Logarithmic intertwiners and P(z)-maps correspond

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(·,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 $[L(j),Y(w1,x)]={\sum }_{i=0}^{j+1}\mathrm{binom}(j+1,i)x^{i}Y(L(j-i)w1,x)$ 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 retains the same integer-string truncation. For ordinary modules (grading-restricted generalized modules with diagonalizable L(0)) over a conformal vertex algebra of CFT type it agrees with the usual ordinary intertwining-operator space; no such identification is asserted for unrestricted generalized modules.

P(z)-intertwining map: Let z∈C× and W1,W2,W3 be generalized V-modules. A P(z)-intertwining map of type (W3;W1,W2) is a linear map $I:W1\otimes W2\to \bar{W}3={\prod }_{h\in C}W{3}_{[h]}$ whose Ã-degrees add, for which ${\pi }_{h-m}I(w1\otimes w2)=0$ for all sufficiently large integers m for each fixed h,w1,w2, and which satisfies the P(z)-Jacobi identity and, for $j=-1,0,1,L(j)I(w1\otimes w2)=I(w1\otimes L(j)w2)+{\sum }_{i=0}^{j+1}\mathrm{binom}(j+1,i)z^{i}I((L(j-i)w1)\otimes w2)$. There is no L(-1)-derivative axiom for I. Formal variables x0,x1 are independent; z is a fixed complex number.

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 has $V={\oplus }_{n\in Z}{V}_{(n)}$, with $V_{(n)}=\{v\in V:L(0)v=nv\}$, and 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 $\alpha ,1\in V_{(0)}^{(0)},v_m{V}^{(\beta )}\subset V^{(\alpha +\beta )}$, 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 Ã⊃A, a generalized module $W={\oplus }_{h\in C,\beta \in \tilde{A}}{W}_{[h]}^{(\beta )}$ is strongly Ã-graded when $W_{[h]}^{(\beta )}=W_{[h]}\cap W^{(\beta )},W^{(\beta )}={\oplus }_h{W}_{[h]}^{(\beta )}$ and $W_{[h]}={\oplus }_{\beta }{W}_{[h]}^{(\beta )}$, 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.