When is the Bloch–Okounkov q-bracket modular?

Ittersum, Jan-Willem M. van; Willem, Michel

doi:10.1007/s11139-019-00144-1

Cited by 3 publications

(12 citation statements)

References 11 publications

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“…Holomorphic anomaly equation. Just as in [17] we answer the question when {g(0)} q is actually modular (rather than quasimodular). Namely, if we, instead, consider c to be a formal variable, the holomorphic anomaly equation of {g} q (determining the failure of modularity) can be expressed as…”

Section: A Criterion For Quasimodularitymentioning

confidence: 88%

Quantum KdV hierarchy and quasimodular forms

Ittersum¹,

Ruzza²

2022

Preprint

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Dubrovin [10] has shown that the spectrum of the quantization (with respect to the first Poisson structure) of the dispersionless Korteweg-de Vries (KdV) hierarchy is given by shifted symmetric functions; the latter are related by the Bloch-Okounkov Theorem [1] to quasimodular forms on the full modular group. We extend the relation to quasimodular forms to the full quantum KdV hierarchy (and to the more general Intermediate Long Wave hierarchy). These quantum integrable hierarchies have been described by Buryak and Rossi [6] in terms of the Double Ramification cycle in the moduli space of curves. The main tool and conceptual contribution of the paper is a general effective criterion for quasimodularity.

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Section: A Criterion For Quasimodularitymentioning

confidence: 88%

Quantum KdV hierarchy and quasimodular forms

Ittersum¹,

Ruzza²

2022

Preprint

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“…The method of proof in this work (i.e. using quasi-Jacobi forms) allowed us to state Theorem 1.5 for many algebras F, whereas the results in [21] could not easily be generalized to other algebras than * . In Sect.…”

Section: When Is the Q-bracket Modular?mentioning

confidence: 99%

“…The functions h λ ∈ * defined in [21] form a basis for the space M( * ). The method of proof in this work (i.e.…”

Section: When Is the Q-bracket Modular?mentioning

confidence: 99%

“…(i) The function ϕ is a strictly meromorphic quasi-Jacobi form of weight k and index M for which the functions ϕ i,j determine its transformation behaviour as in ( 8) and ( 9). (ii) For all X = (λ, μ) ∈ M n,2 (Q) with (τ , λτ + μ) not a pole of ϕ for some τ ∈ h, the function g X 0 (ϕ) is a vector-valued quasimodular form satisfying (21) and transforming as…”

Section: Theorem 235mentioning

confidence: 99%

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The Bloch–Okounkov theorem for congruence subgroups and Taylor coefficients of quasi-Jacobi forms

Ittersum

2022

Res Math Sci

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There are many families of functions on partitions, such as the shifted symmetric functions, for which the corresponding q-brackets are quasimodular forms. We extend these families so that the corresponding q-brackets are quasimodular for a congruence subgroup. Moreover, we find subspaces of these families for which the q-bracket is a modular form. These results follow from the properties of Taylor coefficients of strictly meromorphic quasi-Jacobi forms around rational lattice points.

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“…In case n = 1 the result (i) is trivially true, so we assume n ≥ 2. By definition of the connected product and S k,f (see (23) and (18) respectively), we have…”

Section: Proof For the First Part We Let M Kmentioning

confidence: 99%

A symmetric Bloch–Okounkov theorem

Ittersum

2021

Res Math Sci

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The algebra of so-called shifted symmetric functions on partitions has the property that for all elements a certain generating series, called the q-bracket, is a quasimodular form. More generally, if a graded algebra A of functions on partitions has the property that the q-bracket of every element is a quasimodular form of the same weight, we call A a quasimodular algebra. We introduce a new quasimodular algebra $$\mathcal {T}$$ T consisting of symmetric polynomials in the part sizes and multiplicities.

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When is the Bloch–Okounkov q-bracket modular?

Cited by 3 publications

References 11 publications

Quantum KdV hierarchy and quasimodular forms

Quantum KdV hierarchy and quasimodular forms

The Bloch–Okounkov theorem for congruence subgroups and Taylor coefficients of quasi-Jacobi forms

A symmetric Bloch–Okounkov theorem

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