Spacetime Duality Beyond Conformality

Abstract

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 $\sim m^2(e^{\ddot{I}}•-1)$ to the Liouville action cancels exactly, with the genuine leading deformation being $-\frac{m^2}{16\ddot{I}€}(e^{\ddot{I}}•-1{)}^2$. This breaks self-duality and renders the dual theory for the Lagrange multiplier field $\hat{I}›$ non-local. For the massive Dirac fermion, two independent derivations establish that the fermion mass dresses under conformal scaling as $m\to m{e}^{\ddot{I}•/2}$, reflecting the Weyl weight $\frac{1}{2}$ of the two-dimensional spinor. Via the Coleman-Mandelstam bosonisation, this transfers to the mass bilinear as $\hat{I}¼\cos ({\hat{I}}^2\vartheta )\to \hat{I}¼e^{\ddot{I}•/2}\cos ({\hat{I}}^2\vartheta )$, 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($\hat{I}\pounds$)/Diff($\hat{I}\pounds$).

31 pages, no figures