In that sixty-square-meter dormitory, Xiao Yi sat calmly at the large solid-wood desk, looking at the draft papers spread before him.
The contents of those drafts were the conclusions he had finally completed after all this time.
The theorem he had proposed was also a conjecture: Let G be a compact, simple Lie group, and let A be a Yang-Mills field defined on the four-dimensional sphere S^4. If there exists a nonzero Chern number c, then the lowest-energy excitation state of the Yang-Mills field A has a strictly positive mass gap.
Now, he had proven it completely.
And this officially meant that the existence and mass gap problem for the Yang-Mills equations—one of the Millennium Prize Problems, and one of only two mathematical physics problems among them—had become the second of the seven problems to be solved.
He reached out and sorted the drafts on the desk one by one, reading through them again from beginning to end.
The final proof had still originated from the sudden burst of inspiration he had experienced nearly two months ago because of the Hodge standard conjecture. From his countless previous thoughts, he had finally extracted a crucial mathematical method.
The most essential components of that method were Hodge decomposition and vertex algebras.
In the study of complex algebraic varieties, Hodge theory provided an extremely powerful tool for analyzing their geometric and topological properties.
Hodge decomposition was one of the core concepts of Hodge theory. It decomposed de Rham cohomology on a complex algebraic variety into components of (p,q)-type.
On the other hand, as an important tool in Quantum Field Theory and Algebraic Geometry, vertex algebras could be used to describe the algebraic structures in conformal field theory.
Combining Hodge decomposition with the framework of vertex algebras could provide new perspectives and tools for