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Abstract For a smooth projective surface$$X$$satisfying$$H_1(X,\mathbb{Z}) = 0$$and$$w \in H^2(X,\mu _r)$$, we study deformation invariants of the pair$$(X,w)$$. Choosing a Brauer–Severi variety$$Y$$(or, equivalently, Azumaya algebra$$\mathcal{A}$$) over$$X$$with Stiefel–Whitney class$$w$$, the invariants are defined as virtual intersection numbers on suitable moduli spaces of stable twisted sheaves on$$Y$$constructed by Yoshioka (or, equivalently, moduli spaces of$$\mathcal{A}$$-modules of Hoffmann–Stuhler). We show that the invariants do not depend on the choice of$$Y$$. Using a result of de Jong, we observe that they are deformation invariants of the pair$$(X,w)$$. For surfaces with$$h^{2,0}(X) \gt 0$$, we show that the invariants can often be expressed as virtual intersection numbers on Gieseker–Maruyama–Simpson moduli spaces of stable sheaves on$$X$$. This can be seen as a$${\rm PGL}_r$$–$${\rm SL}_r$$correspondence. As an application, we express$${\rm SU}(r) / \mu _r$$Vafa–Witten invariants of$$X$$in terms of$${\rm SU}(r)$$Vafa–Witten invariants of$$X$$. We also show how formulae from Donaldson theory can be used to obtain upper bounds for the minimal second Chern class of Azumaya algebras on$$X$$with given division algebra at the generic point. 

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.