scholarly journals Modular forms of virtually real-arithmetic type I: Mixed mock modular forms yield vector-valued modular forms

2021 ◽  
Vol 28 (2) ◽  
pp. 511-561
Author(s):  
Michael H. Mertens ◽  
Martin Raum
Keyword(s):  
Modular Forms ◽  
Type I ◽  
Mock Modular Forms ◽  
Mixed Mock Modular Forms ◽  
Vector Valued

scholarly journals Partition identities with mixed mock modular forms

2016 ◽  
Vol 158 ◽  
pp. 356-364 ◽  
Author(s):  
George E. Andrews ◽  
Stephen Hill
Keyword(s):  
Modular Forms ◽  
Partition Identities ◽  
Mock Modular Forms ◽  
Mixed Mock Modular Forms

scholarly journals Sums of class numbers and mixed mock modular forms

2018 ◽  
Vol 167 (02) ◽  
pp. 321-333 ◽  
Author(s):  
KATHRIN BRINGMANN ◽  
BEN KANE
Keyword(s):  
Modular Forms ◽  
Class Numbers ◽  
Explicit Formulas ◽  
Mock Modular Forms ◽  
Mixed Mock Modular Forms

AbstractIn this paper, we consider sums of class numbers of the type ∑m ≡ a (mod p) H (4n − m2), where p is an odd prime, n ∈ ℕ, and a ∈ ℤ. By showing that these are coefficients of mixed mock modular forms, we obtain explicit formulas. Using these formulas for p = 5 and 7, we then prove a conjecture of Brown et al. in the case that n = ℓ is prime.

Construction of vector-valued modular integrals and vector-valued mock modular forms

2014 ◽  
Vol 41 (1-3) ◽  
pp. 51-114 ◽  
Author(s):  
Jose Gimenez ◽  
Tobias Mühlenbruch ◽  
Wissam Raji
Keyword(s):  
Modular Forms ◽  
Mock Modular Forms ◽  
Vector Valued ◽  
Modular Integrals

scholarly journals Secord-order cusp forms and mixed mock modular forms

2012 ◽  
Vol 31 (1-2) ◽  
pp. 147-161 ◽  
Author(s):  
Kathrin Bringmann ◽  
Ben Kane
Keyword(s):  
Modular Forms ◽  
Cusp Forms ◽  
Mock Modular Forms ◽  
Mixed Mock Modular Forms

scholarly journals Higher depth mock theta functions and q-hypergeometric series

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Joshua Males ◽  
Andreas Mono ◽  
Larry Rolen
Keyword(s):  
Theta Function ◽  
Modular Forms ◽  
Hypergeometric Series ◽  
Theta Functions ◽  
Mock Theta Function ◽  
Mock Theta Functions ◽  
Maass Forms ◽  
Mock Modular Forms ◽  
Harmonic Maass Forms

Abstract In the theory of harmonic Maaß forms and mock modular forms, mock theta functions are distinguished examples which arose from q-hypergeometric examples of Ramanujan. Recently, there has been a body of work on higher depth mock modular forms. Here, we introduce distinguished examples of these forms, which we call higher depth mock theta functions, and develop q-hypergeometric expressions for them. We provide three examples of mock theta functions of depth two, each arising by multiplying a classical mock theta function with a certain specialization of a universal mock theta function. In addition, we give their modular completions, and relate each to a q-hypergeometric series.

scholarly journals Mocking the u-plane integral

2021 ◽  
Vol 8 (3) ◽  
Author(s):  
Georgios Korpas ◽  
Jan Manschot ◽  
Gregory W. Moore ◽  
Iurii Nidaiev
Keyword(s):  
Asymptotic Behavior ◽  
Gauge Theory ◽  
Entire Function ◽  
Gauge Group ◽  
Strong Coupling ◽  
Modular Forms ◽  
Correlation Functions ◽  
Coulomb Branch ◽  
Mock Modular Forms ◽  
Donaldson Invariants

AbstractThe u-plane integral is the contribution of the Coulomb branch to correlation functions of $${\mathcal {N}}=2$$ N = 2 gauge theory on a compact four-manifold. We consider the u-plane integral for correlators of point and surface observables of topologically twisted theories with gauge group $$\mathrm{SU}(2)$$ SU ( 2 ) , for an arbitrary four-manifold with $$(b_1,b_2^+)=(0,1)$$ ( b 1 , b 2 + ) = ( 0 , 1 ) . The u-plane contribution equals the full correlator in the absence of Seiberg–Witten contributions at strong coupling, and coincides with the mathematically defined Donaldson invariants in such cases. We demonstrate that the u-plane correlators are efficiently determined using mock modular forms for point observables, and Appell–Lerch sums for surface observables. We use these results to discuss the asymptotic behavior of correlators as function of the number of observables. Our findings suggest that the vev of exponentiated point and surface observables is an entire function of the fugacities.

scholarly journals Modular invariant models of leptons at level 7

2020 ◽  
Vol 2020 (8) ◽  
Author(s):  
Gui-Jun Ding ◽  
Stephen F. King ◽  
Cai-Chang Li ◽  
Ye-Ling Zhou
Keyword(s):  
Modular Forms ◽  
Galois Field ◽  
Special Linear Group ◽  
Type I ◽  
Time Level ◽  
Projective Special Linear Group ◽  
Modular Invariant ◽  
Seesaw Mechanism ◽  
Linearly Independent ◽  
First Time

Abstract We consider for the first time level 7 modular invariant flavour models where the lepton mixing originates from the breaking of modular symmetry and couplings responsible for lepton masses are modular forms. The latter are decomposed into irreducible multiplets of the finite modular group Γ7, which is isomorphic to PSL(2, Z7), the projective special linear group of two dimensional matrices over the finite Galois field of seven elements, containing 168 elements, sometimes written as PSL2(7) or Σ(168). At weight 2, there are 26 linearly independent modular forms, organised into a triplet, a septet and two octets of Γ7. A full list of modular forms up to weight 8 are provided. Assuming the absence of flavons, the simplest modular-invariant models based on Γ7 are constructed, in which neutrinos gain masses via either the Weinberg operator or the type-I seesaw mechanism, and their predictions compared to experiment.

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