8,369 research outputs found
Mirror Maps, Modular Relations and Hypergeometric Series I
Motivated by the recent work of Kachru-Vafa in string theory, we study in
Part A of this paper, certain identities involving modular forms,
hypergeometric series, and more generally series solutions to Fuchsian
equations. The identity which arises in string theory is the simpliest of its
kind. There are nontrivial generalizations of the identity which appear new. We
give many such examples -- all of which arise in mirror symmetry for algebraic
K3 surfaces.
In Part B, we study the integrality property of certain -series, known as
mirror maps, which arise in mirror symmetry.Comment: 24 pages; harvma
Mirror Maps, Modular Relations and Hypergeometric Series II
As a continuation of \lianyaufour, we study modular properties of the
periods, the mirror maps and Yukawa couplings for multi-moduli Calabi-Yau
varieties. In Part A of this paper, motivated by the recent work of
Kachru-Vafa, we degenerate a three-moduli family of Calabi-Yau toric varieties
along a codimension one subfamily which can be described by the vanishing of
certain Mori coordinate, corresponding to going to the ``large volume limit''
in a certain direction. Then we see that the deformation space of the subfamily
is the same as a certain family of K3 toric surfaces. This family can in turn
be studied by further degeneration along a subfamily which in the end is
described by a family of elliptic curves. The periods of the K3 family (and
hence the original Calabi-Yau family) can be described by the squares of the
periods of the elliptic curves. The consequences include: (1) proofs of various
conjectural formulas of physicists \vk\lkm~ involving mirror maps and modular
functions; (2) new identities involving multi-variable hypergeometric series
and modular functions -- generalizing \lianyaufour. In Part B, we study for
two-moduli families the perturbation series of the mirror map and the type A
Yukawa couplings near certain large volume limits. Our main tool is a new class
of polynomial PDEs associated with Fuchsian PDE systems. We derive the first
few terms in the perturbation series. For the case of degree 12 hypersurfaces
in , in one limit the series of the couplings are expressed in
terms of the function. In another limit, they are expressed in terms of
rational functions. The latter give explicit formulas for infinite sequences of
``instanton numbers'' .Comment: 27 pages; harvma
Small Instantons in and Sigma Models
The anomalous scaling behavior of the topological susceptibility in
two-dimensional sigma models for is studied using the
overlap Dirac operator construction of the lattice topological charge density.
The divergence of in these models is traced to the presence of small
instantons with a radius of order (= lattice spacing), which are directly
observed on the lattice. The observation of these small instantons provides
detailed confirmation of L\"{u}scher's argument that such short-distance
excitations, with quantized topological charge, should be the dominant
topological fluctuations in and , leading to a divergent
topological susceptibility in the continuum limit. For the \CP models with
the topological susceptibility is observed to scale properly with the
mass gap. These larger models are not dominated by instantons, but rather
by coherent, one-dimensional regions of topological charge which can be
interpreted as domain wall or Wilson line excitations and are analogous to
D-brane or ``Wilson bag'' excitations in QCD. In Lorentz gauge, the small
instantons and Wilson line excitations can be described, respectively, in terms
of poles and cuts of an analytic gauge potential.Comment: 33 pages, 12 figure
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