高奇数维杨-米尔斯方程的三个自相似解
Three self-similar solutions of Yang-Mills equations in high odd dimensions
AI总结：
本文研究高奇数维闵可夫斯基时空中SO(d)规范群的球对称杨-米尔斯方程，证明存在恰好N个光滑自相似解，并发现对于所有奇数d≥11，N=3，其中两个解具有闭式表达式。
摘要：
我们考虑$d+1$维闵可夫斯基时空中规范群为$SO(d)$的球对称杨-米尔斯方程。对于任意给定的奇数$d\geq 11$，我们证明了恰好存在$N$个光滑自相似解（模反射对称性），其中$N$是显式多项式$P_m(z)$（次数$m=(d-5)/2$）在区间$0<z<1$内零点的个数。$N$可以通过显式计算算法确定。我们对大奇数维的广泛计算表明，对于所有奇数$d\geq 11$，$N=3$。其中两个自相似解具有闭式表达式：一个先前已知，另一个似乎是新的。我们的结果指向高维杨-米尔斯方程可能的爆破情景相对简单的图景。除了纯数学兴趣外，这种自相似爆破的刚性可能也与物理相关，因为它限制了在弦理论启发的额外维设置和全息模型中出现的更高维杨-米尔斯理论中非阿贝尔规范场的可能紫外动力学。
英文摘要：
We consider spherically symmetric Yang-Mills equations with gauge group $SO(d)$ in $d+1$ dimensional Minkowski spacetime. For any given odd $d\geq 11$, we establish existence and uniqueness (modulo reflection symmetry) of exactly $N$ smooth self-similar solutions, where $N$ is the number of zeros of an explicit polynomial $P_m(z)$ of degree $m=(d-5)/2$ in the interval $0<z<1$. The number $N$ can be determined algorithmically by an explicit computation. Our extensive computations for large odd dimensions suggest that $N=3$ for all odd $d\geq 11$. Two of these self-similar solutions admit closed-form expressions: one has been known previously, while the other appears to be new. Our result points toward a relatively simple landscape of possible blowup scenarios for high-dimensional Yang-Mills equations. Beyond its purely mathematical interest, this rigidity of self-similar blowup may also be relevant from a physical perspective, as it constrains the possible ultraviolet dynamics of non-abelian gauge fields in higher-dimensional Yang-Mills theories arising in string-inspired extra-dimensional setups and in holographic models.
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