We define enumerative invariants associated to a hybrid Gauged Linear Sigma Model. We prove that in the relevant special cases these invariants recover both the Gromov-Witten type invariants defined by Chang-Li and Fan-Jarvis-Ruan using cosection localization as well as the FJRW type invariants constructed by Polishchuk-Vaintrob. The invariants are defined by constructing a "fundamental factorization" supported on the moduli space of Landau-Ginzburg maps to a convex hybrid model. This gives the kernel of a Fourier-Mukai transform; the associated map on Hochschild homology defines our theory.
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