Quantizing holomorphic field theories on twistor space

The first example studied in detail in this paper is the four-dimensional WZW model of Donaldson [1] and Losev, Moore, Nekrasov and Shatashvili [2]. The fundamental field of this model is a map

where $G$ is a Lie group. In terms of the current $J=\sigma^{-1}\mathrm{d}\sigma$, the Lagrangian is

Here $A$ is a background $U(1)$ gauge field on $\mathbb{R}^{4}$ whose fields strength is the Kähler form $\omega$.

The first term in the Lagrangian is the standard $\sigma$-model Lagrangian. The $\sigma$-model has a topological $U(1)$ symmetry in dimension $4$, because $\pi_{3}(G)=\mathbb{Z}$. The current for this symmetry is $\operatorname{tr}(J,[J,J])$. The second term in the Lagrangian simply means that we have a background $U(1)$ gauge field for this topological symmetry.

It was stated in [3] that this model arises from holomorphic Chern-Simons on twistor space; this result was proved in [4, 5] with several interesting generalizations presented in [4].

Our results apply to this model with $G=SO(8)$, coupled to some additional “gravitational” fields which we refer to as “closed string fields” because of their origin in a certain twistor string theory. The restriction to $SO(8)$ is forced on us by a Green-Schwarz anomaly cancellation on twistor space.

Without the addition of this closed-string field, the model is not twistorial. In section 2 we show that an anomaly prevents one constructing the model as a local QFT on twistor space. In section 11 we show that property (1) above fails without the addition of the closed-string field: there are logarithmic terms in the two-loop operator product expansion which are canceled by a Green-Schwarz mechanism when we introduce the closed-string field.

The closed string field is a scalar field $\rho$, which we view as the Kähler potential for a closed $(1,1)$-form $\partial\overline{\partial}\rho.$ The field $\rho$ is defined up to the addition of a constant; it might be better to think of $\rho$ as being a closed $1$-form.

The $\sigma$-model can be defined on Kähler manifolds of complex dimension $2$, and so has a Kähler stress-energy tensor $T_{\text{K\"{a}hler}}$ which is a $(1,1)$-form that couples to the variation of the Kähler form (in a fixed complex structure). The Kähler potential $\rho$ couples to this by

This makes it clear that we should treat $\rho$ as as gravitational, because it couples to the stress-energy tensor of the $4d$ WZW model.

The Lagrangian for the Kähler potential, to the order we compute it, is

Note the unusual kinetic term, involving the square of the Laplace operator. The constant $C$ is non-zero, and its value is determined in principle by anomaly cancellation on twistor space, but we do not determine it.

To quadratic order in $\rho$, the equations of motion are that the scalar curvature of the Kähler metric given by $\omega+\partial\overline{\partial}\rho$ vanishes. It is known [6] that every scalar flat Kähler manifold is anti-self-dual, and so has a description in terms of an integrable twistor space²²2I’m very grateful to David Skinner for pointing out the result in [6] and for other helpful discussions. . This suggests that the Lagrangian enforces vanishing of the scalar curvature to all orders. Unfortunately I was unable to prove this.

A simpler, but more physically relevant, twistorial field theory we study is a variant of self-dual Yang-Mills theory.

Self-dual Yang-Mills theory is a degenerate limit of ordinary Yang-Mills theory, where the fields are a gauge field $A\in\Omega^{1}(\mathbb{R}^{4},\mathfrak{g})$ and an adjoint-valued self-dual two-form $B\in\Omega^{2}_{+}(\mathbb{R}^{4},\mathfrak{g})$. The Lagrangian is

It is well-known [7, 8] that, at the classical level, this Lagrangian comes from a local Lagrangian on twistor space.

In section 6 we demonstrate that this fails at the quantum level, because of an anomaly on twistor space.

In some cases, we can use a Green-Schwarz mechanism to cancel the anomaly, making self-dual Yang-Mills into a twistorial theory at the quantum level. We can do this for any Lie algebra $\mathfrak{g}$ for which, for $X\in\mathfrak{g}$,

where $\operatorname{Tr}$, $\operatorname{tr}$ indicate trace in the adjoint and fundamental representations respectively³³3More generally, one can add matter in any real representation $R$, in which case the left hand side must be modified by subtracting $\operatorname{tr}_{R}(X^{4})$. and $\lambda_{\mathfrak{g}}$ is a $\mathfrak{g}$-dependent constant.

This identity holds for $\mathfrak{sl}_{2}$, $\mathfrak{sl}_{3}$, $\mathfrak{so}(8)$ and all exceptional groups, with some values of $\lambda_{\mathfrak{g}}$ being

The Green-Schwarz mechanism also holds for $\mathfrak{g}=\mathfrak{so}(N_{c})$ with matter in $N_{f}=N_{c}-8$ copies of the fundamental representation, and for⁴⁴4We write $\mathfrak{sl}(N)$ instead of $\mathfrak{su}(N)$ as we will work with analytically continued fields; a choice of path integral contour gives us $\mathfrak{su}(N)$ gauge theory, and in perturbation theory all results are independent of the contour. $\mathfrak{g}=\mathfrak{sl}(N_{c})$ with $N_{f}=N_{c}$.

