The String Coffee Table

It is nontrivial to recognize from the keywords in the respective titles a direct relation between these papers. The solution to that is one of the insights of hep-th/0504078.

To roughly see what is going on, consider a 2D $\sigma$-model with flat hermitian target, i.e. something with worldsheet action looking like

together with a nilpotent operator $Q$ defined on the fields, whose commutator acts as

which should be thought of as the antiholomorphic exterior derivative $\bar \partial$ on target space.

If, furthermore, there are equations of motion which imply that $\partial_z \alpha = 0$, it is not too hard to see that the cohomology of $Q$ consists of operators that are functions of the $\partial$-derivatives of $\phi$ and those of $\bar \phi$ as well as of $\phi$ itself but not of $\bar \phi$ itself.

The point of this is

a) that worldsheet theories with $(0,2)$-supersymmetry are examples of this structure if one of the supercharges is identified with $Q$

and

b) in such a situation a natural field-redefinition makes contact with the theory of chiral differential operators (as well as with the free $\beta\gamma$-system which is in particular relevant for the pure spinor string).

This field redifinition is very natural: Since in the $Q$-cohomology which we decided to be interested in nothing depends on $\bar \phi$ but only on its derivatives, we introduce instead of $\bar\phi$ the field variable

Since a $\beta$ likes to have a $\gamma$-partner, let’s also simply rename $\phi$:

Now from the standard OPE of $\phi$ and $\bar\phi$ it follows that the only nontrivial OPE of $\beta$ and $\gamma$ is

Moreover, $\beta$ is obviously of weight $(1,0)$ and $\gamma$ of weight $(0,0)$ and in terms of these variables the above action reads

This is what in the string literature is known as the $\beta\gamma$-system. The vertex algebra generated by such $\beta$ and $\gamma$ is known otherwise as the algebra of chiral differential operators (see text around equation (2.1) of math.AG/9906117).

What one is interested in now is patching such $\beta\gamma$-systems together appropriately in order to describe more nontrivial target spaces than the flat hermitian one we started with. More technically, we want to associate a (‘free’) $\beta\gamma$ system like above to each open contractible set of some target space $X$ such that we can consistently restrict from open subsets to smaller open subsets and such that we can glue $\beta\gamma$-systems on overlapping open subsets. Hence the structure needed here is that of a sheaf of such systems, or equivalently, a sheaf of chiral differential operators.

We could construct a globally well defined CFT on all of $X$ by appropriately gluing all such local systems. But this is not always possible. The obstruction for this to be possible, otherwise known as an anomaly, can be nicely seen in Čech cohomology as the failure of certain objects to glue properly on multiple overlaps of open subsets.

In order to understand gluing of $\beta\gamma$-systems, one needs to understand the symmetries of such systems. These symmetries, again, are determined by the ‘Hamiltonian’.

The energy momentum tensor of the (local, free) $\beta\gamma$ system is

As always in quantum mechanics, a symmetry of the system is something generated by an operator which commutes with this ‘Hamiltonian’ $T$. As usual in CFT, such operators are obtained from integrals of worldsheet currents of unit conformal weight, which in the present case means of weight $(1,0)$.

There are only two classes of such currents in the game, namely

for $V = V(\gamma)$ some holomorphic vector field on target space (i.e. one not depending on $\bar \gamma$), and

for $B$ some holomorphic 1-form on target space.

As always, the first of these generates reparameterizations of target space. Witten therefore calls it the geometric symmetry. On the other hand, $J_B$ acts by mixing the fields of the sigma model without this having a geometric interpretation in terms of target space. Hence this is called the ‘nongeometric symmetry’.

What is interesting is that in computing the OPEs of these currents, one finds that, due to a quantum effect coming from a double contraction at one point, the (integrals of the) $J_V$ and $J_B$ generate a nontrivial central extension of the group of holomorphic diffeomorphisms on target space.

Hence, given a complex target manifold $X$ with holomorphic transition functions, one can ask when the cocycle relations satisfied by these transition functions lift to the above central extension. In such a case it is possible to glue together local free $\beta\gamma$ systems (including their ‘nongeometric’ gluing conditions) on $X$ to a globally defined ‘nonlinear’ $\beta\gamma$-system on all of $X$.

This lifting procedure is formally precisely analogous to how the lifting of the structure group of some bundle to a centrally extended structure group is obstructed by the corresponding ‘lifting gerbe’. In ‘Gerbes of Chiral Differential Operators’ this obstructiuon is shown to be $p_1(X)$.

One important subtlety of this discussion is the following. Even if this obstruction vanishes and a globally defined $\beta\gamma$-system is obtainable by gluing, it does not follow necessarily that the resulting global theory is conformally invariant! None of the above discussion had any dependence on the nature (or existence!) of a stress-energy tensor of the model. (Except maybe the determination of the symmetries of the $\beta\gamma$-system, which assumed that a symmetry is something generated by the integral over a unit weight current.)

In fact, the theory can only be conformal globally if the local stress energy tensors defined for every local $\beta\gamma$ system by $T = :\beta\partial\gamma:$ also glue consistently. The obstruction for this to happen is an old friend, now carrying its Čech-clothes: the conformal anomaly.

Witten gives an instructive example for this issue in section 5 of his paper. Assume target space is $\mathbb{CP}^1$. The obvious choice of open sets covering $\mathbb{CP}^1$ is $U_1$=the complex $\gamma$-plane and $U_2$=the complex $\tilde\gamma$-plane which are glued by the holomorphic transition $\tilde \gamma = 1/\gamma$.

There is a local $\beta\gamma$ system on $U_1$ and a $\tilde\beta \tilde\gamma$-system on $U_2$. One needs to lift the holomorphic transition $\tilde\gamma = 1/\gamma$ to symmetries on these two systems. A nontrivial computation shows that the ‘nongeometric’ symmetries have to be chosen such that the gluing relation between $\tilde \beta$ and $\beta$ is not quite the one expected from target space geometry, but rather

(with normal ordering implicit). Here the second term on the right is the nongeometric quantum correction.

This correction has the effect that the stress-energy tensors $T = \beta\partial\gamma$ and $\tilde T = \tilde \beta\partial\tilde \gamma$ do not coincide on $U_1\cap U_2$. Instead one finds that their difference is

Hence there is no global conformal symmetry in the glued system. This is of course no surprise, since we know that the string on $\mathbb{CP}^1$ has nonvanishing beta-functions. Here one sees this conformal anomaly in terms of Čech cohomology.

Since this little summary of some aspects of Witten’s last paper is supposed to provide some illumination of some of the remarks made in my recent notes on Nikita Nekrasov’s talk, I should maybe remark that one should not confuse the (role played by the) operator $Q$ in Witten’s paper with the (role played by the) operator $Q$ in the context of these notes. Witten’s $Q$ is essentially the bridge between $(0,2)$-susy theories and chiral differential algebras and can be essentially forgotten once one is only interested in gluing local $\beta\gamma$-systems. In Nekrasov’s context there is another $Q$ (the pure spinor BRST operator) which apparently appears in weakening Čech-closure conditions for $\beta\gamma$-systems on the space of pure spinors.