Non-holomorphic Modular Forms Research Articles

A minimal modular invariant neutrino model

We present a neutrino mass model based on modular symmetry with the fewest input parameters to date, which successfully accounts for the 12 lepton masses and mixing parameters through 6 real free parameters including the modulus. The neutrino masses are predicted to be normal ordering, the atmospheric angle θ23 is quite close to maximal value and the Dirac CP phase δCP is about 1.34π. We also study the soft supersymmetry breaking terms due to the modulus F-term in this minimal model, which are constrained to be the non-holomorphic modular forms. The radiative lepton flavor violation process μ → eγ is discussed.

Modular graph forms from equivariant iterated Eisenstein integrals

The low-energy expansion of closed-string scattering amplitudes at genus one introduces infinite families of non-holomorphic modular forms called modular graph forms. Their differential and number-theoretic properties motivated Brown’s alternative construction of non-holomorphic modular forms in the recent mathematics literature from so-called equivariant iterated Eisenstein integrals. In this work, we provide the first validations beyond depth one of Brown’s conjecture that equivariant iterated Eisenstein integrals contain modular graph forms. Apart from a variety of examples at depth two and three, we spell out the systematics of the dictionary and make certain elements of Brown’s construction fully explicit to all orders.

Towards closed strings as single-valued open strings at genus one

We relate the low-energy expansions of world-sheet integrals in genus-one amplitudes of open- and closed-string states. The respective expansion coefficients are elliptic multiple zeta values (eMZVs) in the open-string case and non-holomorphic modular forms dubbed ‘modular graph forms (MGFs)’ for closed strings. By inspecting the differential equations and degeneration limits of suitable generating series of genus-one integrals, we identify formal substitution rules mapping the eMZVs of open strings to the MGFs of closed strings. Based on the properties of these rules, we refer to them as an elliptic single-valued map which generalizes the genus-zero notion of a single-valued map acting on MZVs seen in tree-level relations between the open and closed string.

Basis decompositions and a Mathematica package for modular graph forms

Modular graph forms (MGFs) are a class of non-holomorphic modular forms which naturally appear in the low-energy expansion of closed-string genus-one amplitudes and have generated considerable interest from pure mathematicians. MGFs satisfy numerous non-trivial algebraic- and differential relations which have been studied extensively in the literature and lead to significant simplifications. In this paper, we systematically combine these relations to obtain basis decompositions of all two- and three-point MGFs of total modular weight , starting from just two well-known identities for banana graphs. Furthermore, we study previously known relations in the integral representation of MGFs, leading to a new understanding of holomorphic subgraph reduction as Fay identities of Kronecker–Eisenstein series and opening the door toward decomposing divergent graphs. We provide a computer implementation for the manipulation of MGFs in the form of the Mathematica package ModularGraphForms which includes the basis decompositions obtained.

Maximal U(1)Y-violating n-point correlators in mathcal{N} = 4 super-Yang-Mills theory

This paper concerns a special class of n-point correlation functions of operators in the stress tensor supermultiplet of mathcal{N} = 4 supersymmetric SU(N) Yang-Mills theory. These are “maximal U(1)Y-violating” correlators that violate the bonus U(1)Y charge by a maximum of 2(n − 4) units. We will demonstrate that such correlators satisfy SL(2, ℤ)-covariant recursion relations that relate n-point correlators to (n − 1)-point correlators in a manner analogous to the soft dilaton relations that relate the corresponding amplitudes in flat-space type IIB superstring theory. These recursion relations are used to determine terms in the large-N expansion of n-point maximal U(1)Y-violating correlators in the chiral sector, including correlators with four superconformal stress tensor primaries and (n − 4) chiral Lagrangian operators, starting from known properties of the n = 4 case. We concentrate on the first three orders in 1/N beyond the supergravity limit. The Mellin representations of the correlators are polynomials in Mellin variables, which correspond to higher derivative contact terms in the low-energy expansion of type IIB superstring theory in AdS5× S5 at the same orders as R4, d4R4 and d6R4. The coupling constant dependence of these terms is found to be described by non-holomorphic modular forms with holomorphic and anti-holomorphic weights (n − 4, 4 − n) that are SL(2, ℤ)-covariant derivatives of Eisenstein series and certain generalisations. This determines a number of non-leading contributions to U(1)Y-violating n-particle interactions (n > 4) in the low-energy expansion of type IIB superstring amplitudes in AdS5× S5.

