Modular properties of sl2 Torus 1-point functions

Shafiq, Jamal 2024. Modular properties of sl2 Torus 1-point functions. PhD Thesis, Cardiff University. Item availability restricted.

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Abstract

At least as early as 1970, conformal symmetry had been considered in statistical mechanics in the context of critical phenomena, seen as a generalisation of scale invariance to rescaling by a position-dependent parameter [1]. By 1984, a family of two-dimensional quantum field theories enjoying conformal symmetry known as the minimal models had been described, one of the advantages being correlation functions were entirely determined by conformal symmetry [2]. In the context of string theory, the surface which a string sweeps out in space-time – the world-sheet – is the setting for a two-dimensional conformal field theory. Scattering amplitudes Of increasing order order in perturbation theory correspond to considering conformal field theory on increasing genus Riemann surfaces. An important theme of conformal field theory is well definedness on these, that is, not only on the Riemann sphere, but on all Riemann surfaces in the sense that n-point correlation functions should be well defined [3, 4]. The modern notion of vertex algebras arose in the context of Monstrous Moonshine in [5] from a study of representations of Kac-Moody algebras, instances of infinite-dimensional Lie algebras. Vertex operator algebras were described in [6] by considering an action by the Virasoro algebra, in the context of moonshine, a phenomenon that unexpectedly suggested a link between the modular j-function and the Monster finite group. (From the high energy physics point of view, the modes of the stress-energy tensor of any two-dimensional conformal field theory generate the Virasoro algebra.) In fact, this link is elucidated by the study of vacuum torus 1-point functions at the vacuum vector which give characters (graded dimensions), as the j-function arises as such a character for a vertex operator algebra dubbed the moonshine module [7, 8]. However, vacuum 1-point functions have played a role in other research avenues, including the classification of rational (bulk) conformal field theories from a given vertex operator algebra [9, 10], number theoric and combinatorial problems [11–13] and the classification of certain families of vertex operator algebras [14–16]. On the other hand, general torus 1-point functions have received considerably less attention. To the best of our knowledge, in the context of vertex operator algebras, the only case considered in the literature is the family of minimal models [17]. One of the goals of the thesis is to detail another family of examples: the important class of simple affine vertex operator algebras L(k,0) constructed from the Lie algebra sl2 at non-negative integral levels k. In the physics literature this is referred to as the SU(2) Wess-Zumino-Witten model whose starting point is an action of an SU(2)-valued field on a world-sheet R×S 1 ; a pedagogic introduction to how the theory is set up may be found in [18, Chapter 6]. We note that some relevant, but different work is performed in [19, 20] where generalisations of Jack symmetric polynomials are constructed in the context of affine Lie algebras. To provide more context for readers unfamiliar with n-point correlation functions in conformal field theory, on higher genus Riemann surfaces these are constructed from those on the sphere by gluing together points to add handles. For each pair of points glued in this way, the number of points in the correlation function decreases by two and the genus of the surface increases by one. The configuration of these points determines the complex structure of the resulting surface with many different configurations giving equivalent complex structures. All configurations giving an equivalent complex structure are famously related by the actions of mapping class groups. Due to conformal invariance being closely related to the existence of complex structure, one may be tempted to conclude that a well defined conformal field theory should not be able to distinguish Riemann surfaces with equivalent complex structures. However, this is only true for bulk or full conformal field theory. For chiral conformal field theory, which is the focus here (specifically its algebraic axiomatisation in the form of vertex operator algebras), one merely has that the mapping class groups act on the spaces of chiral correlation functions, as opposed to this action being trivial. All considerations from here on will be purely chiral and so henceforth correlation function or n-point function will refer to the chiral version. In the special case of the torus with either 0 or 1 points, the mapping class groups are, respectively, PSL2(Z) and B3 (the Braid group on three strands). Recall that B3 is the universal central extension of PSL2(Z). It turns out that the action of B3 on torus 1-point functions can always MODULAR PROPERTIES OF sl2 TORUS 1-POINT FUNCTIONS 3 be rescaled using multiplier systems to yield an action by its quotient Γ = SL2(Z). Hence the properties of torus correlation functions are commonly presented in terms of Γ. The groups PSL2(Z) and Γ are somewhat confusingly both commonly referred to as “the modular group” Γ in the literature and so one speaks of modular invariance of torus 1-point functions. While [4] gives a compelling motivation for the role of modular invariance, this has only been rigorously established for conformal field theories constructed from rational C2-cofinite vertex operator algebras. In this setting, torus n-point functions where the n points only take insertions from the vertex operator algebra can be constructed as traces of a product of n copies of the vertex operator algebra action on some module M. We call these n-point functions vacuum torus n-point functions because the vertex operator algebra is sometimes also called the vacuum module (note that the insertions need not be the vacuum vector of the vertex operator algebra). The special case of n = 1 with the insertion being the vacuum vector (this can also be thought of as a 0-point function) is called the character of M. In [21], Zhu proved the modular invariance of such vacuum torus n-point functions and in particular showed that vacuum torus 1-point functions are closed under the action of Γ. The properties of the Γ representations arising from vacuum torus 1-point functions have been heavily studied. Much of this work, for example the congruence property [22] rests on using the theory of tensor categories [23] and Verlinde’s formula [24, 25]. An alternative route to studying congruence now exists in the case of characters due to developments in number theory and a proof of the unbounded denominator conjecture [26]. It states that for a modular form f(τ) ∈ Q[[q 1/N]] on the upper half plane, for a positive integer N, if it is not modular for a congruence subgroup then it has unbounded denominators in its coefficients. However, again, this does not apply to general torus 1-point functions such as those studied in the thesis. While noncongruence subgroups outnumber congruence subgroups of Γ, the modular forms of the former are more poorly understood, partially attributed to the difficulty of defining suitable Hecke operators [27, Section 2.1, Section 2.3]. The transition from vacuum torus n-point functions to general torus n-point functions requires the replacement of vertex operator algebra actions by intertwining operators. This case, as mentioned above, has so far received far less attention within the literature and is the focus here. For rational C2-cofinite vertex operator algebras the modular invariance of general torus 1-point functions was shown in [28]. This was generalised to orbifolds in [29] and to torus n-point functions in [30]. However, as aforementioned, neither insights from Verlinde’s formula nor the unbounded denominator property apply here. Part of the thesis will review packaging these 1-point functions into vectors will yield vector-valued modular forms transforming under a representation of the modular group, and we provide results on the congruence properties of these representations (or non-congruence) for dimensions up to three, including a more general statement on the occurrence of non-congruence for sl2 and results on spaces of vector-valued modular forms given a representation, for dimension up to four. That general torus 1-point functions of vertex operator algebras may be a source of congruence and noncongruence representations has also been observed for the Virasoro minimal models in [17]. A dual goal of the thesis has been to improve the toolkit available for studying torus 1-point functions and this has entailed employing a categorical approach as well, specifically modular tensor categories that enjoy a rich structure in addition to those of an ordinary tensor category. More specifically, it will be shown that not only do non-congruence representations arise, but that our case is a source of infinite families and in the aforementioned dimensions provide explicit formulae for their q-series. This analytic number theoretic data will then be contrasted with the output of the categorical approach from the modular tensor categories formed by L(k,0)-modules. A natural avenue of study beyond the thesis would be considering the same analysis for affine vertex operator algebras from higher rank Lie algebras, such as simply the rank two case, sl3. There are also two ways to further refine the torus 1-point functions studied here. Rather than only tracking the L0 eigenvalues, one can also include an additional variable tracking the h0 eigenvalue from the Cartan subalgebra, which leads to considering Jacobi forms and one can continue this for higher rank. The other means of refinement is considering twisted modules, where the notion of MODULAR PROPERTIES OF sl2 TORUS 1-POINT FUNCTIONS 5 twist depends on a choice of (not necessarily proper) subgroup of the automorphism group of the vertex operator algebra. This also rapidly enlarges the dimensions of representations, since even for a finite cyclic subgroup ⟨g⟩ of order n, one must include in the vector-valued modular form the torus 1-point