C1-cofinite ordinary intertwining spaces are finite dimensional

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

C1-cofinite generalized module in the ungraded HLZ convention: Let V be an integer-graded Möbius or conformal vertex algebra with $A=\tilde{A}=\{0\}$, and W a generalized V-module. Put $V_{+}={⨁}_{n>0}{V}_n$ and . The module is C1-cofinite when dim_C(W/C1(W))<∞.

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.

Logarithmic intertwiners and P(z)-maps correspond: Fix z∈C×, p∈Z, and ℓ_p(z)=log|z|+i arg(z)+2πip with 0≤arg(z)<2π. For strongly graded generalized V-modules W1,W2,W3, $I_{Y,p}(w1\otimes w2)=Y(w1,e^{{\mathrm{\ell }}_p(z)})w2$ is a linear isomorphism from grading-compatible logarithmic intertwiners of type (W3;W1,W2) to P(z)-intertwining maps of that type. Its inverse is $Y_{I,p}(w1,x)w2=(y^{L(0)}{x}^{L(0)}I(y^{-L(0)}{x}^{-L(0)}w1\otimes y^{-L(0)}{x}^{-L(0)}w2)){\mathrm{\mid }}_{y=e^{-{\mathrm{\ell }}_p(z)}}$.

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.

Conformal vertex algebra of CFT type: A conformal vertex algebra of CFT type is a complex vertex algebra $V$ in the sense of def-vertex-algebra-state-field, with a specified vector $\omega \in V$ and scalar $c\in \mathbb{C}$, such that, writing $Y(\omega ,z)={\sum }_{n\in \mathbb{Z}}L(n)z^{-n-2},$ the operators $L(n)$ and scalar central action $c$ satisfy the Virasoro relations of def-virasoro-algebra, and $L(-1)=T$. In addition, $L(0)$ is diagonalizable, $V_n:=\{v\in V:L(0)v=nv\}$ gives $V={⨁}_{n\ge 0}{V}_n$, every $V_n$ is finite dimensional, and $V_0=\mathbb{C}\mathbf{1}$. Under this page's convention, this is a vertex operator algebra of CFT type. The grading and the CFT-type condition are part of the definition; they do not follow from the vertex-algebra axioms alone. The conformal weight of $v\in V_n$ is $n$. A conformal vector, a grading, and the nonnegative grading condition are additional data beyond an ordinary vertex algebra.

The restricted dual with opposite action is a contragredient module: For every grading-restricted generalized module M over a conformal vertex algebra V of CFT type, the opposite-field restricted dual M' is a grading-restricted generalized module, and componentwise evaluation M→M'' is a V-module isomorphism.

Quasi-finite-dimensional generalized module: Let $W={⨁}_{h\in R}{W}_{[}h]$ be a generalized V-module with real generalized L(0)-weights. It is quasi-finite-dimensional if for every real N the strict cutoff sum ${⨁}_{h<N}{W}_{[}h]$ is finite dimensional.

Finite generation of the multipoint Jacobi quotient defines the exact multipoint Ward/Jacobi relations and proves that the quotient is finitely generated and that intertwiner products annihilate the relations.