Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 092, 41 pages      arXiv:2005.11419

Cluster Configuration Spaces of Finite Type

Nima Arkani-Hamed a, Song He bcde and Thomas Lam f
a) School of Natural Sciences, Institute for Advanced Studies, Princeton, NJ, 08540, USA
b) CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, 100190, China
c) School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS, Hangzhou 310024, China
d) ICTP-AP International Centre for Theoretical Physics Asia-Pacific, Beijing/Hangzhou, China
e) School of Physical Sciences, University of Chinese Academy of Sciences, No.19A Yuquan Road, Beijing 100049, China
f) Department of Mathematics, University of Michigan, 530 Church St, Ann Arbor, MI 48109, USA

Received January 05, 2021, in final form October 04, 2021; Published online October 16, 2021

For each Dynkin diagram $D$, we define a ''cluster configuration space'' ${\mathcal{M}}_D$ and a partial compactification ${\widetilde {\mathcal{M}}}_D$. For $D = A_{n-3}$, we have ${\mathcal{M}}_{A_{n-3}} = {\mathcal{M}}_{0,n}$, the configuration space of $n$ points on ${\mathbb P}^1$, and the partial compactification ${\widetilde {\mathcal{M}}}_{A_{n-3}}$ was studied in this case by Brown. The space ${\widetilde {\mathcal{M}}}_D$ is a smooth affine algebraic variety with a stratification in bijection with the faces of the Chapoton-Fomin-Zelevinsky generalized associahedron. The regular functions on ${\widetilde {\mathcal{M}}}_D$ are generated by coordinates $u_\gamma$, in bijection with the cluster variables of type $D$, and the relations are described completely in terms of the compatibility degree function of the cluster algebra. As an application, we define and study cluster algebra analogues of tree-level open string amplitudes.

Key words: configuration space; cluster algebras; generalized associahedron; string amplitudes.

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