Single-valued integration and double copy

Abstract In this paper, we study a single-valued integration pairing between differential forms and dual differential forms which subsumes some classical constructions in mathematics and physics. It can be interpreted as a p-adic period pairing at the infinite prime. The single-valued integration pairing is defined by transporting the action of complex conjugation from singular to de Rham cohomology via the comparison isomorphism. We show how quite general families of period integrals admit canonical single-valued versions and prove some general formulae for them. This implies an elementary “double copy” formula expressing certain singular volume integrals over the complex points of a smooth projective variety as a quadratic expression in ordinary period integrals of half the dimension. We provide several examples, including non-holomorphic modular forms, archimedean Néron–Tate heights on curves, single-valued multiple zeta values and polylogarithms. The results of the present paper are used in [F. Brown and C. Dupont, Single-valued integration and superstring amplitudes in genus zero, preprint 2019, https://arxiv.org/abs/1910.01107] to prove a recent conjecture of Stieberger which relates the coefficients in a Laurent expansion of two different kinds of periods of twisted cohomology on the moduli spaces of curves ℳ 0 , n {\mathcal{M}_{0,n}} of genus zero with n marked points. We also study a morphism between certain rings of “motivic” periods, called the de Rham projection, which provides a bridge between complex periods and single-valued periods in many situations of interest.

On the Modular Completion of Certain Generating Functions

Abstract We complete several generating functions to non-holomorphic modular forms in two variables. For instance, we consider the generating function of a natural family of meromorphic modular forms of weight two. We then show that this generating series can be completed to a smooth, non-holomorphic modular form of weights $\frac 32$ and two. Moreover, it turns out that the same function is also a modular completion of the generating function of weakly holomorphic modular forms of weight $\frac 32$, which prominently appear in work of Zagier [27] on traces of singular moduli.

All-order differential equations for one-loop closed-string integrals and modular graph forms

We investigate generating functions for the integrals over world-sheet tori appearing in closed-string one-loop amplitudes of bosonic, heterotic and type-II theories. These closed-string integrals are shown to obey homogeneous and linear differential equations in the modular parameter of the torus. We spell out the first-order Cauchy-Riemann and second-order Laplace equations for the generating functions for any number of external states. The low-energy expansion of such torus integrals introduces infinite families of non-holomorphic modular forms known as modular graph forms. Our results generate homogeneous first- and second-order differential equations for arbitrary such modular graph forms and can be viewed as a step towards all-order low-energy expansions of closed-string integrals.

Asymptotics and Ramanujan's mock theta functions: then and now.

This article is in commemoration of Ramanujan's election as Fellow of The Royal Society 100 years ago, as celebrated at the October 2018 scientific meeting at the Royal Society in London. Ramanujan's last letter to Hardy, written shortly after his election, surrounds his mock theta functions. While these functions have been of great importance and interest in the decades following Ramanujan's death in 1920, it was unclear how exactly they fit into the theory of modular forms-Dyson called this 'a challenge for the future' at another centenary conference in Illinois in 1987, honouring the 100th anniversary of Ramanujan's birth. In the early 2000s, Zwegers finally recognized that Ramanujan had discovered glimpses of special families of non-holomorphic modular forms, which we now know to be Bruinier and Funke's harmonic Maass forms from 2004, the holomorphic parts of which are called mock modular forms. As of a few years ago, a fundamental question from Ramanujan's last letter remained, on a certain asymptotic relationship between mock theta functions and ordinary modular forms. The author, with Ono and Rhoades, revisited Ramanujan's asymptotic claim, and established a connection between mock theta functions and quantum modular forms, which were not defined until 90 years later in 2010 by Zagier. Here, we bring together past and present, and study the relationships between mock modular forms and quantum modular forms, with Ramanujan's mock theta functions as motivation. In particular, we highlight recent work of Bringmann-Rolen, Choi-Lim-Rhoades and Griffin-Ono-Rolen in our discussion. This article is largely expository, but not exclusively: we also establish a new interpretation of Ramanujan's radial asymptotic limits in the subject of topology. This article is part of a discussion meeting issue 'Srinivasa Ramanujan: in celebration of the centenary of his election as FRS'.

