scholarly journals Energy gap for Yang–Mills connections, II: Arbitrary closed Riemannian manifolds

2017 ◽  
Vol 312 ◽  
pp. 547-587 ◽  
Author(s):  
Paul M.N. Feehan
Keyword(s):  
Riemannian Manifolds ◽  
Energy Gap ◽  
Yang Mills

scholarly journals Local Regularity for the Harmonic Map and Yang–Mills Heat Flows

2021 ◽  
Author(s):  
Ahmad Afuni
Keyword(s):  
Riemannian Manifolds ◽  
Harmonic Map ◽  
Local Regularity ◽  
Yang Mills ◽  
Heat Flows ◽  
Singular Sets ◽  
Regularity Results ◽  
Local Monotonicity ◽  
Partial Differ ◽  
Singular Time

AbstractWe establish new local regularity results for the harmonic map and Yang–Mills heat flows on Riemannian manifolds of dimension greater than 2 and 4, respectively, obtaining criteria for the smooth local extensibility of these flows. As a corollary, we obtain new characterisations of singularity formation and use this to obtain a local estimate on the Hausdorff measure of the singular sets of these flows at the first singular time. Finally, we show that smooth blow-ups at rapidly forming singularities of these flows are necessarily nontrivial and admit a positive lower bound on their heat ball energies. These results crucially depend on some local monotonicity formulæ for these flows recently established by Ecker (Calc Var Partial Differ Equ 23(1):67–81, 2005) and the Afuni (Calc Var 555(1):1–14, 2016; Adv Calc Var 12(2):135–156, 2019).

scholarly journals An Energy Gap for Yang-Mills Connections

2010 ◽  
Vol 298 (2) ◽  
pp. 515-522 ◽  
Author(s):  
Claus Gerhardt
Keyword(s):  
Energy Gap ◽  
Yang Mills

scholarly journals Twistor spaces for real four-dimensional Lorentzian manifolds

1994 ◽  
Vol 134 ◽  
pp. 107-135 ◽  
Author(s):  
Yoshinori Machida ◽  
Hajime Sato
Keyword(s):  
Riemannian Manifolds ◽  
Complex Space ◽  
Lorentzian Manifolds ◽  
Field Equations ◽  
Curvature Condition ◽  
Twistor Spaces ◽  
Analytical Geometry ◽  
Yang Mills ◽  
3 Dimensional ◽  
Conformally Invariant

It is R. Penrose who constructed the twistor theory which gives a correspondence between complex space-times and 3-dimensional complex manifolds called twistor spaces. He and his colleagues investigated conformally invariant equations (e.g. massless field equations, self-dual Yang-Mills equations) on the space-time by transforming them into objects in complex analytical geometry. See e.g. Penrose-Ward [P-W] or Ward-Wells [W-W]. After that, Atiyah-Hitchin-Singer ([A-H-S], cf. [Fr]) constructed the twistor spaces corresponding to real 4-dimensional Riemannian manifolds. Their construction as well as that of Penrose is mainly effective under the condition of the self-duality. In this paper we will construct twistor spaces more geometrically from real 4-dimensional Lorentzian manifolds under a suitable curvature condition.

scholarly journals A SPECTRAL APPROACH TO YANG-MILLS THEORY

2002 ◽  
Vol 17 (06n07) ◽  
pp. 926-935
Author(s):  
GIAMPIERO ESPOSITO
Keyword(s):  
Riemannian Manifolds ◽  
Differential Operators ◽  
Coulomb Gauge ◽  
Point Of View ◽  
Order Differential Operator ◽  
Matrix Elements ◽  
Four Dimensions ◽  
Yang Mills ◽  
First Order ◽  
Pseudo Differential Operators

Yang–Mills theory in four dimensions is studied by using the Coulomb gauge. The Coulomb gauge Hamiltonian involves integration of matrix elements of an operator [Formula: see text] built from the Laplacian and from a first-order differential operator. The operator [Formula: see text] is studied from the point of view of spectral theory of pseudo-differential operators on compact Riemannian manifolds, both when self-adjointness holds and when it is not fulfilled. In both cases, well-defined matrix elements of [Formula: see text] are evaluated as a first step towards the more difficult problems of quantized Yang–Mills theory.

scholarly journals Calabi-Yau Manifolds, Hermitian Yang-Mills Instantons, and Mirror Symmetry

2017 ◽  
Vol 2017 ◽  
pp. 1-27
Author(s):  
Hyun Seok Yang ◽  
Sangheon Yun
Keyword(s):  
Gauge Theory ◽  
Riemannian Manifolds ◽  
Mirror Symmetry ◽  
Vector Spaces ◽  
Spinor Representation ◽  
Yang Mills ◽  
Spin Manifolds ◽  
Mirror Pair ◽  
Irreducible Spinor ◽  
Calabi Yau Manifolds

We address the issue of why Calabi-Yau manifolds exist with a mirror pair. We observe that the irreducible spinor representation of the Lorentz group Spin(6) requires us to consider the vector spaces of two forms and four forms on an equal footing. The doubling of the two-form vector space due to the Hodge duality doubles the variety of six-dimensional spin manifolds. We explore how the doubling is related to the mirror symmetry of Calabi-Yau manifolds. Via the gauge theory formulation of six-dimensional Riemannian manifolds, we show that the curvature tensor of a Calabi-Yau manifold satisfies the Hermitian Yang-Mills equations on the Calabi-Yau manifold. Therefore, the mirror symmetry of Calabi-Yau manifolds can be recast as the mirror pair of Hermitian Yang-Mills instantons. We discuss the mirror symmetry from the gauge theory perspective.

scholarly journals A proof of energy gap for Yang–Mills connections

2017 ◽  
Vol 355 (8) ◽  
pp. 910-913 ◽  
Author(s):  
Teng Huang
Keyword(s):  
Energy Gap ◽  
Yang Mills
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