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
• 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 $y=\Im(\tau)$ 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
• Mathieu moonshine

encyclopedia

• http://en.wikipedia.org/wiki/Appell

articles

• Tohru Eguchi and Kazuhiro Hikami Superconformal Algebras and Mock Theta Functions, 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
• Yutaka Matsuo Character Formula of C<1 Unitary representation of N=2 Superconformal Algebra , Prog. Theor. Phys. Vol. 77 No. 4 (1987) pp. 793-797
• C. Truesdell On a Function Which Occurs in the Theory of the Structure of Polymers, The Annals of Mathematics, Second Series, Vol. 46, No. 1 (Jan., 1945), pp. 144-157