_{[a,b]}(z,Ω) with characteristic [a,b] for Riemann surface of genus g as functions of z and Teichmueller parameters Ω are the basic building blocks of modular invariant vacuum functionals defined in the finite-dimensional moduli space whose points characterize the conformal equivalence class of the induced metric of the partonic 2-surface. Obviously, kind of spinorial variants of theta functions are in question with g+g spinor indices for genus g.

The case of Hurwitz thetas corresponds to g=1 Riemann surface (torus) so that a and b are g=1-component vectors having values 0 or 1/2 and Hurwitz zeta corresponds to θ_{[0,1/2]}. The four Jacobi theta functions listed in Wikipedia must correspond to these thetas for torus. The values for a and b are 0 and 1 for them but this must be a mere convention.

** 1. Series of fractional modular groups and theta functions**

The extensions of modular group to fractional modular groups obtained by replacing integers with integers shifted by multiples of 1/n suggest the existence of new kind of q-theta functions with characteristics [a,b] with a and b being g-component vectors having fractional values k/n, k=0,1...n-1. There exists also a definition of q-theta functions working for 0≤|q|<1 but not for roots of unity. The q-theta functions assigned to roots of unity would be associated with Riemann surfaces with additional Z_{n} conformal symmetry but ** not necessarily ** with generic Riemann surfaces and obtained by simply replacing the value range of characteristics [a,b] with the new value range in the defining formula for theta functions.

If Z_{n} conformal symmetry is relevant it is probably so because it would make the generalized theta functions sections in a bundle with a finite fiber having Z_{n} action. This hierarchy would correspond to the hierarchy of quantum groups for roots of unity and Jones inclusions and one could probably define also corresponding zeta function multiplets. These theta functions would be building blocks of the elementary particle vacuum functionals for dark variants of elementary particles invariant under fractional modular group. They would also define a hierarchy of fractal variants of number theoretic functions: it would be interesting to see what this means from the point of view of Langlands program involving ordinary modular invariance in an essential manner.

** 2. There are three fermions generations in TGD Universe**

Generation-genus correspondence seems to predict infinite number of fermionic families and the basic idea has been that the vanishing of elementary particle vacuum functionals for hyper-elliptic Riemann surfaces with genus larger than 2 explains why the the number of light families is three. I have been refining this idea for at least 15 years now.

The observation related to this is that if fermions correspond to n=2 dark matter with Z_{2} conformal symmetry as strong quantum classical correspondence suggests (see the previous posting), the number of ordinary fermion families is ** three** without any further assumptions (for background see this).

To see this suppose that also the sectors corresponding to M^{4}→M^{4}/Z_{2} and CP_{2}→ CP_{2}/Z_{2} coverings are possible. Z_{2} conformal symmetry implies that partonic Riemann surfaces are hyper-elliptic. For genera g> 2 this means that some theta functions of θ[a,b] appearing in the product of theta functions defining the vacuum functional vanish. Hence fermionic elementary particle vacuum functionals would vanish for g> 2 and only 3 fermion families would be possible for n=2 dark matter.

This results can be strengthened. The existence of space-time correlate for the fermionic 2-valuedness suggests that fermions quite generally to even values of n, so that this result would hold for all fermions. Elementary bosons (actually exotic particles belonging to Kac-Moody type representations) would correspond to odd values of n, and could possess also higher families. There is a nice argument supporting this hypothesis. n-fold discretization provided by covering associated with H corresponds to discretization for angular momentum eigenstates. Minimal discretization for 2j+1 states corresponds to n=2j+1. j=1/2 requires n=2 at least, j=1 requires n=3 at least, and so on. n=2j+1 allows spins j≤n-1/2. This very nice spin-quantum phase connection which very probably follows also from the representations of quantum SU(2).

These rules would hold only for genuinely elementary particles corresponding to single partonic component and all bosonic particles of this kind are exotics (excitations in only "vibrational" degrees of freedom of partonic 2-surface with modular invariance eliminating quite a number of them): ordinary gauge bosons correspond to fermion pairs at throats of a wormhole contact and decompose to SU(3) singlet and octet, whose states are labelled by handle-number pairs (g_{1},g_{2}): they define new kind of heavy bosons giving rise to neutral flavor changing currents (could they be visible in LHC?). Note that gravitons necessarily correspond to pairs of fermions or gauge bosons connected by flux tubes so that they are stringy objects in this sense.

For more details see the revised chapter Construction of Quantum Theory: Symmetries of "Towards S-matrix".

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