A Supergravity Dual of a (1,0) Field Theory in Six Dimensions

We suggest a supergravity dual for the $(1,0)$ superconformal field theory in six dimensions which has $E_8$ global symmetry. Compared to the description of the (2,0) field theory, the 4-sphere is replaced by a 4-hemisphere, or by orbifolding the 4-sphere.Comment: 5 pages, Harvmac. Typos corrected, References correcte

From SYM Perturbation Theory to Closed Strings in Matrix Theory

For the purpose of better understanding the AdS/CFT correspondence it is useful to have a description of the theory for all values of the 't Hooft coupling, and for all $N$. We discuss such a description in the framework of Matrix theory for SYM on D4-branes, which is given in terms of quantum mechanics on the moduli space of solutions of the Nahm equations. This description reduces to both SYM perturbation theory and to closed string perturbation theory, each in its appropriate regime of validity, suggesting a way of directly relating the variables in the two descriptions. For example, it shows explicitly how holes in the world-sheets of the 't Hooft expansion close to give closed surfaces.Comment: 19 page

Relational Width of First-Order Expansions of Homogeneous Graphs with Bounded Strict Width

Solving the algebraic dichotomy conjecture for constraint satisfaction problems over structures first-order definable in countably infinite finitely bounded homogeneous structures requires understanding the applicability of local-consistency methods in this setting. We study the amount of consistency (measured by relational width) needed to solve CSP(?) for first-order expansions ? of countably infinite homogeneous graphs ? := (A; E), which happen all to be finitely bounded. We study our problem for structures ? that additionally have bounded strict width, i.e., for which establishing local consistency of an instance of CSP(?) not only decides if there is a solution but also ensures that every solution may be obtained from a locally consistent instance by greedily assigning values to variables, without backtracking. Our main result is that the structures ? under consideration have relational width exactly (2, ?_?) where ?_? is the maximal size of a forbidden subgraph of ?, but not smaller than 3. It beats the upper bound: (2 m, 3 m) where m = max(arity(?)+1, ?, 3) and arity(?) is the largest arity of a relation in ?, which follows from a sufficient condition implying bounded relational width given in [Manuel Bodirsky and Antoine Mottet, 2018]. Since ?_? may be arbitrarily large, our result contrasts the collapse of the relational bounded width hierarchy for finite structures ?, whose relational width, if finite, is always at most (2,3)

Equidimensional Isometric Extensions

Let $\Sigma$ be a hypersurface in an $n$-dimensional Riemannian manifold $M$, $n\geqslant 2$. We study the isometric extension problem for isometric immersions $f:\Sigma\to\mathbb R^n$, where $\mathbb R^n$ is equipped with the Euclidean standard metric. Using a weak form of convex integration suggested by Sz\'ekelyhidi, we construct "one-sided" isometric Lipschitz-extensions and obtain an accompanying density result.Comment: 15 page

hh-Principle for Curves with Prescribed Curvature

We prove that every immersed $C^2$-curve $\gamma$ in $\mathbb R^n$, $n\geqslant 3$ with curvature $k_{\gamma}$ can be $C^1$-approximated by immersed $C^2$-curves having prescribed curvature $k>k_{\gamma}$. The approximating curves satisfy a $C^1$-dense $h$-principle. As an application we obtain the existence of $C^2$-knots of arbitrary positive curvature in each isotopy class, which generalizes a similar result by McAtee for $C^2$-knots of constant curvature.Comment: Final version, to appear in Geometriae Dedicata, 9 pages, 1 figur

Near Hagedorn Dynamics of NS Fivebranes, or A New Universality Class of Coiled Strings

We analyze the thermodynamics of NS 5-branes as the temperature approaches the NS 5-branes' Hagedorn temperature, and conclude that the dynamics of ``Little String Theory'' is a new universality class of interacting strings. First we point out how to vary the temperature of the near extremal solution by taking into account $g_s$ corrections. The Hagedorn temperature is shown to be a limiting temperature for the theory. We then compare the thermodynamics to that of a toy model made of free strings and find basic discrepancies. This suggests a need for a new class of string interactions. We suggest that this new universality class is characterized by a strong attractive self-intersection interaction, which causes strings to be coiled. This model might also explain why ``Little String Theories'' exist in at most 5+1 dimensions.Comment: 37 pages, important signs in section 6.4 corrected, references adde