Yang-Mills – MathRlife

Van Abel May 10, 2023StatusArXivchern classes, minimizing, Taubes construction, Yang-Mills Leave a comment

DIRECT MINIMIZING METHOD FOR YANG-MILLS ENERGY OVER SO(3) BUNDLE

DIRECT MINIMIZING METHOD FOR YANG-MILLS ENERGY OVER SO(3) BUNDLE In this paper, we use the direct minimizing method to find Yang- Mills connections for SO(3) bundles over closed four manifolds. By constructing test connections, we prove that a minimizing sequence converges strongly to a minimizer under certain assumptions. In case the strong convergence fails, we find an anti-selfdual (or selfdual) connection.

Van Abel May 10, 2023StatusArXivblowup, expansion of connection forms, Yang-Mills Leave a comment

ON THE BLOW-UP OF YANG-MILLS FIELDS IN DIMENSION FOUR

ON THE BLOW-UP OF YANG-MILLS FIELDS IN DIMENSION FOUR In this paper, we study the blow-up of a sequence of Yang-Mills connection with bounded energy on a four manifold. We prove a set of equations relating the geometry of the bubble connection at the infinity with the geometry of the limit connection at the energy concentration point. These equations exclude certain scenarios from happening, for example, there is no sequence of Yang-Mills SU(2) connections on S4 converging to an ASD … Continue reading “ON THE BLOW-UP OF YANG-MILLS FIELDS IN DIMENSION FOUR”

Van Abel May 27, 2017MATHYang-Mills, 椭圆性 Leave a comment

Yang–Mills方程的椭圆性验证

通过计算, 好像Yang–Mills方程即使在库伦规范下也不是严格椭圆的啊? 我记得Yang–Mills方程主项是$dd^*A+d^*dA$, 其中$A=A_idx^i$. 则其弱形式是 $$ \int\langle d^*A,d^*B\rangle+\langle dA,dB\rangle=0,\quad\forall B=B_jdx^j\in C_0^\infty. $$ 直接计算, 我们知道 \begin{align*} dA&=d(A_idx^i)=\partial_jA_idx^j\wedge dx^i\\ dB&=\partial_k B_ld^k\wedge dx^l\\ d^*A&=-*d*A=-*d\left((-1)^{i-1}A_idx^1\wedge\cdots\wedge\widehat{dx^i}\wedge\cdots\wedge dx^n\right)\\ &=-*(\partial_iA_idx^1\wedge\cdots dx^n)\\ &=-\partial_iA_i\\ d^*B&=-\partial_kB_k. \end{align*}