P(z)-intertwining map

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

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 $Y(\cdot ,x)\cdot :W1\otimes W2\to W3[\log x]\{x\}$, 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 $[L(j),Y(w1,x)]={\sum }_{i=0}^{j+1}\mathrm{binom}(j+1,i)x^{i}Y(L(j-i)w1,x)$ 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.

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.

Strongly graded generalized module in the HLZ setting: Let V be an ordinary non-super vertex algebra in the sense of def-vertex-algebra-state-field. A Möbius vertex algebra is V with operators L(-1),L(0),L(1) representing sl(2), with L(-1) equal to translation, and satisfying $[L(j),Y(v,x)]={\sum }_{i=0}^{j+1}\mathrm{binom}(j+1,i)x^{i}Y(L(j-i)v,x)$ for $j=-1,0,1$. A Möbius module has the corresponding sl(2)-action and field commutator identities. A conformal vertex algebra supplies these operators through its Virasoro field. For a generalized module write $L(0{)}_s$ for the diagonal operator acting by h on $W_{[h]}$; the nilpotent part is $L(0)-L(0{)}_s$. Let A be an abelian group and suppose V has compatible integer-weight and A-gradings $V={\oplus }_{n\in Z,\alpha \in A}{V}_{(n)}^{(\alpha )}$, each component finite dimensional and lower truncated in n for each α, $1\in V_{(0)}^{(0)}$, $v_m{V}^{(\beta )}\subset V^{(\alpha +\beta )}$, and $L(j)V^{(\alpha )}\subset V^{(\alpha )}$ for $j=-1,0,1$. In the conformal case require $\omega \in V_{(2)}^{(0)}$. For an abelian group $\tilde{A}\supset A$, a generalized module $W={\oplus }_{h\in C,\beta \in \tilde{A}}{W}_{[h]}^{(\beta )}$ is strongly Ã-graded when the gradings are compatible, each $W_{[h]}^{(\beta )}$ is finite dimensional, $W_{[h+k]}^{(\beta )}=0$ for all sufficiently negative integers k for each h,β, $v_m{W}^{(\beta )}\subset W^{(\alpha +\beta )}$, and $L(j)W^{(\beta )}\subset W^{(\beta )}$ for $j=-1,0,1$. Morphisms preserve both gradings and the V-action.