functions for modules twisted by all powers of the generator up to g n−1 , as modular transformations take us between 1-point functions with different twists. The thesis is organised as follows. Section 3 to Section 7 provide a pedagogic exposition of the less specialised background material for the thesis, based on the references cited therein. Specifically, Section 3 reviews the basic calculus of formal distributions and Section 4 and Section 5 put it to use in formulating the vertex algebra theory needed. Section 6 covers the basics about modularity and modular forms. Finally, Section 7 introduces the categorical notions required for defining modular tensor categories. In Section 8 we review vector-valued modular forms and the analytic number theory to study them. Torus 1-point functions are defined using intertwining operators and it is shown how vector-valued modular forms emerge by constructing vectors whose entries are torus 1-point functions. Section 9 reviews and develops general tools to characterise the space of all torus 1-point functions (as modules over the algebra of holomorphic modular forms and the algebra R of modular differential operators) obtained by varying the insertion vector over an entire simple vertex operator algebra module. The main results are Proposition 9.1, which gives sufficient conditions for the span of torus 1-point functions obtained from Virasoro descendants of certain vectors to be a cyclic R-module, and Theorem 9.3 which gives sufficient conditions for the span of all torus 1-point functions to be a cyclic R-module. Section 10 introduces the simple affine vertex operator algebra associate to sl2 at non-negative integral levels, which was reviewed in Section 5.2. The main result of the section is the multi-part Theorem 10.2, which collects the most important general results surrounding the analysis of torus 1-point functions for the aforementioned case. These include finding vectors to insert giving non-zero torus 1-point functions, establishing linear independence among a certain set of these functions, obtaining that vectors generated from these functions are weakly holomorphic vector-valued modular forms, and providing necessary and sufficient conditions for when these are holomorphic. Section 11 studies the representations of the modular group arising in Section 10 and the associated spaces of vector-valued modular forms. We show that such representations in the case of dimension one and two are always congruence, and upon reaching dimension three, that there exists an infinite family of non-congruence representations. For all the aforementioned dimensions we provide explicit distinguished vector-valued modular forms from which all others are generated and the exact levels for which the space of all holomorphic vector-valued modular forms is obtained from torus 1-point functions, that is, when they provide such generators. For dimension four we describe the space of all torus 1-point functions in those cases that a relevant space of holomorphic vector-valued modular forms is a cyclic module over the algebra of modular differential operators. For general dimensions we identify levels of affine sl2 for which the representation is non-congruence, if it is irreducible. Section 12 uses the fact categories of modules over rational C2-cofinite vertex operator algebras are modular tensor categories. The parallel to torus 1-point functions from the categorical perspective are 3-point coupling spaces and we study the action of the braid group on three strands on these. In the case of vacuum 1-point functions this leads to the familiar S and T matrices of the modular group described in Section 6. These are invariants of modular tensor categories that are, however, known not to be complete invariants [31]. Repeating this procedure for general torus 1-point functions has the potential to yield finer invariants. We derive explicit formulae for this action in terms categorical data (specifically twists and fusing matrices) in Theorem 12.2 and conclude by showing the irreducibility of a representation in the dimension four case, complementing the results of Section 11. We refer to Section 2 for a list of recurring important notation, to facilitate nonsequential readings of the thesis, or those skipping part of the review of material. MODULAR PROPERTIES OF sl2 TORUS 1-POINT FUNCTIONS 7 Appendix A includes an explanation of the code used for the example in Section 12.1 and Appendix B outlines the code used to compute the explicit q-series of the vectorvalued modular forms in Table 1 and Table 2, being the dimension two and dimension three cases respectively. We remark that the original theorems of this thesis appear in author’s own paper [32], but with substantial additional exposition to provide greater accessibility and more context

Item Type:	Thesis (PhD)
Date Type:	Completion
Status:	Unpublished
Schools:	Mathematics
Subjects:	Q Science > QA Mathematics
Date of First Compliant Deposit:	12 September 2024
Last Modified:	13 Sep 2024 10:33
URI:	https://orca.cardiff.ac.uk/id/eprint/172068

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