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An explicit logarithmic triplet module and a nonsplit submodule extension

Statement

Fix an integer $p \geq 2$ . Let $L = Z α$ with $⟨ α, α ⟩ = 2 p$ , and use the explicit oscillator, trivial-cocycle and square-zero lattice conventions below. Let $W (p) \subset V_{L}$ be the vertex subalgebra generated by $ω = (α_{- 1}^{2} / (4 p) + (p - 1) α_{- 2} / (2 p)) 1$ , $F = e^{- α}$ , $H = Q F$ , and $E = Q^{2} F$ , where $Q = (e^{α})_{(0)}$ . Set $v = e^{- α / p}$ and $M = V_{L + α / 2} \oplus V_{L + α / 2 - α / p}, \tilde{Y} (a, z) = Y_{M} (a, z) + \sum_{n \geq 1} \frac{(- 1)^{n + 1}}{n} z^{- n} Y_{M} (v_{(n)} a, z) .$ Then these fields give an actual admissible $W (p)$ -module action on $M$ . Its cyclic submodule $P_{+}^{p - 1} = W (p) e^{α / 2}$ is grading-restricted generalized and logarithmic. Writing $h = (2 - p) / 4$ , its lowest generalized conformal-weight space has basis $u = e^{α / 2}$ , $w = e^{α / 2 - α / p}$ with $\tilde{L} (0) u = h u + w$ and $\tilde{L} (0) w = h w$ . The cyclic submodule $M_{1} = W (p) e^{- α / 2}$ lies in $P_{+}^{p - 1}$ and fits into a nonsplit short exact sequence of admissible $W (p)$ -modules $0 ⟶ Λ (p - 1) ⟶ M_{1} ⟶ Π (1) ⟶ 0.$ Here the endpoint modules use the undeformed lattice actions: $Λ (p - 1) = W (p) e^{α / 2 - α / p}$ in $V_{L + α / 2 - α / p}$ and $Π (1) = W (p) e^{- α / 2}$ in $V_{L + α / 2}$ . The injection identifies $Λ (p - 1)$ with the corresponding cyclic submodule in the second summand of the deformed module. The quotient map is the restriction to $M_{1}$ of projection onto the first summand, with image $Π (1)$ . All three terms use the common admissible grading $d = wt - h$ inherited from the two lattice summands; in particular $Π (1)$ first occurs in degree $p - 1$ , without resetting its bottom degree to zero. The operator $\tilde{L} (0)$ is semisimple on $M_{1}$ ; the Jordan block asserted above is in $P_{+}^{p - 1}$ .

Statement references

Rank-one lattice and square-zero screening data for the triplet construction

Admissible triplet action and the Jordan top of the cyclic logarithmic module

Module for an ordinary vertex algebra

Admissible module for a vertex operator algebra

Grading-restricted generalized module

Logarithmic generalized module

Adamović–Milas, lattice construction, Theorem 5.1 and Lemma 5.2

Adamović–Milas, triplet algebra, §§1, 3 and Proposition 4.2

Facts & Assumptions

Given: The explicit rank-one lattice data for $p \geq 2$ and the module/endpoint conventions in the Statement. Put $N = p - 1$ , $h = (2 - p) / 4$ , $t = e^{- α / 2}$ , $z = Q t$ , $w = e^{α / 2 - α / p}$ , $B_{0} = V_{L + α / 2 - α / p}$ , and $N_{2} = W (p) w$ . Every degree is the inherited weight minus $h$ .

[F1]

Rank-one lattice and square-zero screening data for the triplet construction defines the original actual fields, deformed cyclic spaces and undeformed endpoints. Explicit rank-one square-zero lattice fields gives the earlier generic square-zero lattice spaces and the complete exponential and normal-order formulas used by the boundary and endpoint suppliers.

[F2]

Admissible triplet action and the Jordan top of the cyclic logarithmic module proves the full deformed module action, its finite nonnegative grading, the actual Jordan top of $P_{+}$ , the inclusion $M_{1} \subset P_{+}$ and semisimplicity of the conformal zero mode on $M_{1}$ .

[F3]

Screening coefficient and nonsection obstruction for the actual triplet projection proves $N_{2} \subset M_{1}$ , that the actual projection has image the undeformed cyclic $Π (1)$ and kernel $M_{1} \cap B_{0}$ , and that this projection admits no admissible section. Its explicit coefficient is $C_{p} = 2 \frac{p}{p - 1} (- 1)^{p - 1} (\binom{2 p - 2}{p - 1}) \neq 0$ ; this nonsection result does not assume the kernel identity to be proved here.

[F4]

The actual two-dimensional triplet zero-mode plane modulo the endpoint proves $z \in M_{1}$ and that the independent degree- $N$ classes of $t, z$ in $M_{1} / N_{2}$ span exactly the space generated from $\overset{ˉ}{t}$ under the zero modes of $ω, F, H, E$ .

[F5]

Simple triplet endpoints and the actual lattice quotient proves the actual cyclic endpoint identifications and irreducibility. It proves $B_{0} / N_{2}$ simple with lowest degree $N$ and a two-dimensional lowest space, and proves those same lowest-space facts for the actual $Π (1)$ .

