scholarly journals Infinite-dimensional Lie algebras, theta functions and modular forms

1984 ◽  
Vol 53 (2) ◽  
pp. 125-264 ◽  
Author(s):  
Victor G Kač ◽  
Dale H Peterson
Keyword(s):  
Lie Algebras ◽  
Modular Forms ◽  
Theta Functions ◽  
Infinite Dimensional

scholarly journals Quantum black holes, elliptic genera and spectral partition functions

2014 ◽  
Vol 11 (05) ◽  
pp. 1450048 ◽  
Author(s):  
A. A. Bytsenko ◽  
M. Chaichian ◽  
R. J. Szabo ◽  
A. Tureanu
Keyword(s):  
Black Hole ◽  
Lie Algebras ◽  
Modular Forms ◽  
Partition Functions ◽  
Affine Lie Algebras ◽  
Hilbert Schemes ◽  
Spectral Functions ◽  
Infinite Dimensional ◽  
Elliptic Genera ◽  
Bps Invariants

We study M-theory and D-brane quantum partition functions for microscopic black hole ensembles within the context of the AdS/CFT correspondence in terms of highest weight representations of infinite-dimensional Lie algebras, elliptic genera, and Hilbert schemes, and describe their relations to elliptic modular forms. The common feature in our examples lies in the modular properties of the characters of certain representations of the pertinent affine Lie algebras, and in the role of spectral functions of hyperbolic three-geometry associated with q-series in the calculation of elliptic genera. We present new calculations of supergravity elliptic genera on local Calabi–Yau threefolds in terms of BPS invariants and spectral functions, and also of equivariant D-brane elliptic genera on generic toric singularities. We use these examples to conjecture a link between the black hole partition functions and elliptic cohomology.

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 On complete reducibility for infinite-dimensional Lie algebras

2011 ◽  
Vol 226 (2) ◽  
pp. 1911-1972 ◽  
Author(s):  
Maria Gorelik ◽  
Victor Kac
Keyword(s):  
Lie Algebras ◽  
Infinite Dimensional ◽  
Complete Reducibility

scholarly journals Partial theta functions and mock modular forms as q-hypergeometric series

2012 ◽  
Vol 29 (1-3) ◽  
pp. 295-310 ◽  
Author(s):  
Kathrin Bringmann ◽  
Amanda Folsom ◽  
Robert C. Rhoades
Keyword(s):  
Modular Forms ◽  
Hypergeometric Series ◽  
Theta Functions ◽  
Mock Modular Forms ◽  
Partial Theta Functions

A note on the algebra of Poisson brackets

1984 ◽  
Vol 96 (1) ◽  
pp. 45-60 ◽  
Author(s):  
C. J. Atkin
Keyword(s):  
Poisson Bracket ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Symplectic Manifold ◽  
Vector Fields ◽  
Poisson Brackets ◽  
Infinite Dimensional

In a long sequence of notes in the Comptes Rendus and elsewhere, and in the papers [1], [2], [3], [6], [7], Lichnerowicz and his collaborators have studied the ‘classical infinite-dimensional Lie algebras’, their derivations, automorphisms, co-homology, and other properties. The most familiar of these algebras is the Lie algebra of C∞ vector fields on a C∞ manifold. Another is the Lie algebra of ‘Poisson brackets’, that is, of C∞ functions on a C∞ symplectic manifold, with the Poisson bracket as composition; some questions concerning this algebra are of considerable interest in the theory of quantization – see, for instance, [2] and [3].

scholarly journals Boundary behaviour of special cohomology classes arising from the Weil representation

2012 ◽  
Vol 12 (3) ◽  
pp. 571-634 ◽  
Author(s):  
Jens Funke ◽  
John Millson
Keyword(s):  
Modular Forms ◽  
Symmetric Spaces ◽  
Theta Functions ◽  
Theta Series ◽  
Siegel Modular Forms ◽  
Boundary Behaviour ◽  
Orthogonal Groups ◽  
Local Coefficients ◽  
Generating Series ◽  
Vector Valued

AbstractIn our previous paper [J. Funke and J. Millson, Cycles with local coefficients for orthogonal groups and vector-valued Siegel modular forms, American J. Math. 128 (2006), 899–948], we established a correspondence between vector-valued holomorphic Siegel modular forms and cohomology with local coefficients for local symmetric spaces $X$ attached to real orthogonal groups of type $(p, q)$. This correspondence is realized using theta functions associated with explicitly constructed ‘special’ Schwartz forms. Furthermore, the theta functions give rise to generating series of certain ‘special cycles’ in $X$ with coefficients.In this paper, we study the boundary behaviour of these theta functions in the non-compact case and show that the theta functions extend to the Borel–Sere compactification $ \overline{X} $ of $X$. However, for the $ \mathbb{Q} $-split case for signature $(p, p)$, we have to construct and consider a slightly larger compactification, the ‘big’ Borel–Serre compactification. The restriction to each face of $ \overline{X} $ is again a theta series as in [J. Funke and J. Millson, loc. cit.], now for a smaller orthogonal group and a larger coefficient system.As an application we establish in certain cases the cohomological non-vanishing of the special (co)cycles when passing to an appropriate finite cover of $X$. In particular, the (co)homology groups in question do not vanish. We deduce as a consequence a sharp non-vanishing theorem for ${L}^{2} $-cohomology.

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