Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Modular Invariant Dynamics and Fermion Mass Hierarchies around $\tau =
  i$   We discuss fermion mass hierarchies within modular invariant flavour models.
We analyse the neighbourhood of the self-dual point $\tau=i$, where modular
invariant theories possess a residual $Z_4$ invariance. In this region the
breaking of $Z_4$ can be fully described by the spurion $\epsilon \approx \tau
- i$, that flips its sign under $Z_4$. Degeneracies or vanishing eigenvalues of
fermion mass matrices, forced by the $Z_4$ symmetry at $\tau=i$, are removed by
slightly deviating from the self-dual point. Relevant mass ratios are
controlled by powers of $|\epsilon|$. We present examples where this mechanism
is a key ingredient to successfully implement an hierarchical spectrum in the
lepton sector, even in the presence of a non-minimal K\"ahler potential.

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The categories of this paper are: adap-org
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The categories of this paper are: hep-ex
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The categories of this paper are: mtrl-th
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The categories of this paper are: math
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The categories of this paper are: astro-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: nucl-ex
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The categories of this paper are: econ
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The categories of this paper are: atom-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: supr-con
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The categories of this paper are: eess
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The categories of this paper are: cs
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The categories of this paper are: nlin
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The categories of this paper are: alg-geom
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The categories of this paper are: funct-an