Abstract Let u u be a nontrivial harmonic function in a domain D ⊂ R d D\subset {{\mathbb{R}}}^{d} , which vanishes on an open set of the boundary. In a recent article, we showed that if D D is a C 1 {C}^{1} Dini domain, then, within the open set, the singular set of u u , defined as { X ∈ D ¯ : u ( X ) = 0 = ∣ ∇ u ( X ) ∣ } \left\{X\in \overline{D}:u\left(X)=0= \nabla u\left(X) \right\} , has finite ( d − 2 ) \left(d2) dimensional Hausdorff measure. In this article, we show that the assumption of C 1 {C}^{1} Dini domains is sharp, by constructing a large class of nonDini (but almost Dini) domains whose singular sets have infinite ℋ d − 2 {{\mathcal{ {\mathcal H} }}}^{d2} measures.
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Superpotentials from singular divisors
A bstract We study Euclidean D3branes wrapping divisors D in CalabiYau orientifold compactifications of type IIB string theory. Witten’s counting of fermion zero modes in terms of the cohomology of the structure sheaf $$ {\mathcal{O}}_D $$ O D applies when D is smooth, but we argue that effective divisors of CalabiYau threefolds typically have singularities along rational curves. We generalize the counting of fermion zero modes to such singular divisors, in terms of the cohomology of the structure sheaf $$ {\mathcal{O}}_{\overline{D}} $$ O D ¯ of the normalization $$ \overline{D} $$ D ¯ of D . We establish this by detailing compactifications in which the singularities can be unwound by passing through flop transitions, giving a physical incarnation of the normalization process. Analytically continuing the superpotential through the flops, we find that singular divisors whose normalizations are rigid can contribute to the superpotential: specifically, $$ {h}_{+}^{\bullet}\left({\mathcal{O}}_{\overline{D}}\right)=\left(1,0,0\right) $$ h + • O D ¯ = 1 0 0 and $$ {h}_{}^{\bullet}\left({\mathcal{O}}_{\overline{D}}\right)=\left(0,0,0\right) $$ h − • O D ¯ = 0 0 0 give a sufficient condition for a contribution. The examples that we present feature infinitely many isomorphic geometric phases, with corresponding infiniteorder monodromy groups Γ. We use the action of Γ on effective divisors to determine the exact effective cones, which have infinitely many generators. The resulting nonperturbative superpotentials are Jacobi theta functions, whose modular symmetries suggest the existence of strongweak coupling dualities involving inversion of divisor volumes.
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 NSFPAR ID:
 10434453
 Date Published:
 Journal Name:
 Journal of High Energy Physics
 Volume:
 2022
 Issue:
 11
 ISSN:
 10298479
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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