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November 11, 2006


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=1d=1: A certain supersymmetric quantum mechanics model, to be discussed below.
  • d=2d=2: A topological σ\sigma-model (the “A” model), which is related to Gromov-Witten Theory
  • d=4d=4: Topologically-twisted N=2N=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, QQ. If one computes the expectation value of topological observables (functions on \mathcal{F} which are QQ-invariant, modulo QQ-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=2d=2 case, an “instanton” is a holomorphic map from the worldsheet, ΣM\Sigma\to M. In the d=4d=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=4d=4 case. The Euclidean action S E=12g 2trF*F+iθ8π 2trFF+ 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=i4π(τF F +τ¯F +F +)+... S_E = -\frac{i}{4\pi}\int \left(\tau F^-\wedge F^- +\overline{\tau} F^+\wedge F^+\right) + ... where F ±=12(F±*F)F^\pm = \tfrac{1}{2} (F\pm *F) and τ=θ2π+4πig 2,τ¯=θ2π4πig 2 \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 τ¯i\overline{\tau}\to -i\infty, while holding τ\tau fixed, we localize on the ASD configurations F +=0 F^+=0 But that’s crazy! you say, τ\tau and τ¯\overline{\tau} are complex conjugates of each other. True, they are, if θ\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 θ\theta a large, negative imaginary part, we can we can take τ¯i\overline{\tau}\to -i\infty, with τ\tau fixed, by simultaneously going to weak coupling, g 20g^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+10+1 dimensional supersymmetric σ\sigma-model (with two real supercharges), whose action is

S=12dt [g μν(ϕ)(dϕ μdtdϕ νdt+iπ μDψ νDt)+14R μνλσπ μψ λπ νψ σ λ 2g μνfϕ μfϕ νλD 2fDϕ μDϕ νπ μψ ν] \begin{aligned} S= \frac{1}{2}\int dt & \left[ g_{\mu\nu}(\phi)\left( \frac{d\phi^\mu}{d t}\frac{d\phi^\nu}{d t}+i\pi^\mu \frac{D \psi^\nu}{D t}\right) +\frac{1}{4} R_{\mu\nu\lambda\sigma} \pi^\mu\psi^\lambda\pi^\nu \psi^\sigma \right. \\ &\left. - \lambda^2 g^{\mu\nu} \frac{\partial f}{\partial\phi^\mu}\frac{\partial f}{\partial\phi^\nu} - \lambda \frac{D^2 f}{D\phi^\mu D\phi^\nu}\pi^\mu\psi^\nu \right] \end{aligned}

Frenkel et al specialize to the case of MM a Kähler manifold with a circle action. ff 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=ξ+ξ¯v_f = \xi +\overline{\xi} and the circle action is generated by i(ξξ¯)i(\xi-\overline{\xi}).

Upon quantization of (1), the fermions, ψ μ\psi^\mu transform like 1-forms on MM, and the Hilbert space, =Ω (X)\mathcal{H}= \Omega^\bullet(X), the space of 2\mathcal{L}^2 differential forms on MM, with inner product, α|β= M*α¯β \langle\alpha|\beta\rangle = \int_M *\overline{\alpha}\wedge\beta and the two supercharges have the form Q= e λfde λf=d+λdf Q = e λfd *e λf=d *+i v f \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 fi_{v_f} is the interior product with the vector field, v f=g μνfϕ μϕ νv_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 MM by a factor of λ\lambda): H/λ=12λ{Q,Q }=12(1λΔ+λdf 2+( v f+ v f *)) 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 λ\lambda, the approximate ground states are concentrated near the critical points of ff (which we will assume are isolated).

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

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

S E=dτ[12g μν(dϕ μdτ±λg μσfϕ σ)(dϕ νdτ±λg νλfϕ λ)λdfdτ+fermions] S_E = \int d\tau \left[ \tfrac{1}{2}g_{\mu\nu} \left(\tfrac{d\phi^\mu}{d\tau}\pm\lambda g^{\mu\sigma}\tfrac{\partial f}{\partial\phi^\sigma}\right) \left(\tfrac{d\phi^\nu}{d\tau}\pm\lambda g^{\nu\lambda}\tfrac{\partial f}{\partial\phi^\lambda}\right) \mp \lambda \tfrac{d f}{d\tau} + \text{fermions} \right]

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

dϕ μdτ±λg μνfϕ ν=0 \frac{d\phi^\mu}{d\tau}\pm \lambda g^{\mu\nu} \frac{\partial f}{\partial \phi^\nu}=0

