Abstract
We consider the Whittaker modules \(M_{1}(\varvec{\lambda },\varvec{\mu })\) for the Weyl vertex algebra M (also called \(\beta \gamma \) vertex algebra), constructed in Adamović et al. (J Algebra 539:1–23, 2019), where it was proved that these modules are irreducible for each finite cyclic orbifold \(M^{{\mathbb {Z}}_n}\). In this paper, we consider the modules \(M_{1}(\varvec{\lambda },\varvec{\mu })\) as modules for the \({{\mathbb {Z}}}\)-orbifold of M, denoted by \(M^0\). \(M^0\) is isomorphic to the vertex algebra \({\mathcal {W}}_{1+\infty , c=-1} = {\mathcal {M}}(2) \otimes M_1(1)\) which is the tensor product of the Heisenberg vertex algebra \(M_1(1)\) and the singlet algebra \({\mathcal {M}}(2)\) (cf. Adamović in J Algebra 270:115–132, 2003; Kac and Radul in Transform Groups 1:41–70, 1996; Wang in Commun Math Phys 195:95–111, 1998). Furthermore, these modules are also modules of the Lie algebra \(\widehat{\mathfrak {gl}}\) with central charge \(c=-1\). We prove that they are reducible as \(\widehat{\mathfrak {gl}}\)-modules (and therefore also as \(M^0\)-modules), and we completely describe their irreducible quotients \(L(d,\varvec{\lambda },\varvec{\mu })\). We show that \(L(d,\varvec{\lambda },\varvec{\mu })\) in most cases are not tensor product modules for the vertex algebra \( {\mathcal {M}}(2) \otimes M_1(1)\). Moreover, we show that all constructed modules are typical in the sense that they are irreducible for the Heisenberg–Virasoro vertex subalgebra of \({\mathcal {W}}_{1+\infty , c=-1}\).
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Acknowledgements
We would like to thank C. H. Lam, N. Yu, A. Romanov and K. Zhao for discussions about Whittaker modules. We would also like to thank the referees for their valuable comments.
The authors are partially supported by the QuantiXLie Centre of Excellence, a project cofinanced by the Croatian Government, and European Union through the European Regional Development Fund—the Competitiveness and Cohesion Operational Programme (KK.01.1.1.01.0004).
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Appendix A: \(M_1(\varvec{\lambda }, \varvec{\mu })\) as a module for the Heisenberg–Virasoro algebra
Appendix A: \(M_1(\varvec{\lambda }, \varvec{\mu })\) as a module for the Heisenberg–Virasoro algebra
In this section, we prove a stronger statement that \(M_1(\varvec{\lambda }, \varvec{\mu })\) is a cyclic module for the Heisenberg–Virasoro vertex subalgebra of \({\mathcal {W}}_{1+\infty ,-1}\). Our main argument will be that the action of singlet \({\mathcal {M}}(2)\) on the Whittaker vector \( \textbf{w}_{ \varvec{ \lambda }, \varvec{ \mu } }\) can be replaced by the action of the Virasoro subalgebra of \({\mathcal {M}}(2)\). Next we identify \(M_1(\varvec{\lambda }, \varvec{\mu })\) and \(L(d,\varvec{\lambda }, \varvec{\mu }) \) as restricted modules for the Heisenberg–Virasoro algebra recently classified in [22].
Recall that
The singlet algebra \({\mathcal {M}}(2)\) (cf. [2, 21]) is generated by the Virasoro vector \(L = J^1 + \tfrac{1}{2}:(J^0)^2: + \tfrac{1}{2} D J^0\) and by the primary field H of conformal weight 3. Recall that the \({{\mathbb {Z}}}_2\)–orbifold of M is isomorphic to the simple affine vertex algebra \(L_{-1/2}(\mathfrak {sl}_2)\) generated by
the singlet algebra is isomorphic to the parafermion vertex algebra \(N_{-1/2}(\mathfrak {sl}_2)\) and H is a scalar multiple of the parafermion generator \(W^3\) in [11, Section 2]:
(see also [19, Section 4]).
We have that \({\mathcal {L}}^{HVir} _{c=-1} = {\mathcal {L}}^{Vir} _{c=-2} \otimes M_1(1) \), where \({\mathcal {L}}^{Vir} _{c=-2}\) denotes the simple Virasoro vertex algebra of central charge \(c=-2\) generated by L.
Theorem A.1
We have:
-
(1)
\(M_1(\varvec{\lambda }, \varvec{\mu })\) is an cyclic \({\mathcal {L}}^{HVir} _{c=-1}\)-module.
-
(2)
\(L(d,\varvec{\lambda }, \varvec{\mu }) \) is an irreducible \({\mathcal {L}}^{HVir}_{c=-1}\)-module.
