nLab pillowcase orbifold

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Definition

The pillowcase orbifold is the 2-dimensional flat (complex) orbifold which is the global quotient of the torus /[i]\mathbb{C}/\mathbb{Z}[i] by the reflection involution zzz \mapsto -z:

Pillow(/[i]) refl 2 Pillow \;\coloneqq \; (\mathbb{C}/\mathbb{Z}[i])\sslash_{refl} \mathbb{Z}_2

Properties

Coarse underlying space

The Weierstrass elliptic function \wp, regarded as a holomorphic function with values in the Riemann sphere P 1\mathbb{C}P^1, exhibits the coarse underlying topological space of the pillowcase orbifold as the 2-sphere:

:/[i](/[i])/ refl 2homeoP 1 \wp \;\colon\; \mathbb{C}/\mathbb{Z}[i] \longrightarrow (\mathbb{C}/\mathbb{Z}[i])/_{refl} \mathbb{Z}_2 \overset{homeo}{\longrightarrow} \mathbb{C}P^1

(e.g Mukase 2004, end of §1.4, Eskin & Okounkov 2005, p. 1, Goujard & Moeller 2018, §2.3)

  • spindle orbifold][## References ### General * [[Alex Eskin]?, Andrei Okounkov, Pillowcases and quasimodular forms, In: Ginzburg V. (ed.) Algebraic Geometry and Number Theory, Progress in Mathematics 253 Birkhäuser (2006) [arXiv:math/0505545, doi:10.1007/978-0-8176-4532-8_1]

  • Elise Goujard, Martin Moeller, Pillowcase covers: Counting Feynman-like graphs associated with quadratic differentials [arXiv:1809.05016]

In string theory

On fluxed KK-compactification of D=6 supergravity on pillowcase orbifold fibers:

and in the broader context of non-supersymmetric flat orbifolds of supergravity theories:

Last revised on May 11, 2024 at 08:54:46. See the history of this page for a list of all contributions to it.