Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0701893

Quantisation of twistor theory by cocycle twist

S. Brain·S. Majid
View original source ↗Read source document ↗Export metadata ↓

Abstract

We present the main ingredients of twistor theory leading up to and including the Penrose-Ward transform in a coordinate algebra form which we can then `quantise' by means of a functorial cocycle twist. The quantum algebras for the conformal group, twistor space CP^3, compactified Minkowski space CMh and the twistor correspondence space are obtained along with their canonical quantum differential calculi, both in a local form and in a global *-algebra formulation which even in the classical commutative case provides a useful alternative to the formulation in terms of projective varieties. We outline how the Penrose-Ward transform then quantises. As an example, we show that the pull-back of the tautological bundle on CMh pulls back to the basic instanton on S^4\subset CMh and that this observation quantises to obtain the Connes-Landi instanton on θ-deformed S^4 as the pull-back of the tautological bundle on our θ-deformed CMh. We likewise quantise the fibration CP^3--> S^4 and use it to construct the bundle on θ-deformed CP^3 that maps over under the transform to the θ-deformed instanton.

Explore related subjects

math.QAhep-thmath.AG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

S. Brain, S. Majid. 2007-03-20. Quantisation of twistor theory by cocycle twist. https://doi.org/10.1007/s00220-008-0607-1

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On singular vectors and associated Weyl elements for simply-laced universal affine vertex operator algebras

Given a finite-dimensional complex simple Lie algebra $\mathfrak{g}$ and a complex number $κ$, let $V^κ(\mathfrak{g})$ be the associated universal affine vertex algebra. Gorelik and Kac [GK07] gave a sufficient and necessary condition for $V^κ(\mathfrak{g})$ to be simple. In this paper, for simply-laced $\mathfrak{g}$ and non-critical $κ$, we determine the weights of singular vectors of $V^κ(\mathfrak{g})$ with minimal conformal weights, when $V^κ(\mathfrak{g})$ is not simple. We further determine all the longest Weyl elements in the Kashiwara-Tanisaki character theorem [KT00] which correspond to the weights of the singular vectors.

math.QA↗
arXiv2026

Modular invariants and NIM-reps

Given a pivotal module category over a spherical fusion category, we introduce the encircling module, a module over the fusion algebra defined using the pivotal structure, and prove that it is isomorphic to the NIM-rep as a fusion algebra module. When applied to the $\mathcal{TM}$ realisation of the modular invariant partition function (arXiv:1911.09024), this identifies the NIM-rep multiplicities with the entries $Z_{I,I^\vee}$ of the modular invariant (its diagonal relative to the charge-conjugation invariant, which is the one given by the regular module category), providing a categorical generalisation of Böckenhauer, Evans and Kawahigashi's results (arXiv:math/9907149). We also show that for indecomposable module categories the dimension condition on $\mathcal{TM}$ required for modular invariance is automatically satisfied, and that $\mathcal{TM}$ is, as an algebra, the full centre of the module category in the sense of Davydov, Kong and Runkel (arXiv:1307.5956), which for module categories of algebras is the full centre of Fjelstad, Fuchs, Runkel and Schweigert (arXiv:hep-th/0612306). Consequently $\mathcal{TM}$ gives rise to Cardy algebras in the sense of Kong and Runkel (arXiv:0807.3356).

math.QA↗
arXiv2026

A diagrammatic presentation for every pivotal pointed fusion category

We provide a generators and relations presentation of pivotal pointed fusion categories, $Vec(G,ω,π)$. Unlike the well-known skeletal model, our presentation is strict and allows multiple isomorphic objects. Our main tool is skein theory, which allows us to apply topological tools to understand the relations of morphisms in the category. This expository paper provides details of skein-theoretic work in the setup that allows multiple isomorphic labels in the generators, which didn't previously exist in the literature.

math.QA↗