Counting BPS operators in gauge theories: quivers, syzygies and plethystics
Counting BPS operators in gauge theories: quivers, syzygies and plethystics
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DOI：
10.1088/1126-6708/2007/11/050
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发表时间：
2006-08
影响因子：
5.4
通讯作者：
S. Benvenuti;Bo Feng;A. Hanany;Yang-Hui He
中科院分区：
文献类型：
--
作者：
S. Benvenuti;Bo Feng;A. Hanany;Yang-Hui He
我们开发了一个系统的和有效的方法计数单迹和多迹BPS运营商与两个超荷，为世界体积规范理论的$N$D-膜探针为$N \to \infty$和有限$N$。该技术适用于一般奇点，orbifold，复曲面，非复曲面，完整的交叉点，等等，甚至几何形状的精确场论，尚未知道。所谓的“Plethystic Exponential”在（1）Calabi-Yau的定义方程，（2）单迹BPS算子的生成函数和（3）多迹算子的生成函数之间提供了一个简单的桥梁。在数学上，规范理论、代数几何、组合数学和数论之间迷人而复杂的相互关系以体积学和合朔的形式展现出来。
We develop a systematic and efficient method of counting single-trace and multi-trace BPS operators with two supercharges, for world-volume gauge theories of $N$ D-brane probes for both $N \to \infty$ and finite $N$. The techniques are applicable to generic singularities, orbifold, toric, non-toric, complete intersections, et cetera, even to geometries whose precise field theory duals are not yet known. The so-called ``Plethystic Exponential'' provides a simple bridge between (1) the defining equation of the Calabi-Yau, (2) the generating function of single-trace BPS operators and (3) the generating function of multi-trace operators. Mathematically, fascinating and intricate inter-relations between gauge theory, algebraic geometry, combinatorics and number theory exhibit themselves in the form of plethystics and syzygies.