Title: Quantum dilogarithm identities for the square product of A-type Dynkin quivers,
Abstract: The famous pentagon identity for quantum dilogarithms has a generalization
for every Dynkin quiver, due to Reineke. A more advanced generalization is
associated with a pair of alternating Dynkin quivers, due to Keller. The
description and proof of Keller's identities involves cluster algebras and
cluster categories, and the statement of the identity is implicit. In this
paper we describe Keller's identities explicitly, and prove them by a dimension
counting argument. Namely, we consider quiver representations
$\boldsymbol{\mathrm{Rep}}_\gamma$ together with a superpotential function
$W_\gamma$, and calculate the Betti numbers of the equivariant $W_\gamma$ rapid
decay cohomology algebra of $\boldsymbol{\mathrm{Rep}}_\gamma$ in two different
ways corresponding to two natural stratifications of
$\boldsymbol{\mathrm{Rep}}_\gamma$. This approach is suggested by Kontsevich
and Soibelman in relation with the Cohomological Hall Algebra of quivers, and
the associated Donaldson-Thomas invariants.