Solutions to Yang-Mills equations on four-dimensional de Sitter space

We consider pure SU(2) Yang-Mills theory on four-dimensional de Sitter space dS$_4$ and construct a smooth and spatially homogeneous magnetic solution to the Yang-Mills equations. Slicing dS$_4$ as ${\mathbb R}\times S^3$, via an SU(2)-equivariant ansatz we reduce the Yang-Mills equations to ordinary matrix differential equations and further to Newtonian dynamics in a double-well potential. Its local maximum yields a Yang-Mills solution whose color-magnetic field at time $\tau\in{\mathbb R}$ is given by $\tilde{B}_a=-\frac12 I_a/(R^2\cosh^2\!\tau)$, where $I_a$ for $a=1,2,3$ are the SU(2) generators and $R$ is the de Sitter radius. At any moment, this spatially homogeneous configuration has finite energy, but its action is also finite and of the value $-\frac12j(j{+}1)(2j{+}1)\pi^3$ in a spin-$j$ representation. Similarly, the double-well bounce produces a family of homogeneous finite-action electric-magnetic solutions with the same energy. There is a continuum of other solutions whose energy and action extend down to zero.Comment: 1+7 pages; v2: introduction extended, gauge group representation dependence added, minor clarifications, 3 more references; v3: title change, published versio

Instantons in six dimensions and twistors

Recently, conformal field theories in six dimensions were discussed from the twistorial point of view. In particular, it was demonstrated that the twistor transform between chiral zero-rest-mass fields and cohomology classes on twistor space can be generalized from four to six dimensions. On the other hand, the possibility of generalizing the correspondence between instanton gauge fields and holomorphic bundles over twistor space is questionable. It was shown by Saemann and Wolf that holomorphic line bundles over the canonical twistor space Tw(X) (defined as a bundle of almost complex structures over the six-dimensional manifold X) correspond to pure-gauge Maxwell potentials, i.e. the twistor transform fails. On the example of X=CP^3 we show that there exists a twistor correspondence between Abelian or non-Abelian Yang-Mills instantons on CP^3 and holomorphic bundles over complex submanifolds of Tw(CP^3), but it is not so efficient as in the four-dimensional case because the twistor transform does not parametrize instantons by unconstrained holomorphic data as it does in four dimensions.Comment: 14 pages; v2: discussion of aims and results extended; v3: published versio

A combinatorial approach to the set-theoretic solutions of the Yang-Baxter equation

A bijective map $r: X^2 \longrightarrow X^2$, where $X = \{x_1, ..., x_n \}$ is a finite set, is called a \emph{set-theoretic solution of the Yang-Baxter equation} (YBE) if the braid relation $r_{12}r_{23}r_{12} = r_{23}r_{12}r_{23}$ holds in $X^3.$ A non-degenerate involutive solution $(X,r)$ satisfying $r(xx)=xx$, for all $x \in X$, is called \emph{square-free solution}. There exist close relations between the square-free set-theoretic solutions of YBE, the semigroups of I-type, the semigroups of skew polynomial type, and the Bieberbach groups, as it was first shown in a joint paper with Michel Van den Bergh. In this paper we continue the study of square-free solutions $(X,r)$ and the associated Yang-Baxter algebraic structures -- the semigroup $S(X,r)$, the group $G(X,r)$ and the $k$- algebra $A(k, X,r)$ over a field $k$, generated by $X$ and with quadratic defining relations naturally arising and uniquely determined by $r$. We study the properties of the associated Yang-Baxter structures and prove a conjecture of the present author that the three notions: a square-free solution of (set-theoretic) YBE, a semigroup of I type, and a semigroup of skew-polynomial type, are equivalent. This implies that the Yang-Baxter algebra $A(k, X,r)$ is Poincar\'{e}-Birkhoff-Witt type algebra, with respect to some appropriate ordering of $X$. We conjecture that every square-free solution of YBE is retractable, in the sense of Etingof-Schedler.Comment: 34 page

