Orbifold Lattice Research Articles

Toward QCD on quantum computer: orbifold lattice approach

We propose an orbifold lattice formulation of QCD suitable for quantum simulations. We show explicitly how to encode gauge degrees of freedom into qubits using noncompact variables, and how to write down a simple truncated Hamiltonian in the coordinate basis. We show that SU(3) gauge group variables and quarks in the fundamental representation can be implemented straightforwardly on qubits, for arbitrary truncation of the gauge manifold.

Fourvolutions and Automorphism Groups of Orbifold Lattice Vertex Operator Algebras

Let $L$ be an even positive definite lattice with no roots, i.e., $L(2) = \{ x \in L \mid (x|x) = 2 \} = \emptyset$. Let $g \in O(L)$ be an isometry of order $4$ such that $g^2 = -1$ on $L$. In this article, we determine the full automorphism group of the orbifold vertex operator algebra $V_L^{\widehat{g}}$. As our main result, we show that $\operatorname{Aut}(V_L^{\widehat{g}})$ is isomorphic to $N_{\operatorname{Aut}(V_L)}(\langle \widehat{g} \rangle)/\langle \widehat{g} \rangle$ unless $L \cong \sqrt{2} E_8$ or $BW_{16}$.

Quantum simulation of gauge theory via orbifold lattice

We propose a new framework for simulating U(k) Yang-Mills theory on a universal quantum computer. This construction uses the orbifold lattice formulation proposed by Kaplan, Katz, and Unsal, who originally applied it to supersymmetric gauge theories. Our proposed approach yields a novel perspective on quantum simulation of quantum field theories, carrying certain advantages over the usual Kogut-Susskind formulation. We discuss the application of our constructions to computing static properties and real-time dynamics of Yang-Mills theories, from glueball measurements to AdS/CFT, making use of a variety of quantum information techniques including qubitization, quantum signal processing, Jordan-Lee-Preskill bounds, and shadow tomography. The generalizations to certain supersymmetric Yang-Mills theories appear to be straightforward, providing a path towards the quantum simulation of quantum gravity via holographic duality.

Non-perturbative gauge-Higgs unification: symmetries and order parameters

We consider pure SU(N) gauge theories defined on an orbifold lattice, analogous to the S^1/Z_2 gauge theory orbifolds of the continuum, which according to the perturbative analysis do not have a Higgs phase. Non-perturbatively the conclusion for N even is the opposite, namely that spontaneous symmetry breaking does take place and some of the gauge bosons become massive. We interpret this new, non-perturbative phenomenon both mathematically and physically.

Deformed matrix models, supersymmetric lattice twists and 𝒩 = ¼ supersymmetry

A manifestly supersymmetric nonperturbative matrix regularization for a twisted version of N=(8,8) theory on a curved background (a two-sphere) is constructed. Both continuum and the matrix regularization respect four exact scalar supersymmetries under a twisted version of the supersymmetry algebra. We then discuss a succinct Q=1 deformed matrix model regularization of N=4 SYM in d=4, which is equivalent to a non-commutative $A_4^*$ orbifold lattice formulation. Motivated by recent progress in supersymmetric lattices, we also propose a N=1/4 supersymmetry preserving deformation of N=4 SYM theory on $\R^4$. In this class of N=1/4 theories, both the regularized and continuum theory respect the same set of (scalar) supersymmetry. By using the equivalence of the deformed matrix models with the lattice formulations, we give a very simple physical argument on why the exact lattice supersymmetry must be a subset of scalar subalgebra. This argument disagrees with the recent claims of the link approach, for which we give a new interpretation.

From twisted supersymmetry to orbifold lattices

We show how to derive the supersymmetric orbifold lattices of Cohen et al. \cite{Cohen:2003xe,Cohen:2003qw} and Kaplan et al. \cite{Kaplan:2005ta} by direct discretization of an appropriate twisted supersymmetric Yang-Mills theory. We examine in detail the four supercharge two dimensional theory and the theory with sixteen supercharges in four dimensions. The continuum limit of the latter theory is the well known Marcus twist of ${\cal N}=4$ Yang-Mills. The lattice models are gauge invariant and possess one exact supersymmetry at non-zero lattice spacing.

Exact vacuum energy of orbifold lattice theories

We investigate the orbifold lattice theories constructed from supersymmetric Yang-Mills matrix theories (mother theories) with four and eight supercharges. We show that the vacuum energy of these theories does not receive any quantum correction perturbatively.

Relationship between various supersymmetric lattice models

We comment on the relationships between several supersymmetric lattice models; the ``orbifold lattice theory'' by Cohen-Kaplan-Katz-Unsal (CKKU), lattice regularization of the topological field theory by Sugino and the ``geometrical approach'' by Catterall. We point out that these three models have close relationships; the = (2, 2) model by Catterall [1] and the two-dimensional = (2, 2) lattice theory being similar to Sugino's construction [2] can be derived by appropriate truncation of fields in the two-dimensional = (4, 4) orbifold lattice theory by CKKU [3]. Catterall's = (2, 2) description possesses extra degrees of freedom compared to the target = (2, 2) theory. If we remove those extra degrees of freedom in a way keeping supersymmetry on the lattice, Catterall's description reduces to a model of the Sugino type.

Twisted supersymmetric gauge theories and orbifold lattices

We examine the relation between twisted versions of the extended supersymmetric gauge theories and supersymmetric orbifold lattices. In particular, for the $\mathcal{N}=4$ SYM in $d=4$, we show that the continuum limit of orbifold lattice reproduces the twist introduced by Marcus, and the examples at lower dimensions are usually Blau-Thompson type. The orbifold lattice point group symmetry is a subgroup of the twisted Lorentz group, and the exact supersymmetry of the lattice is indeed the nilpotent scalar supersymmetry of the twisted versions. We also introduce twisting in terms of spin groups of finite point subgroups of $R$-symmetry and spacetime symmetry.

On the moduli space of the T6/Z3 orbifold and its modular group

The authors describe the duality group Gamma =SU(3, 3, Z) for the Narain lattice of the T6/Z3 orbifold and its action on the corresponding moduli space M3.3/ Gamma , where M3.3=SU(3, 3)/(SU(3)(X)SU(3)(X)U(1)). A symplectic embedding of the momenta and winding numbers allows them to connect the orbifold lattice to the special geometry of M3.3. As an application, a formal expression for an automorphic function, which is a candidate for a non-perturbative superpotential, is given.