The closed-string field we need to adjoin is $\rho$ as before, where the full Lagrangian is

where $CS(A)$ is the Chern-Simons three-form normalized so $\mathrm{d}CS(A)=\operatorname{tr}(F(A)^{2})$. The field $\rho$ is a kind of axion; we can take it to be periodic because of the quantization of $\operatorname{tr}(F(A)^{2})$.

Our main result about the $WZW_{4}$ model, plus the closed-string field $\rho$, is that the counter-terms can all be fixed uniquely in a scheme-independent way. That is, the model is well-defined at the quantum level, despite being non-renormalizable.

Since this seems to dramatically contradict the usual philosophy, let me sketch the constraints that fix the counter-terms before discussing the models in more detail.

The most direct way to fix the counter-terms is to use the fact, which we discuss in more detail in section 4, that the classical Lagrangian arises as the dimensional reduction of a local Lagrangian on twistor space, $\mathbb{PT}=\mathscr{O}(1)^{2}\to\mathbb{CP}^{1}$. At the quantum level, we can ask that the counter-terms are expressed as local functionals on twistor space. Here, being local on twistor space is essential: any counter-term on $\mathbb{R}^{4}$ can be lifted to non-local expressions on twistor space, but only very special counter-terms come from local expressions on twistor space. This constraint, which is equivalent to asking that the quantum theory is dimensionally reduced from a local field theory twistor space, turns out to fix all counter-terms uniquely.

The twistor-space theory is a type I topological string [9] , and the topological string version of the Green-Schwarz mechanism fixes the group to be $SO(8)$.

It would be more satisfying to have a direct space-time way to constrain the counter-terms. In section 4 we will argue in detail that any local holomorphic field theory on twistor space gives rise to a theory on $\mathbb{R}^{4}$ satisfying properties (1) and (2) above. Both of these properties constrain the counter terms very strongly, and we conjecture that they are enough to uniquely fix all counter-terms.

As mentioned above, this argument gives a no-go theorem:

This applies, in particular, to Yang-Mills theory ⁵⁵5This result appears, at first sight, to be in tension with the interesting recent paper [10]. There is no contradiction, however. The Lagrangian proposed in [10] is based on holomorphic Chern-Simons for a non-integrable complex structure on twistor space. Because the $\overline{\partial}$ operator for a non-integrable complex structure does not square to zero, the Lagrangian is not invariant under the usual gauge transformations of holomorphic Chern-Simons. If we do not impose the usual gauge transformations, the theory is not holomorphic. Just like a topological theory is one in which all operators are killed by all translations, a holomorphic theory is one in which all operators are killed by $\partial_{\overline{z}_{i}}$ in local holomorphic coordinates. If we study holomorphic Chern-Simons theory without imposing gauge invariance, this holomorphic condition no longer holds. and to the $\phi^{4}$ theory.

We find in section 11 that the property (2) – that the OPEs are entire analytic functions – requires a Green-Schwarz type anomaly cancellation and fails unless we use the closed-string field $\rho$.

We have already stated our main result about self-dual Yang-Mills theory: all correlation functions are rational functions, for the gauge groups mentioned above and when we couple to an axion field as in equation (1.2.4).

We also show that without the axion field there are logarithmic correlation functions. We find that at two loops there is a logarithm appearing in the OPE of the stress-energy tensor with itself. We also find that at three loops there is a logarithm in the two-point function of the operator $B^{2}$:

Such logarithmic two-point function can not appear in a local theory on twistor space. Therefore it is a reflection of the anomaly on twistor space. This logarithmic OPE must be canceled when we couple to the field $\rho$.

This gives us a technique for calculating the precise coefficient of non-rational correlation functions, which we used in the three-loop computation above. Any logarithmic correlation functions in the theory without axions must be canceled by an exchange of axion fields, as in Figure 1. This vastly simplifies computations, because each axion line cancels the log divergences in a loop of gluons.

For the computation of the two-point function of $B^{2}$, there is only one diagram involving an exchange of axion fields which can contribute a logarithm.

Although I didn’t perform any computations of correlation functions beyond the one described above, one can expect a pattern like this to continue. For instance, we can contemplate computing the connected $2n$ point function of the operator $B^{2}$ in self-dual Yang-Mills. This involves Feynman diagrams with $2n+1$ loops. One can expect that the term which is the most transcendental as a function of the position of the operators is canceled, when we include the axion field, by an exchange of the maximum number of axions, which is $n$. Since each axion corresponds to a loop of gluons, this means that we could compute the maximally transcendental term by using diagrams with $n+1$ loops instead of $2n+1$.

Full Yang-Mills theory is obtained by adjoining to the self-dual Lagrangian $BF(A)_{+}$ the term $-\tfrac{1}{4}g^{2}B^{2}$. (The $B^{2}$ term was realized as a non-local term on twistor space in [11, 8]). The field $B$ in first-order Yang-Mills becomes $2g^{-2}F_{+}$. The computation of the two-point function of $B^{2}$ in self-dual Yang-Mills immediately gives us a term in the two-point function of $F_{+}^{2}$ in full Yang-Mills:

In this direction, our results also imply that if we take full Yang-Mills plus the axion field $\rho$, the first non-rational term in the two-point function of $F_{+}^{2}$ occurs at $4$ loops, or order $g^{10}$. More generally, we have:

Another result we find using our method is a surprising statement about the RG flow of self-dual Yang-Mills deformed by $-\tfrac{1}{4}g^{2}B^{2}$. The standard RG flow in Yang-Mills theory occurs at second order in $g^{2}$. This also happens when we view full Yang-Mills as a deformation of the self-dual theory. It has been proven [12] that when we deform self-dual Yang-Mills by $-g^{2}B^{2}$, then the RG flow modifies the deformation to $-\tfrac{1}{4}g^{2}B^{2}+Cg^{4}B^{2}$ for some constant $C$.

