Intertwining-operator space and fusion multiplicity

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

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.

Ordinary module: An ordinary module for a conformal vertex algebra of CFT type is a grading-restricted generalized module in the sense of def-grading-restricted-generalized-vertex-module on which $L_M(0)$ acts semisimply. Equivalently, $M$ is a direct sum of its ordinary eigenspaces $M_h=\{m:L_M(0)m=hm\}$, each eigenspace is finite dimensional, and the real parts of the occurring weights are bounded below. Ordinary-module status is a condition on the module action; it does not assert that the category of such modules is semisimple.

Grading-restricted generalized module: Let $V$ be a conformal vertex algebra of CFT type and $M$ a left $V$-module. Write $L_M(0)$ for the coefficient determined by $Y_M(\omega ,z)={\sum }_{n\in \mathbb{Z}}{L}_M(n)z^{-n-2}\mathrm{.}$ For $h\in \mathbb{C}$, define the generalized $L_M(0)$-eigenspace $M_{[h]}=\{m\in M:(L_M(0)-h{)}^{N}m=0\text{ for some integer }N\ge 1\}\mathrm{.}$ The module is grading-restricted generalized if $M={⨁}_{h\in \mathbb{C}}{M}_{[h]}$, every $M_{[h]}$ is finite dimensional, and the real parts of its nonzero generalized weights are bounded below. The direct sum means each vector has only finitely many generalized-weight components. This convention explicitly adds finite-dimensionality and a lower bound to the generalized-eigenspace decomposition; those conditions are not consequences of the module Jacobi identity.

Logarithmic intertwining operator: Let V be a Möbius or conformal vertex algebra and W1,W2,W3 generalized V-modules. A logarithmic intertwining operator of type (W3;W1,W2) is a linear map Y(·,x)·:W1⊗W2→W3[log x]{x}, written $Y(w1,x)w2={\sum }_{n\in C}{\sum }_{k\ge 0}(w1{)}_{n;k}w2x^{-n-1}(\log x{)}^k$, satisfying: for each w1,w2,n, $(w1{)}_{n+m;k}w2=0$ for all sufficiently large integers m, with the bound independent of k; the intertwining Jacobi identity; ${\mathrm{\partial }}_{x}Y(w1,x)=Y(L(-1)w1,x)$; and $[L(j),Y(w1,x)]={\sum }_{i=0}^{j+1}\left(\genfrac{}{}{0px}{}{j+1}{i}\right)x^{i}Y(L(j-i)w1,x)$ for $j=-1,0,1$. Each coefficient of $x^{-n-1}$ is a finite polynomial in log x; no uniform degree bound over all n is imposed. In the strongly graded setting it is grading-compatible when coefficient degrees add. The no-log subspace consists of logarithmic intertwining operators with all coefficients for $k>0$ zero, retaining the stated integer-string truncation. It agrees with the usual ordinary intertwining-operator space over a conformal vertex algebra of CFT type when W1,W2,W3 are ordinary modules, meaning grading-restricted generalized modules with diagonalizable L(0); without that lower bound, no uniform real-part truncation is asserted.