Narrow Results

The main result is a wall-crossing formula for central projections defined on submanifolds of a Real projective space. Our formula gives the jump of the degree of such a projection when the center of the projection varies. The fact that the degree depends on the projection is a new phenomenon, specific to Real algebraic geometry. We illustrate this phenomenon in many interesting situations. The crucial assumption on the class of maps we consider is relative orientability, a condition which allows us to define a ℤ-valued degree map in a coherent way. We end the article with several examples, e.g. the pole placement map associated with a quotient, the Wronski map, and a new version of the Real subspace problem.

We find sharp bounds on h⁰(F) for one-dimensional semistable sheaves F on a projective variety X. When X is the projective plane ℙ², we study the stratification of the moduli space by the spectrum of sheaves. We show that the deepest stratum is isomorphic to a closed subset of a relative Hilbert scheme. This provides an example of a family of semistable sheaves having the biggest dimensional global section space.

We survey recent progress in the study of moduli of vector bundles on higher-dimensional base manifolds. In particular, we discuss an algebro-geometric construction of an analogue for the Donaldson–Uhlenbeck compactification and explain how to use moduli spaces of quiver representations to show that Gieseker–Maruyama moduli spaces with respect to two different chosen polarizations are related via Thaddeus-flips through other “multi-Gieseker”-moduli spaces of sheaves. Moreover, as a new result, we show the existence of a natural morphism from a multi-Gieseker moduli space to the corresponding Donaldson–Uhlenbeck moduli space.

Let $v_d({\mathbb{P}}^2)\subset |{{𝒪}}_{{\mathbb{P}}^2}(d)|$ denote the $d$-uple Veronese surface. After studying some general aspects of the wall-crossing phenomena for stability conditions on surfaces, we are able to describe a sequence of flips of the secant varieties of $v_d({\mathbb{P}}^2)$ by embedding the blow-up ${\mathrm{\text{bl}}}_{v_d({\mathbb{P}}^2)}|{{𝒪}}_{{\mathbb{P}}^2}(d)|$ into a suitable moduli space of Bridgeland semistable objects on ${\mathbb{P}}^2$.

We outline a comprehensive and first-principle solution to the wall-crossing problem in D = 4N = 2 Seiberg–Witten theories. We start with a brief review of the multi-centered nature of the typical BPS states and of how this allows them to disappear abruptly as parameters or vacuum moduli are continuously changed. This means that the wall-crossing problem is really a bound state formation/dissociation problem. A low energy dynamics for arbitrary collections of dyons is derived, with the proximity to the so-called marginal stability wall playing the role of the small expansion parameter. We discover that the low energy dynamics of such BPS dyons cannot be reduced to one on the classical moduli space, , yet the index can be phrased in terms of . The so-called rational invariant, first seen in Kontsevich–Soibelman formalism of wall-crossing, is shown to incorporate Bose/Fermi statistics automatically. Furthermore, an equivariant version of the index is shown to compute the protected spin character of the underlying D = 4N = 2 theory, where isometry of is identified as a diagonal subgroup of rotation SU(2)_L and R-symmetry SU(2)_R.

We first study the WKB analysis of the third order ODE, which can be regarded as the quantized Seiberg-Witten curve of the (A₂, A_N )-type Argyres-Douglas theory in the Nekrasov-Shatashvili limit of Omega background. We then derive thermodynamic Bethe ansatz (TBA) equations satisfied by the Y-functions from the solutions of the ODE, and identify the Y-function with the WKB period. For the (A₂, A₂)-type ODE, we study the process of wall-crossing of the TBA equation from the minimal chamber to the maximal chamber.

In this note, we discuss the infrared physics of 5d 𝒩 = 1 Yang-Mills theories compactified on 𝕊, with a view toward 4d and 5d limits. The Coulomb phase boundaries in the decompactification limit are given particular attention and related to how the wall-crossings by 5d BPS particles turn off. On the other hand, the elliptic genera of magnetic BPS strings do wall-cross and retain the memory of 4d wall-crossings, which we review with the example of dP₂ theory.

We give a summary of a talk delivered at the ICMP in Aalborg, Denmark, August, 2012. We review d = 4, N = 2 quantum field theory and some of the exact statements which can be made about it. We discuss the wall-crossing phenomenon. An interesting application is a new construction of hyperkähler metrics on certain manifolds. Then we discuss recent geometric constructions which lead to exact results on the BPS spectra for some d = 4, N = 2 field theories and on expectation values of – for example – Wilson line operators. These new constructions have interesting relations to a number of other areas of physical mathematics.