What we find here is that there is another term which appears at a lower order. The very first term in the RG flow changes the coefficient of the toplogical term $F\wedge F$:

If the coupling constant $g$ doesn’t depend on the space-time coordinates, this term in the RG flow is trivial in perturbation theory, because $F^{2}$ is is a total derivative. However, if $g$ is non-constant, then this analysis means that the leading term in the RG flow is given by the Chern-Simons term $\mathrm{d}gCS(A)$.

I am grateful to Roland Bittleston, Lionel Mason and David Skinner for some very helpful conversations; to Greg Moore and Samson Shatashvili for explaining their work on $WZW_{4}$ and for correcting an important error I made in an early talk on this work; and to Yehao Zhou for collaborating in the early stages of this paper. The author is supported by the NSERC Discovery Grant program and by the Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.

Holomorphic Chern-Simons theory, or holomorphic BF theory, has a one-loop anomaly on any Calabi-Yau manifold [18, 9, 19], associated to the diagram in Figure 2. If $\mathrm{c}$ is the parameter for the gauge transformation, then the gauge variation of this diagram is

with $\mathcal{A}$ being the holomorphic Chern-Simons gauge field.

Since anomalies are local, this means that holomorphic Chern-Simons on twistor space also has a one-loop anomaly. The same anomaly appears in holomorphic BF theory on twistor space, meaning that any naive quantization of the Penrose-Ward correspondence suffers from a one-loop anomaly.

Many, if not most, if the twistorial field theories mentioned above have anomalies. One of the few examples that doesn’t is the twistor representation of self-dual $N=4$ Yang-Mills, where the contribution from the bosonic and fermionic fields in the box diagram cancels.

Fortunately, in very special cases, it is possible to cancel the anomaly from the box diagram. In [9] Si Li and the author introduced a topological string version of the Green-Schwarz mechanism which cancels the anomaly for holomorphic Chern-Simons theory for $G=SO(8)$. Let us explain this briefly.

The cancellation involves the closed-string fields of the type I $B$-model, which is a variant of the topological $B$-model whose world-sheet is un-oriented. The closed string fields of the type I $B$-model are a subsector of those of Kodaira-Spencer theory [20]: on a Calabi-Yau manifold $X$, they consist of a Beltrami differential field

which is constrained to satisfy $\operatorname{Div}\mu=0$, where $\operatorname{Div}$ is the holomorphic divergence operator

(In practice, it is often better to implement this constraint homologically, by asking that $\operatorname{Div}\mu$ is $\overline{\partial}$ exact). The field $\mu$ describes deformations of $X$ as a Calabi-Yau manifold.

Under the isomorphism between $\Omega^{0,\ast}(X,TX)$ and $\Omega^{2,\ast}(X)$, the holomorphic divergence operator becomes $\partial$, the holomorphic part of the de Rham operator. As such, we will often use the symbol $\partial$ to refer to it.

The Lagrangian for the closed-string fields of the type I $B$-model is

This is a variant of the Lagrangian introduced in [20].

Note the unusual kinetic term, which only makes sense because of the constraint $\partial\mu=0$. The gauge symmetries of this model are given by vector fields which are sections of the holomorphic tangent bundle $TX$ and which are divergence free.

The Beltrami differential $\mu$ is coupled to the holomorphic Chern-Simons gauge field $\mathcal{A}$ by replacing the $\overline{\partial}$ by the $\overline{\partial}$ operator in the new complex structure. Explicitly, if we write

then the term in the Lagrangian is

(The normalization is computed in the appendix A).

When we work on twistor space, we need to take into account the behaviour at the poles of the meromorphic volume form. Recall that there are second order poles along the divisors $z=0$, $z=\infty$.

The boundary conditions for the Beltrami differential $\mu$ are best written in terms of the $(2,1)$ form $\eta=\mu\vee\Omega$. Our boundary condition is simply that the $(2,1)$ form $\eta$ is regular on all of $\mathbb{PT}$ ⁶⁶6In [21] we studied a related model where the boundary condition asked that $\eta$ has log poles at the poles of $\Omega$. That also works, but we find the boundary condition chosen here a little more convenient.. In terms of the Beltrami differential $\mu$, this condition means that it vanishes to second order near $z=0$ and $z=\infty$ (gauge transformations behave in the same way). Near $z=0$, the cubic term in the action has a factor of $z^{6}$ from the zeroes in the fields, and of $z^{-4}$ from the second-order pole in $\Omega$ (recalling that $\Omega$ appears twice in the cubic term). Therefore, overall, the cubic term has a coefficient of $z^{2}$, and vanishes to order $2$. Similarly, it vanishes to order $2$ at $z=\infty$.

To understand the kinetic term, we note that it can be written entirely in terms of the $(2,1)$ form $\eta$, which is $\partial$ closed. The kinetic term is

which can be written down on any complex three-fold.

Now that we understand the closed string fields, we can discuss the Green-Schwarz mechanism. The following result is proved in [9].

The tree level anomaly is drawn in Figure 3.

