214 research outputs found
Naive boundary strata and nilpotent orbits
We study certain real Lie-group orbits in the compact duals of Mumford-Tate
domains, verifying a prediction made in [Green, Griffiths, Kerr; Mumford-Tate
domains: their geometry and arithmetic] and determining which orbits contain a
limit point of some period map. A variety of examples are worked out for the
groups SU(2,1), Sp_4, and G_2.Comment: 57 pages, 34 figure
Polarized relations on horizontal SL(2)s
We introduce a relation on real conjugacy classes of SL(2)-orbits in a
Mumford-Tate domain D which is compatible with natural partial orders on the
sets of nilpotent orbits in the corresponding Lie algebra and boundary orbits
in the compact dual. A generalization of the SL(2)-orbit theorem to such
domains leads to an algorithm for computing this relation, which is worked out
in several examples and special cases including period domains, Hermitian
symmetric domains, and complete flag domains, and used to define a poset of
equivalence classes of multivariable nilpotent orbits on D.Comment: 65 pages; version to appear in Documenta Mathematic
Local mirror symmetry and the sunset Feynman integral
We study the sunset Feynman integral defined as the scalar two-point
self-energy at two-loop order in a two dimensional space-time.
We firstly compute the Feynman integral, for arbitrary internal masses, in
terms of the regulator of a class in the motivic cohomology of a 1-parameter
family of open elliptic curves. Using an Hodge theoretic (B-model) approach, we
show that the integral is given by a sum of elliptic dilogarithms evaluated at
the divisors determined by the punctures.
Secondly we associate to the sunset elliptic curve a local non-compact
Calabi-Yau 3-fold, obtained as a limit of elliptically fibered compact
Calabi-Yau 3-folds. By considering the limiting mixed Hodge structure of the
Batyrev dual A-model, we arrive at an expression for the sunset Feynman
integral in terms of the local Gromov-Witten prepotential of the del Pezzo
surface of degree 6. This expression is obtained by proving a strong form of
local mirror symmetry which identifies this prepotential with the second
regulator period of the motivic cohomology class.Comment: 67 pages. v2: minor typos corrected and now per-section numbering of
theorems, lemmas, propositions and remarks. v3: minor typos corrected.
Version to appear in Advances in Theoretical and Mathematical Physic
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