[F6]

Four triplet fields strongly generate the complete singular span proves that the four-generator algebra in the Statement is strongly generated by those same four homogeneous states.

[F7]

Strong generators generate the lowest space of a cyclic module proves directly from module Jacobi and strong generation that the lowest space of a cyclic admissible module generated there is generated by the strong generators' zero modes.

[F8]

Module for an ordinary vertex algebra, Admissible module for a vertex operator algebra, Grading-restricted generalized module and Logarithmic generalized module give the precise module, grading, generalized-weight and logarithmic conventions.

Proof

1.1givenF1F2F5F6F8

Between the two data models in [F1], send each polynomial charge vector $f e^{t α}$ in a specified sector to the vector with the same polynomial, charge and sector. This is a linear isomorphism on the algebra and module spaces. The oscillator multiplications and derivatives, charge shifts and their zero products, zero-mode powers, scalar cocycle $1$ , and every summand of the finite normal-order rule agree. Thus it intertwines every coefficient of the state and module fields. It fixes the displayed conformal state and $v$ , intertwines $Q$ , and carries $F, H, E$ and their generated vertex subalgebra to the corresponding original ones. Therefore [F5]–[F6] apply to exactly the original algebra and ordinary lattice endpoints in the Statement. Apply [F2] to the actual data [F1]. It supplies the complete Jacobi action, admissible grading and finite generalized conformal-weight spaces. Its degree-zero space in $P_{+}$ is exactly $C e^{α / 2} \oplus C w$ , with conformal zero-mode action $(\tilde{L} (0) - h) e^{α / 2} = w$ and $(\tilde{L} (0) - h) w = 0$ . This is a nonzero Jordan block, so $P_{+}$ is logarithmic in [F8]. The same supplier proves that the actual $M_{1}$ is a submodule of $P_{+}$ and its conformal zero mode is semisimple.

1.2F3F5F8algebra

By [F3], $N_{2} \subset M_{1}$ and projection onto the first summand gives a surjective graded homomorphism $π : M_{1} \to Π (1)$ with kernel $K = M_{1} \cap B_{0}$ . Put $R = M_{1} / N_{2}$ and $J = K / N_{2}$ . The induced sequence $0 \to J \to R \to Π (1) \to 0$ is exact in every inherited degree, and $J$ embeds as a submodule of the ordinary $B_{0} / N_{2}$ . Both $B_{0} / N_{2}$ and $Π (1)$ have no degrees below $N$ by [F5], so exactness gives $R (d) = 0$ for $d < N$ .

2.1F4F6F7step 1.2algebra

The actual quotient $R$ is cyclic on $\overset{ˉ}{t}$ in degree $N$ . Reindex $R (N + r)$ by $r \geq 0$ and apply [F7], using [F6] for actual strong generation. Thus all of $R (N)$ is generated by the four indicated zero modes on $\overset{ˉ}{t}$ . By [F4] that generated subspace is exactly the two-dimensional span of $\overset{ˉ}{t}, \overset{ˉ}{z}$ : writing $A = (\binom{3 p - 2}{p - 1}) \neq 0$ , its actions are $o (F) \overset{ˉ}{t} = 0$ , $o (F) \overset{ˉ}{z} = A \overset{ˉ}{t}$ , $o (H) \overset{ˉ}{t} = - A \overset{ˉ}{t}$ , $o (H) \overset{ˉ}{z} = A \overset{ˉ}{z}$ , $o (E) \overset{ˉ}{t} = - 2 A \overset{ˉ}{z}$ , $o (E) \overset{ˉ}{z} = 0$ , and $o (ω) = (h + N) I$ . Therefore $\dim R (N) = 2$ .

3.1F5step 1.2step 2.1algebra

Simplicity of $B_{0} / N_{2}$ in [F5] says that its submodule $J$ is zero or the whole module. In the latter case $\dim J (N) = 2$ . The degree- $N$ part of the exact sequence in step 1.2 is a short exact sequence of finite-dimensional vector spaces, and [F5] gives $\dim Π (1) (N) = 2$ . It would force $\dim R (N) = 2 + 2 = 4$ , contradicting step 2.1. Hence $J = 0$ and $K = N_{2}$ . This proves exactly the kernel equality, rather than only the inclusion of the cyclic endpoint in that kernel.

4.1F2F3F5F8step 1.1step 3.1algebra∎

It follows that the original sequence $0 \to Λ (p - 1) = N_{2} \to M_{1} \overset{π}{\to} Π (1) \to 0$ is short exact, with the actual simple endpoint identifications from [F5]. This is precisely the projection studied in [F3], which has no admissible section; consequently the sequence is nonsplit. The proof there uses the explicit nonzero value ${\tilde{E}}_{(3 p - 3)} t = C_{p} w$ and the vanishing of that mode on every possible second-sector correction to a degree- $N$ lift, and requires no Jordan block on $M_{1}$ . All maps retain the common grading, so the quotient's bottom stays in degree $N$ . Together with step 1.1 this proves every assertion of the Statement with the stated module conventions.

An explicit logarithmic triplet module and a nonsplit submodule extension

Statement

Statement references

Facts & Assumptions

Proof

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