As a consequence of this, we find that the one-loop anomaly for holomorphic Chern-Simons cancels for any Lie algebra $\mathfrak{g}$ for which the identity

holds.

This identity holds for a number of Lie algebras, including $\mathfrak{so}(8)$, $\mathfrak{sl}(2)$, $\mathfrak{sl}(3)$, and all exceptional simple Lie algebras. In [9] it was shown that for $\mathfrak{g}=\mathfrak{so}(8)$ all anomalies at all orders in the loop expansion vanish, and further, all higher loop counter-terms are uniquely fixed by the requirement that all anomalies vanish. A small variant of this cohomological argument applies on twistor space:

We will give the proof of this, along with related results, in the appendix C. We have chosen our normalization of the open-closed interaction to cancel the anomaly.

We do not currently know⁷⁷7As in the usual Green-Schwarz mechanism, a one-loop diagram with only closed string fields on the external lines should force the gauge algebra to have dimension $28$. This raises the possibility that the $\mathfrak{g}_{2}\oplus\mathfrak{g}_{2}$ theory might also be anomaly free to all orders. whether or not higher loop anomalies can be canceled for any other groups.

It is worth mentioning that the type I topological string should admit an embedding in the physical string, in a way similar to the standard embedding of the topological A and B models in the type IIA and IIB string theories. For the type I topological string, the relevant set-up is type IIB with an $O7-$ plane and $4$ $D7$’ s. This system is placed in an $\Omega$ background and subject to supersymmetric localization, leaving us with an effective $6$ dimensional theory with gauge group $SO(8)$.

The $WZW_{4}$ model with target a general group $G$ is not a gauge theory, and therefore can not suffer from a gauge anomaly. The gauge anomaly on twistor space can not be canceled by a local counter-term. However, because there is no gauge anomaly on $\mathbb{R}^{4}$, we see that it must be possible to cancel the twistor-space anomaly with a term which is non-local from the point of view of twistor space, but still local from the point of view of $\mathbb{R}^{4}$.

Here we will write an expression that does this. We will coordinatize twistor space as $\mathbb{R}^{4}\times\mathbb{CP}^{1}$, with coordinates $x,z$. We will write down an expression where we integrate over one $x$ variable but two $z$ variables – so it is bi-local on twistor space but local on $\mathbb{R}^{4}$. The expression is

This expression is interpreted by viewing $\mathcal{A}(x,z)$ as a $1$-form on twistor space. We check in the appendix B that the gauge variation of this expression cancels the anomaly.

Holomorphic BF theory on twistor space corresponds to self-dual Yang-Mills theory on $\mathbb{R}^{4}$. We let $\mathcal{B}\in\Omega^{3,1}(\mathbb{PT},\mathfrak{g})$ and $\mathcal{A}\in\Omega^{0,1}(\mathbb{PT},\mathfrak{g})$ be the fields of holomorphic BF theory. The Lagrangian is

However, holomorphic BF theory has a one-loop anomaly, which is the same as that associated to holomorphic Chern-Simons theory, associated to Figure 2.

Take any simple Lie algebra $\mathfrak{g}$, and let $\operatorname{Tr}$ denote trace in the adjoint representation and $\operatorname{tr}$ that in the fundamental representation. Let us assume that there is some $\lambda_{\mathfrak{g}}$ such that

We mentioned above when this happens.

Then, we can cancel the anomaly coupling to the free limit of the Kodaira-Spencer theory we used to cancel the anomaly for holomorphic Chern-Simons theory. The new field we introduce is $\eta\in\Omega^{2,1}(\mathbb{PT})$, satisfying the constraint $\partial\eta=0$ and subject to the gauge transformation $\eta\mapsto\eta+\overline{\partial}\chi$, for $\chi\in\Omega^{2,0}(\mathbb{PT})$. The Lagrangian is

which is the kinetic term of the Lagrangian we used before.

We couple this field to the gauge field $\mathcal{A}$ by

In section 6, we will determine the theory on $\mathbb{R}^{4}$ corresponding to this Lagrangian.

One variant of the model we have introduced is obtained by using the same type I topological string, but on a different geometry. The simplest case is when we take a hyperelliptic genus $3$ curve $\Sigma\to\mathbb{CP}^{1}$. The pull-back of $\mathscr{O}(2)$ is the canonical bundle, so that the three-fold

is Calabi-Yau. This fibres over $\mathbb{R}^{4}$ with fibres $\Sigma$.

The field theory on $\mathbb{R}^{4}$ corresponding to the type I topological string theory with this geometry includes a $\sigma$-model with target the moduli of $SO(8)$-bundles on $\Sigma$. The fields coming from the closed-string sector include a map from $\mathbb{R}^{4}$ to the moduli space of genus $3$ hyperelliptic curves, which is a divisor in the moduli of genus $3$ curves. As always, these theories are analytically-continued theories, and the path integral should be defined by the choice of a contour.

Other examples are obtained by starting with supersymmetric sectors of $10$ dimensional string theories. These supersymmetric sectors are themselves described by $10$-dimensional topological strings [22, 23, 24] . We can often use these holomorphic twists of string theory to build holomorphic field theories on twistor space. By compactifying to $\mathbb{R}^{4}$ we find a mechanism to produce non-supersymmetric field theories on $\mathbb{R}^{4}$ from string theory.

