Musings

Their model is a specialization of the SUSY Quantum Mechanics model used by Witten to give a physics proof of the Morse Inequalities.

Witten studied a $0+1$ dimensional supersymmetric $\sigma$-model (with two real supercharges), whose action is

Frenkel et al specialize to the case of $M$ a Kähler manifold with a circle action. $f$ is taken to be the moment map for the circle action. The gradient vector field can be decomposed into a (1,0) and a (0,1) piece: $v_f = \xi +\overline{\xi}$ and the circle action is generated by $i(\xi-\overline{\xi})$.

Upon quantization of (1), the fermions, $\psi^\mu$ transform like 1-forms on $M$, and the Hilbert space, $\mathcal{H}= \Omega^\bullet(X)$, the space of $\mathcal{L}^2$ differential forms on $M$, with inner product, $$\langle\alpha|\beta\rangle = \int_M *\overline{\alpha}\wedge\beta$$ and the two supercharges have the form $$\begin{aligned} Q =& e^{-\lambda f} d e^{\lambda f} = d +\lambda d f \wedge \\ Q^\dagger =& e^{\lambda f} d^* e^{-\lambda f} = d^* + i_{v_f} \end{aligned}$$ where $i_{v_f}$ is the interior product with the vector field, $v_f = g^{\mu\nu}\tfrac{\partial f}{\partial \phi^\mu}\tfrac{\partial}{\partial \phi^\nu}$ The Hamiltonian (later, we’ll find it convenient to rescale the metric on $M$ by a factor of $\lambda$): $$H/\lambda = \frac{1}{2\lambda} \{Q,Q^\dagger\} = \frac{1}{2}\left(\tfrac{1}{\lambda}\Delta + \lambda {\Vert d f\Vert}^2 +(\mathcal{L}_{v_f}+\mathcal{L}^*_{v_f})\right)$$

For large $\lambda$, the approximate ground states are concentrated near the critical points of $f$ (which we will assume are isolated).

At the critical point, $a$, with Morse index¹ $p_a$, we obtain a state, $|a\rangle$, which is a $p_a$-form. In the tree approximation (indeed, to all orders in perturbation theory), these states are annihilated by $Q$ and $Q^\dagger$. To do better, we need to compute the instanton corrections to the matrix elements of $Q$.

Write the Euclidean action (the Wick-rotated version of (1)) as

and note that $$S_E \geq \lambda |f(\tau=+\infty)- f(\tau= -\infty)|$$ with equality for

We’re interested in computing the matrix element $$\langle b| Q |a \rangle$$ The matrix element will vanish unless the number of fermion zero modes in the instanton background (3) is equal to 1 (and so can be absorbed in $Q$). The Index Theorem says that this number is just the difference in Morse indices of the two critical points. The nonzero modes cancel in the small fluctuation determinant, and one obtains $$Q|a\rangle = \sum_\substack{b \\ p_b = p_a+1} e^{-\lambda(f(b)-f(a))} n(a,b) |b\rangle$$ Here, $n(a,b)$ is an integer ² depending on the two critical points.

The exponential factor can be absorbed by rescaling the incoming and outgoing states

and then $$\langle\tilde{b}|Q|\tilde{a}\rangle = n(a,b)$$

Only the states in the cohomology of the full, instanton-corrected, $Q$ are true ground states of the system.

These were the ingredients of Witten’s proof of the Morse inequalities. To go further, Frenkel et al want to study more general observables in this theory. To do that, we first want to localize on the instanton configurations. So let’s return to the Euclidean action (2) and

1. Pick the “+” sign in (2)
2. Rescale the metric $g_{\mu\nu}= \lambda \tilde{g}_{\mu\nu}$. The instanton equation becomes $$\frac{d\phi^\mu}{d\tau}+ \tilde{g}^{\mu\nu} \frac{\partial f}{\partial \phi^\nu}=0$$
3. Add a term, $\Delta S_E = \lambda\int d\tau \tfrac{d f}{d\tau}$ to the action.
4. Finally, take $\lambda\to \infty$. The instanton configurations have zero action, and survive, but everything else is suppressed.

This non-CPT-invariant modification of the action (the analogue of taking $\theta$ complex in the SYM action) corresponds to the rescaling of the incoming and outgoing states (4). In the limit, $\lambda\to\infty$, $\mathcal{H}_{\text{in}}$ and $\mathcal{H}_{\text{out}}$ are no longer isomorphic. They are still dual to each other, and we can study matrix elements of operators (not just $Q$) between an “in” state and an “out” state. $\mathcal{H}_{\text{in}}$ is a space of distributions (currents, actually). Its dual, $\mathcal{H}_{\text{out}}$, is the space of smooth differential forms on $M$.

In the Kähler case, there are no instantons between critical points of Morse indices differing by 1. So there is no Morse complex to study. There are, however, instantons between critical points whose Morse indices differ by larger amounts, and Frenkel et al propose to study the matrix elements of more general observables between the corresponding “in” and “out” states.