Optimal Transport: A Semi-Discrete Approach

We analyze global anomalies for elementary Type II strings in the presence of D-branes. Global anomaly cancellation gives a restriction on the D-brane topology. This restriction makes possible the interpretation of D-brane charge as an element of K-theory.

https://www.intlpress.com/site/pub/files/_fulltext/journals/ajm/1999/0003/0004/AJM-1999-0003-0004-a006.pdf

Anomalies in string theory with D-branes

Demailly's Holomorphic Morse Inequalities.- Characterization of Moishezon Manifolds.- Holomorphic Morse Inequalities on Non-compact Manifolds.- Asymptotic Expansion of the Bergman Kernel.- Kodaira Map.- Bergman Kernel on Non-compact Manifolds.- Toeplitz Operators.- Bergman Kernels on Symplectic Manifolds.

/pdf/holomorphic-morse-inequalities-and-bergman-kernels-3j0olx2uv9.pdf

Holomorphic Morse Inequalities and Bergman Kernels

Anomalies in String Theory with D-Branes

Symmetry-protected topological (SPT) phases of matter have been interpreted in terms of anomalies, and it has been expected that a similar picture should hold for SPT phases with fermions. Here, we describe in detail what this picture means for phases of quantum matter that can be understood via band theory and free fermions. The main examples we consider are time-reversal invariant topological insulators and superconductors in 2 or 3 space dimensions. Along the way, we clarify the precise meaning of the statement that in the bulk of a 3d topological insulator, the electromagnetic -angle is equal to .

Fermion Path Integrals And Topological Phases

We study the asymptotic of the Bergman kernel of the spin c Dirac operator on high tensor powers of a line bundle.

/pdf/on-the-asymptotic-expansion-of-bergman-kernel-4lnbmjzdx0.pdf

On the asymptotic expansion of Bergman kernel

The η‐invariant of an odd dimensional manifold with boundary is investigated. The natural boundary condition for this problem requires a trivialization of the kernel of the Dirac operator on the boundary. The dependence of the η‐invariant on this trivialization is best encoded by the statement that the exponential of the η‐invariant lives in the determinant line of the boundary. Our main results are a variational formula and a gluing law for this invariant. These results are applied to reprove the formula for the holonomy of the natural connection on the determinant line bundle of a family of Dirac operators, also known as the ‘‘global anomaly formula.’’ The ideas developed here fit naturally with recent work in topological quantum field theory, in which gluing (which is a characteristic formal property of the path integral and the classical action) is used to compute global invariants on closed manifolds from local invariants on manifolds with boundary.

η‐invariants and determinant lines

Inspired by the recent work [HHM03], we prove two stability results for compact Riemannian manifolds with nonzero parallel spinors. Our first result says that Ricci flat metrics which also admit nonzero parallel spinors are stable (in the direction of changes in conformal structures) as the critical points of the total scalar curvature functional. Our second result, which is a local version of the first one, shows that any metric of positive scalar curvature cannot lie too close to a metric with nonzero parallel spinor. We also prove a rigidity result for special holonomy metrics. In the case of SU(m) holonomy, the rigidity result implies that scalar flat deformations of Calabi-Yau metric must be Calabi-Yau. Finally we explore the connection with a positive mass theorem of [D03], which presents another approach to proving these stability and rigidity results.

/pdf/on-the-stability-of-riemannian-manifold-with-parallel-2aengs0oer.pdf

On the stability of Riemannian manifold with parallel spinors

We study eta-invariants on odd dimensional manifolds with boundary. The dependence on boundary conditions is best summarized by viewing the (exponentiated) eta-invariant as an element of the (inverse) determinant line of the boundary. We prove a gluing law and a variation formula for this invariant. This yields a new, simpler proof of the holonomy formula for the determinant line bundle of a family of Dirac operators, also known as the ``global anomaly'' formula. 
This paper is written using AMSTeX 2.1, which can be obtained via ftp from the American Mathematical Society (instructions included). A postscript file with figures was submitted separately in uuencoded tar-compressed format.

Eta-Invariants and Determinant Lines

In this paper, we study the change of the ADM mass of an ALE space along the Ricci flow. Thus we first show that the ALE property is preserved under the Ricci flow. Then, we show that the mass is invariant under the flow in dimension three (similar results hold in higher dimension with more assumptions). A consequence of this result is the following. Let $(M,g)$ be an ALE manifold of dimension $n=3$. If $m(g)\neq 0$, then the Ricci flow starting at $g$ can not have Euclidean space as its (uniform) limit.

Xianzhe Dai

Papers

On the asymptotic expansion of Bergman kernel

η‐invariants and determinant lines

On the stability of Riemannian manifold with parallel spinors

Eta-Invariants and Determinant Lines

Mass under the Ricci flow