Harmonic maps as an integrable system

AbstractThe Heisenberg supermagnet model is an important supersymmetric integrable system in (1+1)-dimensions. We construct two types of the (2+1)-dimensional integrable Heisenberg supermagnet models with the quadratic constraints and investigate the integrability of the systems. In terms of the gage transformation, we derive their gage equivalent counterparts. Furthermore, we also construct new solutions of the supersymmetric integrable systems by means of the Bäcklund transformations.

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Renormalised energies and renormalisable singular harmonic maps into a compact manifold on planar domains

Mathematische Annalen ◽

10.1007/s00208-021-02204-8 ◽

2021 ◽

Author(s):

Antonin Monteil ◽

Rémy Rodiac ◽

Jean Van Schaftingen

Keyword(s):

Compact Manifold ◽

Harmonic Maps ◽

Planar Domains

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Circular quiver gauge theories, isomonodromic deformations and $$W_N$$ fermions on the torus

Letters in Mathematical Physics ◽

10.1007/s11005-020-01343-4 ◽

2021 ◽

Vol 111 (3) ◽

Author(s):

Giulio Bonelli ◽

Fabrizio Del Monte ◽

Pavlo Gavrylenko ◽

Alessandro Tanzini

Keyword(s):

Integrable System ◽

Gauge Theories ◽

The Self ◽

Free Fermion ◽

Two Dimensional ◽

Isomonodromic Deformation ◽

Specific Time ◽

Dimensional Torus ◽

Isomonodromic Deformations ◽

Deformation Problem

AbstractWe study the relation between class $$\mathcal {S}$$ S theories on punctured tori and isomonodromic deformations of flat SL(N) connections on the two-dimensional torus with punctures. Turning on the self-dual $$\Omega $$ Ω -background corresponds to a deautonomization of the Seiberg–Witten integrable system which implies a specific time dependence in its Hamiltonians. We show that the corresponding $$\tau $$ τ -function is proportional to the dual gauge theory partition function, the proportionality factor being a nontrivial function of the solution of the deautonomized Seiberg–Witten integrable system. This is obtained by mapping the isomonodromic deformation problem to $$W_N$$ W N free fermion correlators on the torus.

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Exponentially harmonic maps carrying potential

Boletín de la Sociedad Matemática Mexicana ◽

10.1007/s40590-021-00343-1 ◽

2021 ◽

Vol 27 (1) ◽

Author(s):

Yuan-Jen Chiang

Keyword(s):

Harmonic Maps

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The Adjunction Inequality for Weyl-Harmonic Maps

Complex Manifolds ◽

10.1515/coma-2020-0007 ◽

2020 ◽

Vol 7 (1) ◽

pp. 129-140

Author(s):

Robert Ream

Keyword(s):

Minimal Surfaces ◽

Twistor Space ◽

Harmonic Maps ◽

Holomorphic Curves ◽

Weyl Connection ◽

Almost Kähler ◽

Conformal Manifolds

AbstractIn this paper we study an analog of minimal surfaces called Weyl-minimal surfaces in conformal manifolds with a Weyl connection (M4, c, D). We show that there is an Eells-Salamon type correspondence between nonvertical 𝒥-holomorphic curves in the weightless twistor space and branched Weyl-minimal surfaces. When (M, c, J) is conformally almost-Hermitian, there is a canonical Weyl connection. We show that for the canonical Weyl connection, branched Weyl-minimal surfaces satisfy the adjunction inequality\chi \left( {{T_f}\sum } \right) + \chi \left( {{N_f}\sum } \right) \le \pm {c_1}\left( {f*{T^{\left( {1,0} \right)}}M} \right).The ±J-holomorphic curves are automatically Weyl-minimal and satisfy the corresponding equality. These results generalize results of Eells-Salamon and Webster for minimal surfaces in Kähler 4-manifolds as well as their extension to almost-Kähler 4-manifolds by Chen-Tian, Ville, and Ma.

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