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HLZ convergence and extension property

Definition

Let V be a strongly A-graded Möbius or conformal vertex algebra and C a full category of strongly Ã-graded generalized V-modules. The HLZ convergence and extension property for products and iterates consists of the following two-operator conditions.

For the scalar pairings, use the restricted graded dual $W^{'} = ⨁_{h \in C, β \in \tilde{A}} (W_{[h]}^{(β)})^{*}$ , with its componentwise evaluation pairing; each element has finite support in these gradings.

If $w$ has generalized weight $h \in C$ , set $w t (w) = h$ ; every cutoff inequality below is an inequality of real numbers and uses the real part of the displayed sum of weights.

For the product condition, take $W 1, W 2, W 3, W 4, M 1 \in C$ and grading-compatible logarithmic intertwining operators $Y_{1}, Y_{2}$ of types $(\binom{W_{4}}{W_{1} M_{1}})$ and $(\binom{M_{1}}{W_{2} W_{3}})$ . For every $β \in \tilde{A}$ there is an integer Nβ depending only on $Y_{1}, Y_{2}, β$ . For $\tilde{A}$ - and weight-homogeneous $w_{1} \in W_{1}^{(β_{1})}$ , $w_{2} \in W_{2}^{(β_{2})}$ with $β 1 + β 2 = - β$ , and any $w 3 \in W 3, w 4^{'} \in W 4^{'}$ , there are a finite $M \in Z +$ , real exponents rk,sk, nonnegative integers ik,jk for $1 \leq k \leq M$ , a positive integer K independent of w1,w2 with $i k < K$ , and functions fk holomorphic on $∣ ζ ∣ < 1$ , such that $Re (w t (w_{1}) + w t (w_{2})) + s_{k} > N_{β}$ for each k and the radial product series ${⟨ w_{4}^{'}, Y_{1} (w_{1}, x_{1}) Y_{2} (w_{2}, x_{2}) w_{3} ⟩ ∣}_{x_{1} = z_{1}, x_{2} = z_{2}}$ is absolutely convergent for |z1|>|z2|>0 and analytically continues to $\sum_{k = 1}^{M} z_{2}^{r_{k}} (z_{1} - z_{2})^{s_{k}} (\log z_{2})^{i_{k}} (\log (z_{1} - z_{2}))^{j_{k}} f_{k} ⁣ (\frac{z_{1} - z_{2}}{z_{2}})$ on |z2|>|z1−z2|>0.

For the iterate condition, take W1,W2,W3,W4,M2∈C and grading-compatible logarithmic intertwining operators ${\tilde{Y}}_{1}, {\tilde{Y}}_{2}$ of types $(\binom{W_{4}}{M_{2} W_{3}})$ and $(\binom{M_{2}}{W_{1} W_{2}})$ . For every β∈Ã there is an integer ${\tilde{N}}_{β}$ depending only on these operators and β. For $\tilde{A}$ - and weight-homogeneous $w_{2} \in W_{2}^{(β_{2})}$ , $w_{3} \in W_{3}^{(β_{3})}$ with β2+β3=−β, and any w1∈W1 and w4′∈W4′, there are a finite $\tilde{M}$ , real exponents ${\tilde{r}}_{k}, {\tilde{s}}_{k}$ , nonnegative integers ${\tilde{i}}_{k}, {\tilde{j}}_{k}$ , a positive integer $\tilde{K}$ independent of w2,w3 with ${\tilde{i}}_{k} < \tilde{K}$ , and functions ${\tilde{f}}_{k}$ holomorphic on |ζ|<1, such that $Re (w t (w_{2}) + w t (w_{3})) + {\tilde{s}}_{k} > {\tilde{N}}_{β}$ for each k and the radial iterate series ${⟨ w_{4}^{'}, {\tilde{Y}}_{1} ({\tilde{Y}}_{2} (w_{1}, x_{0}) w_{2}, x_{2}) w_{3} ⟩ ∣}_{x_{0} = z_{1} - z_{2}, x_{2} = z_{2}}$ is absolutely convergent for |z2|>|z1−z2|>0 and analytically continues to $\sum_{k = 1}^{\tilde{M}} z_{1}^{{\tilde{r}}_{k}} z_{2}^{{\tilde{s}}_{k}} (\log z_{1})^{{\tilde{i}}_{k}} (\log z_{2})^{{\tilde{j}}_{k}} {\tilde{f}}_{k} ⁣ (\frac{z_{2}}{z_{1}})$ on |z1|>|z2|>0. All powers and logarithms in these continuation formulas are multivalued; the selected branches continue those of the radial series. Setting all logarithmic exponents to zero gives the corresponding property without logarithms.

