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March 3, 2010

Coupling to Supergravity

There seems to be a certain amount of confusion about the claims of Seiberg and Komargodski in their latest paper. I have to say that I was confused, and there’s at least one recent paper arguing (more-or-less correctly) against claims that I don’t think they’re making.

So here’s my attempt to clear things up.

Consider an 𝒩=1\mathcal{N}=1 supersymmmetric nonlinear σ\sigma-model in d=4d=4, ie a Wess-Zumino model with target space, MM, a Kähler maninfold of complex dimension, nn. When can one couple such a theory to supergravity? A naïve reading of their paper might lead one to think that the possibilities are

  1. One can couple the theory to minimal supergravity if and only if the Kähler form, ω\omega, on MM, is exact.
  2. One can couple the theory to “new minimal” supergravity if and only if the theory has an exact U(1) RU(1)_R symmetry. In this case, ω\omega could be cohomologically nontrivial.
  3. If ω\omega is cohomologically nontrivial, and the theory does not have a U(1) RU(1)_R symmetry, then the only possibility is to couple to non-minimal “16|16” supergravity.

One might think that, but one would be wrong. As Bagger and Witten showed, nearly 30 years ago, coupling to minimal supergravity does not require the Kähler form to be exact. Rather, [ω][\omega] must be an even integral class.

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

(1)S=d 4xd 4θK(Φ,Φ¯)S = {\int d^4 x\, d^4\theta\, K(\Phi,\overline{\Phi})}

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

Unfortunately, the supergravity action

(2)S=3d 4xd 4θEe 13K(Φ,Φ¯)S = -3 {\int d^4 x\, d^4\theta E e^{-\tfrac{1}{3}K(\Phi,\overline{\Phi})}}

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 MM). However, one pays a price: the chiral rotations must be consistent across triple overlaps, U αU βU γU_\alpha\cap U_\beta\cap U_\gamma, which leads to the aforementioned quantization condition on [ω][\omega].

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

(3)e(ϵ μνρσψ¯ νσ¯ ρψ σig iȷ¯χ¯ ȷ¯σ¯ μχ i)𝒜 μ(ϕ,ϕ¯)e \left( \epsilon^{\mu\nu\rho\sigma}\overline{\psi}_\nu\overline{\sigma}_\rho\psi_\sigma -i g_{i\overline{\jmath}} \overline{\chi}^{\overline{\jmath}} \overline{\sigma}^\mu \chi^i\right) \mathcal{A}_\mu(\phi,\overline{\phi})

where 𝒜 μ=14(Kϕ j μϕ jKϕ¯ ȷ¯ μϕ¯ ȷ¯) \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, 𝒜 μ\mathcal{A}_\mu must be the pullback to spacetime of a connection, AA, on LL. 2A2A is a connection of L 2L^2, and ω\omega is its curvature. The fact that the curvature of LL must be an integral class is what gives the quantization condition on [ω][\omega].

The, seemingly troublesome, term (3) is precisely what’s needed to combine with other terms to form e(ϵ μνρσψ¯ νσ¯ ρ𝒟˜ μψ σig iȷ¯χ¯ ȷ¯σ¯ μ𝒟˜ μχ i) 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 𝒟˜μ\tilde{\mathcal{D}}\mu is the proper covariant derivative acting on fermions twisted by the appropriate power of LL.

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

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

What about the “necessity” that ω\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, GG is spontaneously broken to HH. The low energy theory necessarily has Goldstone bosons, and there’s a supersymmetric nonlinear σ\sigma-model at low energies, with M=T *(G/H)M= T^*(G/H). This is a Kähler manifold, but the Kähler form, ω=dσ\omega = d\sigma is exact, because the Darboux form, σ=pdq\sigma=p\cdot d q is globally-defined. Note that the zero section, G/HMG/H \hookrightarrow M can have quite complicated topology. So MM can have quite a lot of interesting homology, but no holomorphic representatives. ω\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 σ\sigma-model anomalies, where the fermion path integral might not be well-defined (as a function) over a 2-parameter family of σ\sigma-model maps from spacetime into MM.

The local anomaly can be obtained from the 6-form piece of A^(X)ch(E) \hat{A}(X)ch(E) where E=TML(TX1)L 1 E = TM\otimes L - (TX-1)\otimes L^{-1} (I’m dropping the ϕ *()\phi^*(\dots) from the notation.) The 6-form piece is

(4)ch 3(M)p 1(X)24c 1(M)+c 1(L)(ch 2(M)+21n24p 1(X))+12c 1(L) 2c 1(M)+n+36c 1(L) 3{\color{red} ch_3(M) -\frac{p_1(X)}{24} c_1(M)} + c_1(L) \left(ch_2(M) + \tfrac{21-n}{24} p_1(X)\right) +\tfrac{1}{2} {c_1(L)}^2 c_1(M) + \tfrac{n+3}{6} {c_1(L)}^3

The terms in red occur, even in the absence of coupling to supergravity. The supersymmetric nonlinear σ\sigma-model, decoupled from gravity, is not a well-defined quantum theory2 unless ch 3(M)=0ch_3(M)= 0. If ω\omega is exact (so that c 1(L)=0c_1(L)=0), the only further condition that follows in supergravity is that we should now also impose c 1(M)=0c_1(M)= 0.

If [ω]0[\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 …

Update (8/5/2010):

Readers of this post might enjoy this paper.

1I should be very precise about which gravitini I mean. The action (2) has a nontrivial function of the ϕ\phi multiplying the Einstein-Hilbert term and the gaugino kinetic term.. We need to perform a Weyl rescaling to put the action in Einstein frame, where the coefficient of these terms are constants. It’s gravitino in Einstein frame that we’re talking about, above.

2 One might suppose that further restrictions might follow from the p 1(X)c 1(M)p_1(X)c_1(M) term. But, at least for the globally supersymmetric theory, there’s no compelling reason to consider spacetimes with nontrivial topology (other than 4\mathbb{R}^4 and S 4S^4). In any case, c 1(M)=ch 3(M)=0c_1(M)=ch_3(M)=0, for the commonly-considered case of M=T *(G/H)M=T^*(G/H).

3We’ve also been discussing the case of U(1)U(1) gauge theories with Fayet-Iliopoulos D-terms. Considerations very similar to the ones I’ve outlined here lead to the quantization of the FI coefficient.

Posted by distler at March 3, 2010 8:53 PM

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