Chern-Simons flows on Aloff-Wallach spaces and Spin(7)-instantons

Due to their explicit construction, Aloff-Wallach spaces are prominent in flux compactifications. They carry G_2-structures and admit the G_2-instanton equations, which are natural BPS equations for Yang-Mills instantons on seven-manifolds and extremize a Chern-Simons-type functional. We consider the Chern-Simons flow between different G_2-instantons on Aloff-Wallach spaces, which is equivalent to Spin(7)-instantons on a cylinder over them. For a general SU(3)-equivariant gauge connection, the generalized instanton equations turn into gradient-flow equations on C^3 x R^2, with a particular cubic superpotential. For the simplest member of the Aloff-Wallach family (with 3-Sasakian structure) we present an explicit instanton solution of tanh-like shape.Comment: 1+17 pages, 1 figur

Dressing Symmetries of Holomorphic BF Theories

We consider holomorphic BF theories, their solutions and symmetries. The equivalence of Cech and Dolbeault descriptions of holomorphic bundles is used to develop a method for calculating hidden (nonlocal) symmetries of holomorphic BF theories. A special cohomological symmetry group and its action on the solution space are described.Comment: 14 pages, LaTeX2

Instantons on sine-cones over Sasakian manifolds

We investigate instantons on sine-cones over Sasaki-Einstein and 3-Sasakian manifolds. It is shown that these conical Einstein manifolds are K"ahler with torsion (KT) manifolds admitting Hermitian connections with totally antisymmetric torsion. Furthermore, a deformation of the metric on the sine-cone over 3-Sasakian manifolds allows one to introduce a hyper-K"ahler with torsion (HKT) structure. In the large-volume limit these KT and HKT spaces become Calabi-Yau and hyper-K"ahler conifolds, respectively. We construct gauge connections on complex vector bundles over conical KT and HKT manifolds which solve the instanton equations for Yang-Mills fields in higher dimensions.Comment: 1+15 pages, 2 figure

Household Sharing for Carbon and Energy Reductions: The Case of EU Countries

As households get smaller worldwide, the extent of sharing within households reduces, resulting in rising per capita energy use and greenhouse gas (GHG) emissions. This article examines for the first time the differences in household economies of scale across EU countries as a way to support reductions in energy use and GHG emissions, while considering differences in effects across consumption domains and urban-rural typology. A country-comparative analysis is important to facilitate the formulation of context-specific initiatives and policies for resource sharing. We find that one-person households are most carbon- and energy-intensive per capita with an EU average of 9.2 tCO2eq/cap and 0.14 TJ/cap, and a total contribution of about 17% to the EU’s carbon and energy use. Two-person households contribute about 31% to the EU carbon and energy footprint, while those of five or more members add about 9%. The average carbon and energy footprints of an EU household of five or more is about half that of a one-person average household, amounting to 4.6 tCO2eq/cap and 0.07 TJ/cap. Household economies of scale vary substantially across consumption categories, urban-rural typology and EU countries. Substantial household economies of scale are noted for home energy, real estate services and miscellaneous services such as waste treatment and water supply; yet, some of the weakest household economies of scale occur in high carbon domains such as transport. Furthermore, Northern and Central European states are more likely to report strong household economies of scale—particularly in sparsely populated areas—compared to Southern and Eastern European countries. We discuss ways in which differences in household economies of scale may be linked to social, political and climatic conditions. We also provide policy recommendations for encouraging sharing within and between households as a contribution to climate change mitigation

Self-dual Yang-Mills fields in pseudoeuclidean spaces

The self-duality Yang-Mills equations in pseudoeuclidean spaces of dimensions $d\leq 8$ are investigated. New classes of solutions of the equations are found. Extended solutions to the D=10, N=1 supergravity and super Yang-Mills equations are constructed from these solutions.Comment: 9 pages, LaTeX, no figure