Intrinsic mirror symmetry for orbifold cubic surfaces and the $A_1$ spherical DAHA

We determine the intrinsic mirror of an orbifold cubic surface with three nodes and interpret the resulting geometry through mirror symmetry between the $PGL(2,\mathbb{C})$ and $SL(2,\mathbb{C})$ character varieties of the once-punctured torus. We further construct a quantization of the mirror using higher genus Gromov-Witten theory and show that it recovers the $A_1$ spherical double affine Hecke algebra.