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Register a new userSymmetric calorons and the rotation map
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Josh Cork
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We study $SU(2)$ calorons, also known as periodic instantons, and consider invariance under isometries of $S^1timesmathbb{R}^3$ coupled with a non-spatial isometry called the rotation map. In particular, we investigate the fixed points under various cyclic symmetry groups. Our approach utilises a construction akin to the ADHM construction of instantons -- what we call the monad matrix data for calorons -- derived from the work of Charbonneau and Hurtubise. To conclude, we present an example of how investigating these symmetry groups can help to construct new calorons by deriving Nahm data in the case of charge $2$.
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We derive a one-parameter family of gauged Skyrme models from Yang-Mills theory on $S^1timesmathbb{R}^3$, in which skyrmions are well-approximated by calorons and monopoles. In particular we study the spherically symmetric solutions to the model with two distinct classes of boundary conditions, and compare them to calorons and monopoles. Calorons interpolate between instantons and monopoles in certain limits, and we observe similar behaviour in the constructed gauged Skyrme model in the weak and strong coupling limits. This comparison of calorons, monopoles, and skyrmions may be a way to further understand the apparent relationships between skyrmions and monopoles on $mathbb{R}^3$.
We construct finite time blow-up solutions to the 3-dimensional harmonic map flow into the sphere $S^2$, begin{align*} u_t & = Delta u + | abla u|^2 u quad text{in } Omegatimes(0,T) u &= u_b quad text{on } partial Omegatimes(0,T) u(cdot,0) &= u_0 quad text{in } Omega , end{align*} with $u(x,t): bar Omegatimes [0,T) to S^2$. Here $Omega$ is a bounded, smooth axially symmetric domain in $mathbb{R}^3$. We prove that for any circle $Gamma subset Omega$ with the same axial symmetry, and any sufficiently small $T>0$ there exist initial and boundary conditions such that $u(x,t)$ blows-up exactly at time $T$ and precisely on the curve $Gamma$, in fact $$ | abla u(cdot ,t)|^2 rightharpoonup | abla u_*|^2 + 8pi delta_Gamma text{ as } tto T . $$ for a regular function $u_*(x)$, where $delta_Gamma$ denotes the Dirac measure supported on the curve. This the first example of a blow-up solution with a space-codimension 2 singular set, the maximal dimension predicted in the partial regularity theory by Chen-Struwe and Cheng.
Associated to each finite dimensional linear representation of a group $G$, there is a vector bundle over the classifying space $BG$. We introduce a framework for studying this construction in the context of infinite discrete groups, taking into account the topology of representation spaces. This involves studying the homotopy group completion of the topological monoid formed by all unitary (or general linear) representations of $G$, under the monoid operation given by block sum. In order to work effectively with this object, we prove a general result showing that for certain homotopy commutative topological monoids $M$, the homotopy groups of $Omega BM$ can be described explicitly in terms of unbased homotopy classes of maps from spheres into $M$. Several applications are developed. We relate our constructions to the Novikov conjecture; we show that the space of flat unitary connections over the 3-dimensional Heisenberg manifold has extremely large homotopy groups; and for groups that satisfy Kazhdans property (T) and admit a finite classifying space, we show that the reduced $K$-theory class associated to a spherical family of finite dimensional unitary representations is always torsion.
How close is the Dirichlet-to-Neumann (DtN) map to the square root of the corresponding boundary Laplacian? This question has been actively investigated in recent years. Somewhat surprisingly, a lot of techniques involved can be traced back to a newly rediscovered manuscript of Hormander from the 1950s. We present Hormanders approach and its applications, with an emphasis on eigenvalue estimates and spectral asymptotics. In particular, we obtain results for the DtN maps on non-smooth boundaries in the Riemannian setting, the DtN operators for the Helmholtz equation and the DtN operators on differential forms.
We look at the interior operator reconstruction from the point of view of Petz map and study its complexity. We show that Petz maps can be written as precursors under the condition of perfect recovery. When we have the entire boundary system its complexity is related to the volume / action of the wormhole from the bulk operator to the boundary. When we only have access to part of the system, Pythons lunch appears and its restricted complexity depends exponentially on the size of the subsystem one loses access to.
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