The AdS/CFT Correspondence and the Witt and Virasoro Algebras

In physics, there are many phenomenon which seem to exist on multiple scales or even every scale. For example, if we look at how electrons align in a ferromagnetic material, at certain critical temperatures, the electrons will align with each other in patches. This means that within a patch of material, all the electrons are basically oriented in the same direction whereas in a neighboring patch, they’ll align in a different direction. You can model this type of behavior using statistical mechanics and surprisingly, the sorts of patterns one sees at these critical temperatures seem to appear at all scales. One can “renormalize” which is sort of like “zooming in and out” and find these phenomena. This type of behavior also happens for phase diagrams. For example, if you follow the boundary between gaseous and liquid water, eventually the boundary disappears and there is no hard line between gas and liquid phase. Instead, you will see patterns that persist even as you renormalize. So despite the fact that individual particles behave randomly, macroscopic behavior emerges.

We say that something is scale invariant if it is a fixed point under the renormalization group. As we’ve said, this is a common and natural symmetry that appears in physics. A conformal field theory is a quantum field theory that is invariant under conformal transformations. This is a stronger condition than scale invariance since scale invariance preserves angles at a global level while conformal transformations are locally angle-preserving. So conformal transformations include scaling but also rotations, translations, etc.

The AdS/CFT Correspondence

This correspondence was first proposed by Juan Maldacena in 1997 and is conjectural. The basic idea is that there is a “dictionary” for translating calculations in a particular gravitational theory (that of anti-de Sitter space, hence AdS) into calculations in a conformal field theory. Every entity in one theory should have a counterpart in the other theory. For example, a single particle in the gravitational theory might correspond to some collection of particles in the boundary theory. In addition, the predictions in the two theories are quantitatively identical so that if two particles have a 30% chance of colliding in the gravitational theory, then the corresponding collections in the boundary theory would also have a 30% chance of colliding.

This follows something called a holography principle which I wrote about last week. The idea is that quantum gravity in the interior of the spacetime manifold (in the bulk) can be understood by just understanding the boundary quantum theory that is without gravity, only special relativity. This is akin to how one can understand a lot about a black hole (in the bulk) by understanding the event horizon (the boundary). I believe that gravity in AdS is quite strong as we have vacuum energy permeating spacetime everywhere and strongly; so it’s not a good model for, say, the solar system. The original and main motivation for the holographic principle itself was the fact that the apparent black hole entropy in Einstein gravity scales with the area of the event horizon instead of the black hole’s bulk volume (which is not even well-defined), suggesting that gravity encodes or is encoded by some boundary field theory associated with horizons.

This intuition about holographic black hole entropy has found remarkably detailed reflection in (mathematically fairly rigorous) analysis of holographic entanglement entropy, specifically via holographic tensor networks, which turn out to embody key principles of the AdS/CFT correspondence in the guise of quantum information theory, with concrete applications such as to quantum error correcting codes.

So back to the AdS/CFT correspondence. The heart of it is the observation that the classical action functionals for various fields coupled to Einstein gravity on a anti-de Sitter spacetime $M$ are, when expressed as functions of the asymptotic boundary-values of the fields, of the form of generating functions for correlators/$n$-point functions of a conformal field theory on that asymptotic boundary $\partial M$, in a large $N$ limit. If we want to boil this down to one equation, one might write it as (see Witten 1998): $Z_{CFT}[\gamma] = \exp(-I(g))$ or perhaps $Z_{CFT}[\gamma] = \sum \exp(-I(g))$.

Here, $Z$ is the partition function for the conformal field theory. In statistical mechanics, partition functions give a full understanding of the theory so one definitely is motivated to understands $Z$. They are similarly important for CFTs. Moving on, $[\gamma]$ is a conformal structure on $\partial M$. $I(g)$ is the normalized Einstein-Hilbert action of an Einstein metric $g$ that is in the conformal class $[\gamma]$ at infinity. In the second equation, the sum would be over all Einstein manifolds $(W,g)$ such that $\partial W = \partial M$ and $g$ is, at conformal infinity, in the class $[\gamma]$. The sum might not make sense since it might not be finite. In the Riemannian setting (rather than Lorentzian), the PDE for Einstein metrics is such that the moduli space should be finitely many points but this might not be the case with AdS spaces.

Another important aspect is that the AdS/CFT correspondence crucially involves the exceptional isomorphism between the isometry group of anti de Sitter spacetime $\text{AdS}_{d+1}$ and the conformal group of Minkowski spacetime of dimension $d$: the connected component of both is the special orthogonal group $SO(d,2)$. But the AdS/CFT correspondence is deeper and more subtle than this group theory underlying it, in particular in how it puts fields and states on the gravity side in correspondence with sources and correlators on the field theory side, respectively.

