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On non-linear chiral 4-form theories in D=10

We consider properties of non-linear theories of a chiral 4-form gauge field $A_4$ in ten space-time dimensions with an emphasis on a subclass of these theories which are invariant under the $D = 10$ conformal symmetry. We show that general results regarding a peculiar structure of duality-invariant abelian gauge theories in four and six space-time dimensions do not extend to non-linear chiral 4-form theories in ten dimensions. This discrepancy arises primarily from the large number 81 of independent invariants constructible from the self-dual part of the five-form field strength $F_5=dA_4$ in $D=10$, in stark contrast to the single independent (fourth-order) invariants which are building blocks of the actions in the lower dimensional cases. In particular, unlike the $D=4$ and $D=6$ cases, where non-linear duality-invariant theories can be viewed as stress-tensor ($T\overline T$-like) deformations of seed theories, the flow equations in $D=10$ generally involve both stress-tensor invariants and additional higher-order structures constructed from $F_5$. In passing, we prove the equivalence of three Lagrangian formulations of non-linear duality-invariant $p$-form theories: the PST, the Ivanov-Nurmagambetov-Zupnik and the ``clone" one.