Appell-Lerch sums

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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

\[ \mu(u,v;\tau) = \frac{i a^{1/2}}{\theta_{11}(v;\tau)}\sum_{n\in Z}\frac{(-b)^nq^{n(n+1)/2}}{1-aq^n} \] where \[\displaystyle q= e^{2\pi i \tau},\quad a= e^{2\pi i u},\quad b= e^{2\pi i v}\] and \[\theta_{11}(v,\tau) = \sum_{n\in Z}(-1)^n b^{n+1/2}q^{(n+1/2)^2/2}\]

completion by adding a non-holomorphic part

  • definte \(\hat\mu(u,v;\tau)\) by

\[\hat\mu(u,v;\tau) = \mu(u,v;\tau)-R(u-v;\tau)/2\] where \[R(z;\tau) = \sum_{\nu\in \mathbb{Z}+1/2}(-1)^{\nu-1/2}[{\rm sign}(\nu)-E\left((\nu+\frac{\Im(z)}{y})\sqrt{2y}\right)]e^{-2\pi i \nu z}q^{-\nu^2/2}\] and \(y=\Im(\tau)\) and \[E(z) = 2\int_0^ze^{-\pi u^2}\,du\]


Mordell integral

\[ \mu(u,v;\tau)+\sqrt{\frac{i}{\tau}}e^{\pi i \frac{(u-v)^2}{\tau}}\mu(\frac{u}{\tau},\frac{v}{\tau};-\frac{1}{\tau})=\frac{1}{2}M(u-v;\tau) \] where \[M(v;\tau)=\int_{-\infty}^{\infty} \frac{e^{\pi i \tau x^2-2\pi v x}}{\cosh (\pi x)} dx\]

  • non-holomorpic part is an incomplete period integral of the modular form \(\eta^3\)

\[ iR(0;\tau)=\int_{-\bar{\tau}}^{i\infty}\frac{\eta(z)^3}{\sqrt{\frac{z+\tau}{i}}}\,dz \]

  • property

\[ R(z;\tau)+\sqrt{\frac{i}{\tau}}R(\frac{z}{\tau};-\frac{1}{\tau})=M(z;\tau) \]


modularity

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

  • 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.


special case

  • mock theta function

\[ \mu(z;\tau):= \mu(z,z;\tau)= \frac{i e^{\pi i z}}{\theta_{11}(z;\tau)}\sum_{n\in Z}\frac{(-1)^nq^{n(n+1)/2}e^{2\pi i n z}}{1-q^ne^{2\pi i z}} \]

  • Mordell integrals

\[ \mu(z;\tau)+\sqrt{\frac{i}{\tau}}\mu(\frac{z}{\tau};-\frac{1}{\tau})=\frac{1}{2}M(0;\tau)=\frac{1}{2}\int_{-\infty}^{\infty} \frac{e^{\pi i \tau x^2}}{\cosh (\pi x)} dx \]

  • completion

\[ \hat{\mu}(z;\tau)=\mu(z;\tau)-R(0;\tau)/2 \]

  • modularity

\[ \hat{\mu}(\frac{z}{\tau};\frac{-1}{\tau})=-\sqrt{\frac{\tau}{i}}\hat{\mu}(z;\tau) \]

higher level Appell function

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



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