Dynamics of SU(N)SU(N) Supersymmetric Gauge Theory

We study the physics of the Seiberg-Witten and Argyres-Faraggi-Klemm-Lerche-Theisen-Yankielowicz solutions of $D=4$, $\mathcal{N}=2$ and $\mathcal{N}=1$ $SU(N)$ supersymmetric gauge theory. The $\mathcal{N}=1$ theory is confining and its effective Lagrangian is a spontaneously broken $U(1)^{N-1}$ abelian gauge theory. We identify some features of its physics which see this internal structure, including a spectrum of different string tensions. We discuss the limit $N\rightarrow\infty$, identify a scaling regime in which instanton and monopole effects survive, and give exact results for the crossover from weak to strong coupling along a scaling trajectory. We find a large hierarchy of mass scales in the scaling regime, including very light $W$ bosons, and the absence of weak coupling. The light $W$'s leave a novel imprint on the effective dual magnetic theory. The effective Lagrangian appears to be inadequate to understand the conventional large $N$ limit of the confining $\mathcal{N}=1$ theory.Comment: 28 pages, harvmac, 4 eps figures in separate uuencoded file. We have extended this paper considerably, adding new results, discussion and figures. In particular, we give exact formulas for masses and couplings along a scaling trajectory appropriate to the large $N$ limit. These formulas display a novel effect due to light electric $W$ bosons down to energy scales $\sim e^{-N}$, deep in the weak coupling magnetic regim

FQHE and tt * geometry

Cumrun Vafa [1] has proposed a microscopic description of the Fractional Quantum Hall Effect (FQHE) in terms of a many-body Hamiltonian H invariant under four supersymmetries. The non-Abelian statistics of the defects (quasi-holes and quasi-particles) is then determined by the monodromy representation of the associated tt* geometry. In this paper we study the monodromy representation of the Vafa 4-susy model. Modulo some plausible assumption, we find that the monodromy representation factors through a Temperley-Lieb/Hecke algebra with q = \ub1 exp (\u3c0i/\u3bd) as predicted in [1]. The emerging picture agrees with the other predictions of [1] as well. The bulk of the paper is dedicated to the development of new concepts, ideas, and techniques in tt* geometry which are of independent interest. We present several examples of these geometric structures in various contexts

Higher S-dualities and Shephard-Todd groups

Abstract: Seiberg and Witten have shown that in N=2$$\mathcal{N}=2$$ SQCD with Nf = 2Nc = 4 the S-duality group PSL2\u2124$$\mathrm{P}\mathrm{S}\mathrm{L}\left(2,\mathrm{\mathbb{Z}}\right)$$ acts on the flavor charges, which are weights of Spin(8), by triality. There are other N=2$$\mathcal{N}=2$$ SCFTs in which SU(2) SYM is coupled to strongly-interacting non-Lagrangian matter: their matter charges are weights of E6, E7 and E8 instead of Spin(8). The S-duality group PSL2\u2124$$\mathrm{P}\mathrm{S}\mathrm{L}\left(2,\mathrm{\mathbb{Z}}\right)$$ acts on these weights: what replaces Spin(8) triality for the E6, E7, E8root lattices? In this paper we answer the question. The action on the matter charges of (a finite central extension of) PSL2\u2124$$\mathrm{P}\mathrm{S}\mathrm{L}\left(2,\mathrm{\mathbb{Z}}\right)$$ factorizes trough the action of the exceptional Shephard-Todd groups G4 and G8 which should be seen as complex analogs of the usual triality group S3 43WeylA2$${\mathfrak{S}}_3\simeq \mathrm{Weyl}\left({A}_2\right)$$. Our analysis is based on the identification of S-duality for SU(2) gauge SCFTs with the group of automorphisms of the cluster category of weighted projective lines of tubular type. \ua9 2015, The Author(s)

c--Map,very Special Quaternionic Geometry and Dual Ka\"hler Spaces

We show that for all very special quaternionic manifolds a different N=1 reduction exists, defining a Kaehler Geometry which is ``dual'' to the original very special Kaehler geometry with metric G_{a\bar{b}}= - \partial_a \partial_b \ln V (V={1/6}d_{abc}\lambda^a \lambda^b \lambda^c). The dual metric g^{ab}=V^{-2} (G^{-1})^{ab} is Kaehler and it also defines a flat potential as the original metric. Such geometries and some of their extensions find applications in Type IIB compactifications on Calabi--Yau orientifolds.Comment: 9 pages, LaTeX, typos corrected, comments adde

Analogue mouse pointer control via an online steady state visual evoked potential (SSVEP) brain-computer interface