This will give a broad class of non-renormalizable and non-supersymmetric four-dimensional theories whose UV properties are nonetheless under control, because they come from holomorphic anomaly-free theories on twistor space.

The simplest such models arise by taking the holomorphic twist [22] of the type I or type IIB strings, compactified to twistor space. The geometry we need to compactify these models to twistor space is a complex $3$-fold $X$, together with a non-constant map $\pi:X\to\mathbb{CP}^{1}$, and an isomorphism of line bundles

(Note that this implies that the fibres of $\pi$ are Calabi-Yau surfaces). We can then build a Calabi-Yau $5$-fold

As a real manifold, there is an isomorphism

Complexified space-time $\mathbb{C}^{4}$ consists of the moduli space of the $3$-fold $X$ in the $5$-fold $\mathbb{PT}_{X}$. Thus, $\mathbb{PT}_{X}$ is a higher-dimensional variation of twistor space, in which the $\mathbb{CP}^{1}$ twistor lines are replaced by the $3$-fold $X$.

One way to construct such a variety $X$ (suggested to me by Davesh Maulik) is to start with a Calabi-Yau $3$-fold $X_{0}$, fibred over $\mathbb{CP}^{1}$ in $K3$ surface. Then, we take $X$ to be the pull-back of $X_{0}$ along a double branched cover $\mathbb{CP}^{1}\to\mathbb{CP}^{1}$, whose branch points consist of smooth fibres of the fibration $X_{0}\to\mathbb{CP}^{1}$. It is then elementary to check that $\pi^{\ast}O(2)$ is $K_{X}$.

The construction of field theories on $\mathbb{R}^{4}$ from field theories on $\mathbb{PT}$ carries through when we compactify holomorphic twists of string theories from the $5$-fold $\mathbb{PT}_{X}$ to $\mathbb{R}^{4}$. I will leave it as an open problem to carefully work out the Lagrangian of the theory on $\mathbb{R}^{4}$ corresponding to a holomorphic twist of a ten-dimensional string theory in some geometry. The computation seems to me to be quite challenging.

Let us first emphasize that passing from a theory on twistor space to one on $\mathbb{R}^{4}$ is somewhat better-behaved than the familiar Kaluza-Klein reduction. Consider, for example, KK reduction of a field theory on $S^{1}\times\mathbb{R}^{d}$ to $\mathbb{R}^{d}$. We obtain a local field theory on $\mathbb{R}^{d}$ only in the limit as the radius of the circle goes to zero, or equivalently when we work in the far IR. Without taking this limit, we find a field theory on $\mathbb{R}^{d}$ with an infinite number of massive KK modes. These disappear in the infra-red, so that the local field theory on $\mathbb{R}^{d}$ is an effective low-energy description of the $d+1$ dimensional theory.

When passing from a theory on twistor space to one on $\mathbb{R}^{4}$, the procedure is more robust. The description as a local field theory on $\mathbb{R}^{4}$ is valid at all energy scales. Without passing to the IR, there are only a finite number of fields.

Let us explain how this happens, first using physical language and then formalizing the idea using the concept of factorization algebra.

It is helpful to choice a simple concrete example. We will choose the free scalar field, which is represented on twistor space by a field

with Lagrangian $\mathcal{A}\overline{\partial}\mathcal{A}$ and gauge symmetry $\mathcal{A}\mapsto\mathcal{A}+\overline{\partial}\chi$, where $\chi\in\Omega^{0,0}(\mathbb{PT},K^{\tfrac{1}{2}})$.

Using the isomorphism $\mathbb{PT}\cong\mathbb{R}^{4}\times\mathbb{CP}^{1}$, we can write this as a field theory on $\mathbb{R}^{4}$ with an infinite number of fields. We will see that these fields are all (except for a single scalar field) either auxiliary fields, i.e. fields whose kinetic term contains only a mass term and no space-time derivatives; or else, they are unphysical fields, because there are gauge transformations which shift their values at every point.

To see this, let us decompose the field $\mathcal{A}$ into $\mathcal{A}_{\overline{z}}$ and a doublet $\mathcal{A}_{\overline{v}_{i}}$. The component $\mathcal{A}_{\overline{z}}$ is a $(0,1)$ form on each twistor fibre twisted by $\mathscr{O}(-2)$, so we can view it as a $(1,1)$ form. Under the $SU(2)$ symmetry which rotates $\mathbb{CP}^{1}$, the bundle of $(1,1)$-forms is trivial so we can view $\mathcal{A}_{\overline{z}}$ as a scalar.

The fields $\mathcal{A}_{\overline{v}_{i}}$ are sections of the anti-holomorphic co-normal bundle $\overline{\mathscr{O}(-1)}$, tensored with $\mathscr{O}(-2)$. We can use a metric to identify $\overline{\mathscr{O}(-1)}$ with $\mathscr{O}(1)$, so that $\mathcal{A}_{\overline{v}_{i}}$ are a doublet of sections of $\mathscr{O}(-1)$.

We can decompose $\mathcal{A}_{\overline{z}}$ into its Fourier modes $\mathcal{A}_{\overline{z}}^{(2j)}$, which lives in the spin $2j$ representations of $SU(2)$ for $j\geq 0$. Similarly we can decompose $\mathcal{A}_{\overline{v}_{i}}$ into Fourier modes $\mathcal{A}_{\overline{v}_{i}}^{(2j+\tfrac{1}{2})}$, which live in the half-integer spin representations of $SU(2)$. We can of course do the same with the generator of gauge transformations $\chi$, whose Fourier components $\chi^{(2j)}$ live in the integer spin representations of $SU(2)$ with spin $\geq 1$.

