Opening Mirror Symmetry on the Quintic

Aided by mirror symmetry, we determine the number of holomorphic disks ending on the real Lagrangian in the quintic threefold. The tension of the domainwall between the two vacua on the brane, which is the generating function for the open Gromov-Witten invariants, satisfies a certain extension of the Picard-Fuchs differential equation governing periods of the mirror quintic. We verify consistency of the monodromies under analytic continuation of the superpotential over the entire moduli space. We reproduce the first few instanton numbers by a localization computation directly in the A-model, and check Ooguri-Vafa integrality. This is the first exact result on open string mirror symmetry for a compact Calabi-Yau manifold.Comment: 26 pages. v2: minor corrections and improvement

Enumerative geometry of Calabi-Yau 4-folds

Gromov-Witten theory is used to define an enumerative geometry of curves in Calabi-Yau 4-folds. The main technique is to find exact solutions to moving multiple cover integrals. The resulting invariants are analogous to the BPS counts of Gopakumar and Vafa for Calabi-Yau 3-folds. We conjecture the 4-fold invariants to be integers and expect a sheaf theoretic explanation. Several local Calabi-Yau 4-folds are solved exactly. Compact cases, including the sextic Calabi-Yau in CP5, are also studied. A complete solution of the Gromov-Witten theory of the sextic is conjecturally obtained by the holomorphic anomaly equation.Comment: 44 page

Electromagnetic fields at the sea bottom induced by a line of immersed electric dipoles

The analysis of electromagnetic fields caused by alternate or transient electric currents flowing along a cable in sea water has several applications. It supports the interpretation of electromagnetic geophysical data and safety procedures against the threat of sea mines. The approach to the problem employs a magnetic vector potential in the frequency domain due to a pulse source electric dipole, and performs Laplace and Hankel transforms and integration along the cable, to describe the variation of the magnetic induction field due to an electric dipole of finite length. The result is applicable to shallow or deep sea water environments, adaptable to any transmitting current waveform and useful for wave-field separation. The prospects relate to a horizontal receiving coil at the sea bottom and simulate: a minesweeper campaign with a current source at the sea surface or a geophysical survey with a current source close to the sea floor. Therefore, the present analysis may serve: to define parameters in counter-sweeping of submarine mines; to map the conductivity of sediments under shallow waters for the prevention and control of contamination; and as a first approach in the characterization of offshore mineral and oil economic deposits.A análise de campos eletromagnéticos causados por correntes alternadas ou transientes fluindo ao longo de um cabo na água do mar tem várias aplicações. Ela prove suporte à interpretação de dados geofísicos eletromagnéticos e aos procedimentos de segurança contra a ameaça de minas submarinas. A abordagem do problema emprega um potencial vetorial magnético, no domínio da frequência, devido a um dipolo elétrico com uma fonte tipo pulso e calcula transformações de Laplace e de Hankel e integração ao longo do comprimento do cabo, para descrever a variação temporal do campo magnético de indução devido a um dipolo elétrico de comprimento finito. O resultado é aplicável em ambientes de água do mar rasa ou profunda, adaptável para qualquer forma de onda de corrente transmitida e útil para separação de campos de ondas. As configurações se referem a uma bobina receptora horizontal no fundo do mar e simulam: uma campanha de varredura de minas submarinas com uma fonte de corrente na superfície ou um levantamento geofísico com uma fonte de corrente próxima ao assoalho do mar. Portanto, a presente análise pode servir: para definir parâmetros em contra-varredura de minas submarinas; para mapear a condutividade de sedimentos sob águas rasas em prevenção e controle de contaminação e como uma primeira abordagem na caracterização de depósitos minerais e de hidrocarbonetos no mar

Equivariant Riemann-Roch theorems for curves over perfect fields

We prove an equivariant Riemann-Roch formula for divisors on algebraic curves over perfect fields. By reduction to the known case of curves over algebraically closed fields, we first show a preliminary formula with coefficients in Q. We then prove and shed some further light on a divisibility result that yields a formula with integral coefficients. Moreover, we give variants of the main theorem for equivariant locally free sheaves of higher rank