Claus Jeschek, Frederik Witt
In the context of generalised geometry we investigate reductions to $SU(m)\times SU(m)$ together with an integrability condition which in dimension 6 describes the geometry of type II supergravity compactifications.
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The goal of this paper is to re-examine D-brane Ramond-Ramond field couplings in the presence of a B-field. We will argue that the generalised geometry induced on the world volume by the B-field results in an important but subtle change on the coupling. In order to explain this, we use the language of differential K-theory. The expression determining the coupling is then seen to be a consequence of the Riemann-Roch theorem. Our key assertion is that the appropriate Riemann-Roch theorem changes in the presence of the B-field. In particular, the A-hat forms appearing in the theorem are now const…
We define $(p,q)$ hermitian geometry as the target space geometry of the two dimensional $(p,q)$ supersymmetric sigma model. This includes generalised Kähler geometry for $(2,2)$, generalised hyperkähler geometry for $(4,2)$, strong Kähler with torsion geometry for $(2,1)$ and strong hyperkähler with torsion geometry for $(4,1)$. We provide a generalised complex geometry formulation of hermitian geometry, generalising Gualtieri's formulation of the $(2,2)$ case. Our formulation involves a chiral version of generalised complex structure and we provide explicit formulae for the map to generalise…
We examine how generalised geometries can be associated with a labelled Dynkin diagram built around a gravity line. We present a series of new generalised geometries based on the groups $\mathit{Spin}(d,d)\times\mathbb{R}^+$ for which the generalised tangent space transforms in a spinor representation of the group. In low dimensions these all appear in subsectors of maximal supergravity theories. The case $d=8$ provides a geometry for eight-dimensional backgrounds of M theory with only seven-form flux, which have not been included in any previous geometric construction. This geometry is also o…
New Examples of Translating Solitons in Generalised Robertson-Walker Geometries
本文研究了广义罗伯逊-沃克时空中的翻译子，识别出三个一参数家族的挤压函数，并分类了Grim Reapers的类比。
In this short paper, we will review the proposal of a correspondence between the doubled geometry of Double Field Theory and the higher geometry of bundle gerbes. Double Field Theory is T-duality covariant formulation of the supergravity limit of String Theory, which generalises Kaluza-Klein theory by unifying metric and Kalb-Ramond field on a doubled-dimensional space. In light of the proposed correspondence, this doubled geometry is interpreted as an atlas description of the higher geometry of bundle gerbes. In this sense, Double Field Theory can be interpreted as a field theory living on th…
This is the second in a series of papers discussing in the framework of gerbe theory canonical and geometric aspects of the 2d nonlinear sigma model in the presence of conformal defects in the worldsheet. Employing the formal tools worked out in the first paper of the series, 1101.1126 [hep-th], a thorough analysis of rigid symmetries of the sigma model is carried out with emphasis on algebraic structures on generalised tangent bundles over the target space of the theory and over its state space that give rise to a realisation of the symmetry algebra on states. The analysis leads to a proposal…
In this paper we define the analogue of Calabi--Yau geometry for generic $D=4$, $\mathcal{N}=2$ flux backgrounds in type II supergravity and M-theory. We show that solutions of the Killing spinor equations are in one-to-one correspondence with integrable, globally defined structures in $E_{7(7)}\times\mathbb{R}^+$ generalised geometry. Such "exceptional Calabi--Yau" geometries are determined by two generalised objects that parametrise hyper- and vector-multiplet degrees of freedom and generalise conventional complex, symplectic and hyper-Kahler geometries. The integrability conditions for both…