The fields $\mathcal{A}_{\overline{v}_{1}}$, $\mathcal{A}_{\overline{v}_{2}}$ are coupled only by a $z$-derivative, and not by derivative on $\mathbb{R}^{4}$. The fields $\mathcal{A}_{\overline{v}_{i}}$, $\mathcal{A}_{\overline{z}}$ are coupled by a derivative on $\mathbb{R}^{4}$. Therefore the Lagrangian is schematically of the form

where $D$ indicates some derivative on $\mathbb{R}^{4}$.

The fields $\mathcal{A}_{\overline{v}_{i}}^{(2j+\tfrac{1}{2})}$ are an infinite tower of auxiliary fields – that is, fields with only a mass and not a kinetic term. They can be integrated out without destroying locality, because the propagator is a $\delta$-function.

The fields $\mathcal{A}_{\overline{z}}^{(2j)}$ for $j>0$ are unphysical, as the are shifted by modes of the gauge transformation $\chi^{(2j)}$:

It is important that there is no space-time derivative appearing in this gauge variation. We can set all the fields $\mathcal{A}_{\overline{z}}^{(2j)}$ for $j>0$ to zero by a gauge choice, leaving us with simply the zero mode $\mathcal{A}_{\overline{z}}^{(0)}$, which is the scalar field $\phi$. The $\phi\bigtriangleup\phi$ Lagrangian on $\mathcal{A}_{\overline{z}}^{(0)}$ appears from integrating out the fields $\mathcal{A}_{\overline{v}_{i}}^{(1/2)}$.

This completes the proof that the theory on $\mathbb{R}^{4}$ corresponding to a theory on twistor space is just a scalar field, without an infinite tower of Kaluza-Klein modes.

Passing from a field theory on twistor space to one on $\mathbb{R}^{4}$ can be done in a simple and clean way using the language of factorization algebras [25] . Let us first briefly recall what a factorization algebra is, and how they appear in field theory.

A factorization algebra $\mathcal{F}$ on a manifold $M$ is a structure which assigns to every open subset $U\subset M$ a graded vector space $\mathcal{F}(U)$. These graded vector spaces are related for different opens as follows. First, we have linear maps $\mathcal{F}(U)\to\mathcal{F}(U^{\prime})$ if $U\subset U^{\prime}$, so that $\mathcal{F}$ is a pre-cosheaf. We also have a factorization product

whenever $U$, $V$ are disjoint and contained in $W$. These structures satisfy natural commutativity and associativity constraints detailed in [25], together with a “gluing” relation expressing $\mathcal{F}(U)$ in terms of the values of $\mathcal{F}$ on a sufficiently fine open cover of $U$. The gluing relation will not be important for us.

Any classical or quantum field theory on a manifold $M$ gives rise to a factorization algebra. At the classical level, the factorization algebra $\operatorname{Obs}^{cl}$ of classical observables sends $U$ to the algebra of functions on the space of solutions to the equations of motion of $U$. If we are dealing with a gauge theory, we take gauge-invariant functions, where gauge invariance and the equations of motion are both defined homologically, and then we take cohomology ⁸⁸8 We are abusing notation slightly hear: what we refer to as $\operatorname{Obs}^{cl}$ and $\operatorname{Obs}^{q}$ are the cohomology of the cochain complexes referred to as $\operatorname{Obs}^{cl}$ and $\operatorname{Obs}^{q}$ in [25]..

At the quantum level, $\operatorname{Obs}^{q}(U)$ is given by the functionals of the field which only depend on its behaviour on $U$, modulo divergences with respect to the functional measure. Removing divergences is the quantum version of working on-shell, because the classical limit of the divergence associated to a field redefinition is simply the corresponding Euler-Lagrange equation. Again, we should also impose gauge invariance homologically.

Of course, defining the divergence with respect to the functional measure is technically difficult and requires renormalization. This is done in [26, 25].

The factorization algebras of classical field theories are commutative: for every open, $\operatorname{Obs}^{cl}(U)$ is a commutative algebra and the structure maps are maps of commutative algebras. The factorization algebras of quantum field theories do not have this property. To leading order, the deformation away from commutative can be measured by a kind of Poisson bracket, built from the one-loop OPE (see [25]). Factorization algebras thus provide a deformation-quantization perspective on constructing quantum field theory.

The language of factorization algebras generalizes standard algebraic constructions in field theory. For quantum mechanics, the factorization algebra encodes the usual associative algebra of operators; for two-dimensional chiral CFTs, factorization algebras are vertex algebras [25, 27]; for Lorentzian theories [28, 29], factorization algebras reproduce the axioms of algebraic quantum field theory; and for Euclidean theories, factorization algebras encode the OPE [25].

If $g:M\to N$ is a smooth map between manifolds, and $\mathcal{F}$ is a factorization algebra on $M$, then we can build a factorization algebra on $N$ by the formula

This expression for the push-forward factorization algebra is the same as that for the push-forward of sheaves.

A classical or quantum field theory on twistor space gives rise to a factorization algebra on twistor space, say $\operatorname{Obs}^{cl}_{\mathbb{PT}}$ or $\operatorname{Obs}^{q}_{\mathbb{PT}}$. The corresponding factorization algebra on $\mathbb{R}^{4}$ is simply the push-forward along the fibration $\pi:\mathbb{PT}\to\mathbb{R}^{4}$.

