### Localized

Frenkel, Losev and Nekrasov have put out Part I of a huge project to study topological field theories “beyond the topological sector.”

It sounds like we will spend some time discussing their work in the Geometry and String Theory Seminar, so it might be good to give a little summary here.

They’re interested in a set of related theories in various dimensions

- $d=1$: A certain supersymmetric quantum mechanics model, to be discussed below.
- $d=2$: A topological $\sigma$-model (the “A” model), which is related to Gromov-Witten Theory
- $d=4$: Topologically-twisted $N=2$ SYM, which is related to Donaldson Theory.

In each case, the field space, $\mathcal{F}$, is an infinite dimensional supermanifold (of bosonic and fermionic fields), with a nilpotent odd involution, $Q$. If one computes the expectation value of topological observables (functions on $\mathcal{F}$ which are $Q$-invariant, modulo $Q$-exact), one finds that the computation localized on a finite dimensional subspace of $\mathcal{F}$ which, in each case, is called “instanton moduli space.” In the $d=2$ case, an “instanton” is a holomorphic map from the worldsheet, $\Sigma\to M$. In the $d=4$ case, an “instanton” is an anti-self-dual connection (modulo gauge transformations).

But there’s another way in which this localization can occur. Consider the $d=4$ case. The Euclidean action $S_E = \int \frac{1}{2g^2} \tr F\wedge *F +\frac{i\theta}{8\pi^2} \tr F\wedge F +\dots$ can be written $S_E = -\frac{i}{4\pi}\int \left(\tau F^-\wedge F^- +\overline{\tau} F^+\wedge F^+\right) + ...$ where $F^\pm = \tfrac{1}{2} (F\pm *F)$ and $\tau = \frac{\theta}{2\pi} +\frac{4\pi i}{g^2},\quad \overline{\tau} = \frac{\theta}{2\pi} -\frac{4\pi i}{g^2}$

If we send $\overline{\tau}\to -i\infty$, while holding $\tau$ fixed, we localize on the ASD configurations
$F^+=0$
*But that’s crazy!* you say,

*$\tau$ and $\overline{\tau}$ are complex conjugates of each other.*True, they are, if $\theta$ is real, as required by CPT invariance (more precisely, Reflection-Positivity). However, if we are willing to deform the theory, in a CPT-violating fashion, by giving $\theta$ a large, negative imaginary part, we can we can take $\overline{\tau}\to -i\infty$, with $\tau$ fixed, by simultaneously going to weak coupling, $g^2\to 0$.

The price we will pay, in a canonical formalism, is that the spaces of “in” states and “out” states will no longer be isomorphic. The computation of topological observables is unaffected. But Frenkel *et al* want to go beyond the topological sector and compute the matrix elements of arbitrary observables, in this localized limit.

The first (100 page) installment is about the SUSY Quantum Mechanics case.

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 index^{1} $p_a$, 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

- Pick the “+” sign in (2)
- Rescale the metric $g_{\mu\nu}= \lambda \tilde{g}_{\mu\nu}$. The instanton equation becomes $\frac{d\phi^\mu}{d\tau}+ \tilde{g}^{\mu\nu} \frac{\partial f}{\partial \phi^\nu}=0$
- Add a term, $\Delta S_E = \lambda\int d\tau \tfrac{d f}{d\tau}$ to the action.
- 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.

^{1} The Morse index is the number of negative eigenvalues of the Hessian $\frac{D^2 f}{D\phi^\mu D\phi^\nu}$.

^{2} Let $V_a$ and $V_b$ be the negative eigenspaces of the Hessian at the respective critical points. Let $v$ be the tangent vector to the “instanton” trajectory, $\gamma$, between the critical points, and $\tilde{V}_b$ the subspace of $V_b$ orthogonal to $v$. Parallel transport along the trajectory gives us a map $\tilde{V}_b\to V_a$, and the sign $n_\gamma=\pm 1$, depending on whether the orientations agree. We then take $n(a,b)=\sum n_\gamma$ for all instantons connecting the two critical points.

## Re: Localized

Frenkel, Losev and Nekrasov have put out Part I of a huge project to study topological field theories “beyond the topological sector.”I confess to having only made it through the introduction of this particular opus, but do you have any idea why they want to do this? I understand the idea that the dga has more information than just the physical states, but, under my usual belief that correlation functions are going to be related to an A$_\infty$ structure, I’d guess that no information has been missed for topological stuff (although I have no good argument why the “Massey products” correspond to the correlation functions in general, it seems like something that ought to be true, and is definitely true for the open topological string as shown by Katz and Aspinwall.)

Assuming the above isn’t completely wrong and we don’t get any new topological invariants, then, what do they hope to learn from this study that we don’t already know? They’re all really smart guys – it must be something I’ve missed.