Edward Witten gave two talks at MIT last week. The first was on gauge theory and wild ramification – very similar to earlier work he did with Kapustin and Gukov on geometric Langlands, but with some clever use of nineteenth century technology (namely, Stokes matrices) to deal with irregular singularities. I won’t say much about it, except to mention that his use of the term “wild ramification” employs a tacit conjectural dictionary between irregularity for differential equations and the Swan conductor for Galois representations. The second talk was on some calculations in pure 3D gravity he did with Alex Maloney, and even though I didn’t understand much of it, I’m going to write about it. Perhaps people with more background in phyisics or three-manifold topology can make illuminating comments and corrections.

Witten began by introducing the Anti-de Sitter space as the basic classical solution to Einstein’s equations in three dimensions with negative cosmological constant. It is topologically with Poincaré’s hyperbolic metric on , and it is typically presented as the quotient , which is the same as by an exceptional Lie group isomorphism. The topological decomposition seems to be a manifestation of Iwasawa decomposition. Witten takes the universal cover , with the time direction. A rather curious fact about the metric is that a lightlike trajectory can go to the boundary and back in finite time, even though in a spacelike slice it is infinitely distant.

Witten’s goal was to study quantum gravity on spaces that are only asymptotically Anti-de Sitter, and find its spectrum. In four dimensions, one can work in asymptotically Minkowskian spacetimes, and one can find S-matrices with graviton poles, like photon poles in QED. [I have no idea what that meant, except it sounded like you can get excitations that behave like plain old massless particles.] In three dimensions, pure gravity has no particles and no gravity waves, and there is no S-matrix and no observables – this seems to be bad news if you want an interesting theory. However, for negative cosmological constant, Bañados, Teitelboim, and Zanelli found black hole solutions, and Brown and Henneaux found that they have boundary modes that behave like gravitons.

We consider two distinguished infinitesimal symmetries: the Hamiltonian and the momentum given by rotations of the disc. There are a few other symmetries, but this is a maximal commuting set. The partition function is given by , where , and the letter has been repurposed as a gluing parameter rather than an angular coordinate on the disc.

We want to take a trace in the time direction, so we make time imaginary, switch to a Euclidean path integral, and glue the top and bottom of the torus. [This seems to be a standard technique, but it sounded very unmotivated.] This quotient is a complete metric space, but it has a conformal boundary, which is a 2-torus with a complex structure, determined by the time we allow to pass before gluing and the angle by which we twist the boundary circle when gluing. In other words, the conformal boundary is an elliptic curve of the form . For a given torus, we compute the partition function by a path integral over all possible interiors.

If a given interior has no classical solution, then it should give a negligible contribution to the path integral. [I think this principle is called stationary phase or steepest descent, depending on the signature.] For each topology, we want to enumerate the saddle points of the Einstein-Hilbert action, i.e., the metrics that obey the equation , and take the sum of over all saddles. A very convenient fact is that one can compute the 1-loop corrections exactly and show that the higher order corrections are zero. Furthermore, one can enumerate the saddles in a very straighforward manner [Witten remarked that you can get a very quick answer if you phrase this question correctly to a hyperbolic 3-manifold theorist. It would have been nice if he had explained what a loop was in this context, but maybe the physicists in the audience already knew.]

There is exactly one saddle for each topology and one can transition between the topologies by transformations of the boundary. [The uniqueness is false for boundary genus greater than one by a theorem of Thurston, but I don’t know which theorem.] For example, when , one gets the obvious classical saddle from Anti-de Sitter space, and acting by yields a Euclidean black hole. If we write for the contribution of the obvious saddle point, then , where the summation is over the coset space . [This summation is similar to the construction of real-analytic Eisenstein series.]

The project is now reduced to two goals: computation of and summation over the coset space. is the partition function for small perturbations around Anti-de Sitter space, for which the vacuum is the only obvious state. The fact that there are more states is due to Brown and Henneaux, around 1990. We get the spectrum by quantizing the classical phase space, meaning we do a semiclassical approximation, then argue that the solution is exact.

Naïvely, a point in phase space is a class of classical solutions under the equivalence relation of asymptotically trivial diffeomorphisms. In particular, time translation is not trivial at infinity, so it acts nontrivially on phase space. There is a good group of diffeomorphisms that acts on it. For four dimensional general relativity with zero cosmological constant, this group is the Poincaré group. Brown and Henneaux showed that in our case, the good group is , which acts transitively. Therefore, the phase space is a quotient of this group by some subgroup, namely the stabilizer of Anti-de Sitter space, . The phase space is then a product of two copies of .

