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# Communications in Number Theory and Physics

## Volume 2 (2008)

### Number 4

### On the space of elliptic genera

Pages: 803 – 833

DOI: http://dx.doi.org/10.4310/CNTP.2008.v2.n4.a4

#### Author

#### Abstract

Invariance under modular transformations and spectral flowrestrict the possible spectra of superconformal fieldtheories (SCFTs). This paper presents a technique tocalculate the number of constraints on the polar spectra of$\mathcal{N}=(2,2)$ and $\CN=(4,0)$ SCFTs by analyzingtheir elliptic genera. The polar spectrum corresponds tothe principal part of a Laurent expansion derived from theelliptic genus. From the point of view of theAdS$_3$/CFT$_2$ correspondence, these are the states whichlie below the cosmic censorship bound in classical gravity.The dimension of the space of elliptic genera is obtainedas the number of coefficients of the principal part minusthe number of constraints. As an additional illustration ofthe technique, the constraints on the spectrum of $\CN=4$topologically twisted Yang–Mills on $\mathbb{CP}^2$ arediscussed.