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# Communications in Number Theory and Physics

## Volume 3 (2009)

### Number 4

### The automorphic NS5-brane

Pages: 697 – 754

DOI: http://dx.doi.org/10.4310/CNTP.2009.v3.n4.a5

#### Authors

#### Abstract

Understanding the implications of ${\rm SL}(2,\mathbb{Z})$S-duality for the hypermultiplet moduli space of type IIstring theories has led to much progress recently inuncovering D-instanton contributions. In this work, wesuggest that the extended duality group ${\rm SL}(3,\mathbb{Z})$,which includes both S-duality andEhlers symmetry, may determine the contributions of D5 andNS5-branes. We support this claim by automorphizing theperturbative corrections to the “extended universalhypermultiplet,” a five-dimensional universal${\rm SO}(3)\backslash {\rm SL}(3,\mathbb{R})$ subspace whichincludes the string coupling, overall volume, Ramondzero-form and six-form and NS axion. Using the non-AbelianFourier expansion of the Eisenstein series attached to theprincipal series of ${\rm SL}(3,\mathbb{R})$, workedout many years ago by Vinogradov, Takhtajan and Bump,we extract the contributions of D(-1)–D5and NS5-brane instantons, corresponding to the Abelian andnon-Abelian coefficients, respectively. In particular, thecontributions of $k$ NS5-branes can be summarized into avector of wave functions $\Psi_{k,\ell}$, $\ell=0,\dots ,k-1$,as expected on general grounds. We also point outthat for more general models with a symmetric moduli space$K\backslash G$, the minimal theta series of $G$ generatesan infinite series of exponential corrections of the formrequired for “small” D(-1)–D1–D3–D5–NS5 instantonbound states. As a mathematical spin-off, we make contactwith earlier results in the literature about the sphericalvectors for the principal series of ${\rm SL}(3,\mathbb{R})$and for minimal representations.