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P(z)-compatibility subspace and local grading restriction

Definition

Let A be an abelian group, let Ã contain A as a subgroup, and let V be a strongly A-graded Möbius or conformal vertex algebra. Fix z∈C× and strongly Ã-graded generalized V-modules W1,W2, using the decompositions and operators of def-cft-hlz-strongly-graded-generalized-module. Fix a full category C of strongly Ã-graded generalized V-modules, with morphisms preserving both gradings and, in the Möbius case, all three L(j). For any strongly $\tilde{A}$ -graded module $M$ , give its restricted dual the grading $M^{'}_{[h]}^{(β)} = (M_{[h]}^{(- β)})^{*}$ . A functional in $(W_{1} \otimes W_{2})^{*}$ is homogeneous of degree $β$ when it vanishes on every input component except those of total degree $- β$ . For λ∈(W1⊗W2)* define the transported field by

$(Y_{P (z)}^{'} (v, x_{1}) λ) (w_{1} \otimes w_{2}) = λ (w_{1} \otimes Y_{W_{2}}^{o} (v, x_{1}) w_{2}) + {Res}_{x_{0}} z^{- 1} δ ⁣ (\frac{x_{1}^{- 1} - x_{0}}{z}) λ ⁣ (Y_{W_{1}} (e^{x_{1} L (1)} (- x_{1}^{- 2})^{L (0)} v, x_{0}) w_{1} \otimes w_{2}) .$ The transported action of $x_{0}^{- 1} δ ((x_{1}^{- 1} - z) / x_{0}) Y_{t} (v, x_{1})$ on λ is the formal series whose evaluation at w1⊗w2 is $z^{- 1} δ ⁣ (\frac{x_{1}^{- 1} - x_{0}}{z}) λ ⁣ (Y_{W_{1}} (e^{x_{1} L (1)} (- x_{1}^{- 2})^{L (0)} v, x_{0}) w_{1} \otimes w_{2}) + x_{0}^{- 1} δ ⁣ (\frac{z - x_{1}^{- 1}}{- x_{0}}) λ (w_{1} \otimes Y_{W_{2}}^{o} (v, x_{1}) w_{2}),$ where $Y_{t}$ is the diagonal field and formal delta expansions follow the module conventions. Let ${C O M P}_{P (z)}$ consist of λ for which $Y_{P (z)}^{'} (v, x_{1}) λ$ has finitely many negative powers and the transported series equals $x_{0}^{- 1} δ ((x_{1}^{- 1} - z) / x_{0}) Y_{P (z)}^{'} (v, x_{1}) λ$ for every v∈V. For $j = - 1, 0, 1$ define the transported operators by $(L_{P (z)}^{'} (j) λ) (w_{1} \otimes w_{2}) = λ (w_{1} \otimes L (- j) w_{2}) + \sum_{i = 0}^{1 - j} (\binom{1 - j}{i}) z^{i} λ (L (- j - i) w_{1} \otimes w_{2}) .$ In the conformal case these are also the corresponding coefficients in $Y_{P (z)}^{'} (ω, x) = \sum_{j \in Z} L_{P (z)}^{'} (j) x^{- j - 2}$ . The P(z)-local grading restriction requires λ to be a finite sum of generalized $L_{P (z)}^{'} (0)$ -eigenvectors, each homogeneous for the $\tilde{A}$ -grading. Let Wλ be the smallest doubly graded subspace containing λ and stable under every mode of $Y_{P (z)}^{'} (v, x)$ and the operators $L_{P (z)}^{'} (- 1), L_{P (z)}^{'} (0), L_{P (z)}^{'} (1)$ . For every $β \in \tilde{A}$ and generalized weight $h \in C$ , require $\dim (W_{λ})_{[h]}^{(β)} < \infty$ and $(W_{λ})_{[h + k]}^{(β)} = 0$ for all sufficiently negative integers k. For a P(z)-map I with target W, define $I^{'} (w^{'}) (w 1 \otimes w 2) = ⟨ w^{'}, I (w 1 \otimes w 2) ⟩$ . Define W1 P(z) W2 as the sum of these images I'(W') over all W∈C and all such I; transposed Jacobi and grading conditions put each image in the stated compatibility and local-grading subspace.

Facts & Assumptions

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

[F1]

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 $ω \in V$ and scalar $c \in C$ , such that, writing $Y (ω, z) = \sum_{n \in Z} L (n) z^{- n - 2},$ 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, $V_{n} : = {v \in V : L (0) v = n v}$ gives $V = ⨁_{n \geq 0} V_{n}$ , every $V_{n}$ is finite dimensional, and $V_{0} = C 1$ . 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.

[F2]

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 $Y_{M} (ω, z) = \sum_{n \in Z} L_{M} (n) z^{- n - 2} .$ For $h \in C$ , define the generalized $L_{M} (0)$ -eigenspace $M_{[h]} = {m \in M : (L_{M} (0) - h)^{N} m = 0 for some integer N \geq 1} .$ The module is grading-restricted generalized if $M = ⨁_{h \in C} M_{[h]}$ , 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.

