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The restricted dual with opposite action is a contragredient module

Statement

For every grading-restricted generalized module $M$ over a conformal vertex algebra $V$ of CFT type, the restricted dual action defined by the opposite field forms a grading-restricted generalized $V$ -module $M^{'}$ . The componentwise evaluation $M \to M^{''}$ is a $V$ -module isomorphism.

Statement references

Contragredient module of a generalized module

Opposite fields under formal coordinate inversion

Facts & Assumptions

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

[F1]

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.

[F2]

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.

[F3]

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.

[F4]

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.

[F5]

Opposite fields under formal coordinate inversion proves opposite Jacobi, translation, the three opposite Möbius brackets, the explicit opposite modes and their generalized-weight shifts, and $A (x^{- 1}) A (x) = 1$ .

Proof

1.1givenF3F5algebra

For $v \in V_{d}$ the opposite mode $v_{n}^{o}$ maps $M_{[h]}$ into $M_{[h + n + 1 - d]}$ . Thus its transpose sends $M_{[k]}^{*}$ into $M_{[k + d - n - 1]}^{*}$ . If $R e (h) \geq B$ for all weights of $M$ , the latter space vanishes when $R e (k) + d - n - 1 < B$ . A restricted-dual vector has finite weight support, so one bound works for that vector. This proves both preservation of $M^{'}$ by every mode and lower truncation of its field. This bound comes from the lower weight bound on $M$ , rather than from upper truncation of the opposite field on a fixed vector of $M$ .

2.1givenF1F2F4F5step 1.1algebra

Pair each candidate module identity against $w \in M$ . Vacuum follows from opposite vacuum. Transposition reverses the two products in opposite Jacobi, giving exactly the ordinary module Jacobi identity in F4; every matrix coefficient and delta expansion is legitimate by step 1.1 and F5. The derivative identity transposes directly. Since $A (x) ω = x^{- 4} ω$ , one has $⟨ L_{M^{'}} (j) w^{'}, w ⟩ = ⟨ w^{'}, L_{M} (- j) w ⟩$ . The Virasoro bracket is preserved: transposing reverses a commutator and reverses its two indices, so the central term remains $c (j^{3} - j) δ_{j + k, 0} / 12$ . The opposite Möbius brackets likewise transpose to the ordinary module brackets. Hence this is a conformal generalized module.

3.1F3step 2.1algebra

On $M_{[h]}^{*}$ the operator $L_{M^{'}} (0)$ is the transpose of $L_{M} (0)$ , so its only generalized eigenvalue is $h$ . Distinct generalized eigenspaces remain distinct, and the restricted direct sum is exactly their decomposition. Their dimensions and their common lower real weight bound are the same as for $M$ . Thus $M^{'}$ is grading restricted.

4.1F1F3F5step 3.1algebra∎

Evaluation identifies $M_{[h]}$ with $M_{[h]}^{* *}$ because each is finite dimensional. Pair the action on its double dual against $w^{'} \in M^{'}$ : twice applying the defining transpose gives the original action with state $A (x^{- 1}) A (x) v = v$ . Thus evaluation commutes coefficientwise with every $Y_{M} (v, x)$ . The direct sum of the finite-dimensional evaluation isomorphisms is therefore a $V$ -module isomorphism $M ≅ M^{''}$ .

Depends on

Dependency tree · 2 levels

Sources

Huang, Lepowsky, and Zhang, Logarithmic tensor category theory, Part I