Counting $\mathcal{N} = 8$ Black Holes as Algebraic Varieties

We calculate the helicity trace index $B_{14}$ for $\mathcal{N} = 8$ pure D-brane black holes using various techniques of computational algebraic geometry and find perfect agreement with the existing results in the literature. For these black holes, microstate counting is equivalent to finding the number of supersymmetric vacua of a multi-variable supersymmetric quantum mechanics which in turn is equivalent to solving a set of multi-variable polynomial equations with syzygies. We explore four different techniques to solve a set of polynomial equations, namely Newton Polytopes, Homotopy continuation, Monodromy and Hilbert series. Unlike other schemes, these methods are almost exact with a very high success rate for finding all solutions. A gauge invariant analysis is also possible using the symbolic Hilbert series. Furthermore, exploiting various exchange symmetries, we show the non-existence of a quartic and(or) higher order terms in the potential which if present would have spoiled the counting. Incorporating recent developments in mathematics, algorithms and multi-threading, we have extended the scope of one of the author's previous works to other charges and presented a new perspective for the microstate counting problem. This further establishes the pure D-brane system as a consistent model, bringing us a step closer to $\mathcal{N} = 2$ black hole counting.