Revisiting the gauge fields of strained graphene

We show that, when graphene is only subject to strain, the spin connection gauge field that arises plays no measurable role, but when intrinsic curvature is present and strain is small, spin connection dictates most the physics. We do so by showing that the Weyl field associated with strain is a pure gauge field and no constraint on the $(2+1)$-dimensional spacetime appears. On the other hand, for constant intrinsic curvature that also gives a pure-gauge Weyl field, we find a classical manifestation of a quantum Weyl anomaly, descending from a constrained spacetime. We are in the position to do this because we find the equations that the conformal factor in $(2+1)$-dimensions has to satisfy, that is a nontrivial generalization to $(2+1)$-dimensions of the classic Liouville equation of differential geometry of surfaces. Finally, we comment on the peculiarities of the only gauge field that can describe strain, that is the well known {\it pseudogauge field} $A_1 \sim u_{11} - u_{22}$ and $A_2 \sim u_{12}$, and conclude by offering some scenarios of fundamental physics that this peculiar field could help to realize.