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Register a new userWavefunctions and Yukawa Couplings on Resolutions of $T^2/mathbb{Z}_N$ Orbifolds
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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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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 $n_+ - n_- = (2M - V_+ + V_-)/2N$.
Generalizations of the AGT correspondence between 4D $mathcal{N}=2$ $SU(2)$ supersymmetric gauge theory on ${mathbb {C}}^2$ with $Omega$-deformation and 2D Liouville conformal field theory include a correspondence between 4D $mathcal{N}=2$ $SU(N)$ supersymmetric gauge theories, $N = 2, 3, ldots$, on ${mathbb {C}}^2/{mathbb {Z}}_n$, $n = 2, 3, ldots$, with $Omega$-deformation and 2D conformal field theories with $mathcal{W}^{, para}_{N, n}$ ($n$-th parafermion $mathcal{W}_N$) symmetry and $widehat{mathfrak{sl}}(n)_N$ symmetry. In this work, we trivialize the factor with $mathcal{W}^{, para}_{N, n}$ symmetry in the 4D $SU(N)$ instanton partition functions on ${mathbb {C}}^2/{mathbb {Z}}_n$ (by using specific choices of parameters and imposing specific conditions on the $N$-tuples of Young diagrams that label the states), and extract the 2D $widehat{mathfrak{sl}}(n)_N$ WZW conformal blocks, $n = 2, 3, ldots$, $N = 1, 2, ldots, .$
We discuss an effective way for analyzing the system on the magnetized twisted orbifolds in operator formalism, especially in the complicated cases $T^{2}/Z_{3}$, $T^{2}/Z_{4}$ and $T^{2}/Z_{6}$. We can obtain the exact and analytical results which can be applicable for any larger values of the quantized magnetic flux M, and show that the (non-diagonalized) kinetic terms are generated via our formalism and the number of the surviving physical states are calculable in a rigorous manner by simply following usual procedures in linear algebra in any case. Our approach is very powerful when we try to examine properties of the physical states on (complicated) magnetized orbifolds $T^{2}/Z_{3}$, $T^{2}/Z_{4}$, $T^{2}/Z_{6}$ (and would be in other cases on higher-dimensional torus) and could be an essential tool for actual realistic model construction based on these geometries.
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.
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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