### Topological G2 Sigma Models

de Boer, Naqvi and Shomer have a very interesting paper, in which they claim to construct a topological version of the supersymmetric $\sigma$-model on a 7-manifold of $G_2$ holonomy. The construction is quite a bit more delicate than the usual topologically-twisted $\sigma$-model. The latter are local 2D field theories, in which the spins of the fields have been shifted in such a way that one of the (nilpotent) supercharges becomes a scalar. If you wish, you can think of them as a 3-stage process:

- Start with the original “untwisted” $\sigma$-model.
- Twist, to form a local, but nonunitary field theory.
- Pass to the $Q$-cohomology, which finally yields a unitary theory (with, in fact, a finite-dimensional Hilbert space of states).

In their construction, the observables (and, for that matter, the nilpotent “scalar” supercharge itself) are nonlocal operators, defined as projections onto particular conformal blocks in the underlying CFT. So there is no intermediate “step 2”, at least not one that is recognizable as a local field theory.

The idea that there might be a topological version of the supersymmetric $\sigma$-model on a $G_2$ manifold dates back to Shatashvili and Vafa. They noticed that, in addition to the $N=1$ superconformal algebra (generated by $T(z)$ and $G(z)$), the theory has an extended chiral algebra, with the additional generators forming supermultiplets of spin 3/2 and 2. In (1,1)superspace (with $D= \partial_\theta + \theta\partial_z$), we have the $N=1$ supercurrent and stress tensor $G+\theta T = - \frac{1}{2} g_{i j}(X) D X^i \partial_z X^j$ as well as $G_I + \theta K = \frac{i}{15} \phi^{(3)}_{i j k}(X) D X^i D X^j D X^k$ and $T_I + \theta M = \frac{1}{5} ( \phi^{(4)}_{i j k l} D X^i D X^j D X^k D X^l + \frac{1}{2} g_{i j}(X) D X^i \partial_z D X^j)$ formed out of the covariantly-constant 3-form, $\phi^{(3)}$, and its Hodge dual, $\phi^{(4)}=*\phi^{(3)}$, associated to the existence of a $G_2$ structure. The key facts are

- $G_I$ and $T_I$ form a second superconformal algebra, with central charge $c= 7/10$,
*i.e.*there’s a hidden Tricritical Ising Model in this theory. - If we write $T=T_I+T_r$, then $T_I(z) T_r(w)=$ nonsingular, which is to say that, thought of as a conformal (as opposed to superconformal) theory, this model is the tensor product of a $c=7/10$ Tricritical Ising Model and a second theory with $c= 49/5$, whose stress tensor is $T_r$.

de Boer *et al* show that there’s a BPS bound on the conformal weight
$h = h_I + h_r \geq \frac{1+\sqrt{1+80 h_I}}{8}$
which is saturated for the following conformal primaries in the NS sector: $|h_I,h_r\rangle=|0,0\rangle$, $|1/10, 2/5\rangle$, $|6/10,2/5\rangle$ and $|3/2,0\rangle$, whose Tricritical Ising components are just the primaries $\Phi_{n,1}$, $n=1,2,3,4$ in the Kač table.

Defining $P_n$ to be the projection (in the NS sector) onto the Virasoro representations corresponding to $\Phi_{n,1}$, the fusion rules of the Tricritical Ising Model allow us to decompose
$G(z) = G^\uparr(z) + G^\darr(z)$
in the NS sector, where
$G^\uparr(z) = \sum_n P_{n+1} G(z) P_n,\qquad
G^\darr(z) = \sum_n P_{n} G(z) P_{n=1}$
de Boer *et al* define
$Q = G^\uparr_{-1/2}$
which is nilpotent, by virtue of
$Q^2 = \sum_n P_{n+2} G_{-1/2}^2 P_n = \sum_n P_{n+2} L_{-1} P_n =0$
where we used the above decomposition of $G(z)$ and the fact that $L_n$ has vanishing matrix elements between different Virasoro representations.

The spin field, which creates the ground state of the Ramond sector, has $h=7/16$, and lies entirely in the Tricritical Ising sector of the theory (it is $\Phi_{1,2}$ in the Kač table). We can decompose it into two conformal blocks $\Phi_{1,2} = \Phi_{1,2}^+ + \Phi_{1,2}^-$ defined by its action on the two Virasoro representations that comprise the R-sector ($\mathcal{H}_{1,2}$, with $h_I=7/16$ and $\mathcal{H}_{2,2}$, with $h_I=3/80$): $\Phi_{1,2}^+ :\, \left\{ \array{ \mathcal{H}_{1,2}\to \mathcal{H}_{4,1}\\ \mathcal{H}_{2,2}\to \mathcal{H}_{3,1}}\right.\qquad \Phi_{1,2}^- :\, \left\{ \array{ \mathcal{H}_{1,2}\to \mathcal{H}_{1,1}\\ \mathcal{H}_{2,2}\to \mathcal{H}_{2,1}}\right.$ Let $\mathcal{O}_{n,\alpha}$ be the operators corresponding to the “special” NS conformal primary states introduced above, $|h_I = \frac{(2n-3)(n-1)}{10}, h_r = \frac{(4-n)(n-1)}{5},\alpha\rangle$, where $\alpha$ is some discrete index labeling the possibly distinct operators with these conformal weights. ($\mathcal{O}_1=𝟙$ and $\mathcal{O}_4=G_I(z)$ presumably don’t need such a label if the “internal” $c=49/5$ theory is unitary.) The $\mathcal{O}_{n,\alpha}$ don’t commute with $Q$, but $\mathcal{A}_{n,\alpha}(z)= \sum_m P_{n+m-1} \mathcal{O}_{n,\alpha}(z) P_m$ do. The observables of the “topological” theory are defined as

where I’ve suppressed the right-movers, as in the rest of my summary. The claim is that precisely these amplitudes, for $n=1,\dots,4$, are independent of the insertion points, and constitute a 2D topological field theory.

de Boer *et al* also have a proposal for a higher-genus “topological string theory” generalization, but I have to say that I don’t really understand it. So, maybe I’d better stop here.