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1. Geometrizing $$ T\overline{T} $$

   https://doi.org/10.1007/JHEP03(2021)140  

   Caputa, Pawel; Datta, Shouvik; Jiang, Yunfeng; Kraus, Per (March 2021, Journal of High Energy Physics)

   null (Ed.)

   A bstract The $$ T\overline{T} $$ T T ¯ deformation can be formulated as a dynamical change of coordinates. We establish and generalize this relation to curved spaces by coupling the undeformed theory to 2d gravity. For curved space the dynamical change of coordinates is supplemented by a dynamical Weyl transformation. We also sharpen the holographic correspondence to cutoff AdS 3 in multiple ways. First, we show that the action of the annular region between the cutoff surface and the boundary of AdS 3 is given precisely by the $$ T\overline{T} $$ T T ¯ operator integrated over either the cutoff surface or the asymptotic boundary. Then we derive dynamical coordinate and Weyl transformations directly from the bulk. Finally, we reproduce the flow equation for the deformed stress tensor from the cutoff geometry. 

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2. Supersymmetry and T T ¯ $$ T\overline{T} $$ deformations

   https://doi.org/10.1007/JHEP04(2019)131  

   Chang, Chih-Kai; Ferko, Christian; Sethi, Savdeep (April 2019, Journal of High Energy Physics)
3. dS/dS and T T ¯ $$ T\overline{T} $$

   https://doi.org/10.1007/JHEP03(2019)085  

   Gorbenko, Victor; Silverstein, Eva; Torroba, Gonzalo (March 2019, Journal of High Energy Physics)
4. Refining the cutoff 3d gravity/$$ T\overline{T} $$ correspondence

   https://doi.org/10.1007/JHEP10(2022)094  

   Kraus, Per; Monten, Ruben; Roumpedakis, Konstantinos (October 2022, Journal of High Energy Physics)

   A bstract Pure gravity in AdS 3 is a theory of boundary excitations, most simply expressed as a constrained free scalar with an improved stress tensor that is needed to match the Brown-Henneaux central charge. Excising a finite part of AdS gives rise to a static gauge Nambu-Goto action for the boundary graviton. We show that this is the $$ T\overline{T} $$ T T ¯ deformation of the infinite volume theory, as the effect of the improvement term on the deformed action can be absorbed into a field redefinition. The classical gravitational stress tensor is reproduced order by order by the $$ T\overline{T} $$ T T ¯ trace equation. We calculate the finite volume energy spectrum in static gauge and find that the trace equation imposes sufficient constraints on the ordering ambiguities to guarantee agreement with the light-cone gauge prediction. The correlation functions, however, are not completely fixed by the trace equation. We show how both the gravitational action and the $$ T\overline{T} $$ T T ¯ deformation allow for finite improvement terms, and we match these to the undetermined total derivative terms in Zamolodchikov’s point splitting definition of the $$ T\overline{T} $$ T T ¯ operator. 

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5. q,t$q,t$‐Catalan measures

   https://doi.org/10.1112/blms.12989  

   Cavey, Ian (March 2024, Bulletin of the London Mathematical Society)

   Abstract We introduce the ‐Catalan measures, a sequence of piece‐wise polynomial measures on . These measures are defined in terms of suitable area, dinv, and bounce statistics on continuous families of paths in the plane, and have many combinatorial similarities to the ‐Catalan numbers. Our main result realizes the ‐Catalan measures as a limit of higher ‐Catalan numbers as . We also give a geometric interpretation of the ‐Catalan measures. They are the Duistermaat–Heckman measures of the punctual Hilbert schemes parametrizing subschemes of supported at the origin. 

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