The Decomposition of Global Conformal Invariants: Some Technical Proofs. I

关于整体共形不变量的分解

On the decomposition of global conformal invariants I

这是两篇论文中的第一篇，我们在其中讨论并部分证实了Deser和Schwimmer最初在高能物理中假设的猜想。研究的对象是由曲率及其共变导数构造的标量黎曼量，其在紧致流形上的积分在基本度量的共形变化下是不变的。我们的主要结论是，每一个这样的量，局部只依赖于曲率张量（没有共变导数）可以写为一个线性组合的Chern-Gauss-Bonnet被积函数和一个标量共形不变量。

The decomposition of global conformal invariants VI: The proof of the proposition on local Riemannian invariants

This is the last in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of ``global conformal invariants''; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed as a linear combination of a local conformal invariant, a divergence and of the Chern-Gauss-Bonnet integrand. The present paper, jointly with [6,7] gives a proof of an algebraic Proposition regarding local Riemannian invariants, which lies at the heart of our resolution of the Dese…

The decomposition of Global Conformal Invariants I: On a conjecture of Deser and Schwimmer

This is the first in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of ``global confor- mal invariants"; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed as a linear combination of a local conformal invariant, a divergence and of the Chern-Gauss-Bonnet integrand. In this paper we set up an iterative procedure that proves the decom- position. We then derive the iterative step in the first of two cases, subject to a purely algebraic…

关于全局共形不变量的分解II

On the decomposition of global conformal invariants II

本文是文[2]的继续，在文[2]中，我们完成了关于“整体共形不变量”结构的S-Schwimmer猜想的部分证明。我们的定理处理这样的不变量P（g^n），局部只依赖于曲率张量R_{ijkl}（没有共变导数）。在[2]中，我们开发了一个强大的工具，“超发散公式”，它适用于任何黎曼算子，总是积分为零的紧流形。特别地，它适用于测量P（g^n）的“非共形不变部分”的算子I_{g^n}（n）。本文解决了利用我们在I_{g^n}（n）结构上获得的这些信息来理解P（g^n）结构的问题。

Conformal operators on weighted forms; their decomposition and null space on Einstein manifolds

There is a class of Laplacian like conformally invariant differential operators on differential forms $L^\ell_k$ which may be considered the generalisation to differential forms of the conformally invariant powers of the Laplacian known as the Paneitz and GJMS operators. On conformally Einstein manifolds we give explicit formulae for these as explicit factored polynomials in second order differential operators. In the case the manifold is not Ricci flat we use this to provide a direct sum decomposition of the null space of the $L^\ell_k$ in terms of the null spaces of mutually commuting second…

Anti-invariant Riemannian submersions from locally conformal Kaehler manifolds

B. Sahin [9] introduced the notion of anti-invariant Riemannian submersions from almost Hermitian manifolds onto Riemannian manifolds. In the present paper we extend the notion of anti-invariant and Lagrangian Riemannian submersions (a special anti-invariant Riemannian submersion) to the case of locally conformal Kaehler manifolds. We discuss the geometry of foliation and obtain some decomposition theorems for the total manifold of such submersions.

Conformal slant submersions in contact geometry

Akyol M.A. [Conformal anti-invariant submersions from cosymplectic manifolds, Hacettepe Journal of Mathematics and Statistic, 46(2), (2017), 177-192.] defined and studied conformal anti-invariant submersions from cosymplectic manifolds. The aim of the present paper is to define and study the notion of conformal slant submersions (it means the Reeb vector field $ξ$ is a vertical vector field) from almost contact metric manifolds onto Riemannian manifolds as a generalization of Riemannian submersions, horizontally conformal submersions, slant submersions and conformal anti-invariant submersions.…

带边界的Liouville CFT中的共形自举。第二部分：谱分解与自举

Conformal Bootstrap for surfaces with boundary in Liouville CFT. Part II: spectral resolution and bootstrap

本文通过散射方法建立半环半群的谱分解，将边界态空间分解为Virasoro不可约表示，并利用Ward恒等式将关联函数表示为谱参数积分，完成带边界Liouville CFT的共形自举。

Some Properties of the Higher Spin Laplace Operator

The higher spin Laplace operator has been constructed recently as the generalization of the Laplacian in higher spin theory. This acts on functions taking values in arbitrary irreducible representations of the Spin group. In this paper, we first provide a decomposition of the higher spin Laplace operator in terms of Rrita-Schwinger operators. With such a decomposition, a connection between the fundamental solutions for the higher spin Laplace operator and the fundamental solutions for the Rarita-Schwinger operators is provided. Further, we show that the two components in this decomposition are…

Poincaré-Einstein 流形中极小子流形重整化面积的高斯-博内公式

A Gauss-Bonnet formula for the renormalized area of minimal submanifolds of Poincaré-Einstein manifolds

本文证明偶维极小子流形在 Poincaré-Einstein 流形中的重整化面积可由欧拉示性数与共形不变量积分线性表示，并在二维和四维验证了外在 Q-曲率的分解猜想。

Virasoro 共形块与来自Liouville CFT的模函子

Virasoro Conformal Blocks and modular functor from Liouville CFT

本文从Liouville CFT构造了亏格g、m个标记点黎曼曲面上Virasoro共形块的全局全纯截面，满足Ward恒等式，并证明其承载映射类群的射影酉表示，对Moore-Seiberg移动有酉同构。