Musings

Exotic Instanton Effects

I’ve been reading the recent Beasley-Witten paper on instanton effects in string theory.

As you know, instantons can have dramatic effects on the vacuum structure of supersymmetric gauge theories. They can induce a superpotential that lifts degeneracies that are present to all orders in perturbation theory. More subtly, as found by Seiberg, in the case of $SU(N_c)$ QCD">SQCD, with $N_f=N_c$ flavours, instanton effects can change the topology of the vacuum manifold. The singular affine variety,

$$\det(M)- B\tilde B = 0$$ (where $M$ is an $N_c\times N_c$ complex matrix) is deformed to

$$\det(M)- B\tilde B = \Lambda^{2 N_c}$$

In their previous paper, Beasley and Witten argued that this effect can be understood as the generation of an exotic sort of “superpotential” of the form

$$W = \omega_{\overline{\imath}\overline{\jmath}}(\Phi,\overline{\Phi}) D_{\dot{\alpha}}\Phi^{\overline{\imath}}D^{\dot{\alpha}}\Phi^{\overline{\jmath}}$$ where $\Phi^{\overline{\imath}}$ are anti-chiral superfields, parametrizing the moduli space, $M$, and $$\omega_{\overline{\imath}\overline{\jmath}} = \frac{1}{2} \left(g_{\overline{\imath}k} \tensor{\omega}{_\overline{\imath}_^k} + g_{\overline{\jmath}k} \tensor{\omega}{_\overline{\imath}_^k}\right)$$ where $g_{\overline{\imath}j}$ is the Kähler form on $M$ and $\tensor{\omega}{_\overline{\imath}_^j}\in H^1(M, T_M)$ is the deformation of complex structure of $M$ that deforms (1) into (2)¹.

In this particular case, this fancy formalism is somewhat superfluous. The constraint (2) can simply be imposed by introducing an additional chiral superfield, $S$, and a garden-variety superpotential

$$W= S(\det(M)- B\tilde B - \Lambda^{2 N_c})$$ For $\Lambda\neq0$ (and, even for $\Lambda=0$, away from the origin in field space), $S$ and some particular combination of the other fields are massive, and can be integrated out. When $\Lambda=0$, all the fields are massless at the origin, and so integrating out $S$ leads to the singular Lagrangian, of the form (3), constructed in detail by Beasley and Witten for $SU(2)$.

In String Theory, alas, you can’t willy-nilly integrate-in fields. So there are situations where, presumably, you need to represent the instanton-induced deformation of the classical moduli space by an exotic superpotential of the form (3), instead of the more transparent (4).

Beasley and Witten lay out a couple of instances where they argue that’s the case, and show how you can calculate a superpotential of the form (3), induced by worldsheet instantons in heterotic string theory.

What’s quite interesting is that some of their heterotic computations are closely related to the higher-genus worldsheet instanton computations in the topological A-model. These compute what are, by now, quite well-known corrections to the $N=2$ supergravity action that one obtains from a Type-IIA compactification on a Calabi-Yau. Beasley and Witten compute the analogous corrections to the $N=1$ theory arising from a heterotic compactification on the same Calabi-Yau (they’re F-terms involving higher powers of the supersymmetric gauge field strength squared, $W_\alpha W^\alpha$).