# L-series and Modular Forms in String Theory?

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On the border of algebraic geometry and number theory, there is what is known as the "modularity conjecture" for certain algebraic varieties defined over $\mathbb{Q}$.  Roughly speaking, this is the conjecture that there is a modular object whose Dirichlet series coincides with the L-series of the variety.

A little more specifically, if $X$ is a Calabi-Yau threefold defined over $\mathbb{Q}$ with $h^{2,1}(X)=0$, then one can form its L-series:

$$L(X, s) = \prod_{p-\text{prime}} \big(1-t_{3}(p)p^{-s} + p^{3-2s}\big)^{-1}.$$

(I'm lying slightly...the product excludes finitely many "bad" primes).  Here, $t_{3}(p)$ are integers closely related to the number of points over the finite field $\mathbb{F}_{p}$ in $X$.  The modularity conjecture is that $t_{3}(p)=a_{p}$ where $a_{p}$ are the Fourier coefficients of a weight 4 modular cusp form for congruence subgroup $\Gamma_{0}(N)$, for some $N$.  For a nice summary, one can see (https://projecteuclid.org/download/pdf_1/euclid.kjm/1250517640).

I know in many areas in enumerative geometry/string theory various partition functions have non-trivial automorphic properties.  Quite simply, I'm curious if there's any well-known or conjectural relationship of these L-series in string theory?  Particularly, perhaps topological string theory on $X$ or something closely related?

I suggest searching the work of Noriko Yui, she is a guru of modularity in string theory.

@Mitchell Porter Yes!  I've been reading a lot of her work.  But can I ask you where she mentions applications to string theory?  Any sources, slides, or talks that you know of by chance?  Applications of her work on modularity to string theory is exactly what I'm looking for.

Actually you have a point, she studies Calabi-Yaus more than string theory per se. And from Rolf Schimmrigk e.g. https://arxiv.org/abs/hep-th/0603234 I have a glimpse of the barrier that must be crossed, you need to find a relationship between the properties of the CY, and the worldsheet description. See the paragraph beginning "In general the L-function of a variety..." Schimmrigk has a number of interesting papers on this, up to the start of 2013.

@Mitchell Porter Thanks a lot, indeed his papers look very interesting, and will take me quite a while to digest.  In that paragraph you mention, can I ask if you happen to know what he means by "string theoretic modular form"?

I don't know how broad a meaning he intends, but https://arxiv.org/abs/hep-th/0211284 describes a specific kind of modular form obtained from the string worldsheet CFT, which is the subject of subsequent papers.

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