Explore the vast range of titles available.

A bstract We study the topological properties of Calabi-Yau threefold hypersurfaces at large h 1 , 1 . We obtain two million threefolds X by triangulating polytopes from the Kreuzer-Skarke list, including all polytopes with 240 ≤ h 1 , 1 ≤ 491. We show that the Kähler cone of X is very narrow at large h 1 , 1 , and as a consequence, control of the α′ expansion in string compactifications on X is correlated with the presence of ultralight axions. If every effective curve has volume ≥ 1 in string units, then the typical volumes of irreducible effective curves and divisors, and of X itself, scale as ( h 1 , 1 ) p , with 3 ≲ p ≲ 7 depending on the type of cycle in question. Instantons from branes wrapping these cycles are thus highly suppressed.

A bstract We study Euclidean D3-branes wrapping divisors D in Calabi-Yau orientifold compactifications of type IIB string theory. Witten’s counting of fermion zero modes in terms of the cohomology of the structure sheaf O D applies when D is smooth, but we argue that effective divisors of Calabi-Yau 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 O D ¯ of the normalization 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 + • O D ¯ = 1 0 0 and 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 infinite-order 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 strong-weak coupling dualities involving inversion of divisor volumes.

A bstract We show that the Weak Gravity Conjecture (WGC) implies a nontrivial upper bound on the volumes of the minimal-volume cycles in certain homology classes that admit no calibrated representatives. In compactification of type IIB string theory on an orientifold X of a Calabi-Yau threefold, we consider a homology class [Σ] ∈ H 4 ( X, ℝ ) represented by a union Σ ∪ of holomorphic and antiholomorphic cycles. The instanton form of the WGC applied to the axion charge [Σ] implies an upper bound on the action of a non-BPS Euclidean D3-brane wrapping the minimal-volume representative Σ min of [Σ]. We give an explicit example of an orientifold X of a hypersurface in a toric variety, and a hyperplane H ⊂ H 4 ( X, ℝ ), such that for any [Σ] ∈ H that satisfies the WGC, the minimal volume obeys Vol (Σ min ) ≪ Vol(Σ ∪ ): the holomorphic and antiholomorphic components recombine to form a much smaller cycle. In particular, the sub-Lattice WGC applied to X implies large recombination, no matter how sparse the sublattice. Non-BPS instantons wrapping Σ min are then more important than would be predicted from a study of BPS instantons wrapping the separate components of Σ ∪ . Our analysis hinges on a novel computation of effective divisors in X that are not inherited from effective divisors of the toric variety.

Entries from a physician's diary documenting his experience treating patients and dealing with death are presented. Stillman hopes that when it's his time to go he won't give up needlessly nor fight too long.

Stillman considers the meaning of professionalism in today's primary care setting and discusses how this became real to him after treating two patients. He learned that medicine is a profession in which fear of failure and self-doubt are our close companions and which at times stretches the very physicians to their limits.

We study the linear syzygies of a homogeneous ideal I ̱ϲ S = Sym k (V), focussing on the graded betti numbers b i,i+1 =dim k Tor i (S/I,K) i+1 For a variety X and divisor D with V = H O (D), what conditions on D ensure that bi,i+1 what conditions on D ensure that b i,i 1 0 Eisenbud has shown that a decomposition D ~ A + B such that A and B have at two sections gives rise to determinantal equations (and corresponding syzygies) in I x ; and conjectured that if I₂ is generated by quadrics of rank = 4, then the last nonvanishing b i,i 1 is a consequence of such equations. We describe obstructions to this conjecture and prove a variant. The obstuctions arise from toric specializations of the Rees algebra of Koszul cycles, and we give and explicit construction of toric varieties with minimal linear syzygies of arbitrarily high rank. This gives one answer to a question posed by Eisenbud and Koh.

For a vector space VV of homogeneous forms of the same degree in a polynomial ring, we investigate what can be said about the generic initial ideal of the ideal generated by VV, from the form of the generic initial space ginV\\mathrm {gin}{V} for the revlex order. Our main result is a considerable generalisation of a previous result by the first author.

We study Euclidean D3-branes wrapping divisors \\(D\\) in Calabi-Yau 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\\) applies when \\(D\\) is smooth, but we argue that effective divisors of Calabi-Yau 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}}\\) of the normalization \\(\\overline{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}_{+}(\\mathcal{O}_{\\overline{D}})=(1,0,0)\\) and \\(h^{\\bullet}_{-}(\\mathcal{O}_{\\overline{D}})=(0,0,0)\\) give a sufficient condition for a contribution. The examples that we present feature infinitely many isomorphic geometric phases, with corresponding infinite-order monodromy groups \\(\\Gamma\\). We use the action of \\(\\Gamma\\) 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 strong-weak coupling dualities involving inversion of divisor volumes.

We investigate algebraic and topological signatures of networks of coupled oscillators. Translating dynamics into a system of algebraic equations enables us to identify classes of network topologies that exhibit unexpected behaviors. Many previous studies focus on synchronization of networks having high connectivity, or of a specific type (e.g. circulant networks). We introduce the Kuramoto ideal; an algebraic analysis of this ideal allows us to identify features beyond synchronization, such as positive dimensional components in the set of potential solutions (e.g. curves instead of points). We prove sufficient conditions on the network structure for such solutions to exist. The points lying on a positive dimensional component of the solution set can never correspond to a linearly stable state. We apply this framework to give a complete analysis of linear stability for all networks on at most eight vertices. Furthermore, we describe a construction of networks on an arbitrary number of vertices having linearly stable states that are not twisted stable states.

We show that the Weak Gravity Conjecture (WGC) implies a nontrivial upper bound on the volumes of the minimal-volume cycles in certain homology classes that admit no calibrated representatives. In compactification of type IIB string theory on an orientifold \\(X\\) of a Calabi-Yau threefold, we consider a homology class \\([\\Sigma] \\in H_4(X,\\mathbb{Z})\\) represented by a union \\(\\Sigma_{\\cup}\\) of holomorphic and antiholomorphic cycles. The instanton form of the WGC applied to the axion charge \\([\\Sigma]\\) implies an upper bound on the action of a non-BPS Euclidean D3-brane wrapping the minimal-volume representative \\(\\Sigma_{\\mathrm{min}}\\) of \\([\\Sigma]\\). We give an explicit example of an orientifold \\(X\\) of a hypersurface in a toric variety, and a hyperplane \\(\\mathcal{H} \\subset H_4(X,\\mathbb{Z})\\), such that for any \\([\\Sigma] \\in H\\) that satisfies the WGC, the minimal volume obeys \\(\\mathrm{Vol}(\\Sigma_{\\mathrm{min}}) \\ll \\mathrm{Vol}(\\Sigma_{\\cup})\\): the holomorphic and antiholomorphic components recombine to form a much smaller cycle. In particular, the sub-Lattice WGC applied to \\(X\\) implies large recombination, no matter how sparse the sublattice. Non-BPS instantons wrapping \\(\\Sigma_{\\mathrm{min}}\\) are then more important than would be predicted from a study of BPS instantons wrapping the separate components of \\(\\Sigma_{\\cup}\\). Our analysis hinges on a novel computation of effective divisors in \\(X\\) that are not inherited from effective divisors of the toric variety.