P(1)-based braided monoidal structure from HLZ intertwiners

Given: Work over C with the formal-variable, grading, and branch conventions in the linked definitions.

Intertwining-operator space and fusion multiplicity: For ordinary modules W1,W2,W3 of a conformal vertex algebra V, let I( W3 ; W1 W2 ) be the vector space of logarithm-free formal intertwining operators of that type, with the Jacobi, lower-truncation, and L(-1)-derivative conditions of def-cft-logarithmic-intertwining-operator and the ordinary grading convention. In a specified finite semisimple category its fusion multiplicity is dim I( Wk ; Wi Wj ) when that dimension is finite. Finite-dimensionality is a separate theorem, not part of this definition.

Module for an ordinary vertex algebra: Let $V$ be a vertex algebra over $\mathbb{C}$ in the ordinary, non-super convention of def-vertex-algebra-state-field. A left $V$-module is a complex vector space $M$ and a linear map $Y_M(-,z):V⟶\{\text{formal fields on }M\},a⟼Y_M(a,z)={\sum }_{n\in \mathbb{Z}}{a}_{(n)}^M{z}^{-n-1},$ with the following properties: 1. Lower truncation. For every $a\in V$ and $m\in M$, $a_{(n)}^{M}m=0$ for all sufficiently large integers $n$, so $Y_M(a,z)m\in M((z))$ in the support convention of def-formal-distribution-on-a-vector-space. 2. Vacuum. $Y_M(\mathbf{1},z)={\mathrm{id}}_M$. 3. Module Jacobi identity. For every $a,b\in V$, $z_0^{-1}\delta \text{\hspace{0.17em}⁣}\left(\frac{z_1-z_2}{z_0}\right)Y_M(a,z_1)Y_M(b,z_2)-z_0^{-1}\delta \text{\hspace{0.17em}⁣}\left(\frac{z_2-z_1}{-z_0}\right)Y_M(b,z_2)Y_M(a,z_1)=z_2^{-1}\delta \text{\hspace{0.17em}⁣}\left(\frac{z_1-z_0}{z_2}\right)Y_M(Y(a,z_0)b,z_2)\mathrm{.}$ Here $\delta (x)={\sum }_{r\in \mathbb{Z}}{x}^r$; the first expansion is in nonnegative powers of $z_2$, the second in nonnegative powers of $z_1$, and the right-hand expansion is in nonnegative powers of $z_0$. They use the formal binomial conventions of def-formal-distribution-on-a-vector-space. The $Y$ inside the right-hand side is the state-field map of $V$; it is not the action map $Y_M$. A $V$-module homomorphism $f:M\to N$ is a linear map such that $f{Y}_M(a,z)=Y_N(a,z)f$ for every $a\in V$. This definition does not impose a conformal grading, finite-dimensional weight spaces, semisimplicity of $L_0$, a right-module convention, or a twist by an automorphism. For a conformal vertex algebra, write $Y_M(\omega ,z)={\sum }_{n\in \mathbb{Z}}{L}_M(n)z^{-n-2}$ when a conformal vector $\omega$ is specified. A $V$-submodule of $M$ is a linear subspace stable under every mode $a_{(n)}^M$. A module is simple (or irreducible) if it is nonzero and has no proper nonzero $V$-submodule. A module is completely reducible (or semisimple) if it is an internal direct sum of simple $V$-submodules; the empty direct sum is the zero module.

Fusion and multifusion categories: A multifusion category is a finite semisimple multitensor category. A fusion category is a finite semisimple tensor category. In particular the unit of a fusion category is simple, whereas a multifusion unit may split.

Braiding: Let $(\mathcal{C},\otimes ,\mathbf{1},\alpha ,\lambda ,\rho )$ be a monoidal category (def-monoidal-category). A braiding on $\mathcal{C}$ is a natural isomorphism $c_{X,Y}:X\otimes Y⟶Y\otimes X$ natural in both variables (def-natural-isomorphism) such that, for all objects $X,Y,Z$, ${\alpha }_{Y,Z,X}\circ c_{X,Y\otimes Z}\circ {\alpha }_{X,Y,Z}=(1_Y\otimes c_{X,Z})\circ {\alpha }_{Y,X,Z}\circ (c_{X,Y}\otimes 1_Z),$ and ${\alpha }_{Z,X,Y}^{-1}\circ c_{X\otimes Y,Z}\circ {\alpha }_{X,Y,Z}^{-1}=(c_{X,Z}\otimes 1_Y)\circ {\alpha }_{X,Z,Y}^{-1}\circ (1_X\otimes c_{Y,Z})\mathrm{.}$ These are the two hexagon identities. The invertibility of each $c_{X,Y}$ is part of the data, not a consequence of the hexagons.

P(z)-tensor product in a module category gives the exact P(z)-representing bijection; Logarithmic reconstruction and parallel transport of P(z)-products gives the path transport maps and their composition and homotopy laws; HLZ convergence and extension property specifies the binary product and iterate analytic continuation forms.

Monoidal category defines a monoidal category by a tensor bifunctor, a unit object, natural associator and unit isomorphisms satisfying the pentagon and triangle. Its pentagon is ${\alpha }_{W,X,Y\otimes Z}\circ {\alpha }_{W\otimes X,Y,Z}=(1_W\otimes {\alpha }_{X,Y,Z})\circ {\alpha }_{W,X\otimes Y,Z}\circ ({\alpha }_{W,X,Y}\otimes 1_Z),$ and its triangle is $(1_X\otimes {\lambda }_Y)\circ {\alpha }_{X,\mathbf{1},Y}={\rho }_X\otimes 1_Y\mathrm{.}$