Identities of cycle integrals of weak Maass forms

We prove identities between cycle integrals of non-holomorphic modular forms arising from applications of various differential operators to weak Maass forms.

Modular forms and SL(2, \u2124)-covariance of type IIB superstring theory

The local higher-derivative interactions that enter into the low-energy expansion of the effective action of type IIB superstring theory with constant complex modulus generally violate the U(1) R-symmetry of IIB supergravity by qU units. These interactions have coefficients that transform as non-holomorphic modular forms under SL(2, ℤ) transformations with holomorphic and anti-holomorphic weights (w, −w), where qU = −2w.In this paper SL(2, ℤ)-covariance and supersymmetry are used to determine first-order differential equations on moduli space that relate the modular form coefficients of classes of BPS-protected maximal U(1)-violating interactions that arise at low orders in the lowenergy expansion. These are the moduli-dependent coefficients of BPS interactions of the form d2p mathcal{P} n in linearised approximation, where mathcal{P} n is the product of n fields that has dimension = 8 with qU = 8 − 2n, and p = 0, 2 or 3. These first-order equations imply that the coefficients satisfy SL(2, ℤ)-covariant Laplace eigenvalue equations on moduli space with solutions that contain information concerning perturbative and non-perturbative contributions to superstring amplitudes. For p = 3 and n ≥ 6 there are two independent modular forms, one of which has a vanishing tree-level contribution.The analysis of super-amplitudes for U(1)-violating processes involving arbitrary numbers of external fluctuations of the complex modulus leads to a diagrammatic derivation of the first-order differential relations and Laplace equations satisfied by the coefficient modular forms. Combining this with a SL(2, ℤ)-covariant soft axio-dilaton limit that relates amplitudes with different values of n determines most of the modular invariant coefficients, leaving a single undetermined constant.

Heterotic-string amplitudes at one loop: modular graph forms and relations to open strings

We investigate one-loop four-point scattering of non-abelian gauge bosons in heterotic string theory and identify new connections with the corresponding open-string amplitude. In the low-energy expansion of the heterotic-string amplitude, the integrals over torus punctures are systematically evaluated in terms of modular graph forms, certain non-holomorphic modular forms. For a specific torus integral, the modular graph forms in the low-energy expansion are related to the elliptic multiple zeta values from the analogous open-string integrations over cylinder boundaries. The detailed correspondence between these modular graph forms and elliptic multiple zeta values supports a recent proposal for an elliptic generalization of the single-valued map at genus zero.

On two arithmetic theta lifts

Our aim is to clarify the relationship between Kudla’s and Bruinier’s Green functions attached to special cycles on Shimura varieties of orthogonal and unitary type, which play a key role in the arithmetic geometry of these cycles in the context of Kudla’s program. In particular, we show that the generating series obtained by taking the differences of the two families of Green functions is a non-holomorphic modular form and has trivial (cuspidal) holomorphic projection. Along the way, we construct a section of the Maaß lowering operator for moderate growth forms valued in the Weil representation using a regularized theta lift, which may be of independent interest, as it in particular has applications to mock modular forms. We also consider arithmetic-geometric applications to integral models of $U(n,1)$ Shimura varieties. Each family of Green functions gives rise to a formal arithmetic theta function, valued in an arithmetic Chow group, that is conjectured to be modular; our main result is the modularity of the difference of the two arithmetic theta functions. Finally, we relate the arithmetic heights of the special cycles to special derivatives of Eisenstein series, as predicted by Kudla’s conjecture, and describe a refinement of a theorem of Bruinier, Howard and Yang on arithmetic intersections against CM points.

