# Correlation Functions as Morphisms

In https://arxiv.org/abs/1911.07895, the authors consider a generalization of correlation functions to make sense of the $$O(n)$$ symmetry for $$n \in \mathbb{R}$$. As explained in Sec. 7, each field $$\phi_i(x)$$ corresponds to an object $$\mathbf{a_i}$$ in a braided tensor category $$\mathcal{C}$$. The authors say that, with abuse of terminology, $$\phi_i(x)$$ transforms as $$\mathbf{a_i}$$ which I interpret as $$\mathbf{a_i}$$ should become a vector space for usual theories. The correlation function in general is defined as (Eq. (7.1)) $$\left<\phi_1(x_1) \cdots \phi_n(x_n)\right> = Hom\left(\mathbf{a_1} \otimes \cdots \otimes \mathbf{a_n} \to \mathbf{1}\right)$$

where $$\mathbf{1}$$ acts as the identity in the tensor product. To get a number of this, one needs another morphism $$f \in Hom\left(\mathbf{1} \to \mathbf{a_1} \otimes \cdots \otimes \mathbf{a_n} \right)$$ so that $$C \circ f:\mathbf{1} \to \mathbf{1}$$ can be identified with a real (or complex) number.

I am confused by this statement because for standard field theories where $$\mathbf{a_i}$$ becomes a vector space, I believe the correlation function should become a map $$\mathbf{\bar a_1} \otimes \cdots \otimes \mathbf{\bar a_n} \to \mathbb{R}$$ where $$\mathbf{\bar a_i}$$ is the dual vector space of $$\mathbf{a_i}$$. As an example, if $$\phi_i \in \mathbf{a_i}$$ and one integrates over $$\phi$$ in a path integral definition, to get a number out of it one needs to act on $$\phi$$ with a dual vector. A definition could then be $$C_{x_1,\cdots,x_n}(\omega_1,\cdots,\omega_n) = \int [\mathcal{D} \phi_x] e^{-S[\phi]}\omega_1(\phi_{x_1}) \cdots \omega_n(\phi_{x_n})$$ with $$C_{x_1,\cdots,x_n}:\mathbf{\bar a_1} \otimes \cdots \otimes \mathbf{\bar a_n} \to \mathbb{R}$$. The choice of $$\omega_i$$ can be identified with $$f:\mathbf{1} \to \mathbf{\bar a_1} \otimes \cdots \otimes \mathbf{\bar a_n}$$ so that $$C_{x1,\cdots,x_n}\circ f:\mathbf{1} \to \mathbb{R}$$ becomes a number. For example, the correlation function $$\left<\phi^{I_1}_1(x_1) \cdots \phi^{I_n}_n(x_n)\right>$$ can be obtained by choosing $$\omega_k(\phi_{x_k}) = \phi^{I_k}_{x_k}$$.

Therefore, the question is how one can obtain the usual correlation functions from the authors' definition and/or what is wrong with my definition. I believe the two definitions are equivalent when there is a natural identification $$\mathbf{a} \simeq \mathbf{\bar a}$$ (as in the case of $$O(N)$$ theories) but this is not true generally.

I confirm that our choice of terminology was indeed unfortunate. When we said '$$\phi_i(x)$$ transforms as $$\mathbf{a}_i$$', what we really meant to say is that '$$\phi_i(x)$$ is associated with the object $$\mathbf{a}_i$$', in a way which is made explicit with other axioms. We added footnote 21 in a revised version of the paper to reflect that. One should remember about this terminological clash when translating statements of the paper from Deligne-categorical symmetries to usual group-theoretical symmetries.