Abstract
We revisit the results of Zamolodchikov and others on the deformation of two-dimensional quantum field theory by the determinant det T of the stress tensor, commonly referred to as Toverline{T} . Infinitesimally this is equivalent to a random coordinate transformation, with a local action which is, however, a total derivative and therefore gives a contribution only from boundaries or nontrivial topology. We discuss in detail the examples of a torus, a finite cylinder, a disk and a more general simply connected domain. In all cases the partition function evolves according to a linear diffusion-type equation, and the deformation may be viewed as a kind of random walk in moduli space. We also discuss possible generalizations to higher dimensions.
Highlights
Flauger and Gorbenko [2] argued that an example corresponds to an exponential dressing of a free boson S-matrix by CDD factors which describes world-sheet scattering for the NambuGoto string, and Dubovsky, Gorbenko and Mirbabayi [3] gave an expression for such a dressing of the full S-matrix in the more general non-integrable case . (That irrelevant operators should generate CDD factors was pointed out by Mussardo and Simon [4].) Dubovsky, Gorbenko, and Mirbabayi [5] argued that the deformation corresponds to the dressing of the theory by Jackiw-Teitelboim [6, 7] gravity, a correspondence which has recently been demonstrated explicitly by the exact computation of the torus partition function by Dubovsky, Gorbenko, and Hernandez-Chifflet [8]
We point out that the infinitesimal deformation of the action is proportional to the determinant det T of the stress tensor, which reduces to T T only in the limit of a conformal field theory (CFT). In two dimensions, det T is quadratic in Tij, and this term in the action may be decoupled by introducing an auxiliary field hij coupled linearly to Tij, itself with a local quadratic action proportional to det h
We show that (a) it is sufficient to restrict this to flat metrics, which correspond to infinitesimal diffeomorphisms xi → xi + αi(x), and (b) the resulting quadratic action for αi is a total derivative
Summary
We consider a sequence of two-dimensional euclidean field theories T (t), parametrized by a real parameter t. The above argument shows that the metric is flat and that the action is a total derivative just at the saddle point. At the saddle point, the metric is flat and we may take Φ = 0 and αi,j = αj,i ( in Cartesian coordinates) This means that we could parametrize αi = ∂iφ where φ is a scalar potential, but in general it is more convenient not to do so. The fact that the effective action for α is a total derivative is, from this point of view, at the heart of the property that the T T deformation is ‘solvable’, even when the undeformed theory is not integrable. Many of the known results follow from this, as well as new ones in situations where arguments based on translational invariance fail
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