Rigidification of algebras over multi-sorted theories

We define the notion of a multi-sorted algebraic theory, which is a generalization of an algebraic theory in which the objects are of different "sorts." We prove a rigidification result for simplicial algebras over these theories, showing that there is a Quillen equivalence between a model category structure on the category of strict algebras over a multi-sorted theory and an appropriate model category structure on the category of functors from a multi-sorted theory to the category of simplicial sets. In the latter model structure, the fibrant objects are homotopy algebras over that theory. Our two main examples of strict algebras are operads in the category of simplicial sets and simplicial categories with a given set of objects.Comment: This is the version published by Algebraic & Geometric Topology on 14 November 200

Abstracting Asynchronous Multi-Valued Networks: An Initial Investigation

Multi-valued networks provide a simple yet expressive qualitative state based modelling approach for biological systems. In this paper we develop an abstraction theory for asynchronous multi-valued network models that allows the state space of a model to be reduced while preserving key properties of the model. The abstraction theory therefore provides a mechanism for coping with the state space explosion problem and supports the analysis and comparison of multi-valued networks. We take as our starting point the abstraction theory for synchronous multi-valued networks which is based on the finite set of traces that represent the behaviour of such a model. The problem with extending this approach to the asynchronous case is that we can now have an infinite set of traces associated with a model making a simple trace inclusion test infeasible. To address this we develop a decision procedure for checking asynchronous abstractions based on using the finite state graph of an asynchronous multi-valued network to reason about its trace semantics. We illustrate the abstraction techniques developed by considering a detailed case study based on a multi-valued network model of the regulation of tryptophan biosynthesis in Escherichia coli.Comment: Presented at MeCBIC 201

On Matrix Models of M5-branes

We compare the $(0,2)$ theory of the single M5 brane decoupled from gravity in the lightcone with transverse $R^4$, and a matrix model description in terms of quantum mechanics on instanton moduli space. We give some tests of the Matrix model in the case of multi fivebranes on $R^4$. We extract constraints on the operator content of the field theory of the multi-fivebrane system by analyzing the Matrix model. We also begin a study of compactifications of the $(0,2)$ theory in this framework, arguing that for large compactification scale the $(0,2)$ theory is described by super-quantum mechanics on appropriate instanton moduli spaces.Comment: 30 pages, Refs added. Discussion of compactification on four-torus is modifie

Hubbard model description of silicon spin qubits: charge stability diagram and tunnel coupling in Si double quantum dots

We apply the recently introduced Hubbard model approach to quantitatively describe the experimental charge stability diagram and tunnel coupling of silicon double quantum dot systems. The results calculated from both the generalized Hubbard model and the microscopic theory are compared with existing experimental data, and excellent agreement between theory and experiment is found. The central approximation of our theory is a reduction of the full multi-electron multi-band system to an effective two-electron model, which is numerically tractable. In the microscopic theory we utilize the Hund-Mulliken approximation to the electron wave functions and compare the results calculated with two different forms of confinement potentials (biquadratic and Gaussian). We discuss the implications of our work for future studies.Comment: 11 pages, 3 figure

Goldstone's Theorem and Hamiltonian of Multi-galileon Modified Gravity

The galileon model was recently proposed to locally describe a class of modified gravity theories, including the braneworld DGP model. We discuss spontaneous symmetry breaking of the self-accelerating branch in a multi-galileon theory with internal global symmetries. We show a modified version of Goldstone's theorem is applicable to the symmetry breaking pattern and discuss its implications. We also derive the Hamiltonian of a general multi-galileon theory and discuss its implications.Comment: 13 pages, 1 figure; To appear in PR

Multi-Trace Superpotentials vs. Matrix Models

We consider N = 1 supersymmetric U(N) field theories in four dimensions with adjoint chiral matter and a multi-trace tree-level superpotential. We show that the computation of the effective action as a function of the glueball superfield localizes to computing matrix integrals. Unlike the single-trace case, holomorphy and symmetries do not forbid non-planar contributions. Nevertheless, only a special subset of the planar diagrams contributes to the exact result. Some of the data of this subset can be computed from the large-N limit of an associated multi-trace Matrix model. However, the prescription differs in important respects from that of Dijkgraaf and Vafa for single-trace superpotentials in that the field theory effective action is not the derivative of a multi-trace matrix model free energy. The basic subtlety involves the correct identification of the field theory glueball as a variable in the Matrix model, as we show via an auxiliary construction involving a single-trace matrix model with additional singlet fields which are integrated out to compute the multi-trace results. Along the way we also describe a general technique for computing the large-N limits of multi-trace Matrix models and raise the challenge of finding the field theories whose effective actions they may compute. Since our models can be treated as N = 1 deformations of pure N =2 gauge theory, we show that the effective superpotential that we compute also follows from the N = 2 Seiberg-Witten solution. Finally, we observe an interesting connection between multi-trace local theories and non-local field theory.Comment: 35 pages, LaTeX, 6 EPS figures. v2: typos fixed, v3: typos fixed, references added, Sec. 5 added explaining how multi-trace theories can be linearized in traces by addition of singlet fields and the relation of this approach to matrix model