Musings

The point is the following. In rigid supersymmetry, the nonlinear $\sigma$-model Lagrangian is most-easily written in terms of the Kähler potential,

where $\Phi^i$ are chiral superfields, whose lowest components are local complex coordinates on $M$. The Kähler potential, however is not globally-defined on $M$. On patch overlaps, $U_\alpha\cap U_\beta$, it differs by Kähler transformations $$K_\alpha = K_\beta + f_{\alpha\beta}(\phi) + \overline{f}_{\alpha\beta}(\overline{\phi})$$ Still, its integral $d^4\theta$ is globally-defined, and thus so is (1).

Unfortunately, the supergravity action

does not seem to be invariant under Kähler transformations, and so does not appear to be globally-defined.

This apparent inconsistency can, however, be resolved. As Bagger and Witten showed, we can accompany the Kähler transformations by a chiral rotation of the fermions — both the fermions in the chiral multiplets, and the gravitinos transform — in such a way that the action (2) is invariant (and hence well-defined across patch-overlaps on $M$). However, one pays a price: the chiral rotations must be consistent across triple overlaps, $U_\alpha\cap U_\beta\cap U_\gamma$, which leads to the aforementioned quantization condition on $[\omega]$.

More geometrically, one can phrase what’s happening as follows. Naïvely, the fermions, $\chi^i$, in the chiral multiplets, transform as spinor-valued sections of $\phi^*(TM)$ (where $TM$ is the holomorphic tangent bundle of $M$). Instead, there’s a holomorphic line bundle, $L\to M$, and the fermions transform as sections of $\phi^*(TM\otimes L)$. Similarly, the gravitino¹, $\psi_{\alpha\mu}$, transforms as a section of $\phi^*(L^{-1})$. When you look at the Einstein frame Lagrangian, in components, there is a term which is explicitly not Kähler-invariant.

where $$\mathcal{A}_\mu = \tfrac{1}{4}\left(\frac{\partial K}{\partial\phi^j}\partial_\mu\phi^j -\frac{\partial K}{\partial\overline{\phi}^{\overline{\jmath}} } \partial_\mu\overline{\phi}^{\overline{\jmath}}\right)$$ is the lowest component of the Fayet-Ferrara multiplet (whose upper components are the supercurrent and the stress tensor). It is precisely this term which, Seiberg and Komargodski point out, makes the coupling to supergravity potentially ill-defined.

In fact, $\mathcal{A}_\mu$ must be the pullback to spacetime of a connection, $A$, on $L$. $2A$ is a connection of $L^2$, and $\omega$ is its curvature. The fact that the curvature of $L$ must be an integral class is what gives the quantization condition on $[\omega]$.

The, seemingly troublesome, term (3) is precisely what’s needed to combine with other terms to form $$e \left( \epsilon^{\mu\nu\rho\sigma}\overline{\psi}_\nu\overline{\sigma}_\rho\tilde{\mathcal{D}}_\mu\psi_\sigma -i g_{i\overline{\jmath}} \overline{\chi}^{\overline{\jmath}} \overline{\sigma}^\mu\tilde{\mathcal{D}}_\mu\chi^i\right)$$ where $\tilde{\mathcal{D}}\mu$ is the proper covariant derivative acting on fermions twisted by the appropriate power of $L$.

Moreover, if the theory has a superpotential, $W(\phi)$, when couple to supergravity, the superpotential transforms not as a holomorphic function on $M$, but as a holomorphic section of $L^2$.

The quantization of $[\omega]$ gives, immediately, a non-renormalization theorem. The cohomology class $[\omega]$ cannot be renormalized. All corrections to $\omega$ are exact (in fact, they are corrections to $K$ by globally-defined functions on $M$.) It also gave Bagger and Witten their memorable slogan that, when coupling the nonlinear $\sigma$-model to supergravity, Newton’s constant is quantized in units of $f_\pi$.

What about the “necessity” that $\omega$ be exact? While it isn’t necessarily the case, it often obtains in examples. For instance, consider a theory where a continuous global symmetry, $G$ is spontaneously broken to $H$. The low energy theory necessarily has Goldstone bosons, and there’s a supersymmetric nonlinear $\sigma$-model at low energies, with $M= T^*(G/H)$. This is a Kähler manifold, but the Kähler form, $\omega = d\sigma$ is exact, because the Darboux form, $\sigma=p\cdot d q$ is globally-defined. Note that the zero section, $G/H \hookrightarrow M$ can have quite complicated topology. So $M$ can have quite a lot of interesting homology, but no holomorphic representatives. $\omega$ vanishes when restricted to any homology cycle.

All of the above considerations were classical. Further restrictions come in the quantum theory. We have a theory with chiral fermions and, at one loop, we need to worry about anomalies. Even before coupling to supergravity, there are potential $\sigma$-model anomalies, where the fermion path integral might not be well-defined (as a function) over a 2-parameter family of $\sigma$-model maps from spacetime into $M$.

The local anomaly can be obtained from the 6-form piece of $$\hat{A}(X)ch(E)$$ where $$E = TM\otimes L - (TX-1)\otimes L^{-1}$$ (I’m dropping the $\phi^*(\dots)$ from the notation.) The 6-form piece is

The terms in red occur, even in the absence of coupling to supergravity. The supersymmetric nonlinear $\sigma$-model, decoupled from gravity, is not a well-defined quantum theory² unless $ch_3(M)= 0$. If $\omega$ is exact (so that $c_1(L)=0$), the only further condition that follows in supergravity is that we should now also impose $c_1(M)= 0$.

If $[\omega]\neq 0$ (and there are, it seems, examples of this in string theory), then the implications of coupling to supergravity become quite interesting. I’ve been talking with Brian Wecht³ about what this might all mean for string theory …

---