The degeneration formula for stable log maps

We give a direct proof for the degeneration formula of Gromov–Witten invariants including its cycle version for degenerations with smooth singular locus in the setting of stable log maps of Abramovich-Chen, Chen, Gross–Siebert.

INTERSECTION COHOMOLOGY OF THE MODULI SPACE OF HIGGS BUNDLES ON A GENUS 2 CURVE

Let C be a smooth projective curve of genus $2$ . Following a method by O’Grady, we construct a semismall desingularisation $\tilde {\mathcal {M}}_{Dol}^G$ of the moduli space $\mathcal {M}_{Dol}^G$ of semistable G-Higgs bundles of degree 0 for $G=\mathrm {GL}(2,\mathbb {C}), \mathrm {SL}(2,\mathbb {C})$ . By the decomposition theorem of Beilinson, Bernstein and Deligne, one can write the cohomology of $\tilde {\mathcal {M}}_{Dol}^G$ as a direct sum of the intersection cohomology of $\mathcal {M}_{Dol}^G$ plus other summands supported on the singular locus. We use this splitting to compute the intersection cohomology of $\mathcal {M}_{Dol}^G$ and prove that the mixed Hodge structure on it is pure, in analogy with what happens to ordinary cohomology in the smooth case of coprime rank and degree.

Monodromy of projections of hypersurfaces

Let X be an irreducible, reduced complex projective hypersurface of degree d. A point P not contained in X is called uniform if the monodromy group of the projection of X from P is isomorphic to the symmetric group $$S_d$$ S d . We prove that the locus of non-uniform points is finite when X is smooth or a general projection of a smooth variety. In general, it is contained in a finite union of linear spaces of codimension at least 2, except possibly for a special class of hypersurfaces with singular locus linear in codimension 1. Moreover, we generalise a result of Fukasawa and Takahashi on the finiteness of Galois points.

Numerator seagull and extended Symmetries of Feynman Integrals

The Symmetries of Feynman Integrals (SFI) method is extended for the first time to incorporate an irreducible numerator. This is done in the context of the so-called vacuum and propagator seagull diagrams, which have 3 and 2 loops, respectively, and both have a single irreducible numerator. For this purpose, an extended version of SFI (xSFI) is developed. For the seagull diagrams with general masses, the SFI equation system is found to extend by two additional equations. The first is a recursion equation in the numerator power, which has an alternative form as a differential equation for the generating function. The second equation applies only to the propagator seagull and does not involve the numerator. We solve the equation system in two cases: over the singular locus and in a certain 3 scale sector where we obtain novel closed-form evaluations and epsilon expansions, thereby extending previous results for the numerator-free case.

Global propagation of singularities for discounted Hamilton-Jacobi equations

<p style='text-indent:20px;'>The main purpose of this paper is to study the global propagation of singularities of the viscosity solution to discounted Hamilton-Jacobi equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE333"> \begin{document}$ \begin{align} \lambda v(x)+H( x, Dv(x) ) = 0 , \quad x\in \mathbb{R}^n. \quad\quad\quad (\mathrm{HJ}_{\lambda})\end{align} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>with fixed constant <inline-formula><tex-math id="M1">\begin{document}$ \lambda\in \mathbb{R}^+ $\end{document}</tex-math></inline-formula>. We reduce the problem for equation <inline-formula><tex-math id="M2">\begin{document}$(\mathrm{HJ}_{\lambda})$\end{document}</tex-math></inline-formula> into that for a time-dependent evolutionary Hamilton-Jacobi equation. We prove that the singularities of the viscosity solution of <inline-formula><tex-math id="M3">\begin{document}$(\mathrm{HJ}_{\lambda})$\end{document}</tex-math></inline-formula> propagate along locally Lipschitz singular characteristics <inline-formula><tex-math id="M4">\begin{document}$ {{\bf{x}}}(s):[0,t]\to \mathbb{R}^n $\end{document}</tex-math></inline-formula> and time <inline-formula><tex-math id="M5">\begin{document}$ t $\end{document}</tex-math></inline-formula> can extend to <inline-formula><tex-math id="M6">\begin{document}$ +\infty $\end{document}</tex-math></inline-formula>. Essentially, we use <inline-formula><tex-math id="M7">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula>-compactness of the Euclidean space which is different from the original construction in [<xref ref-type="bibr" rid="b4">4</xref>]. The local Lipschitz issue is a key technical difficulty to study the global result. As a application, we also obtain the homotopy equivalence between the singular locus of <inline-formula><tex-math id="M8">\begin{document}$ u $\end{document}</tex-math></inline-formula> and the complement of Aubry set using the basic idea from [<xref ref-type="bibr" rid="b9">9</xref>].</p>

