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Strongly graded generalized module in the HLZ setting

Definition

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 an integer grading $V = \oplus_{n \in Z} V_{(n)}$ satisfying $V_{(n)} = {v \in V : L (0) v = n v}$ , and a representation of sl(2) by L(-1), L(0), L(1), satisfying $[L (0), L (- 1)] = L (- 1)$ , $[L (0), L (1)] = - L (1)$ , and $[L (- 1), L (1)] = - 2 L (0)$ , with $L (- 1) = T$ and, for every $v \in V$ and $j = - 1, 0, 1$ , $[L (j), Y (v, x)] = \sum_{i = 0}^{j + 1} (\binom{j + 1}{i}) x^{i} Y (L (j - i) v, x) .$ A conformal vertex algebra supplies these operators through its Virasoro field. A generalized V-module uses the left-module action of def-vertex-algebra-module: a lower-truncated field map with vacuum action and the module Jacobi identity. In the Möbius case it also has a generalized L(0)-eigenspace decomposition $W = \oplus_{h \in C} W_{[h]}$ , with $(L (0) - h)^{N} w = 0$ for each $w \in W_{[h]}$ and some N depending on w; its additional sl(2)-action is a representation with the same three Lie brackets and satisfies, for every $v \in V$ and $j = - 1, 0, 1$ , $[L_{W} (j), Y_{W} (v, x)] = \sum_{i = 0}^{j + 1} (\binom{j + 1}{i}) x^{i} Y_{W} (L (j - i) v, x), \partial_{x} Y_{W} (v, x) = Y_{W} (L (- 1) v, x) .$ In the conformal case these operators are the modes of the module action of the conformal vector, and the module and Virasoro axioms supply the same identities. Write $L (0)_{s}$ for the diagonal operator acting by h on $W_{[h]}$ ; $L (0) - L (0)_{s}$ acts locally nilpotently on each generalized weight space, with no uniform exponent required.

Let A be an abelian group. V has integer-weight and A-gradings with $V = \oplus_{n \in Z} V_{(n)} = \oplus_{α \in A} V^{(α)}$ . Compatibility means $V_{(n)}^{(α)} = V_{(n)} \cap V^{(α)}$ and $V^{(α)} = \oplus_{n \in Z} V_{(n)}^{(α)}$ , so $V = \oplus_{n \in Z, α \in A} V_{(n)}^{(α)}$ . Each $V_{(n)}^{(α)}$ is finite dimensional and zero for all sufficiently negative n for each fixed α, $1 \in V_{(0)}^{(0)}$ , $v_{m} V^{(β)} \subset V^{(α + β)}$ for $v \in V^{(α)}$ and $m \in Z$ , and $L (j) V^{(α)} \subset V^{(α)}$ for $j = - 1, 0, 1$ . In the conformal case also require $ω \in V_{(2)}^{(0)}$ .

For an abelian group Ã containing A, a generalized module W is strongly Ã-graded when W has an auxiliary decomposition $W = \oplus_{β \in \tilde{A}} W^{(β)}$ compatible with its generalized-weight decomposition: $W_{[h]}^{(β)} : = W_{[h]} \cap W^{(β)}$ , $W^{(β)} = \oplus_{h \in C} W_{[h]}^{(β)}$ , and $W_{[h]} = \oplus_{β \in \tilde{A}} W_{[h]}^{(β)}$ . Thus $W = \oplus_{h \in C, β \in \tilde{A}} W_{[h]}^{(β)}$ . Each $W_{[h]}^{(β)}$ is finite dimensional, and for each fixed h,β one has $W_{[h + k]}^{(β)} = 0$ for all sufficiently negative integers k. In addition, $v_{m} W^{(β)} \subset W^{(α + β)}$ for $v \in V^{(α)}$ , L(j) preserves $W^{(β)}$ for $j = - 1, 0, 1$ , and, in the Möbius case, the action is the specified sl(2)-representation above. A morphism $f : W \to U$ is a V-module map preserving each Ã-degree and generalized weight; in the Möbius case it also satisfies $f L_{W} (j) = L_{U} (j) f$ for $j = - 1, 0, 1$ . In the conformal case these last identities follow from V-linearity because the $L_{W} (j)$ are modes of the action of ω.

Definition references

Vertex algebra in the state–field convention

Module for an ordinary vertex algebra

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]

Vertex algebra in the state–field convention: A (non-super) vertex algebra over $C$ is a complex vector space $V$ equipped with a nonzero vector $1 \in V$ , a linear operator $T : V \to V$ , and a linear state–field map $Y (-, z) : V ⟶ ({End}_{C} (V))^{Z}, a ⟼ Y (a, z) = \sum_{n \in Z} a_{(n)} z^{- n - 1},$ where the codomain denotes all endomorphism-valued coefficient families indexed by $Z$ . satisfying the following axioms. 1. Field and truncation. For each $a, v \in V$ , there is an integer $N (a, v)$ such that $a_{(n)} v = 0$ for all $n \geq N (a, v)$ . Thus $Y (a, z) v \in V ((z))$ in the lower-bounded support sense of def-formal-laurent-series-and-residue. 2. Vacuum and creation. $Y (1, z) = {id}_{V}$ , $T 1 = 0$ , and $Y (a, z) 1 = e^{z T} a : = \sum_{r \geq 0} \frac{z^{r} T^{r} a}{r!} .$ This is a formal power series; no analytic convergence is asserted. 3. Translation covariance. For every $a \in V$ , $[T, Y (a, z)] = \partial_{z} Y (a, z) .$ 4. Locality. For every $a, b \in V$ , some integer $N \geq 0$ satisfies $(z - w)^{N} [Y (a, z), Y (b, w)] = 0.$ The coefficient $a_{(n)} b$ is called the $n$ th product of the states $a, b$ . The mode and formal-distribution conventions in def-formal-distribution-on-a-vector-space are in force. The axioms are algebraic and do not impose a grading, conformal vector, simplicity, unitarity, or analytic OPE convergence.

Proof

1.1givenF1F2F3F4algebra

If $v \in V_{(n)}^{(α)}$ , then $L (0) L (j) v = (n - j) L (j) v$ by the sl(2) brackets, and L(j) preserves α. Thus $L (j) V_{(n)}^{(α)} \subset V_{(n - j)}^{(α)}$ . The fixed-α lower cutoff makes $L (1)^{k} v = 0$ for sufficiently large k. Every state has finite homogeneous support, so L(1) is locally nilpotent on V.

2.1givenF1F2F3F4step 1.1algebra∎

For $v \in V_{(n)}$ , the $j = 0$ module commutator and derivative identity give $[L_{W} (0), v_{m}] = (n - m - 1) v_{m}$ . Hence $(L_{W} (0) - (h + n - m - 1))^{N} v_{m} w = v_{m} (L_{W} (0) - h)^{N} w$ , which vanishes for some N whenever $w \in W_{[h]}$ . Together with auxiliary-degree compatibility this gives $v_{m} W_{[h]}^{(β)} \subset W_{[h + n - m - 1]}^{(α + β)}$ for $v \in V_{(n)}^{(α)}$ . These are the required integer and auxiliary grading compatibilities; no uniform nilpotence exponent on W is imposed.

Strongly graded generalized module in the HLZ setting

Definition

Definition references

Facts & Assumptions

Proof

Depends on

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