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A cooriented circle immersion into the plane can be extended to a stable map of the disk which is an immersion in a neighborhood of the boundary and with outward normal vector field along the boundary equal to the given coorienting normal vector field. We express the minimal number of fold components of such a stable map as a function of its number of cusps and of the normal degree of its boundary. We also show that this minimum is attained for any cooriented circle immersion of normal degree not equal to one.

String theory contains solutions with SL(2,R)_R x U(1)_L-invariant warped AdS3 (WAdS3) factors arising as continuous deformations of ordinary AdS3 factors. We propose that some of these are holographically dual to the IR limits of nonlocal dipole-deformed 2D D-brane gauge theories, referred to as "dipole CFTs". Neither the bulk nor boundary theories are currently well-understood, and consequences of the proposed duality for both sides is investigated. The bulk entropy-area law suggests that dipole CFTs have (at large N) a high-energy density of states which does not depend on the deformation parameter. Putting the boundary theory on a spatial circle leads to closed timelike curves in the bulk, suggesting a relation of the latter to dipole-type nonlocality.

In this paper, we give a uniform construction of irreducible genuine characters of the Pin cover W of a Weyl group W, and put them into the context of theory of Springer representations. In the process, we provide a direct connection between Springer theory, via Green polynomials, the irreducible representations of W, and an extended Dirac operator for graded Hecke algebras. We also introduce a q-elliptic pairing for W with respect to the reflection representation V. These constructions are of independent interest. The q-elliptic pairing is a generalization of the elliptic pairing of W introduced by Reeder, and it is also related to S. Kato's notion of (graded) Kostka systems for the semidirect product A W= C [W] S (V).

In this note, we give a list of Calabi-Yau hypersurfaces in weighted projective 4-spaces with the property that a hypersurface contains naturally a pencil of K3 variety. For completeness we also obtain a similar list in the case K3 hypersurfaces in weighted projective 3-spaces. The first list significantly enlarges the list of K3-fibrations of\KlemmLercheMayr~ which has been obtained on some assumptions on the weights. Our lists are expected to correspond to examples of the so-called heterotic-type II duality\KachruVafa\KachruSilverstein.