Appell-Lerch sums

introduction

• one way to construct mock theta functions
• characters of representations in (nonrational) conformal field theory models based on Lie superalgebras\
• 3rd order mock theta functions

Appell-Lerch sum

Appell–Lerch sums were first studied by Paul Émile Appell (1884) and Mathias Lerch (1892). Watson studied the order 3 mock theta functions by expressing them in terms of Appell–Lerch sums, and Zwegers used them to show that mock theta functions are essentially mock modular forms.

The Appell–Lerch series is

\(\mu(u,v;\tau) = \frac{a^{1/2}}{\theta(v;\tau)}\sum_{n\in Z}\frac{(-b)^nq^{n(n+1)/2}}{1-aq^n}\)

where

\(\displaystyle q= e^{2\pi i \tau},\quad a= e^{2\pi i u},\quad b= e^{2\pi i v}\)

and

\(\theta(v,\tau) = \sum_{n\in Z}(-1)^n b^{n+1/2}q^{(n+1/2)^2/2}\)

The modified series

\[\hat\mu(u,v;\tau) = \mu(u,v;\tau)-R(u-v;\tau)/2\]

where

\[R(z;\tau) = \sum_{\nu\in Z+1/2}(-1)^{\nu-1/2}({\rm sign}(\nu)-E((\nu+\Im(z)/y)\sqrt{2y}))e^{-2\pi i \nu z}q^{-\nu^2/2}\]

and y = Im(τ) and

\[E(z) = 2\int_0^ze^{-\pi u^2}\,du\]

satisfies the following transformation properties

\[\hat\mu(u+1,v;\tau) = a^{-1}bq^{-1/2}\hat\mu(u+\tau,v;\tau) = -\hat\mu(u,v;\tau),\]

\[e^{2\pi i/8}\hat\mu(u,v;\tau+1) = \hat\mu(u,v;\tau) = -(\tau/i)^{-1/2}e^{\pi i (u-v)^2/\tau}\hat\mu(u/\tau,v/\tau;-1/\tau).\]

In other words the modified Appell–Lerch series transforms like a modular form with respect to τ. Since mock theta functions can be expressed in terms of Appell–Lerch series this means that mock theta functions transform like modular forms if they have a certain non-analytic series added to them.

higher level Appell function

• higher-level Appell functions
  
  • a particular instance of indefinite theta series

history

• http://www.google.com/search?hl=en&tbs=tl:1&q=

related items

• Kac-Wakimoto modules
• indefinite theta functions

encyclopedia

• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/Appell
• http://en.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/
• Princeton companion to mathematics(Companion_to_Mathematics.pdf)

question and answers(Math Overflow)

• http://mathoverflow.net/search?q=
• http://mathoverflow.net/search?q=
• http://mathoverflow.net/search?q=

articles

• Superconformal Algebras and Mock Theta Functions Tohru Eguchi
  
  • Kazuhiro Hikami, 2009
• Some characters of Kac and Wakimoto and nonholomorphic modular functions.
  
  • K. Bringmann and K. Ono, Math. Annalen 345, pages 547-558 (2009)
• Appell-Lerch sums as mock modular forms
  
  • Zwegers, 2008
• Higher String Functions, Higher-Level Appell Functions, and the Logarithmic sℓ︿2k/u(1)  CFT Model
  
  • A. M. Semikhatov
• Higher-Level Appell Functions, Modular Transformations, and Characters
  
  • A.M. Semikhatov

• Mock Theta Functions
  
  • Sander Zwegers, 2002
• Integrable highest weight modules over affine superalgebras and Appell’s function
  
  • Kac V.G., Wakimoto M, Commun. Math. Phys. 215(3), 631–682 (2001)
    
• N = 2 superconformal minimal models
  
• Character Formula of C<1 Unitary representation of N=2 Superconformal Algebra
  
  • 1987
    
• On a Function Which Occurs in the Theory of the Structure of Polymers
  
  • C. Truesdell, The Annals of Mathematics, Second Series, Vol. 46, No. 1 (Jan., 1945), pp. 144-157
    

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• http://dx.doi.org/10.1007/s00220-004-1280-7