Abstract
We find a new infinite class of infinite-dimensional algebras acting on BPS states for non-compact toric Calabi-Yau threefolds. In Type IIA superstring compactification on a toric Calabi-Yau threefold, the D-branes wrapping holomorphic cycles represent the BPS states, and the fixed points of the moduli spaces of BPS states are described by statistical configurations of crystal melting. Our algebras are “bootstrapped” from the molten crystal configurations, hence they act on the BPS states. We discuss the truncation of the algebra and its relation with D4-branes. We illustrate our results in many examples, with and without compact 4-cycles.
Highlights
The counting of Bogomol’nyi-Prasad-Sommerfield (BPS) states [1, 2] has been one of the most central questions in quantum field theories, black holes, and string theory
Toric Calabi-Yau manifolds provide an ideal setup for addressing this problem — the geometry of a toric Calabi-Yau manifold in itself is described by the combinatorial data of the toric diagram, and the BPS state counting problem can be recast as the statistical counting problem of crystal melting [3, 4]
We show in figure 7 the example of the BPS crystal for the Suspended Pinched Point geometry discussed in figure 2
Summary
The counting of Bogomol’nyi-Prasad-Sommerfield (BPS) states [1, 2] has been one of the most central questions in quantum field theories, black holes, and string theory. The crystal melting configuration was subsequently generalized to arbitrary toric Calabi-Yau threefold [5],1 based on earlier works [7, 8]. While the BPS counting problem generates an infinite set of numbers (BPS degeneracies), there are clearly some structures in them, and it has long been expected that there is an underlying algebra, the algebra of BPS states acting on BPS states [19]. One hopes such an algebra will provide a better organizing principle for the BPS state counting problem. There was, little discussion of this algebra, in particular not for general toric Calabi-Yau threefolds
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