We also show that the divergences and the boundary modes are both related to the topology of global AdS$_4$.

Quantum Inequivalence.
----------------------

A classical antisymmetric tensor in four dimensions can be mapped into a scalar field via the duality transformation $$\label{eq:Bphiduality}
H_{\mu\nu\sigma} = 3\nabla_{[\mu}B_{\nu\sigma]} = \epsilon_{\mu\nu\sigma\lambda}\nabla^\lambda \phi~.$$ These fields are therefore classically equivalent. However, one loop corrections in curved space do not respect this equivalence. For example, the trace anomaly coefficients for these two fields differ as displayed in table \[table:centralchargesAs\]. This leads to the conclusion that these fields are quantum inequivalent [@Duff:1980qv].

                                $c$                $a$                $c-a$
  ---------------------- ----------------- -------------------- ------------------
   Antisymmetric Tensor   $\frac{1}{120}$   $-\frac{179}{360}$   $\frac{91}{180}$
       Real Scalar        $\frac{1}{120}$    $\frac{1}{360}$     $\frac{1}{180}$
    ${\rm As} - \phi$           $0$           -$\frac{1}{2}$      $\frac{1}{2}$

  : Central charges $c$ and $a$ for the $2$-form, the real scalar, and their evanescent difference.[]{data-label="table:centralchargesAs"}

However, in some sense the dual fields do not differ by terribly much. They have identical physical spectra as captured by propagating on-shell degrees of freedom: the “evanescent” field defined by their difference has no propagating degrees of freedom. Although the $a$-anomaly coefficients do indeed differ, the $c$-anomaly coefficients do not; and the $a$-anomaly is the coefficient of the Euler density which is topological. Many researchers therefore 