We’re interested in computing the matrix element b|Q|a \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 QQ). 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= b p b=p a+1e λ(f(b)f(a))n(a,b)|b 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)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

|a=e λf(a)|a˜,b|=e λf(b)b˜| |a\rangle = e^{\lambda f(a)} |\tilde{a}\rangle,\quad \langle b|= e^{-\lambda f(b)} \langle \tilde{b} |

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

Only the states in the cohomology of the full, instanton-corrected, QQ 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 μν=λg˜ μνg_{\mu\nu}= \lambda \tilde{g}_{\mu\nu}. The instanton equation becomes dϕ μdτ+g˜ μνfϕ ν=0 \frac{d\phi^\mu}{d\tau}+ \tilde{g}^{\mu\nu} \frac{\partial f}{\partial \phi^\nu}=0
  3. Add a term, ΔS E=λdτdfdτ\Delta S_E = \lambda\int d\tau \tfrac{d f}{d\tau} to the action.
  4. Finally, take λ\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 θ\theta complex in the SYM action) corresponds to the rescaling of the incoming and outgoing states (4). In the limit, λ\lambda\to\infty, in\mathcal{H}_{\text{in}} and out\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 QQ) between an “in” state and an “out” state. in\mathcal{H}_{\text{in}} is a space of distributions (currents, actually). Its dual, out\mathcal{H}_{\text{out}}, is the space of smooth differential forms on MM.

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 D 2fDϕ μDϕ ν\frac{D^2 f}{D\phi^\mu D\phi^\nu}.

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

Posted by distler at November 11, 2006 11:19 PM

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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.

Posted by: Aaron Bergman on November 11, 2006 11:42 PM | Permalink | Reply to this

Re: Localized

Although I suppose an explicit A_\infty structure might be too much to ask for here – I need to think about it some more –, but the analogy ought to hold, I’d think.

Posted by: Aaron Bergman on November 12, 2006 12:12 AM | Permalink | Reply to this

Re: Localized

Well, they do want to go a bit beyond just matrix elements between the ground states labelled by the critical points.

in\mathcal{H}_{\text{in}} breaks up into subspaces labeled by the critical points (and similarly for out\mathcal{H}_{\text{out}}). And one can study matrix elements, not just between ground states, but between excited states associated to different critical points.

Maybe that contains some new information not contained in the matrix elements of arbitrary operators between ground states.

Posted by: Jacques Distler on November 12, 2006 1:10 AM | Permalink | PGP Sig | Reply to this

Re: Localized


I was just wondering what your thoughts are on this paper.

Posted by: Mike on November 14, 2006 9:34 AM | Permalink | Reply to this

Re: Localized

  1. Why is this, even vaguely, on-topic?
  2. The “naïve” generalization of the Bekenstein-Hawking entropy bound, which Jacobson uses, is known to fail in examples. Indeed, it was that failure that motivated Bousso to formulate his Covariant Entropy Bound.
  3. I have no idea whether Jacobson’s analysis could be redone, using Bousso’s bound.
  4. Even if it could, I have a philosophical problem with elevating this to a fundamental principle, and relegating the Einstein equations to a derived concept.

Ever since Boltzmann, we have come to realized that the 2nd Law of Thermodynamics is not fundamental but, rather, emerges from the microphysics. It seems to me to be somewhat backwards to assume the reverse in the case of blackhole (or Bousso) entropy.

Indeed, we have made great strides in understanding the microphysical origin of blackhole entropy (see, e.g., here for one striking recent example). Moreover, we know that, even in the case of blackholes, there are corrections to the Bekenstein-Hawking formula, and there is even a very beautiful conjecture for the form of those corrections in the N=2N=2 case (see, e.g., here and here).

Now, I know that a lot of very smart people have wanted to use holography (as embodied, for instance, in these entropy bound) as a guiding principle for understanding quantum gravity. I’m sceptical, and I have expressed that scepticism before, on numerous occasion. But, since others are (still?) more hopeful, you might want to take my scepticism with a grain of salt.

Posted by: Jacques Distler on November 14, 2006 2:54 PM | Permalink | PGP Sig | Reply to this

Re: Localized

This is on topic because it involves quantum gravity and general relativity, both of which are topics discussed in this post of yours.

By the way, Jacques, the string wars are as intense as ever. You are needed over at cosmic variance and not even wrong. That’s an order, soldier!

Posted by: Mike on November 15, 2006 11:57 AM | Permalink | Reply to this

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