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Stable Yang–Mills–Higgs pair

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Concept in differential geometry

In differential geometry and especially Yang–Mills theory, a (weakly) stable Yang–Mills–Higgs (YMH) pair is a Yang–Mills–Higgs pair around which the Yang–Mills–Higgs action functional is positively or even strictly positively curved. Yang–Mills–Higgs pairs are solutions of the Yang–Mills–Higgs equations following from them being local extrema of the curvature of both fields, hence critical points of the Yang–Mills-Higgs action functional, which are determined by a vanishing first derivative of a variation. (Weakly) stable Yang–Mills-Higgs pairs furthermore have a positive or even strictly positive curved neighborhood and hence are determined by a positive or even strictly positive second derivative of a variation.

Definition

Let G {\displaystyle G} be a compact Lie group with Lie algebra g {\displaystyle {\mathfrak {g}}} and E ↠ B {\displaystyle E\twoheadrightarrow B} be a principal G {\displaystyle G} -bundle with a compact orientable Riemannian manifold B {\displaystyle B} having a metric g {\displaystyle g} and a volume form vol g {\displaystyle \operatorname {vol} _{g}} . Let Ad ⁡ ( E ) = E × G g {\displaystyle \operatorname {Ad} (E)=E\times _{G}{\mathfrak {g}}} be its adjoint bundle. Ω Ad 1 ( E , g ) ≅ Ω 1 ( B , Ad ⁡ ( E ) ) {\displaystyle \Omega _{\operatorname {Ad} }^{1}(E,{\mathfrak {g}})\cong \Omega ^{1}(B,\operatorname {Ad} (E))} is the space of connections, which are either under the adjoint representation Ad {\displaystyle \operatorname {Ad} } invariant Lie algebra–valued or vector bundle–valued differential forms. Since the Hodge star operator ⋆ {\displaystyle \star } is defined on the base manifold B {\displaystyle B} as it requires the metric g {\displaystyle g} and the volume form vol g {\displaystyle \operatorname {vol} _{g}} , the second space is usually used.

The Yang–Mills–Higgs action functional is given by:

YMH : Ω 1 ( B , Ad ⁡ ( E ) ) × Γ ∞ ( B , Ad ⁡ ( E ) ) → R , YMH ⁡ ( A , Φ ) := ∫ B ‖ F A ‖ 2 + ‖ d A Φ ‖ 2 d vol g . {\displaystyle \operatorname {YMH} \colon \Omega ^{1}(B,\operatorname {Ad} (E))\times \Gamma ^{\infty }(B,\operatorname {Ad} (E))\rightarrow \mathbb {R} ,\operatorname {YMH} (A,\Phi ):=\int _{B}\|F_{A}\|^{2}+\|\mathrm {d} _{A}\Phi \|^{2}\mathrm {d} \operatorname {vol} _{g}.}

A Yang–Mills–Higgs pair A ∈ Ω 1 ( B , Ad ⁡ ( E ) ) {\displaystyle A\in \Omega ^{1}(B,\operatorname {Ad} (E))} and Φ ∈ Γ ∞ ( B , Ad ⁡ ( E ) ) {\displaystyle \Phi \in \Gamma ^{\infty }(B,\operatorname {Ad} (E))} , hence which fulfill the Yang–Mills–Higgs equations, is called stable if:

d 2 d t 2 YMH ⁡ ( α ( t ) , φ ( t ) ) | t = 0 > 0 {\displaystyle {\frac {\mathrm {d} ^{2}}{\mathrm {d} t^{2}}}\operatorname {YMH} (\alpha (t),\varphi (t))\vert _{t=0}>0}

for every smooth family α : ( − ε , ε ) → Ω 1 ( B , Ad ⁡ ( E ) ) {\displaystyle \alpha \colon (-\varepsilon ,\varepsilon )\rightarrow \Omega ^{1}(B,\operatorname {Ad} (E))} with α ( 0 ) = A {\displaystyle \alpha (0)=A} and φ : ( − ε , ε ) → Γ ∞ ( B , Ad ⁡ ( E ) ) {\displaystyle \varphi \colon (-\varepsilon ,\varepsilon )\rightarrow \Gamma ^{\infty }(B,\operatorname {Ad} (E))} with φ ( 0 ) = Φ {\displaystyle \varphi (0)=\Phi } . It is called weakly stable if only ≥ 0 {\displaystyle \geq 0} holds. A Yang–Mills–Higgs pair, which is not weakly stable, is called instable. For comparison, the condition to be a Yang–Mills–Higgs pair is:

d d t YMH ⁡ ( α ( t ) , φ ( t ) ) | t = 0 = 0. {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\operatorname {YMH} (\alpha (t),\varphi (t))\vert _{t=0}=0.}

Properties

  • Let ( A , Φ ) {\displaystyle (A,\Phi )} be a weakly stable Yang–Mills–Higgs pair on S n {\displaystyle S^{n}} , then the following claims hold:
    • If n = 4 {\displaystyle n=4} , then A {\displaystyle A} is a Yang–Mills connection ( d A ⋆ F A = 0 {\displaystyle \mathrm {d} _{A}\star F_{A}=0} ) as well as d A Φ = 0 {\displaystyle \mathrm {d} _{A}\Phi =0} and ‖ Φ ‖ = 1 {\displaystyle \|\Phi \|=1} .
    • If n ≥ 5 {\displaystyle n\geq 5} , then A {\displaystyle A} is flat ( F A = 0 {\displaystyle F_{A}=0} ) as well as d A Φ = 0 {\displaystyle \mathrm {d} _{A}\Phi =0} and ‖ Φ ‖ = 1 {\displaystyle \|\Phi \|=1} .

See also

References

  1. de los Ríos, Santiago Quintero (2020-12-16). "Connections on principal bundles" (PDF). homotopico.com. Theorem 3.7. Retrieved 2024-11-09.
  2. "Lecture 3: The Yang–Mills equations" (PDF). empg.maths.ed.ac.uk. Retrieved 2024-11-24.
  3. Hu, Zhi; Hu, Sen (2015-02-06). "Degenerate and Stable Yang-Mills-Higgs Pairs". arXiv:1502.01791 .
  4. Cheng, Da Rong (2021). "Stable Solutions to the Abelian Yang–Mills–Higgs Equations on S2 andT2" (PDF). The Journal of Geometric Analysis. 31: 9551–9572, Definition 3.1. doi:10.1007/s12220-021-00619-y. Retrieved 2024-10-27.
  5. ^ Han, Xiaoli; Jin, Xishen; Wen, Yang (2023-03-01). "Stability and energy identity for Yang-Mills-Higgs pairs". Journal of Mathematical Physics. 64 (2). arXiv:2303.00270. doi:10.1063/5.0130905.

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