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.


1 Answer 1


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.


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