Renormalization for Holomorphic Field Theories

Abstract

We introduce the concept of a holomorphic field theory on any complex manifold in the language of the Batalin–Vilkovisky formalism. When the complex dimension is one, this setting agrees with that of chiral conformal field theory. Our main result concerns the behavior of holomorphic theories under renormalization group flow. Namely, we show that holomorphic theories are one-loop finite. We use this to completely characterize holomorphic anomalies in any dimension. Throughout, we compare our approach to holomorphic field theories to more familiar approaches including that of supersymmetric field theories.

Appendix A: Some Functional Analysis

Homological algebra plays a paramount role in our approach to quantum field theory. We immediately run into a subtle issue, which is that the underlying graded spaces of the complexes of fields we are interested in are infinite dimensional, so care must be taken when defining constructions such as duals and homomorphism spaces. A common approach to dealing with issues of infinite dimensional linear algebra is to consider vector spaces equipped with a topology. A problem with this is that the category of topological vector spaces is not an abelian category, so doing any homological algebra in this naive category is utterly hopeless. It is therefore advantageous to enlarge this to the category of differentiable vector spaces. The details of this setup are carried out in the Appendix of [CG17], but we will recall some key points for completeness of exposition. In this appendix we also set up our notation for duals and function spaces.

Let $\mathrm{Mfld}$ be the site of smooth manifolds. The covers defining the Grothendieck topology are given by surjective local diffeomorphisms. There is a natural sheaf of algebras on this site given by smooth functions $C^\infty : M \mapsto C^\infty (M)$.

For any p the assignment $\Omega ^p: M \mapsto \Omega ^p (M)$ defines a $C^\infty$-module. Similarly, if F is any $C^\infty$-module we have the $C^\infty$-module of p-forms with values on F defined by the assignment

$$\begin{aligned} \Omega ^1(F): M \in \mathrm{Mfld} \mapsto \Omega ^p(M, F) = \Omega ^p(M) \otimes _{C^\infty (M)} F(M). \end{aligned}$$

Definition A.1

A differentiable vector space is a $C^\infty$-module equipped with a map of sheaves on $\mathrm{Mfld}$

$$\begin{aligned} \nabla : F \rightarrow \Omega ^1(F) \end{aligned}$$

such that for each M, $\nabla (M)$ defines a flat connection on the $C^\infty (M)$-module F(M). A map of differentiable vector spaces is one of $C^\infty$-modules that intertwines the flat connections. This defines a category that we denote $\mathrm{DVS}$.

Our favorite example of differentiable vector spaces are imported directly from geometry.

Example A.2

Suppose E is a vector bundle on a manifold X. Let denote the space of smooth global sections. Let be the space of sections of the bundle $\pi _X^*E$ on $M \times X$ where $\pi _X: M \times X \rightarrow X$ is projection. The assignment is a $C^\infty$-module with flat connection, so defines a differentiable vector space. Similarly, the space of compactly supported sections is a DVS.

Many familiar categories of topological vector spaces embed inside the category of differentiable vector spaces. Consider the category of locally convex topological vector spaces $\mathrm{LCTVS}$. If V is such a vector space, there is a notion of a smooth map $f: U \subset {\mathbb {R}}^n \rightarrow V$. One can show, Proposition B.3.0.6 of [CG17], that this defines a functor $\mathrm{dif}_t: \mathrm{LCTVS} \rightarrow \mathrm{DVS}$ sending V to the $C^\infty$-module $M \mapsto C^\infty (M, V)$. If $\mathrm{BVS} \subset \mathrm{LCTVS}$ is the subcategory with the same objects but whose morphisms are bounded linear maps, this functor restricts to embed $\mathrm{BVS}$ as a full subcategory $\mathrm{BVS} \subset \mathrm{DVS}$.

There is a notion of completeness that is useful when discussing tensor products. A topological vector space $V \in \mathrm{BVS}$ is complete if every smooth map $c: {\mathbb {R}}\rightarrow V$ has an anti-derivative [KM97]. There is a full subcategory $\mathrm{CVS} \subset \mathrm{BVS}$ of complete topological vector spaces. The most familiar example of a complete topological vector space will be the smooth sections of a vector bundle $E \rightarrow X$.

We let $\mathrm{Ch}(\mathrm{DVS})$ denote the category of cochain complexes in differentiable vector spaces (we will refer to objects as differentiable vector spaces). It is enriched over the category of differential graded vector spaces in the usual way. We say that a map of differentiable cochain complexes $f: V \rightarrow W$ is a quasi-isomorphism if and only if for each M the map $f: C^\infty (M, V) \rightarrow C^\infty (M,W)$ is a quasi-isomorphism.

Theorem A.3

(Appendix B [CG17]). The full subcategory $\mathrm{dif}_c: \mathrm{CVS} \subset \mathrm{DVS}$ is closed under limits, countable coproducts, and sequential colimits of closed embeddings. Furthermore, CVS has the structure of a symmetric monoidal category with respect to the completed tensor product ${\widehat{\otimes }}_{\beta }$.

We will not define the tensor product ${\widehat{\otimes }}_\beta$here, but refer the reader the cited reference for a complete exposition. We will recall its key properties below. Often times we will write $\otimes$ for ${\widehat{\otimes }}_\beta$ where there is no potential conflict of notation. The fundamental property of the tensor product that we use is the following. Suppose that E, F are vector bundles on manifolds X, Y respectively. Then, lie in $\mathrm{CVS}$, so it makes sense to take their tensor product using ${\widehat{\otimes }}_\beta$. There is an isomorphism

(29)

where $E \boxtimes F$ denotes the external product of bundles, and $\Gamma$ is smooth sections.

If E is a vector bundle on a manifold X, then the spaces both lie in the subcategory $\mathrm{CVS} \subset \mathrm{DVS}$. The differentiable structure arises from the natural topologies on the spaces of sections.

We will denote by () the space of (compactly supported) distributional sections. It is useful to bear in mind the following inclusions

When X is compact the bottom left and top right arrows are equalities.

Denote by $E^\vee$ the dual vector bundle whose fiber over $x \in X$ is the linear dual of $E_x$. Let $E^!$ denote the vector bundle $E^\vee \otimes \mathrm{Dens}_X$, where $\mathrm{Dens}_X$ is the bundle of densities. In the case X is oriented, $\mathrm{Dens}_X$ is isomorphic to the top wedge power of $T^*X$. Let denote the space of sections of $E^!$. The natural pairing

that pairs sections of E with the evaluation pairing and integrates the resulting compactly supported top form exhibits as the continuous dual to . Likewise, is the continuous dual to . In this way, the topological vector spaces and obtain a natural differentiable structure.

If V is any differentiable vector space then we define the space of linear functionals on V to be the space of maps $V^* = \mathrm{Hom}_\mathrm{DVS}(V, {\mathbb {R}})$. Since $\mathrm{DVS}$ is enriched over itself this is again a differentiable vector space. Similarly, we can define the polynomial functions of homogeneous degree n to be the space

$$\begin{aligned} \mathrm{Sym}^n(V^*) = \mathrm{Hom}^{multi}_\mathrm{DVS}(V \times \cdots \times V, {\mathbb {R}})_{S_n} \end{aligned}$$

where the hom-space denotes multi-linear maps, and we have taken $S_n$-coinvariants on the right-hand side. The algebra of functions on V is defined by

As an application of Equation (29) we have the following identification.

Lemma A.4

Let E be a vector bundle on X. Then, there is an isomorphism

where is the space of compactly supported distributional sections of the vector bundle $(E^!)^{\boxtimes n}$. Again, we take $S_n$-coinvariants on the right hand side.