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Clean isotropic quantum Hall fluids in the continuum possess a host ofsymmetry-protected quantized invariants, such as the Hall conductivity, shiftand Hall viscosity. Here we develop a theory of symmetry-protected quantizedinvariants for topological phases defined on a lattice, where quantizedinvariants with no continuum analog can arise. We develop topological fieldtheories using discrete crystalline gauge fields to fully characterizequantized invariants of (2+1)D Abelian topological orders with symmetry group$G = U(1) \times G_{\text{space}}$, where $G_{\text{space}}$ consists oforientation-preserving space group symmetries on the lattice. We show howdiscrete rotational and translational symmetry fractionalization can becharacterized by a discrete spin vector, a discrete torsion vector which has noanalog in the continuum or in the absence of lattice rotation symmetry, and anarea vector, which also has no analog in the continuum. The discrete torsionvector implies a type of crystal momentum fractionalization that is onlynon-trivial for $2$, $3$, and $4$-fold rotation symmetry. The quantizedtopological response theory includes a discrete version of the shift, whichbinds fractional charge to disclinations and corners, a fractionally quantizedangular momentum of disclinations, rotationally symmetric fractional chargepolarization and its angular momentum counterpart, constraints on charge andangular momentum per unit cell, and quantized momentum bound to dislocationsand units of area. The fractionally quantized charge polarization, which isnon-trivial only on a lattice with $2$, $3$, and $4$-fold rotation symmetry,implies a fractional charge bound to lattice dislocations and a fractionalcharge per unit length along the boundary. An important role is played by afinite group grading on Burgers vectors, which depends on the point groupsymmetry of the lattice.

Crystalline gauge fields and quantized discrete geometric response for Abelian topological phases with lattice symmetry