Canonical analysis of the gravitational description of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>T</mml:mi> <mml:mover accent="true"> <mml:mi>T</mml:mi> <mml:mo stretchy="false">¯</mml:mo> </mml:mover> </m

The description of the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>T</a:mi> <a:mover accent="true"> <a:mi>T</a:mi> <a:mo stretchy="false">¯</a:mo> </a:mover> </a:math> deformation in terms of two-dimensional gravity is analyzed from the Hamiltonian point of view, in a manner analogous to the Arnowitt-Deser-Misner description of general relativity. We find that the Hamiltonian constraints of the theory imply relations between target-space momentum at finite volume that are equivalent to the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>T</e:mi> <e:mover accent="true"> <e:mi>T</e:mi> <e:mo stretchy="false">¯</e:mo> </e:mover> </e:math> finite-volume flow equations. This fully quantum <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mi>T</i:mi> <i:mover accent="true"> <i:mi>T</i:mi> <i:mo stretchy="false">¯</i:mo> </i:mover> </i:math> result emerges already at the classical level within the gravitational theory. We exemplify the analysis for the case when the undeformed sector is a collection of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mi>D</m:mi> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:math> free massless scalars, where it is shown that—somewhat nontrivially—the target-space two-dimensional Poincaré symmetry is extended to <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:mi>D</o:mi> </o:math> dimensions. The connection between canonical quantization of this constrained Hamiltonian system and previous path integral quantizations is also discussed. We extend our analysis to the “gravitational” description of <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:mi>J</q:mi> <q:mover accent="true"> <q:mi>T</q:mi> <q:mo stretchy="false">¯</q:mo> </q:mover> </q:math> -type deformations, where it is found that the flow equations obtained involve deformations that twist the spatial boundary conditions.