[F3]

Module for an ordinary vertex algebra: Let $V$ be a vertex algebra over $C$ 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 $Y_{M} (-, z) : V ⟶ {formal fields on M}, a ⟼ Y_{M} (a, z) = \sum_{n \in Z} a_{(n)}^{M} z^{- n - 1},$ 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. $Y_{M} (1, z) = {id}_{M}$ . 3. Module Jacobi identity. For every $a, b \in V$ , $z_{0}^{- 1} δ ⁣ (\frac{z_{1} - z_{2}}{z_{0}}) Y_{M} (a, z_{1}) Y_{M} (b, z_{2}) - z_{0}^{- 1} δ ⁣ (\frac{z_{2} - z_{1}}{- z_{0}}) Y_{M} (b, z_{2}) Y_{M} (a, z_{1}) = z_{2}^{- 1} δ ⁣ (\frac{z_{1} - z_{0}}{z_{2}}) Y_{M} (Y (a, z_{0}) b, z_{2}) .$ Here $δ (x) = \sum_{r \in Z} x^{r}$ ; 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 $Y_{M} (ω, z) = \sum_{n \in Z} L_{M} (n) z^{- n - 2}$ when a conformal vector $ω$ 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.

[F4]

Contragredient module of a generalized module: For a grading-restricted generalized module M over a conformal vertex algebra V, its contragredient candidate is $M^{'} = \oplus_{h \in C} M_{[h]}^{*}$ , with action defined by $⟨ Y_{M^{'}} (v, x) w^{'}, w ⟩ = ⟨ w^{'}, Y_{M}^{o} (v, x) w ⟩$ , where $Y_{M}^{o} (v, x) = Y_{M} (e^{x L (1)} (- x^{- 2})^{L (0)} v, x^{- 1})$ . On homogeneous integer-weight vectors, $(- x^{- 2})^{L (0)}$ contributes $(- 1)^{w t (v)} x^{- 2 w t (v)}$ . Whether this formula satisfies all module axioms and gives M''≅M is a separate theorem; this definition alone does not assert it. V is self-contragredient when V≅V' as V-modules.

[F5]

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 $I : W 1 \otimes W 2 \to \overset{ˉ}{W} 3 = \prod_{h \in C} W 3_{[h]}$ whose Ã-degrees add, for which $π_{h - m} I (w 1 \otimes w 2) = 0$ for all sufficiently large integers m for each fixed h,w1,w2, and which satisfies the P(z)-Jacobi identity and, for $j = - 1, 0, 1, L (j) I (w 1 \otimes w 2) = I (w 1 \otimes L (j) w 2) + \sum_{i = 0}^{j + 1} (\binom{j + 1}{i}) z^{i} I ((L (j - i) w 1) \otimes w 2)$ . There is no L(-1)-derivative axiom for I. Formal variables x0,x1 are independent; z is a fixed complex number.

[F6]

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 $[L (j), Y (v, x)] = \sum_{i = 0}^{j + 1} (\binom{j + 1}{i}) x^{i} Y (L (j - i) v, x)$ 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 $V = \oplus_{n \in Z, α \in A} V_{(n)}^{(α)}$ , each component finite dimensional and lower truncated in n for each α, $1 \in V_{(0)}^{(0)}, v_{m} V^{(β)} \subset V^{(α + β)}$ , and $L (j) V^{(α)} \subset V^{(α)}$ for $j = - 1, 0, 1$ . In the conformal case require $ω \in V_{(2)}^{(0)}$ . For an abelian group Ã⊃A, a generalized module $W = \oplus_{h \in C, β \in \tilde{A}} W_{[h]}^{(β)}$ is strongly Ã-graded when the gradings are compatible, each $W_{[h]}^{(β)}$ is finite dimensional, $W_{[h + k]}^{(β)} = 0$ for all sufficiently negative integers k for each h,β, $v_{m} W^{(β)} \subset W^{(α + β)}$ , and $L (j) W^{(β)} \subset W^{(β)}$ for $j = - 1, 0, 1$ . Morphisms preserve both gradings and the V-action.

[F7]

Strongly graded contragredient duality for Möbius and conformal modules proves the opposite-field restricted dual and double-dual evaluation for arbitrary strongly graded generalized modules over strongly graded Möbius or conformal vertex algebras. Its lower truncation uses a fixed auxiliary-degree integer weight string, and does not assume a globally bounded grading or CFT type.

Proof

1.1givenF1F2F3F4F6algebra

The residue action is the transpose of the diagonal vertex action together with the opposite second-input field; module truncation and the finite L(1)-sum make its coefficients defined on homogeneous inputs.

2.1givenF1F2F3F4F5F6F7step 1.1algebra∎

Let $I : W_{1} \otimes W_{2} \to \overline{W}$ be a P(z)-map with $W \in C$ , and define $I^{'} : W^{'} \to (W_{1} \otimes W_{2})^{*}$ by transposition. Transposing its P(z)-Jacobi identity gives the defining transported compatibility identity, while the three $L (j)$ identities give compatibility with the transported $L_{P (z)}^{'} (j)$ . The transposed field identity makes $I^{'}$ intertwine the contragredient action on $W^{'}$ with the transported action on its image. The strong grading of W then gives finite-dimensional components and lower truncation for W'. Since every $w^{'} \in W^{'}$ is a finite sum of doubly homogeneous generalized-weight vectors, $I^{'} (w^{'})$ is a finite sum of homogeneous generalized $L_{P (z)}^{'} (0)$ -eigenvectors. The subspace generated from $I^{'} (w^{'})$ by the transported modes is the image under $I^{'}$ of the submodule generated by $w^{'}$ in $W^{'}$ . Its doubly graded components therefore remain finite dimensional and lower truncated. Thus each image lies in the stated compatibility and local-grading subspace, so their sum does also.

Depends on

Dependency tree · 2 levels

Sources

Huang, Lepowsky, and Zhang, Logarithmic tensor category theory, Part IV, Section 5.2, Proposition 5.33, P(z)-local grading restriction condition and Notation 5.43