brane-count

Methods to count the number of intersecting D6-brane vacua of type IIA on the $\mathbb{T}^6/\mathbb{Z}_2\times\mathbb{Z}_2$ orientifold. Vacua consist of stacks of $N_a$ coincident D6-branes wrapping factorized 3-cycles described by three pairs of coprime winding numbers, $(n_a^i,m_a^i)$. The homology classes can be written as $[\Pi_a] = \sum_I(\widehat{X}_a^I[\pi_I^+] + \widehat{Y}_a^I[\pi_I^-])$, where $\widehat{X}^0=n^1n^2n^3$, $\widehat{X}^1=-n^1\widehat{m}^2\widehat{m}^3,\ldots$ with $\widehat{m}^i=m^i+2b_i(n^i+m^i)\in\mathbb{Z}$. These topological data are subject to several consistency conditions which can be written as follows:

• Tadpole cancellation: $\sum_aN_a\widehat{X}_a^I = 8$ for each $I=0,1,2,3$
• K-theory charge cancellation: $\sum_aN_a\widehat{Y}_a^I \in 2\mathbb{Z}$ for each $I=0,1,2,3$
• Supersymmetry: $\sum_I\widehat{X}_a^I\widehat{U}_I > 0$ and $\sum_I\frac{\widehat{Y}_a^I}{\widehat{U}_I} = 0$ for all $a$

The discrete NSNS fluxes take values $b_{1,2,3}\in\{0,\frac{1}{2}\}$ and the moduli $\widehat{U}_I$ must be positive.

That there are only a finite number of solutions to the above system of Diophantine equations has been known since the work of Douglas and Taylor ('06). In

Gregory J. Loges and Gary Shiu, 134 billion intersecting brane models, J. High Energy Phys. 2022, 97 (2022). doi:10.1007/JHEP12(2022)097 arXiv:2206.03506

we develop techniques to provide an exact count of gauge-inequivalent vacua.