# Indefinite theta series and generalized error functions

@article{Alexandrov2018IndefiniteTS,
title={Indefinite theta series and generalized error functions},
author={Sergei Yu. Alexandrov and Sibasish Banerjee and J. F. M. Manschot and Boris Pioline},
journal={Selecta Mathematica},
year={2018},
volume={24},
pages={3927-3972}
}
• Published 17 June 2016
• Mathematics
• Selecta Mathematica
Theta series for lattices with indefinite signature $$(n_+,n_-)$$(n+,n-) arise in many areas of mathematics including representation theory and enumerative algebraic geometry. Their modular properties are well understood in the Lorentzian case ($$n_+=1$$n+=1), but have remained obscure when $$n_+\ge 2$$n+≥2. Using a higher-dimensional generalization of the usual (complementary) error function, discovered in an independent physics project, we construct the modular completion of a class of…
$r$-Tuple Error Functions and Indefinite Theta Series of Higher-Depth
This work constructs an indefinite theta series for signature $(r,n-r)$ lattices and shows they can be completed to modular forms by using these $r-tuple error functions. Indefinite zeta functions We define generalised zeta functions associated with indefinite quadratic forms of signature $$(g-1,1)$$ ( g - 1 , 1 ) —and more generally, to complex symmetric matrices whose imaginary part has Theta integrals and generalized error functions Recently Alexandrov, Banerjee, Manschot and Pioline constructed generalizations of Zwegers theta functions for lattices of signature ($$n-2,2$$n-2,2). Their functions, which depend on two pairs of On some incomplete theta integrals • Mathematics Compositio Mathematica • 2019 In this paper we construct indefinite theta series for lattices of arbitrary signature$(p,q)$as ‘incomplete’ theta integrals, that is, by integrating the theta forms constructed by the second Refinement and modularity of immortal dyons • Mathematics Journal of High Energy Physics • 2021 Extending recent results in N $$\mathcal{N}$$ = 2 string compactifications, we propose that the holomorphic anomaly equation satisfied by the modular completions of the generating functions of An exact formula for$\mathbf {U (3)}$Vafa-Witten invariants on$\mathbb {P}^2$• Mathematics Transactions of the American Mathematical Society • 2019 Topologically twisted$\mathcal{N} = 4$super Yang-Mills theory has a partition function that counts Euler numbers of instanton moduli spaces. On the manifold$\mathbb{P}^2$and with gauge group Higher depth quantum modular forms, multiple Eichler integrals, and $$\mathfrak {sl}_3$$sl3 false theta functions • Mathematics Research in the Mathematical Sciences • 2019 We introduce and study higher depth quantum modular forms. We construct two families of examples coming from rank two false theta functions, whose “companions” in the lower half-plane can be also Rank$N$Vafa–Witten invariants, modularity and blow-up • S. Alexandrov • Mathematics Advances in Theoretical and Mathematical Physics • 2021 We derive explicit expressions for the generating functions of refined Vafa-Witten invariants$\Omega(\gamma,y)$of$\mathbb{P}^2$of arbitrary rank$N$and for their non-holomorphic modular Vafa–Witten Theory and Iterated Integrals of Modular Forms • J. Manschot • Mathematics Communications in Mathematical Physics • 2019 Vafa–Witten (VW) theory is a topologically twisted version of $${\mathcal{N}=4}$$N=4 supersymmetric Yang–Mills theory. S-duality suggests that the partition function of VW theory with gauge group Theta integrals and generalized error functions, II • Mathematics • 2017 The theory of theta series attached to integral lattices L in rational quadratic spaces L⊗Z Q with bilinear form ( , ) of signature (p, q), pq > 0, has a long history including fundamental work of ## References SHOWING 1-10 OF 66 REFERENCES $${\mathrm {H}}$$H-Harmonic Maaß-Jacobi forms of degree 1 It was shown in previous work that the one-variable $$\widehat{\mu }$$μ^- function defined by Zwegers (and Zagier) and his indefinite theta series attached to lattices of signature$r$-Tuple Error Functions and Indefinite Theta Series of Higher-Depth This work constructs an indefinite theta series for signature$(r,n-r)$lattices and shows they can be completed to modular forms by using these$r-tuple error functions.
Theta integrals and generalized error functions
Recently Alexandrov, Banerjee, Manschot and Pioline constructed generalizations of Zwegers theta functions for lattices of signature ($$n-2,2$$n-2,2). Their functions, which depend on two pairs of
Sheaves on P2 and generalised Appell functions
A closed expression is given for the generating function of (virtual) Poincar\'e polynomials of moduli spaces of semi-stable sheaves on the projective plane $\mathbb{P}^2$ with arbitrary rank $r$ and
On some incomplete theta integrals
• Mathematics
Compositio Mathematica
• 2019
In this paper we construct indefinite theta series for lattices of arbitrary signature $(p,q)$ as ‘incomplete’ theta integrals, that is, by integrating the theta forms constructed by the second
H-Harmonic Maass-Jacobi Forms of Degree 1: The Analytic Theory of Some Indefinite Theta Series
It was shown in previous work that the one-variable $\widehat\mu$-function defined by Zwegers (and Zagier) and his indefinite theta series attached to lattices of signature $(r\!+\!1,1)$ are both
Vafa–Witten Theory and Iterated Integrals of Modular Forms
• J. Manschot
• Mathematics
Communications in Mathematical Physics
• 2019
Vafa–Witten (VW) theory is a topologically twisted version of $${\mathcal{N}=4}$$N=4 supersymmetric Yang–Mills theory. S-duality suggests that the partition function of VW theory with gauge group
REPRESENTATIONS OF AFFINE SUPERALGEBRAS AND MOCK THETA FUNCTIONS
• Mathematics
• 2013
We show that the normalized supercharacters of principal admissible modules over the affine Lie superalgebra $${{\widehat{{s\ell}}}_{2|1 }}$$ (resp. $${{\widehat{{ps\ell}}}_{2|2 }}$$) can be
The Betti Numbers of the Moduli Space of Stable Sheaves of Rank 3 on $${\mathbb{P}^2}$$
This article computes the generating functions of the Betti numbers of the moduli space of stable sheaves of rank 3 on $${\mathbb{P}^2}$$ and its blow-up $${\tilde{\mathbb{P}}^2}$$. Wall-crossing is
Some cubic modular identities of Ramanujan
• Mathematics
• 1994
There is a beautiful cubic analogue of Jacobi's fundamental theta function identity: θ⁴₃ = θ⁴₄ + θ⁴₂. It is \$(\sum_{n,m=-\infty}^{\infty} q^{n^2+nm+m^2})³ = (\sum_{n,m=-\infty}^{\infty}