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Register a new user$SL(2,mathbb{Z})$ action on QFTs with $mathbb{Z}_2$ symmetry and the Brown-Kervaire invariants
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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.
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We know that $mathbb{Z}_n$ is a finite field for a prime number $n$. Let $m,n$ be arbitrary natural numbers and let $mathbb{Z}^m_n= mathbb{Z}_n timesmathbb{Z}_ntimes...timesmathbb{Z}_n$ be the Cartesian product of $m$ rings $mathbb{Z}_n$. In this note, we present the action of $SL(m, mathbb{Z}_n)={A in mathbb{Z}^{m,m}_{n} : det A equiv 1 (modsimn)}$, where $SL(m, mathbb{Z}_n)$ for $ngeq 2$ is a group under matrix multiplication modulo $n$, on the ring $mathbb{Z}^m_n$ as a right multiplication of a row vector of $mathbb{Z}^m_n$ by a matrix of $SL(m, mathbb{Z}_n)$ to determine the orbits of the ring $mathbb{Z}^m_n$. This work is an extension of [1]
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.
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 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 compute the partition function for 6d $mathcal{N}=1$ $SO(2N)$ gauge theories compactified on a circle with $mathbb{Z}_2$ outer automorphism twist. We perform the computation based on 5-brane webs with two O5-planes using topological vertex with two O5-planes. As representative examples, we consider 6d $SO(8)$ and $SU(3)$ gauge theories with $mathbb{Z}_2$ twist. We confirm that these partition functions obtained from the topological vertex with O5-planes indeed agree with the elliptic genus computations.
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