The Witt and Virasoro Algebras

Anyways, setting aside the correspondence and just focusing on the CFT aspect, there are some very structured algebras that are important to CFT. The Witt algebra is an infinite dimensional Lie algebra and can be viewed as the Lie algebra of derivations on Laurent polynomials $\mathbb{C}[z,z^{-1}]$ or equivalently as complex vector fields on $S^1$. This is because vector fields on $S^1$ are sections of $S^1 \times \mathbb{R} \cong \mathbb{C}^*$; we can view this is as $\text{Spec}\, \mathbb{C}[z,z^{-1}]$. Under this view, it’s not surprising that string theorists are interested in the Witt algebra. Since derivations are both linear and satisfy the Leibniz rule $D(fg) = D(f)g + fD(g)$, we can fairly easily show that the derivations are of the form $p(z) \partial_z$ where we differentiate, then multiply by a Laurent polynomial $p(z)$.

If we try to define a multiplication via composition, we’ll get 2nd order terms. But the commutator $[D_1,D_2]=D_1D_2-D_2D_1$ has the 2nd order terms cancel. Thus we can define the Lie algebra structure. Let’s use generators $L_n = -z^{n+1}\partial_z$ for $n \in \mathbb{Z}$. These are chosen because they satisfy $[L_m,L_n] = (m-n)L_{m+n}$.

This Lie algebra is not quite the correct one to use for conformal field theory, however. Rather, we want to use a central extension $V$; that is there is an abelian subalgebra $A$ such that $V/A =W$. In order to construct this, we add in another generator $c$ to generate $A$. Since it’s 1-dim, then of course $[c,c] =0$ by the anti-symmetry relation. The overall relations for $V$ are now $[c,L_n] = 0$, $[L_m,L_n] = (m-n)L_{m+n}+\frac{c}{12}(m^3-m)\delta_{m+n,0}$.

The second relation is pretty much determined up to a constant by the axioms of a Lie algebra (anti-symmetry and Jacobi identity). The $\delta_{m+n,0}$ is only equal to 1 when $m=-n$. Note that in this case, $[L_n,L_{-n}] = 2nL_0 - \frac{c}{12}(n^3-n) = -[L_{-n},L_n]$. The $-1/12$ appears as a way to “renormalize” $1+2+3+4+…$ (this is very wrong, of course). Remember that if we analytically continue the Riemann-Zeta function and evaluate at $z=-1$, then we do get $-1/12$.

This is called the Virasoro algebra and the state space of a CFT is a representation of the product of the two Virasoro algebras. As I understand, we need two of them because of chirality: we have left and right-handed particles. The reason for considering this algebra is that its (usual) representations correspond to projective representations of $W$. A projective representation is a morphism $W \to \mathfrak{pgl}(H)$ where $H$ is a vector space and $\mathfrak{pgl}$ is the Lie algebra of the Lie group $PGL(H) \cong GL(H)/\mathbb{C}^*$. In some sense, projective geometry is the right setting for studying quantum field theory, see my post on anomalies.

Aside: One might ask if there are Lie groups whose Lie algebras are the Witt or Virasoro algebras. Since they are both infinite dimensional, we have to carefully consider what topology we want on these algebras in order to understand what kind of infinite dimensional Lie group we’re looking for. For $W$, it is natural to take the Fréchet topology of compact convergence of the vector fields on $S^1$ and all their derivatives. The Lie group for the real vector fields is orientation-preserving diffeomorphisms of $S^1$ so we just need to find a complexification of such a Lie group. As it turns out:

Theorem (Lempert): No complexification exists for the Lie group $\text{Diff}_+(S^1)$. There is not even a real Lie group which has Lie algebra isomorphic to $W$.

Similarly, there is no complex Lie group with Lie algebra being the (completion of) Virasoro algebra. By completion, I mean a procedure which makes the Lie algebra Cauchy complete and also algebraically complete. There’s a series called the Baker-Campbell-Hausdorff series which Lie algebras always have and we want it to converge.

I would imagine that there are some correspondences between the Virasoro algebra and related objects with gravity. For example, Virasoro symmetries correspond to so-called Brown-Henneaux tansformations which are special 3D coordinate transformations of anti-de Sitter space that, at the boundary, are conformal transformations. By analogy, remember that $PSL(2,\mathbb{C})$ acts on the Riemann sphere which we can view as the boundary of hyperbolic 3-space $\mathbb{H}^3$ and are in fact, isometries of $\mathbb{H}^3$. We can extend conformal transformations on the boundary into isometries of the bulk.

Axiomatic Field Theory

Graeme Segal defined some axioms for a symmetric monoidal functor form a cobordism category of manifolds with conformal structure to an algebraic category. In the 1+1 setting, the objects of the cobordism category are simply disjoint unions of circles and the morphisms are Riemann surfaces with boundary; these Riemann surfaces are equipped with conformal structures.

The Witt algebra $W$ can be viewed as infinitesimal generators of deformations on the boundary circles. We are interested in taking two copies of $W$ because we have left and right handed particles and studying representations. But also, since we’re interested in quantum theory, we really want projective representations or equivalently, we can use the two copies of the Virasoro algebra instead. The Hilbert space in question is a direct sum of irreducible highest-weight Verma modules and it is where a single circle is sent by the functor. The morphisms in the algebraic category are linear maps (called conformal blocks). We want the functor to be such that if we glue to cobordisms together, this amounts to taking a partial trace of the two corresponding linear maps. The linear maps should also be intertwined with the Virasoro symmetry.