A Characterization of Secant Varieties of Severi Varieties Among Cubic Hypersurfaces

It is shown that an irreducible cubic hypersurface with nonzero Hessian and smooth singular locus is the secant variety of a Severi variety if and only if its Lie algebra of infinitesimal linear automorphisms admits a nonzero prolongation.

Towards the classification of rank-r $$ \mathcal{N} $$ = 2 SCFTs. Part I. Twisted partition function and central charge formulae

We derive explicit formulae to compute the a and c central charges of four dimensional $$ \mathcal{N} $$ N = 2 superconformal field theories (SCFTs) directly from Coulomb branch related quantities. The formulae apply at arbitrary rank. We also discover general properties of the low-energy limit behavior of the flavor symmetry of $$ \mathcal{N} $$ N = 2 SCFTs which culminate with our $$ \mathcal{N} $$ N = 2 UV-IR simple flavor condition. This is done by determining precisely the relation between the integrand of the partition function of the topologically twisted version of the 4d $$ \mathcal{N} $$ N = 2 SCFTs and the singular locus of their Coulomb branches. The techniques developed here are extensively applied to many rank-2 SCFTs, including new ones, in a companion paper.This manuscript is dedicated to the memory of Rayshard Brooks, George Floyd, Breonna Taylor and the countless black lives taken by US police forces and still awaiting justice. Our hearts are with our colleagues of color who suffer daily the consequences of this racist world.

Towards a classification of rank r $$ \mathcal{N} $$ = 2 SCFTs. Part II. Special Kahler stratification of the Coulomb branch

We study the stratification of the singular locus of four dimensional $$ \mathcal{N} $$ N = 2 Coulomb branches. We present a set of self-consistency conditions on this stratification which can be used to extend the classification of scale-invariant rank 1 Coulomb branch geometries to two complex dimensions, and beyond. The calculational simplicity of the arguments presented here stems from the fact that the main ingredients needed — the rank 1 deformation patterns and the pattern of inclusions of rank 2 strata — are discrete topological data which satisfy strong self-consistency conditions through their relationship to the central charges of the SCFT. This relationship of the stratification data to the central charges is used here, but is derived and explained in a companion paper [1] by one of the authors. We illustrate the use of these conditions by re-analyzing many previously-known examples of rank 2 SCFTs, and also by finding examples of new theories. The power of these conditions stems from the fact that for Coulomb branch stratifications a conjecturally complete list of physically allowed “elementary slices” is known. By contrast, constraining the possible elementary slices of symplectic singularities relevant for Higgs branch stratifications remains an open problem.

Algebraic analysis of the hypergeometric function $$\,{_1F_{\!\!\;1}}\,$$ of a matrix argument

In this article, we investigate Muirhead’s classical system of differential operators for the hypergeometric function $$\,{_1F_{\!\!\;1}}\,$$ 1 F 1 of a matrix argument. We formulate a conjecture for the combinatorial structure of the characteristic variety of its Weyl closure which is both supported by computational evidence as well as theoretical considerations. In particular, we determine the singular locus of this system.