. Then we apply our results to the HPD of f corresponding to a Veronese embedding of projective space and the projective embedding of Fano complete intersections in P^n.