The free field theory on twistor space with field as in (4.1.1) corresponds, under this push-forward, to a free scalar field theory. At the classical level, this follows from a result of Penrose [30], as we will now see.

The factorization algebra on twistor space assigns to any open subset $V\subset\mathbb{PT}$ the vector space

Here $\mathscr{O}$ indicates “functions”: so $\mathscr{O}(H^{\ast}_{\overline{\partial}}(V,\mathscr{O}(-2))[1])$ is the symmetric algebra⁹⁹9One must be careful to complete the tensor product appropriately when taking the symmetric algebra. This point is discussed in detail in [25]. of the dual of $H^{\ast}_{\overline{\partial}}(V,\mathscr{O}(-2))[1]$. The symbol $[1]$ indicates a shift of cohomological degree, so that classes of $(0,1)$-forms are placed in degree $0$.

Similarly, for an open subset $U\subset\mathbb{R}^{4}$, the factorization algebra of classical observables of the free scalar field theory is

where $\operatorname{Harm}(U)$ is the vector space of harmonic functions on $U$.

Penrose showed that there is an isomorphism

This isomorphism implies that on the right hand side only $H^{1}_{\overline{\partial}}(\pi^{-1}(U),\mathscr{O}(-2))$ is non-vanishing. Penrose’s isomorphism is compatible with inclusions of open subsets (it is part of an isomorphism of sheaves). This immediately implies that we have

A small variation of this construction shows that

i.e. that the push forward of the quantum factorization algebra from twistor space to $\mathbb{R}^{4}$ on twistor space gives rise to the quantum factorization algebra for a free scalar.

For interacting theories, the push forward of the quantum factorization algebra from twistor space to $\mathbb{R}^{4}$ can be taken as a definition of the quantum theory on $\mathbb{R}^{4}$ corresponding to that on twistor space.

The theory on twistor space we focus on most in this paper is the type I topological string. In the introduction, we stated that this becomes the $WZW_{4}$ model for $SO(8)$ together with a Kähler potential field. In the language of factorization algebras, this means that the push-forward of the factorization algebra of the type I topological string is that of $WZW_{4}$, plus the Kähler potential. We will prove this statement in sections 5.

At the quantum level, this push-forward provides a definition of the quantum factorization algebra of $WZW_{4}$ plus the Kähler potential.

In this section we will prove that for any holomorphic theory on twistor space, the corresponding theory on $\mathbb{R}^{4}$ has periodic RG flow.

Intuitively, this is clear: the scaling action of $\mathbb{R}_{>0}$ complexifies to a $\mathbb{C}^{\times}$ action on twistor space, scaling the fibres of the map $\mathscr{O}(1)^{2}\to\mathbb{CP}^{1}$. For any local theory on twistor space, we can apply an element $\mu\in\mathbb{C}^{\times}$ to find a family of such theories. For a holomorphic theory, this must depend holomorphically on $\mu$, so that the renormalization group flow analytically continues to $\mathbb{C}^{\times}$.

Something very similar happens for a chiral theory in dimension $2$. Typically, such theories are classically scale-invariant, and there are never any logarithmic divergences which spoil this.

We find it worthwhile to carefully phrase this intuitive argument as a theorem.

A classical theory on a complex manifold is holomorphic if the space of fields, in the BV formalism, is of the form

where $E$ is some graded holomorphic vector bundle equipped with a non-degenerate symmetric pairing of holomorphic bundles

This map extends to a map

This map, when composed with integration, is the BV odd symplectic pairing.

We require that the kinetic term in the Lagrangian is

for $e\in\Omega^{0,\ast}(\mathbb{PT},E)[1]$. The interaction terms must only involve holomorphic derivatives and must simply wedge and integrate the $(0,\ast)$ forms appearing in the fields.

Under these hypothesis, the linearized BRST operator is the $\overline{\partial}$ operator.

These conditions can be relaxed a little to include the case when the BV anti-bracket is degenerate, as it is for Kodaira-Spencer theory [31, 32] . We still require that the linearized BRST operator is $\overline{\partial}$, and interactions are as above.

The symmetries of such a theory include all anti-holomorphic vector fields $V$ on $\mathbb{PT}$. (The fact that $E$ is a holomorphic bundle is enough data to define their action on the space of fields). Contraction $\iota_{V}$ with an anti-holomorphic vector field $V$ is an operator of ghost number $-1$ on the fields which preserves the interaction terms. The Cartan homotopy formula,

tells us that the Lie derivative $\mathcal{L}_{V}$ is BRST exact.

Now let us turn to proving that the renormalization group flow is periodic for a holomorphic theory on twistor space. We will first prove that no holomorphic theory on twistor space can give rise to a theory on $\mathbb{R}^{4}$ with a one-loop log divergence. This is a consequence of the following lemma.

Now we will use this to prove the following.

This theorem implies that most familiar theories, such as Yang-Mills theory, can not be constructed from a local theory on twistor space.

