We conclude the construction of $r$-spin theory in genus zero for Riemann surfaces with boundary. In particular, we define open $r$-spin intersection numbers, and we prove that their generating function is closely related to the wave function of the $r$th Gelfand–Dickey integrable hierarchy. This provides an analogue of Witten’s $r$-spin conjecture in the open setting and a first step toward the construction of an open version of Fan–Jarvis–Ruan–Witten theory. As an unexpected consequence, we establish a mysterious relationship between open $r$-spin theory and an extension of Witten’s closed theory.

We introduce a moment map picture for holomorphic string algebroids where the Hamiltonian gauge action is described by means of inner automorphisms of Courant algebroids. The zero locus of our moment map is given by the solutions of the Calabi system, a coupled system of equations which provides a unifying framework for the classical Calabi problem and the Hull–Strominger system. Our main results are concerned with the geometry of the moduli space of solutions, and assume a technical condition which is fulfilled in examples. We prove that the moduli space carries a pseudo-Kähler metric with Kähler potential given by the dilaton functional, a topological formula for the metric, and an infinitesimal Donaldson–Uhlenbeck–Yau type theorem.

We introduce a new flow to the LYZ equation on a compact Kähler manifold. We first show the existence of the longtime solution of the flow. We then show that under the Collins–Jacob–Yau’s condition on the subsolution, the longtime solution converges to the solution of the LYZ equation, which was solved by Collins–Jacob–Yau $\href{https://dx.doi.org/10.4310/CJM.2020.v8.n2.a4}{[5]}$ by the continuity method. Moreover, as an application of the flow, we show that on a compact Kähler surface, if there exists a semi-subsolution of the LYZ equation, then our flow converges smoothly to a singular solution to the LYZ equation away from a finite number of curves of negative self-intersection. Such a solution can be viewed as a boundary point of the moduli space of the LYZ solutions for a given Kähler metric.

We address the construction of smooth bundles of fermionic Fock spaces, a problem that appears frequently in fermionic gauge theories. Our main motivation is the spinor bundle on the free loop space of a string manifold, a structure anticipated by Killingback and Stolz–Teichner. We develop a general framework for constructing smooth Fock bundles, and obtain as an application a complete and well-founded construction of smooth spinor bundles on loop space. We also develop a projective version using bundle gerbes and twisted bundles, appropriate for the situation that the base manifold is not string.

We prove a Riemannian positive mass theorem which allows for incompleteness and negative scalar curvature. The manifolds are assumed to have one asymptotically Schwarzschild end, but the complement of this end is otherwise arbitrary. The incompleteness and negativity is compensated for by large positive scalar curvature on an annulus, in a quantitative fashion. In the complete noncompact case with nonnegative scalar curvature, we have no extra assumption and hence prove a long-standing conjecture of Schoen and Yau.

We prove a surgery formula and an excision formula for the Furuta–Ohta invariant $\lambda_{FO}$ defined on homology $S^1 \times S^3$, which provides more evidence on its equivalence with the Casson–Seiberg–Witten invariant $\lambda_{SW}$. These formulae are applied to compute $\lambda_{FO}$ of certain families of manifolds obtained as mapping tori under diffeomorphisms of $3$-manifolds. In the course of the proof, we establish the existence of asymptotic values for finite energy instantons over end-cylindrical manifolds perturbed by exponentially decay holonomy perturbations, and give a complete description of the structure of the moduli space of charge-zero instantons over any non-compact $4$-manifold whose end is modeled on $[0,\infty) \times T^3$, and whose homology is given by $H_\ast (D^2 \times T^2; \mathbb{Z})$.

This paper concerns the behavior of the eigenfunctions and eigenvalues of the round sphere’s Laplacian acting on the space of sections of a real line bundle which is defined on the complement of an even numbers of points in ${S}^2$. Of particular interest is how these eigenvalues and eigenvectors change when viewed as functions on the configuration spaces of points.