Cyclic stratum of Frobenius manifolds, Borel-Laplace (α,β)(\boldsymbol\alpha,\boldsymbol\beta)-multitransforms, and integral representations of solutions of quantum differential equations

In the first part of this paper, we introduce the notion of cyclic stratum of a Frobenius manifold M\smash{M}M. This is the set of points of the extended manifold C∗×M\smash{\mathbb C^*\times M}C∗×M at which the unit vector field is a cyclic vector for the isomonodromic system defined by the flatness condition of the extended deformed connection. The study of the geometry of the complement of the cyclic stratum is addressed. We show that at points of the cyclic stratum, the isomonodromic system attached to M\smash{M}M can be reduced to a scalar differential equation, called the master differential equation of M\smash{M}M. In the case of Frobenius manifolds coming from Gromov–Witten theory, namely quantum cohomologies of smooth projective varieties, such a construction reproduces the notion of quantum differential equation.In the second part of the paper, we introduce two multilinear transforms, called Borel–Laplace (α,β)(\boldsymbol{\alpha}, \boldsymbol{\beta})(α,β)-multitransforms, on spaces of Ribenboim formal power series with exponents and coefficients in an arbitrary finite-dimensional C\smash{\mathbb C}C-algebra A\smash{A}A. When A\smash{A}A is specialized to the cohomology of smooth projective varieties, the integral forms of the Borel–Laplace (α,β)(\boldsymbol{\alpha}, \boldsymbol{\beta})(α,β)-multitransforms are used in order to rephrase the Quantum Lefschetz theorem. This leads to explicit Mellin–Barnes integral representations of solutions of the quantum differential equations for a wide class of smooth projective varieties, including Fano complete intersections in projective spaces.In the third and final part of the paper, as an application, we show how to use the new analytic tools, introduced in the previous parts, in order to study the quantum differential equations of Hirzebruch surfaces. For Hirzebruch surfaces diffeomorphic to P1×P1\smash{\mathbb P^1\times \mathbb P^1}P1×P1, this analysis reduces to the simpler quantum differential equation of P1\smash{\mathbb P^1}P1. For Hirzebruch surfaces diffeomorphic to the blow-up of P2\smash{\mathbb P^2}P2 in one point, the quantum differential equation is integrated via Laplace (1,2;12,13)\smash{(1,2;\frac{1}{2},\frac{1}{3})}(1,2;21​,31​)-multitransforms of solutions of the quantum differential equations of P1\smash{\mathbb P^1}P1 and P2\smash{\mathbb P^2}P2, respectively. This leads to explicit integral representations for the Stokes bases of solutions of the quantum differential equations, and finally to the proof of the Dubrovin conjecture for all Hirzebruch surfaces

The ∗*-Markov equation for Laurent polynomials

We consider the $*$-Markov equation for the symmetric Laurent polynomials in three variables with integer coefficients, which is an equivariant analog of the classical Markov equation for integers. We study how the properties of the Markov equation and its solutions are reflected in the properties of the $*$-Markov equation and its solutions

Equivariant quantum differential equation and qKZqKZ equations for a projective space: Stokes bases as exceptional collections, Stokes matrices as Gram matrices, and B-Theorem

In the previous paper by Tarasov and Varchenko the equivariant quantum differential equation ($qDE$) for a projective space was considered and a compatible system of difference $qKZ$ equations was introduced; the space of solutions to the joint system of the $qDE$ and $qKZ$ equations was identified with the space of the equivariant $K$-theory algebra of the projective space; Stokes bases in the space of solutions were identified with exceptional bases in the equivariant $K$-theory algebra. This paper is a continuation of the paper by Tarasov and Varchenko. We describe the relation between solutions to the joint system of the $qDE$ and $qKZ$ equations and the topological-enumerative solution to the $qDE$ only, defined as a generating function of equivariant descendant Gromov-Witten invariants. The relation is in terms of the equivariant graded Chern character on the equivariant $K$-theory algebra, the equivariant Gamma class of the projective space, and the equivariant first Chern class of the tangent bundle of the projective space. We consider a Stokes basis, the associated exceptional basis in the equivariant $K$-theory algebra, and the associated Stokes matrix. We show that the Stokes matrix equals the Gram matrix of the equivariant Grothendieck-Euler-Poincar\'{e} pairing wrt to the basis, which is the left dual to the associated exceptional basis. We identify the Stokes bases in the space of solutions with explicit full exceptional collections in the equivariant derived category of coherent sheaves on the projective space, where the elements of those exceptional collections are just line bundles on the projective space and exterior powers of the tangent bundle of the projective space. These statements are equivariant analogs of results of G. Cotti, B. Dubrovin, D. Guzzetti, and S. Galkin, V. Golyshev, H. Iritani

A Note on Solid-State Maxwell Demon

Starting from 2002, at least two kinds of laboratory-testable, solid-state Maxwell demons have been proposed that utilize the electric field energy of an open-gap n-p junction and that seem to challenge the validity of the Second Law of Thermodynamics. In the present paper we present some arguments against the alleged functioning of such devices.Comment: 9 pages, 4 figures. Foundations of Physics, forthcoming. arXiv admin note: substantial text overlap with arXiv:1101.505

Constraints on the parameters of the Left Right Mirror Model

We study some phenomenological constraints on the parameters of a left right model with mirror fermions (LRMM) that solves the strong CP problem. In particular, we evaluate the contribution of mirror neutrinos to the invisible Z decay width (\Gamma_Z^{inv}), and we find that the present experimental value on \Gamma_Z^{inv}, can be used to place an upper bound on the Z-Z' mixing angle that is consistent with limits obtained previously from other low-energy observables. In this model the charged fermions that correspond to the standard model (SM) mix with its mirror counterparts. This mixing, simultaneously with the Z-Z' one, leads to modifications of the \Gamma(Z --> f \bar{f}) decay width. By comparing with LEP data, we obtain bounds on the standard-mirror lepton mixing angles. We also find that the bottom quark mixing parameters can be chosen to fit the experimental values of R_b, and the resulting values for the Z-Z' mixing angle do not agree with previous bounds. However, this disagreement disappears if one takes the more recent ALEPH data.Comment: 7 pages, 2 figures, REVTe

New Higgs signals induced by mirror fermion mixing effects

We study the conditions under which flavor violation arises in scalar-fermion interactions, as a result of the mixing phenomena between the standard model and exotic fermions. Phenomenological consequences are discussed within the specific context of a left-right model where these additional fermions have mirror properties under the new SU(2)_R gauge group. Bounds on the parameters of the model are obtained from LFV processes; these results are then used to study the LFV Higgs decays (H --> tau l_j, l_j = e, mu), which reach branching ratios that could be detected at future colliders.Comment: 12 pages, 2 figures, ReVTex4, graphicx, to be published in Phys. Rev.