103 research outputs found
On -derivatives and biextensions of Calabi-Yau motives
We prove that certain differential operators of the form with
hypergeometric and are of Picard-Fuchs type. We give closed
hypergeometric expressions for minors of the biextension period matrices that
arise from certain rank 4 weight 3 Calabi-Yau motives presumed to be of
analytic rank 1. We compare their values numerically to the first derivative of
the -functions of the respective motives at
Quadratic Q-curves, units and Hecke L-values
Abstract We show that if K is a quadratic field, and if there exists a quadratic Q-curve E/K
of prime degree N, satisfying weak conditions, then any unit u of OK satisfies a congruence
ur ≡ 1 (mod N), where r = g.c.d.(N − 1, 12). If K is imaginary quadratic, we prove a
congruence, modulo a divisor of N, between an algebraic Hecke character ψ˜ and, roughly
speaking, the elliptic curve. We show that this divisor then occurs in a critical value L(ψ , ˜ 2),
by constructing a non-zero element in a Selmer group and applying a theorem of Kato
Quantum cohomology and the Satake isomorphism
We prove that the geometric Satake correspondence admits quantum corrections
for minuscule Grassmannians of Dynkin types and . We find, as a
corollary, that the quantum connection of a spinor variety can be
obtained as the half-spinorial representation of that of the quadric
. We view the (quantum) cohomology of these Grassmannians as endowed
simultaneously with two structures, one of a module over the algebra of
symmetric functions, and the other, of a module over the Langlands dual Lie
algebra, and investigate the interaction between the two. In particular, we
study primitive classes in the cohomology of a minuscule Grassmannian
that are characterized by the condition that the operator of cup product by
is in the image of the Lie algebra action. Our main result states that quantum
correction preserves primitivity. We provide a quantum counterpart to a result
obtained by V. Ginzburg in the classical setting by giving explicit formulas
for the quantum corrections to homogeneous primitive elements
Macroscopic Quantum Tunneling in Small Antiferromagnetic Particles: Effects of a Strong Magnetic Field
We consider an effect of a strong magnetic field on the ground state and
macroscopic coherent tunneling in small antiferromagnetic particles with
uniaxial and biaxial single-ion anisotropy. We find several tunneling regimes
that depend on the direction of the magnetic field with respect to the
anisotropy axes. For the case of a purely uniaxial symmetry and the field
directed along the easy axis, an exact instanton solution with two different
scales in imaginary time is constructed. For a rhombic anisotropy the effect of
the field strongly depends on its orientation: with the field increasing, the
tunneling rate increases or decreases for the field parallel to the easy or
medium axis, respectively. The analytical results are complemented by numerical
simulations.Comment: 11 pages, 6 figure
Dimensional interpolation and the Selberg integral
We show that a version of dimensional interpolation for the Riemann–Roch–Hirzebruch formalism in the case of a Grassmannian leads to an expression for the Euler characteristic of line bundles in terms of a Selberg integral. We propose a way to interpolate higher Bessel equations, their wedge powers, and monodromies thereof to non-integer orders, and link the result with the dimensional interpolation of the RRH formalism in the spirit of the gamma conjectures
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