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Indefinite theta series and generalized error functions

Theta series for lattices with indefinite signature $(n_+,n_-)$ arise in many areas of mathematics including representation theory and enumerative algebraic geometry. Their modular properties are well understood in the Lorentzian case ($n_+=1$), but have remained obscure when $n_+\geq 2$. Using a hi...

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Autores principales: Alexandrov, Sergei, Banerjee, Sibasish, Manschot, Jan, Pioline, Boris
Lenguaje:eng
Publicado: 2016
Materias:
Acceso en línea:https://dx.doi.org/10.1007/s00029-018-0444-9
http://cds.cern.ch/record/2162118
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author Alexandrov, Sergei
Banerjee, Sibasish
Manschot, Jan
Pioline, Boris
author_facet Alexandrov, Sergei
Banerjee, Sibasish
Manschot, Jan
Pioline, Boris
author_sort Alexandrov, Sergei
collection CERN
description Theta series for lattices with indefinite signature $(n_+,n_-)$ arise in many areas of mathematics including representation theory and enumerative algebraic geometry. Their modular properties are well understood in the Lorentzian case ($n_+=1$), but have remained obscure when $n_+\geq 2$. 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 `conformal' holomorphic theta series ($n_+=2$). As an application, we determine the modular properties of a generalized Appell-Lerch sum attached to the lattice ${\operatorname A}_2$, which arose in the study of rank 3 vector bundles on $\mathbb{P}^2$. The extension of our method to $n_+>2$ is outlined.
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spelling cern-21621182022-07-08T07:00:12Zdoi:10.1007/s00029-018-0444-9http://cds.cern.ch/record/2162118engAlexandrov, SergeiBanerjee, SibasishManschot, JanPioline, BorisIndefinite theta series and generalized error functionsMathematical Physics and Mathematicsmath.AGMathematical Physics and Mathematicshep-thParticle Physics - Theorymath.NTTheta series for lattices with indefinite signature $(n_+,n_-)$ arise in many areas of mathematics including representation theory and enumerative algebraic geometry. Their modular properties are well understood in the Lorentzian case ($n_+=1$), but have remained obscure when $n_+\geq 2$. 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 `conformal' holomorphic theta series ($n_+=2$). As an application, we determine the modular properties of a generalized Appell-Lerch sum attached to the lattice ${\operatorname A}_2$, which arose in the study of rank 3 vector bundles on $\mathbb{P}^2$. The extension of our method to $n_+>2$ is outlined.Theta series for lattices with indefinite signature $(n_+,n_-)$ arise in many areas of mathematics including representation theory and enumerative algebraic geometry. Their modular properties are well understood in the Lorentzian case ($n_+=1$), but have remained obscure when $n_+\geq 2$. 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 `conformal' holomorphic theta series ($n_+=2$). As an application, we determine the modular properties of a generalized Appell-Lerch sum attached to the lattice $A_2$, which arose in the study of rank 3 vector bundles on $\mathbb{P}^2$. The extension of our method to $n_+>2$ is outlined.arXiv:1606.05495L2C:16-078IPHT-T16/058TCDMATH 16-09CERN-TH-2016-142IPHT-T16-058TCDMATH-16-09L2C:16-078IPHT-T16-058CERN-TH-2016-142oai:cds.cern.ch:21621182016-06-17
spellingShingle Mathematical Physics and Mathematics
math.AG
Mathematical Physics and Mathematics
hep-th
Particle Physics - Theory
math.NT
Alexandrov, Sergei
Banerjee, Sibasish
Manschot, Jan
Pioline, Boris
Indefinite theta series and generalized error functions
title Indefinite theta series and generalized error functions
title_full Indefinite theta series and generalized error functions
title_fullStr Indefinite theta series and generalized error functions
title_full_unstemmed Indefinite theta series and generalized error functions
title_short Indefinite theta series and generalized error functions
title_sort indefinite theta series and generalized error functions
topic Mathematical Physics and Mathematics
math.AG
Mathematical Physics and Mathematics
hep-th
Particle Physics - Theory
math.NT
url https://dx.doi.org/10.1007/s00029-018-0444-9
http://cds.cern.ch/record/2162118
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AT banerjeesibasish indefinitethetaseriesandgeneralizederrorfunctions
AT manschotjan indefinitethetaseriesandgeneralizederrorfunctions
AT piolineboris indefinitethetaseriesandgeneralizederrorfunctions