In quantum mechanics, quantizing a homogeneous phase space typically yields an irreducible representation of some central extension of the group, so our quantum Hilbert space is an irreducible representation of a product of two Virasoro groups. We ask that the energy be bounded below to be physically reasonable, so we have a lowest weight representation, i.e., a vacuum representation with vacuum state given by Anti-de Sitter space. The partition function is then , where the central charge is , which diverges as the cosmological constant approaches zero. This evaluates to , where the asymptotically constant multiplier is from the one-loop corrections. Witten argues that there are no further corrections, because the symmetry group acts on the perturbative Hilbert space, fixing energy levels.

There is a minor problem wen summing over the coset space, which is that the sum diverges. This is regularized in a fairly standard way, by multiplying each term by and analytically continuing to zero. The derivatives of the summands give a convergent series, which matches with the derivative of the regularized sum. Without the one-loop corrections, one gets a pole at zero.

At this point, Witten pointed out that the partition function is physically unrealistic, because the spectrum is not discrete. He pointed out that continuous spectra arise only when free energy is infinite, meaning space is infinite. [Presumably this reflected on the compact conformal boundary rather than the infinite size anti-de Sitter space.] There are two possible interpretations of this result: either the theory doesn’t exist (although it may embed in some bigger theory) or there are unknown contributions to the path integral. Two possibilities for unknown contributions are long strings (which make the integral diverge) and solutions called complex saddle points. Witten said that he couldn’t justify the use of complex saddles, but that they gave physically reasonable answers – in particular, he got the functions that showed up in his recent paper suggesting the existence of a duality between three dimensional gravity and extremal conformal field theories, including exceptional objects like the Moonshine Module.

At the end, he made some comment that normal questions are dominated by real saddles, while abnormal questions are dominated by complex saddles, but didn’t elaborate, to somewhat universal frustration. Some guy kept asking whether adding higher order terms to the Einstein-Hilbert action would resolve his problems, and Witten kept saying that he didn’t know.

Of course complex saddles can be expected to be significant. Already in the case of finite dimensional quantum mechanics they are.

Lots of interesting work has been done by physicists Shudo and coworkers, Berry and Howls, and mathematicians such as Ecalle, Voros, Delabaere and Pham etc. which should non-trivially extend to field theories…

For more on Maloney-Witten, see here.

Thank you for the references. I was actually hoping to find out what complex saddle means in gravity, though. It doesn’t seem to be a manifold with a metric satisfying Einstein’s equations, since those are covered by real saddles. Is it some kind of complex 3-fold with a complex-valued symmetric 2-form satisfying the same equations? I’d be rather surprised if such a naïve generalization actually worked.

I’d guess that part of the computation is the same as the one done by Maldacena and Strominger, in the case of Strings on AdS_3✕S^3✕T^4.

An Einstein-Hilbert metric on a (Riemannian) 3-manifold is a

metric of constant curvature, in this context, a hyperbolic metric,

which is unique up to isometry for a manifold with incompressible

torus boundary, by Mostow-Prasad rigidity. In general, the space

of hyperbolic metrics (if one exists at all) are parameterized by

the Teichmuller space of the boundary components of negative

Euler characteristic (I’m lying – there are more metrics than this,

but these are a big family of them); so when the boundary is a

surface of higher genus, Thurston’s geometrization theorem for

Haken manifolds tells you that under suitable topological

conditions a hyperbolic metric exists, and then Ahlfors-Bers

uniformization theorem (actually the part proved by Morrey)

tells you that it has lots of deformations.

Thanks Danny. That helped a lot.

Forgive me if this is a naïve, but how do I know that the boundary is incompressible? If I’m not mistaken, AdS_3 has a compressing disk.

Indeed, AdS_3 has a compressing disk, and a solid torus admits

no complete hyperbolic metric (of finite volume). Also, changing

the gluing doesn’t change the topology (you still get a solid torus.

(well, I suppose you could also get a solid Klein bottle if the

gluing is orientation-reversing.).)

Without having seen the talk, I would guess that the point is that

the spaces in question are only *asymptotically* de Sitter, and

that the interior is (in interesting cases) topologically complicated,

and will (typically) have incompressible boundary. Perhaps the

terms with compressible boundary make no contribution . . ?