-
(3)
\(M_1(\varvec{\lambda }, \varvec{\mu })\) and \(L(d,\varvec{\lambda }, \varvec{\mu }) \) are not tensor product \({\mathcal {L}}^{HVir}_{c=-1}\)-modules. Moreover, there is a nilpotent subalgebra \({\mathcal {P}}\) of \({\mathcal {H}}\) and one-dimensional \({\mathcal {P}}\)-module U such that \(M_1(\varvec{\lambda }, \varvec{\mu }) = \text{ Ind } _{{\mathcal {P}} } ^{{\mathcal {H}}} U\).
Proof
The field H can be expressed using \(J^0, J^1\) and it contains the non-trivial summand \(J^0(-1)^3\textbf{1}\). Set \(H(i):= H_{i+2}\), so we have
By a direct calculation, we get for \(s \in {{\mathbb {Z}}}_{\ge 0}\):
for certain \(q \in {{\mathbb {C}}}\), \( q \ne 0\).
Using the following relation in \({\mathcal {M}}(2)\)
and using the arguments as in [20, Lemma 3.1], we get that
so \({\mathcal {M}}(2). \textbf{w}_{ \varvec{ \lambda }, \varvec{ \mu } }\) is isomorphic to the Virasoro submodule obtained by the action of the Virasoro generator L on the Whittaker vector \( \textbf{w}_{ \varvec{ \lambda }, \varvec{ \mu } }\). Since by Theorem 5.2\(M_1(\varvec{\lambda }, \varvec{\mu })\) is a cyclic \({\mathcal {M}}(2) \otimes M_1(1)\)-module, we conclude that \(M_1(\varvec{\lambda }, \varvec{\mu })\) is a cyclic \({\mathcal {L}}^{HVir} _{c=-1}\)-module. This proves assertion (1). The assertion (2) is a consequence of (1).
Let us prove assertion (3). For simplicity, we consider the case \(n + 1 \ge m\). Other cases can be treat similarly.
By a direct calculation, we get for \(s \in {{\mathbb {Z}}}_{>0}\):
where \(a_i, b _i \in {{\mathbb {C}}}\), \(a_{m}, b_{ m+1} \ne 0\), and \(J^0(n) \textbf{w}_{ \varvec{ \lambda }, \varvec{ \mu } }\), \(J^1(n-1) \textbf{w}_{ \varvec{ \lambda }, \varvec{ \mu } }\) are not proportional to \( \textbf{w}_{ \varvec{ \lambda }, \varvec{ \mu } }\).
Consider the nilpotent subalgebra \({\mathcal {P}}^{(n)}\) of \({\mathcal {H}}\) spanned by \(J^0(r+1), J^1(r)\), \(r \ge n\) and \(C_1, C_2\). Then \({{\mathbb {C}}} \textbf{w}_{ \varvec{ \lambda }, \varvec{ \mu } }\) is the one-dimensional \({\mathcal {P}}^{(n)} \)-module satisfying relations (A.1)-(A.3) and \(C_1 \textbf{w}_{ \varvec{ \lambda }, \varvec{ \mu } }= 2 \textbf{w}_{ \varvec{ \lambda }, \varvec{ \mu } }\), \(C_2 \textbf{w}_{ \varvec{ \lambda }, \varvec{ \mu } }= - \textbf{w}_{ \varvec{ \lambda }, \varvec{ \mu } }\). Then \(M_1(\varvec{\lambda }, \varvec{\mu })\) and \(L(d, \varvec{\lambda }, \varvec{\mu })\) are quotients of the universal \({\mathcal {H}}\)-module:
These universal modules and their simple quotients appeared in a slightly general form in [22, Section 7], where it was proved that they are not tensor product modules (see [22, Example 7.5, Remark A.4]). In particular, it was shown that for each \(d \in {{\mathbb {C}}}\):
is a simple, restricted \({\mathcal {H}}\)-module. This easily implies that as \({\mathcal {H}}\)-modules:
The proof follows. \(\square \)
Corollary A.2
\({\mathcal {W}}_{1+\infty , c=-1}\)-modules \({{\widetilde{\rho }}}_s (L(d,\varvec{\lambda }, \varvec{\mu }))\) are typical.
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Adamović, D., Pedić Tomić, V. Whittaker modules for \(\widehat{\mathfrak {gl}}\) and \({\mathcal {W}}_{1+ \infty }\)-modules which are not tensor products. Lett Math Phys 113, 39 (2023). https://doi.org/10.1007/s11005-023-01663-1
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DOI: https://doi.org/10.1007/s11005-023-01663-1