"Appell-Lerch sums"의 두 판 사이의 차이

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==introduction==
  
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* one way to construct mock theta functions
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* characters of representations in (nonrational) conformal field theory models based on Lie superalgebras\
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* [[3rd order mock theta functions]]
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==Appell-Lerch sum==
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* Appell–Lerch sums were first studied by [http://en.wikipedia.org/wiki/Paul_%C3%89mile_Appell Paul Émile Appell] ([http://en.wikipedia.org/wiki/Mock_theta_function#CITEREFAppell1884 1884]) and [http://en.wikipedia.org/wiki/Mathias_Lerch Mathias Lerch] ([http://en.wikipedia.org/wiki/Mock_theta_function#CITEREFLerch1892 1892]).
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* Watson studied the order 3 mock theta functions by expressing them in terms of Appell–Lerch sums
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* Zwegers used them to show that mock theta functions are essentially mock modular forms.
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* The Appell–Lerch series is
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:<math>
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\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}
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</math>
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where
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:<math>\displaystyle q= e^{2\pi i \tau},\quad a= e^{2\pi i u},\quad b= e^{2\pi i v}</math>
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and
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:<math>\theta_{11}(v,\tau) = \sum_{n\in Z}(-1)^n b^{n+1/2}q^{(n+1/2)^2/2}</math>
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===completion by adding a non-holomorphic part===
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* definte <math>\hat\mu(u,v;\tau)</math> by
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:<math>\hat\mu(u,v;\tau) = \mu(u,v;\tau)-R(u-v;\tau)/2</math>
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where
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:<math>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}</math>
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and <math>y=\Im(\tau)</math> and
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:<math>E(z) = 2\int_0^ze^{-\pi u^2}\,du</math>
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===Mordell integral===
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* [[Mordell integrals]]
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:<math>
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\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)
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</math>
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where
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:<math>M(v;\tau)=\int_{-\infty}^{\infty} \frac{e^{\pi i \tau x^2-2\pi v x}}{\cosh (\pi x)} dx</math>
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* non-holomorpic part is an incomplete period integral of the modular form <math>\eta^3</math>
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:<math>
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iR(0;\tau)=\int_{-\bar{\tau}}^{i\infty}\frac{\eta(z)^3}{\sqrt{\frac{z+\tau}{i}}}\,dz
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</math>
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* property
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:<math>
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R(z;\tau)+\sqrt{\frac{i}{\tau}}R(\frac{z}{\tau};-\frac{1}{\tau})=M(z;\tau)
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</math>
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===modularity===
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:<math>\hat\mu(u+1,v;\tau) = a^{-1}bq^{-1/2}\hat\mu(u+\tau,v;\tau) = -\hat\mu(u,v;\tau),</math>
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:<math>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).</math>
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* In other words the modified Appell–Lerch series transforms like a modular form with respect to τ.
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* 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.
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===special case===
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* mock theta function
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:<math>
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\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}}
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</math>
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* Mordell integrals
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:<math>
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\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
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</math>
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* completion
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:<math>
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\hat{\mu}(z;\tau)=\mu(z;\tau)-R(0;\tau)/2
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</math>
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* modularity
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:<math>
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\hat{\mu}(\frac{z}{\tau};\frac{-1}{\tau})=-\sqrt{\frac{\tau}{i}}\hat{\mu}(z;\tau)
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</math>
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==higher level Appell function==
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*  higher-level Appell functions
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** a particular instance of indefinite theta series
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==history==
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* http://www.google.com/search?hl=en&tbs=tl:1&q=
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==related items==
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* [[Kac-Wakimoto modules]]
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* [[indefinite theta functions]]
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* [[Mathieu moonshine]]
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==encyclopedia==
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* http://en.wikipedia.org/wiki/Appell
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==articles==
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* Mortenson, Eric. 2012. “On the Dual Nature of Partial Theta Functions and Appell-Lerch Sums.” arXiv:1208.6316 [math], August. http://arxiv.org/abs/1208.6316.
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* Tohru Eguchi and Kazuhiro Hikami [http://dx.doi.org/10.1088/1751-8113/42/30/304010 Superconformal Algebras and Mock Theta Functions], 2009
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* [http://www.math.wisc.edu/~ono/reprints/122.pdf Some characters of Kac and Wakimoto and nonholomorphic modular functions.]
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** K. Bringmann and K. Ono, Math. Annalen 345, pages 547-558 (2009)
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* Zwegers [http://mathsci.ucd.ie/~zwegers/presentations/002.pdf Appell-Lerch sums as mock modular forms], 2008
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* A. M. Semikhatov [http://dx.doi.org/10.1007/s00220-008-0677-0 Higher String Functions, Higher-Level Appell Functions, and the Logarithmic sℓ︿2k/u(1) &nbsp;CFT Model]
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* A.M. Semikhatov [http://dx.doi.org/10.1007/s00220-004-1280-7 Higher-Level Appell Functions, Modular Transformations, and Characters]
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* Sander [http://front.math.ucdavis.edu/author/S.Zwegers Zwegers], [http://front.math.ucdavis.edu/0807.4834 Mock Theta Functions], 2002
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* [http://dx.doi.org/10.1007/s002200000315 Integrable highest weight modules over affine superalgebras and Appell’s function]
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** Kac V.G., Wakimoto M, Commun. Math. Phys. '''215'''(3), 631–682 (2001)
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*  N = 2 superconformal minimal models
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* Yutaka Matsuo [http://ptp.ipap.jp/link?PTP/77/793/ Character Formula of C<1 Unitary representation of N=2 Superconformal Algebra] , Prog. Theor. Phys. Vol. 77 No. 4 (1987) pp. 793-797
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* C. Truesdell [http://www.jstor.org/stable/1969153 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
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[[분류:개인노트]]
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[[분류:math and physics]]
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[[분류:mock modular forms]]
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[[분류:math]]
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[[분류:migrate]]

2020년 11월 13일 (금) 09:49 기준 최신판

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



history



related items


encyclopedia



articles

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