The steady state visual evoked protocol has recently become a popular paradigm in brain–computer interface (BCI) applications. Typically (regardless of function) these applications offer the user a binary selection of targets that perform correspondingly discrete actions. Such discrete control systems are appropriate for applications that are inherently isolated in nature, such as selecting numbers from a keypad to be dialled or letters from an alphabet to be spelled. However motivation exists for users to employ proportional control methods in intrinsically analogue tasks such as the movement of a mouse pointer. This paper introduces an online BCI in which control of a mouse pointer is directly proportional to a user's intent. Performance is measured over a series of pointer movement tasks and compared to the traditional discrete output approach. Analogue control allowed subjects to move the pointer faster to the cued target location compared to discrete output but suffers more undesired movements overall. Best performance is achieved when combining the threshold to movement of traditional discrete techniques with the range of movement offered by proportional control

Scaling Behavior in String Theory

In Calabi--Yau compactifications of the heterotic string there exist quantities which are universal in the sense that they are present in every Calabi--Yau string vacuum. It is shown that such universal characteristics provide numerical information, in the form of scaling exponents, about the space of ground states in string theory. The focus is on two physical quantities. The first is the Yukawa coupling of a particular antigeneration, induced in four dimensions by virtue of supersymmetry. The second is the partition function of the topological sector of the theory, evaluated on the genus one worldsheet, a quantity relevant for quantum mirror symmetry and threshold corrections. It is shown that both these quantities exhibit scaling behavior with respect to a new scaling variable and that a scaling relation exists between them as well.Comment: 10pp, 4 eps figures (essential

Lattice Models with N=2 Supersymmetry

We introduce lattice models with explicit N=2 supersymmetry. In these interacting models, the supersymmetry generators Q^+ and Q^- yield the Hamiltonian H={Q^+,Q^-} on any graph. The degrees of freedom can be described as either fermions with hard cores, or as quantum dimers. The Hamiltonian of our simplest model contains a hopping term and a repulsive potential, as well as the hard-core repulsion. We discuss these models from a variety of perspectives: using a fundamental relation with conformal field theory, via the Bethe ansatz, and using cohomology methods. The simplest model provides a manifestly-supersymmetric lattice regulator for the supersymmetric point of the massless 1+1-dimensional Thirring (Luttinger) model. We discuss the ground-state structure of this same model on more complicated graphs, including a 2-leg ladder, and discuss some generalizations.Comment: 4 page

Cosmological attractor models and higher curvature supergravity

We study cosmological \u3b1-attractors in superconformal/supergravity models, where \u3b1 is related to the geometry of the moduli space. For \u3b1 = 1 attractors [1] we present a generalization of the previously known manifestly superconformal higher curvature supergravity model [2]. The relevant standard 2-derivative supergravity with a minimum of two chiral multiplets is shown to be dual to a 4-derivative higher curvature supergravity, where in general one of the chiral superfields is traded for a curvature superfield. There is a degenerate case when both matter superfields become non-dynamical and there is only a chiral curvature superfield, pure higher derivative supergravity. Generic \u3b1-models [3] interpolate between the attractor point at \u3b1 = 0 and generic chaotic inflation models at large \u3b1, in the limit when the inflaton moduli space becomes flat. They have higher derivative duals with the same number of matter fields as the original theory or less, but at least one matter multiplet remains. In the context of these models, the detection of primordial gravity waves will provide information on the curvature of the inflaton submanifold of the K\ue4hler manifold, and we will learn if the inflaton is a fundamental matter multiplet, or can be replaced by a higher derivative curvature excitation. \ua9 2014 The Author(s)

Instanton effects and linear-chiral duality

We discuss duality between the linear and chiral dilaton formulations, in the presence of super-Yang-Mills instanton corrections to the effective action. In contrast to previous work on the subject, our approach appeals directly to explicit instanton calculations and does not rely on the introduction of an auxiliary Veneziano-Yankielowicz superfield. We discuss duality in the case of an axion that has a periodic scalar potential, and find that the bosonic fields of the dual linear multiplet have a modified interpretation. We note that symmetries of the axion potential manifest themselves as symmetries of the equations of motion for the linear multiplet. We also make some brief remarks regarding dilaton stabilization. We point out that corrections recently studied by Dijkgraaf and Vafa can be used to stabilize the axion in the case of a single super-Yang-Mills condensate.Comment: 1+18 pages, 1 figure, comments and references adde

Enumerative geometry of Calabi-Yau 4-folds

Gromov-Witten theory is used to define an enumerative geometry of curves in Calabi-Yau 4-folds. The main technique is to find exact solutions to moving multiple cover integrals. The resulting invariants are analogous to the BPS counts of Gopakumar and Vafa for Calabi-Yau 3-folds. We conjecture the 4-fold invariants to be integers and expect a sheaf theoretic explanation. Several local Calabi-Yau 4-folds are solved exactly. Compact cases, including the sextic Calabi-Yau in CP5, are also studied. A complete solution of the Gromov-Witten theory of the sextic is conjecturally obtained by the holomorphic anomaly equation.Comment: 44 page