straightforward extension of the arguments presented for abelian [@Benichou:2008it] and fermionic T-duality [@Sfetsos:2010xa].[^3]

Finally let us turn to the transformation of the dilaton field under non-abelian T-duality. For the non-abelian duality to preserve conformality the dualisation procedure must avoid the introduction of a mixed gravitational-gauge anomaly [@Alvarez:1994np; @Elitzur:1994ri] and the structure constants of the algebra in which we dualise should satisfy $$n_a \equiv f_{ab}{}^b = 0 \ .$$ When this is the case the dual dilaton comes from the Gaussian integration in the path integral [@Buscher:1987qj] $$\label{eq:diltrans}
\widehat{\Phi} = \Phi - \frac{1}{2}\log \det M \ .$$

On the other hand if $n_a \neq 0$ the dual model is not expected to be conformal, however it will be globally scale invariant. In this case we still define the dual “dilaton” to be given by . The global scale invariance then implies that, for example, the one-loop metric and $B$-field beta-functions (defined in of appendix \[app:sugra\]) only vanish up to diffeomorphisms and gauge transformations. This is in contrast to the conformal case, for which the beta-functions of the metric, $B$-field and dilaton vanish identically, while the RR fluxes solve the first order equations in eq. of appendix \[app:sugra\].

It transpires that the globally scale invariant models that arise from dualising with $n_a \neq 0$ satisfy a stronger set of equations than those of global scale invariance [@Hoare:2016wsk]. These are a modification of the Type II supergravity equations [@Arutyunov:2015mqj; @Wulff:2016tju; @Sakatani:2016fvh] that depend on a particular Killing vector $I$ of the background such that when $I = 0$ standard Type II supergravity is recovered. These equations are given in eqs. of appendix \[app:sugra\].

As mentioned above we take the dual “dilaton” field in these equations to still be defined in terms of the original dilaton via the transformation , while the one-forms $X$, $Z$ and $W$ are defined in terms of $\