We can also use these methods to prove that the theory on $\mathbb{R}^{4}$ built from our twistor-string theory has periodic RG flow. In this case, the string coupling constant $\lambda$ has charge $-2$ under the $\mathbb{C}^{\times}$ action on twistor space which scales the $\mathscr{O}(1)^{2}$ fibres. We can see this because the holomorphic Chern-Simons action appears with a factor of $\lambda^{-1}\Omega$ where $\Omega$ is the meromorphic volume form, and $\Omega$ has charge $-2$.

We can ask that the holomorphic $\mathbb{C}^{\times}$ action on the classical theory (giving $\lambda$ charge $-2$) persists to the quantum level. We will give a proof of this by obstruction theory in section C:

This implies one of the main results of this paper, that $WZW_{4}$ for gauge group $SO(8)$, coupled to the gravitation theory described above, has a quantization with periodic RG flow.

At this point I should point out a mistake I made in the talk [3]. There, I asserted that for $G\neq SO(8)$, there was a one-loop log divergence. This is incorrect, as pointed out by Greg Moore and Samson Shatashvili. In their paper [2] with Losev and Nekrasov, they showed that there are no one-loop divergences for any group.

This correct statement is consistent with the general picture one gets from twistor space. A classical theory which is local on twistor space can not have any one-loop log divergences, as we saw above. However, suppose the theory has a one-loop gauge anomaly, that can be canceled by a non-local term on twistor space. This non-local term at one loop can contribute to two-loop, and higher, log divergences.

The fact that non-local terms on twistor space lead to log divergences further on in the loop expansion is clear from the example of $\phi^{4}$, where the $\phi^{4}$ interaction is a non-local expression on twistor space. Two copies of this non-local vertex lead to the familiar one-loop logarithmic divergence of $\phi^{4}$ theory.

This discussion tells us that we would expect $WZW_{4}$, for $G\neq SO(8)$, to have a two-loop or higher logarithmic divergence. And, for $G=SO(8)$, such a divergence should be canceled by a Green-Schwartz mechanism.

I was unable to determine whether or not two-loop log divergences arise, but I hope to revisit this question.

Quantum field theories which arise from local theories on twistor space have another remarkable property: the correlation functions extend analytically to meromorphic functions $\mathbb{C}^{4}$, with poles when the locations of the operators are lightlike separated.

Intuitively, this is clear. Let us suppose we have a classical or quantum holomorphic theory on $\mathbb{PT}$. The translation action of $\mathbb{R}^{4}$ lifts to a holomorphic action of $\mathbb{C}^{4}$ on $\mathbb{PT}$. These are the holomorphic vector fields pointing along the fibres of the fibration $\mathbb{PT}=\mathscr{O}(1)^{2}\to\mathbb{CP}^{1}$. Let us suppose that our theory on $\mathbb{PT}$ has a holomorphic action of $\mathbb{C}^{4}$.

A local operator $O_{1}$ at the origin in $\mathbb{R}^{4}$ corresponds to an operator in the holomorphic theory on twistor space localized at the twistor line $\mathbb{CP}^{1}\subset\mathbb{PT}$ corresponding to the zero section of the fibration $\mathscr{O}(1)^{2}\to\mathbb{CP}^{1}$. By using the action of $\mathbb{C}^{4}$ we can move the operator $O_{1}$ to any twistor line $\mathbb{CP}^{1}\subset\mathbb{PT}$. The collection of such $\mathbb{CP}^{1}$’s is $\mathbb{C}^{4}$, so that we get an operator $O_{1}(x)$ corresponding to $x\in\mathbb{C}^{4}$.

If $O_{i}$ is a collection of local operators in the theory on $\mathbb{R}^{4}$, then we can place $O_{i}$ at $x_{i}\in\mathbb{C}^{4}$. As long as the corresponding twistor lines $\mathbb{CP}^{1}_{x_{i}}$ do not touch, we can define the correlation functions

(Of course, the correlation functions depend on a state at $\infty$).

The twistor lines $\mathbb{CP}^{1}_{x_{i}}$, $\mathbb{CP}^{1}_{x_{j}}$ touch whenever $\left\|x_{i}-x_{j}\right\|^{2}=0$ (using the holomorphic metric on $\mathbb{C}^{4}$). Thus, the correlation functions $\left\langle O_{1}(x_{1})\dots O_{n}(x_{n})\right\rangle$ are defined whenever the $x_{i}$ are such that no two are null separated.

This shows that, for any local theory on twistor space, the correlation functions (and OPEs) extend analytically to meromorphic functions on $(\mathbb{C}^{4})^{n}$, with poles on the divisors $D_{ij}$ where $\left\|x_{i}-x_{j}\right\|^{2}$.

This observation gives another very strong constraint on which theories can arise from twistor space. It even constrains which classical theories on $\mathbb{R}^{4}$ can arise as local theories on twistor space. This is because the one-loop correlation functions between local operators only depend on the classical Lagrangian. Any one-loop counter-terms, such as non-local counter-terms introduced to cancel an anomaly, can only contribute to two-loop correlation functions.

We use this later in the calculation of Lagrangian on $\mathbb{R}^{4}$ corresponding to Kodaira-Spencer theory on twistor space. In principle this Lagrangian can be computed directly, but we find it more efficient to constrain the terms in the Lagrangian by asking that the one-loop OPE extends analytically to $\mathbb{C}^{4}$.

It is helpful to formulate these ideas in the language of factorization algebras. This will allow us to prove a theorem about the analytic continuation of the factorization algebra of the quantum theory corresponding to our type I twistor string theory.

First, let us make a preliminary definition: