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Uncovering infinite symmetries on $$[p,q]$$ 7-branes: Kac-Moody algebras and beyond. (English) Zbl 0967.81052
Summary: In a previous paper (see the paper reviewed above, Zbl 0967.81051) we explored how conjugacy classes of the modular group classify the symmetry algebras that arise on type IIB [$$p,q$$] 7-branes. The Kodaira list of finite Lie algebras completely fills the elliptic classes as well as some parabolic classes. Loop algebras of $$E_N$$ fill additional parabolic classes, and exotic finite algebras, hyperbolic extensions of $$E_N$$ and more general indefinite Lie algebras fill the hyperbolic classes. Since they correspond to brane configurations that cannot be made into strict singularities, these non-Kodaira algebras are spectrum generating and organize towers of massive BPS states into representations. The smallest brane configuration with unit monodromy gives rise to the loop algebra $$\hat{E}_9$$ which plays a central role in the theory. We elucidate the patterns of enhancement relating $$E_8, E_9, \hat{E}_9$$ and $$E_10$$. We examine configurations of 24 7-branes relevant to type IIB compactifications on a two-sphere, or F-theory on $$K3$$. A particularly symmetric configuration separates the 7-branes into two groups of twelve branes and the massive BPS spectrum is organized by $$E_10 + E_10$$.

##### MSC:
 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory 81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, $$W$$-algebras and other current algebras and their representations
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