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Register a new userType 0 $mathbb{Z}_2times mathbb{Z}_2$ Heterotic String Orbifolds and Misaligned Supersymmetry
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The $mathbb{Z}_2times mathbb{Z}_2$ heterotic string orbifold yielded a large space of phenomenological three generation models and serves as a testing ground to explore how the Standard Model of particle physics may be incorporated in a theory of quantum gravity. In this paper we explore the existence of type 0 models in this class of string compactifications. We demonstrate the existence of type 0 $mathbb{Z}_2times mathbb{Z}_2$ heterotic string orbifolds, and show that there exist a large degree of redundancy in the space of GGSO projection coefficients when the type 0 restrictions are implemented. We explore the existence of such configurations in several constructions. The first correspond to essentially a unique configuration out of a priori $2^{21}$ discrete GGSO choices. We demonstrate this uniqueness analytically, as well as by the corresponding analysis of the partition function. A wider classification is performed in $tilde S$--models and $S$--models, where the first class correspond to compactifications of a tachyonic ten dimensional heterotic string vacuum, whereas the second correspond to compactifications of the ten dimensional non--tachyonic $SO(16)times SO(16)$. We show that the type 0 models in both cases contain physical tachyons at the free fermionic point in the moduli space. These vacua are therefore necessarily unstable, but may be instrumental in exploring the string dynamics in cosmological scenarios. we analyse the properties of the string one--loop amplitude. Naturally, these are divergent due to the existence of tachyonic states. We show that once the tachyonic states are removed by hand the amplitudes are finite and exhibit a form of misaligned supersymmetry.
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The $Z_2times Z_2$ heterotic string orbifold gives rise to a large space of phenomenological three generation models that serves as a testing ground to explore how the Standard Model of particle physics may be incorporated in a theory of quantum gravity. Recently, we demonstrated the existence of type 0 $Z_2times Z_2$ heterotic string orbifolds in which there are no massless fermionic states. In this paper we demonstrate the existence of non--supersymmetric tachyon--free $Z_2times Z_2$ heterotic string orbifolds that do not contain any massless bosonic states from the twisted sectors. We dub these configurations type ${bar 0}$ models. They necessarily contain untwisted bosonic states, producing the gravitational, gauge and scalar moduli degrees of freedom, but possess an excess of massless fermionic states over bosonic ones, hence producing a positive cosmological constant. Such configurations may be instrumental when trying to understand the string dynamics in the early universe.
We investigate chiral zero modes and winding numbers at fixed points on $T^2/mathbb{Z}_N$ orbifolds. It is shown that the Atiyah-Singer index theorem for the chiral zero modes leads to a formula $n_+-n_-=(-V_++V_-)/2N$, where $n_{pm}$ are the numbers of the $pm$ chiral zero modes and $V_{pm}$ are the sums of the winding numbers at the fixed points on $T^2/mathbb{Z}_N$. This formula is complementary to our zero-mode counting formula on the magnetized orbifolds with non-zero flux background $M eq 0$, consistently with substituting $M = 0$ for the counting formula $n_+ - n_- = (2M - V_+ + V_-)/2N$.
We consider an analogue of Wittens $SL(2,mathbb{Z})$ action on three-dimensional QFTs with $U(1)$ symmetry for $2k$-dimensional QFTs with $mathbb{Z}_2$ $(k-1)$-form symmetry. We show that the $SL(2,mathbb{Z})$ action only closes up to a multiplication by an invertible topological phase whose partition function is the Brown-Kervaire invariant of the spacetime manifold. We interpret it as part of the $SL(2,mathbb{Z})$ anomaly of the bulk $(2k+1)$-dimensional $mathbb{Z}_2$ gauge theory.
The incorporation of Wilson lines leads to an extension of the modular symmetries of string compactification beyond $mathrm{SL}(2,mathbb Z)$. In the simplest case with one Wilson line $Z$, Kahler modulus $T$ and complex structure modulus $U$, we are led to the Siegel modular group $mathrm{Sp}(4,mathbb Z)$. It includes $mathrm{SL}(2,mathbb Z)_Ttimesmathrm{SL}(2,mathbb Z)_U$ as well as $mathbb Z_2$ mirror symmetry, which interchanges $T$ and $U$. Possible applications to flavor physics of the Standard Model require the study of orbifolds of $mathrm{Sp}(4,mathbb Z)$ to obtain chiral fermions. We identify the 13 possible orbifolds and determine their modular flavor symmetries as subgroups of $mathrm{Sp}(4,mathbb Z)$. Some cases correspond to symmetric orbifolds that extend previously discussed cases of $mathrm{SL}(2,mathbb Z)$. Others are based on asymmetric orbifold twists (including mirror symmetry) that do no longer allow for a simple intuitive geometrical interpretation and require further study. Sometimes they can be mapped back to symmetric orbifolds with quantized Wilson lines. The symmetries of $mathrm{Sp}(4,mathbb Z)$ reveal exciting new aspects of modular symmetries with promising applications to flavor model building.
We propose matter wavefunctions on resolutions of $T^2/mathbb{Z}_N$ singularities with constant magnetic fluxes. In the blow-down limit, the obtained wavefunctions of chiral zero-modes result in those on the magnetized $T^2/mathbb{Z}_N$ orbifold models, but the wavefunctions of $mathbb{Z}_N$-invariant zero-modes receive the blow-up effects around fixed points of $T^2/mathbb{Z}_N$ orbifolds. Such blow-up effects change the selection rules and Yukawa couplings among the chiral zero-modes as well as the modular symmetry, in contrast to those on the magnetized $T^2/mathbb{Z}_N$ orbifold models.
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