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 Incoming links: Buscher procedure

A local Buscher procedure sets a flat auxiliary connection equal to a coordinate derivative only on a simply connected patch. A full compact-circle Polyakov path integral must also sum flat-connection holonomies, match the period of the dual multiplier, and include momentum and winding modes. A free circle isometry then gives the radius relation

R^{'} = α^{'} / R

. Fixed points, noncompact orbits and global identifications require additional analysis; a local spacelike Killing vector does not specify these data.

Integrating by parts makes the

Z

dependence

- i \int Z \partial_{α} p^{α}

. Its functional integral imposes the constraint

\partial_{α} p^{α} = 0

. On the simply connected plane, the Poincaré lemma therefore permits the local and global parametrization

p^{α} = \frac{1}{2 π α ^{'}} ϵ^{α β} \partial_{β} Z .

Substitution into

S_{2}

gives

S_{dual} [Y, Z] = \frac{1}{4 π α ^{'}} \int d^{2} σ (G_{ij} (Y) \partial^{α} Y^{i} \partial_{α} Y^{j} + V (Y)^{- 1} \partial^{α} Z \partial_{α} Z) .

z

, and its T-duality replaces

G_{zz} = V

G_{z z} = V^{- 1}