A class of non-holomorphic modular forms III: real analytic cusp forms for mathrm {SL}_2(mathbb {Z})

We define canonical real analytic versions of modular forms of integral weight for the full modular group, generalising real analytic Eisenstein series. They are harmonic Maass waveforms with poles at the cusp, whose Fourier coefficients involve periods and quasi-periods of cusp forms, which are conjecturally transcendental. In particular, we settle the question of finding explicit ‘weak harmonic lifts’ for every eigenform of integral weight k and level one. We show that mock modular forms of integral weight are algebro-geometric and have Fourier coefficients proportional to n^{1-k}(a_n^{prime } + rho a_n) for nne 0, where rho is the normalised permanent of the period matrix of the corresponding motive, and a_n, a_n^{prime } are the Fourier coefficients of a Hecke eigenform and a weakly holomorphic Hecke eigenform, respectively. More generally, this framework provides a conceptual explanation for the algebraicity of the coefficients of mock modular forms in the CM case.

A class of non-holomorphic modular forms I

This introductory paper studies a class of real analytic functions on the upper half plane satisfying a certain modular transformation property. They are not eigenfunctions of the Laplacian and are quite distinct from Maass forms. These functions are modular equivariant versions of real and imaginary parts of iterated integrals of holomorphic modular forms and are modular analogues of single-valued polylogarithms. The coefficients of these functions in a suitable power series expansion are periods. They are related both to mixed motives (iterated extensions of pure motives of classical modular forms) and to the modular graph functions arising in genus one string perturbation theory.

Modular forms and a generalized Cardy formula in higher dimensions

We derive a formula which applies to conformal field theories on a spatial torus and gives the asymptotic density of states solely in terms of the vacuum energy on a parallel plate geometry. The formula follows immediately from global scale and Lorentz invariance, but to our knowledge has not previously been made explicit. It can also be understood from the fact that $\log Z$ on $\mathbb{T}^2\times \mathbb{R}^{d-1}$ transforms as the absolute value of a non-holomorphic modular form of weight $d-1$, which we show. The results are extended to theories which violate Lorentz invariance and hyperscaling but maintain a scaling symmetry. The formula is checked for the cases of a free scalar, free Maxwell gauge field, and free $\mathcal{N}=4$ super Yang-Mills. The case of a Maxwell gauge field gives Casimir's original calculation of the electromagnetic force between parallel plates in terms of the entropy of a photon gas.

Euler-like recurrences for smallest parts functions

We obtain recurrences for smallest parts functions which resemble Euler’s recurrence for the ordinary partition function. The proofs involve the holomorphic projection of non-holomorphic modular forms of weight \(2\).

Aspects of the zeta function originating from pseudodifferential analysis

Pseudodifferential analysis helps linking the one-dimensional and two-dimensional analyses in more than one way. Taking benefit from special features of the two-dimensional case, in particular the fact that homogeneous distributions, invariant under the action by linear changes of coordinates of \(SL(2,{\mathbb Z})\), are essentially disguised versions of nonholomorphic modular forms, we are led to introducing interesting two-dimensional distributions and some associated one-dimensional parts. This results in a collection of necessary and sufficient conditions for the Riemann hypothesis to hold, some of which, but not all, are of a more or less classical type.

Comments on non-holomorphic modular forms and non-compact superconformal field theories

We extend our previous work arXiv:1012.5721 [hep-th] on the non-compact N=2 SCFT_2 defined as the supersymmetric SL(2,R)/U(1)-gauged WZW model. Starting from path-integral calculations of torus partition functions of both the axial-type (`cigar') and the vector-type (`trumpet') models, we study general models of the Z_M-orbifolds and M-fold covers with an arbitrary integer M. We then extract contributions of the degenerate representations (`discrete characters') in such a way that good modular properties are preserved. The `modular completion' of the extended discrete characters introduced in arXiv:1012.5721 [hep-th] are found to play a central role as suitable building blocks in every model of orbifolds or covering spaces. We further examine a large M-limit (the `continuum limit'), which `deconstructs' the spectral flow orbits while keeping a suitable modular behavior. The discrete part of partition function as well as the elliptic genus is then expanded by the modular completions of irreducible discrete characters, which are parameterized by both continuous and discrete quantum numbers modular transformed in a mixed way. This limit is naturally identified with the universal cover of trumpet model. We finally discuss a classification of general modular invariants based on the modular completions of irreducible characters constructed above.

Behavior of theta-series under modular substitutions

In this article we provide a transformation formula of certain theta series, and apply it to obtain non-holomorphic vector-valued modular forms.