HLZ's multipoint extension is a separate conclusion, not part of the binary definition. In the ungraded auxiliary-group case $A = \tilde{A} = {0}$ , suppose every generalized module in C is C1-cofinite in the sense of def-cft-c1-cofinite-module and quasi-finite-dimensional in the sense of def-cft-quasi-finite-dimensional-module. For any $n \geq 1$ and objects W0,W1,…,Wn+1 in C, the $n = 1$ case uses one intertwining operator of type $(\binom{W_{0}}{W_{1} W_{2}})$ ; for $n \geq 2$ also take intermediate modules $M 1, \dots, M_{n - 1}$ in C and a composable chain of operators of types $(\binom{W_{0}}{W_{1} M_{1}}), (\binom{M_{1}}{W_{2} M_{2}}), \dots, (\binom{M_{n - 1}}{W_{n} W_{n + 1}})$ . For all these data, every scalar coefficient $⟨ w_{0}^{'}, Y_{1} (w_{1}, z_{1}) \dots Y_{n} (w_{n}, z_{n}) w_{n + 1} ⟩$ is absolutely convergent for |z1|>⋯>|zn|>0 and extends multivalued analytically to the configuration space zi≠0 and zi≠zj for i≠j. Near any possible boundary divisor zi=0, zi=∞ or zi=zj, including their permitted intersections, it has the local expansion form of a solution of a regular-singular differential system in the precise sense specified by HLZ Part VII, Remark 11.7 and Theorem 11.8 (using Knapp, Appendix B). This is the separate multipoint consequence of HLZ Part VII §11.2, Theorem 11.8; it is not inferred from the binary definition without those finiteness hypotheses.

Definition references

C1-cofinite generalized module in the ungraded HLZ convention

Quasi-finite-dimensional generalized module

Multipoint analytic extension with a bounded exterior dual

Facts & Assumptions

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

[F1]

Logarithmic intertwining operator supplies the formal logarithmic series, Jacobi, lower truncation, derivative, Möbius brackets and grading-compatibility convention; these formal axioms do not assert analytic convergence.

[F2]

Strongly graded contragredient duality for Möbius and conformal modules constructs the actual restricted dual in the full strong Möbius/conformal setting, with $W^{'}_{[h]}^{(β)} = (W_{[h]}^{(- β)})^{*}$ , opposite-field action, fixed-string truncation, and double-dual evaluation. No CFT-type or global grading bound is assumed in that theorem.

[F3]

Strongly graded generalized module in the HLZ setting supplies the strongly graded module hypotheses and grading-compatible morphism convention.

[F4]

Multipoint analytic extension with a bounded exterior dual proves exactly the separate multipoint conclusion from the stated objectwise hypotheses. In a given chain every leaf and intermediate module is C1-cofinite and quasi-finite; its output is quasi-finite. Its finite quotient keeps the output dual in a fixed finite weight cutoff rather than assuming that dual C1-cofiniteness follows from the output. Global dilation handles infinity without reflecting that dual into an unbounded input. No category closure, real-weight/nilpotence qualification beyond the local quasi-finite definition, or relative tensor-product existence is added. The source HLZ VII §§11.1–11.2, Theorems11.6/11.8 and Remark11.7 supplies literature attribution and the expansion convention; its published theorem is not an unproved local premise.

Proof

1.1givenF1F2F3algebra

Pairing a composable product with a restricted-dual vector gives a scalar series whose nested region is fixed by the order of the insertions.

2.1givenF1F2F3F4step 1.1algebra∎

The binary product/iterate conditions are the stated definitions. For the separate multipoint assertion, the actual chain has precisely the leaves, intermediate modules and quasi-finite output required by F4. Apply that locally proved theorem to each restricted-dual scalar coefficient. It proves all-index absolute convergence, continuation, infinity and simultaneous boundary expansions with the same branches, so the conclusion is not being inferred from formal operator axioms or smooth single-divisor pole bounds alone.

HLZ convergence and extension property

Definition

Definition references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources