Bosonization can be viewed as an instance of S-duality. Indeed, the S-duality of the Thirring and sine-Gordon models is the direct field theory analogy; the sine-Gordon model is the S-dual bosonization of the Thirring model [6].

These duality insights have been to numerous to do justice to here, but to convey the broad scope, let us mention just two seemingly disparate examples: Kramers-Wannier duality in locating the critical point in the two-dimensional Ising model [10] and U-duality in the microscopic derivation of the Bekenstein-Hawking entropy of black holes in string theory [23].

Again, there are too many instances to catalog, and we merely mention three examples indicating the diverse applications: constraining the bound state spectrum of confining gauge theories via ’t Hooft anomaly matching conditions [37], critical dimensions [38] and Einstein’s equations [39, 40] from the vanishing of Weyl anomaly [41, 42] on the string world sheet, and the identification of five consistent superstring theories via the Green-Schwarz mechanism [43].

There also exists a “double-dual” graviton, where one dualizes both indices rather than a single index; in four dimensions, this is, however, a mere algebraic relabeling [52].

There are obstructions to going beyond the linear case [54, 55]. These are evaded in Ref. [47] by the fact that the local dual graviton equation of motion includes the graviton itself. In this case, the dual graviton appears as an auxiliary field, which may be eliminated to recover the Einstein-Hilbert action. Integrating out, instead, the graviton would yield a nonlocal action, violating one of the assumptions entering [54].

This is analogous to recovering the “linearized instanton moduli space” $H^2(M;\mathbb{Z})\otimes \mathfrak{g}$ from the moduli space of $\mathfrak{g}$-valued Yang-Mills instantons on a spacetime $M$.

In particular, when $M$ is parallelizable, then $\chi (M;{\mathrm{T}}^{*}M)=\chi (M)d$.

Here, $⌣$ is the cup product between cohomology classes, and $⌢$ is the cap product between homology and cohomology classes [85].

Indeed, this is equivalent to starting from the frame-bundle picture and linearizing. The addition symmetries and ghosts that arise here correspond, in this picture, to the local Lorentz ghosts.

That is, as dual $p$-form and $(d-p-2)$-form theories. On the other hand, $B$ and $\tilde{B}$, for example, should be regarded as dual fields once one recalls that the ($d-2$)-form is a ${\mathrm{T}}^{*}M$-valued ($d-3$)-form with a symmetry constraint and, so, dual to a ${\mathrm{T}}^{*}M$-valued 1-form with a symmetry constraint.

More generally, we can consider any $\frac{d}{2}=p+1$ for arbitrary ${\mathrm{T}}^{*}M$-valued $p$-form and $(d-2-p)$-form dual formulations of linearized gravity.

The dual second Bianchi identities are rotated among themselves [28].

That is, $M$ is a flat Riemannian manifold.

The same observation was made for $p$-forms in Ref. [35].

The details will be given in Ref. [107].