### 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, $W_\alpha W^\alpha$).

^{1} We’re identifying a finite deformation of the complex structure with an infinitesimal tangent vector (an element of $H^1(M, T_M)$) to the space of deformations, at the point corresponding to the complex structure of the classical moduli space, $M_0$ (which, typically, but not in the example above, is a singular point in the space of complex structures). Moreover, to write this “F-term”, we need to use the Kähler metric on $M_0$, which is singular. It’s not obvious that this makes sense. However, when you *can* integrate-in some fields and write the deformation as an *ordinary* superpotential (as above), you can check this procedure reproduces the correct result.

## Higher dimensions of fiction?

Hi there. Recently, I’ve been searching for a credible physicist or scholar to take a look at my novel “The Pink Room,” which was published last month. Briefly, the world’s leading physicist attempts to use the science of string theory to bring his daughter back from the dead. Government agents and a bestselling novelist race to find out if he was succesful.