Spacetime Duality Beyond Conformality

We extend the spacetime duality programme of Burgess \textit{et.al.} to massive theories in 1+1 dimensions. For the massive scalar, a heat-kernel computation tracking three contributions to the conformal-mode effective action reveals that the naive leading correction $\sim m^2(e^{\phi} -1)$ to the Liouville action cancels exactly, with the genuine leading deformation being $-\frac{m^2}{16\pi}(e^{\phi}-1)^{2}$. This breaks self-duality and renders the dual theory for the Lagrange multiplier field $\Lambda$ non-local. For the massive Dirac fermion, two independent derivations establish that the fermion mass dresses under conformal scaling as $m \to m\, e^{\phi/2}$, reflecting the Weyl weight $\frac{1}{2}$ of the two-dimensional spinor. Via the Coleman-Mandelstam bosonisation, this transfers to the mass bilinear as $\mu\cos(\beta\vartheta) \to \mu e^{\phi/2}\cos(\beta\vartheta)$, producing a coupled Liouville-sine-Gordon system as the natural starting point for the fermionic construction. Both results are interpreted in terms of the determinant line bundle over Met($\Sigma$)/Diff($\Sigma$).