We show that generalised geometry gives a unified description of maximally supersymmetric consistent truncations of ten- and eleven-dimensional supergravity. In all cases the reduction manifold admits a "generalised parallelisation" with a frame algebra with constant coefficients. The consistent truncation then arises as a generalised version of a conventional Scherk-Schwarz reduction with the frame algebra encoding the embedding tensor of the reduced theory. The key new result is that all round-sphere $S^d$ geometries admit such generalised parallelisations with an $SO(d+1)$ frame algebra. Th…
Generalised Einstein condition and cone construction for parabolic geometries
本文试图定义一个推广的标准爱因斯坦条件（共形/度量几何）的任何抛物几何。为了做到这一点，它表明，任何保留的对合$σ$的伴随丛$\mc{A}$引起，给定一定的代数条件下，一个唯一的首选仿射联络$\nabla$与协变常数ρ张量$\mathsf{P}$，兼容的代数括号$\mc{A}$。这些条件可以合理地认为是爱因斯坦条件的推广，并在共形几何中重新建立了标准的爱因斯坦条件。这样一个对合的存在是由一些简单的结构所暗示的：当整个代数$\mf{g}$是$\mf{sl}(m,\mbb{F})$时保持度量，对于$\mf{g}=\mf{sp}(2m,\mbb{F})$保持与斜形式反交换的复结构，以及对于所有其他非例外单李代数保持切丛的一定秩的子。爱因斯坦对合的例子构造或引用几个几何。然后证明了某些爱因斯坦对合的锥结构的存在性。
We describe the geometry of generic heterotic backgrounds preserving minimal supersymmetry in four dimensions using the language of generalised geometry. They are characterised by an $SU(3)\times Spin(6+n)$ structure within $O(6,6+n)\times\mathbb{R}^+$ generalised geometry. Supersymmetry of the background is encoded in the existence of an involutive subbundle of the generalised tangent bundle and the vanishing of a moment map for the action of diffeomorphisms and gauge symmetries. We give both the superpotential and the Kähler potential for a generic background, showing that the latter defines…
Calibrated cycles and T-duality
对于希钦的广义几何，我们介绍和分析的概念，结构子流形封装的经典概念的校准子流形。在一个合适的可积性条件下的环境几何，这些广义校准周期最小化功能发生D-膜能量在II型弦理论，涉及两个所谓的NS-NS-和R-R-领域。此外，我们调查的T-对偶下的校准周期的行为，并构造非平凡的例子。
In recent work N. Hitchin introduced the concept of "generalised geometry". The key feature of generalised structures is that that they can be acted on by both diffeomorphisms and 2-forms, the so-called $B$-fields. In this lecture, we give a basic introduction and explain some of the fundamental ideas. Further, we discuss some examples of generalised geometries starting from the usual notion of a Calabi-Yau manifold, as well as applications to string theory.
Hidden symmetries of generalised gravitational instantons
提出一个框架，用于求解具有Killing场的共形Kähler黎曼四流形上对应共形自对偶和宇宙学Einstein-Maxwell的广义引力瞬子场方程，推导曲率恒等式，获得三类几何的SU(∞)Toda表述并求解，找到其广义化。
Optical geometries
该研究从内在挠率视角探究配备零直线分布的洛伦兹流形定义的光学几何，分析其非可积完整约化特性、共形性质及在广义相对论的作用，并扩展至Robinson和Trautman提出的广义光学几何。
We introduce T-duality invariant versions of D-branes in doubled geometry using a global covariant framework based on para-Hermitian geometry and metric algebroids. We define D-branes as conformal boundary conditions for the open string version of the Born sigma-model, where they are given by maximally isotropic vector bundles which do not generally admit the standard geometric picture in terms of submanifolds. When reduced to the conventional sigma-model description of a physical string background as the leaf space of a foliated para-Hermitian manifold, integrable branes yield D-branes as lea…
We present a detailed study of a new mathematical object in $\mathrm{E}_{6(6)}\times \mathbb{R}^{+}$ generalised geometry called an `exceptional complex structure' (ECS). It is the extension of a conventional complex structure to one that includes all the degrees of freedom of M-theory or type IIB supergravity in six or five dimensions, and as such characterises, in part, the geometry of generic supersymmetric compactifications to five-dimensional Minkowkski space. We define an ECS as an integrable $\mathrm{U}^{*}(6)\times \mathbb{R}^{+}$ structure and show it is equivalent to a particular for…
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