Multi-centered black holes, scaling solutions and pure-Higgs indices from localization

The Coulomb Branch Formula conjecturally expresses the refined Witten index for N=4 Quiver Quantum Mechanics as a sum over multi-centered collinear black hole solutions, weighted by so-called `single-centered' or `pure-Higgs' indices, and suitably modified when the quiver has oriented cycles. On the other hand, localization expresses the same index as an integral over the complexified Cartan torus and auxiliary fields, which by Stokes' theorem leads to the famous Jeffrey-Kirwan residue formula. Here, by evaluating the same integral using steepest descent methods, we show the index is in fact given by a sum over deformed multi-centered collinear solutions, which encompasses both regular and scaling collinear solutions. As a result, we confirm the Coulomb Branch Formula for Abelian quivers in the presence of oriented cycles, and identify the origin of the pure-Higgs and minimal modification terms as coming from collinear scaling solutions. For cyclic Abelian quivers, we observe that part of the scaling contributions reproduce the stacky invariants for trivial stability, a mathematically well-defined notion whose physics significance had remained obscure.

Local index theory for operators associated with Lie groupoid actions

We develop a local index theory for a class of operators associated with non-proper and non-isometric actions of Lie groupoids on smooth submersions. Such actions imply the existence of a short exact sequence of algebras, relating these operators to their non-commutative symbol. We then compute the connecting map induced by this extension on periodic cyclic cohomology. When cyclic cohomology is localized at appropriate isotropic submanifolds of the groupoid in question, we find that the connecting map is expressed in terms of an explicit Wodzicki-type residue formula, which involves the jets of non-commutative symbols at the fixed-point set of the action.

Holomorphic vector field and topological sigma model on ℂP1 worldsheet

Witten suggested that fixed-point theorems can be derived by the supersymmetric sigma model on a Riemann manifold [Formula: see text] with potential terms induced from a Killing vector on [Formula: see text].3. One of the well-known fixed-point theorems is the Bott residue formula9 which represents the intersection number of Chern classes of holomorphic vector bundles on a Kähler manifold [Formula: see text] as the sum of contributions from fixed point sets of a holomorphic vector field [Formula: see text] on [Formula: see text]. In this paper, we derive the Bott residue formula by using the topological sigma model (A-model) that describes dynamics of maps from [Formula: see text] to [Formula: see text], with potential terms induced from the vector field [Formula: see text]. Our strategy is to restrict phase space of path integral to maps homotopic to constant maps. As an effect of adding a potential term to the topological sigma model, we are forced to modify the BRST symmetry of the original topological sigma model. Our potential term and BRST symmetry are closely related to the idea used in the paper by Beasley and Witten2 where potential terms induced from holomorphic section of a holomorphic vector bundle and corresponding supersymmetry are considered.

ON THE RELATIVE FREQUENCY OF RESIDUE CLASSES IN PASCAL’S TRIANGLE MODULO A PRIME

When the entries of Pascal’s triangle which are congruent to a given nonzero residue modulo a fixed prime are mapped to corresponding locations of the unit square, a fractal-like structure emerges. In a previous publication, Bradley, Khalil, Niemeyer and Ossanna [The box-counting dimension of Pascal’s triangle [Formula: see text] mod [Formula: see text], Fractals 26(5) (2018) 1850071] showed that this mapping yields a nonempty compact set which can be realized as a limit of a sequence of sets representing incrementally refined approximations. Moreover, it was shown therein that for any fixed prime, the sequence converges to the same set, regardless of the nonzero residue or combination of nonzero residues considered. Consequently, the fractal (box-counting) dimension of the limiting set is independent of the residue. To study the relative frequency of various residue classes in the sequence of approximating sets, it would be desirable to have a closed-form formula for the number of entries in the first [Formula: see text] rows of Pascal’s triangle which are congruent to a given nonzero residue [Formula: see text] modulo the prime [Formula: see text]. Unfortunately, the numerical evidence presented in this paper supports the contention that there is no such formula. Nevertheless, the evidence indicates that for sufficiently large primes [Formula: see text], the number of entries congruent to [Formula: see text] for [Formula: see text], [Formula: see text], and [Formula: see text] is well approximated by the respective linear functions [Formula: see text], [Formula: see text], and [Formula: see text]. In particular, for large primes [Formula: see text] there are approximately six times as many occurrences of the residue [Formula: see text] in the first [Formula: see text] rows of Pascal’s triangle reduced modulo [Formula: see text] than there are of any other residue [Formula: see text] in the range [Formula: see text], and three times as many as [Formula: see text]. On the other hand, if we let the nonnegative integer [Formula: see text] vary while keeping the prime [Formula: see text] fixed, and look at the relative frequency of various residue classes that occur in the first [Formula: see text] rows, the seemingly substantial differences in frequency between [Formula: see text], [Formula: see text], and [Formula: see text] when [Formula: see text] are increasingly dissipated as [Formula: see text] grows without bound. We show that in the limit as [Formula: see text] tends to infinity, all nonzero residues are equally represented with asymptotic proportion [Formula: see text].

Thom Polynomials and the Green–Griffiths–Lang Conjecture for Hypersurfaces with Polynomial Degree

Green and Griffiths [25] and Lang [29] conjectured that for every complex projective algebraic variety X of general type there exists a proper algebraic subvariety of X containing all nonconstant entire holomorphic curves $f:{\mathbb{C}} \to X$. We construct a compactification of the invariant jet differentials bundle over complex manifolds motivated by an algebraic model of Morin singularities and we develop an iterated residue formula using equivariant localisation for tautological integrals over it. Using this we show that the polynomial Green–Griffiths–Lang conjecture for a generic projective hypersurface of degree $\deg (X)>2n^{9}$ follows from a positivity conjecture for Thom polynomials of Morin singularities.