Counting BPS operators in gauge theories: quivers, syzygies and plethystics

By S. Benvenuti, Bo Feng, A. Hanany, Yang-Hui He. 2006

455 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/a64321ac6e1fafafd9738aa0ff16f98e5d077157

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. Alberto Zaffaroni
2. A. Hanany
3. Yang-Hui He
4. Bo Feng
5. S. Benvenuti
6. D. Forcella
7. Cyril Closset
8. Y. Imamura

In Nuclear and High Energy Physics

1. Alberto Zaffaroni
2. A. Hanany
3. Hee-Cheol Kim
4. F. Benini
5. Cyril Closset
6. D. Martelli
7. Y. Imamura
8. Seok Kim

In Statistical and Nonlinear Physics

1. Alberto Zaffaroni
2. A. Hanany
3. Adam Martin
4. Yang-Hui He
5. Y. Imamura
6. Cyril Closset
7. D. Forcella
8. Seok Kim

Citations in the corpus

• THE MASTER SPACE OF N=1 GAUGE THEORIES (2008, methodology): “Because the dummy variable t keeps track of the charge, we can think of the components as chemical potentials [9, 12].”
• SQCD: A Geometric Apercu (2008, background): “14), the generating function can be easily computed from the knowledge that the modul space is a complete intersection (See [13] for a detailed discussion on this).”
• Supersymmetry on curved spaces and holography (2012, unknown): “[152, 159, 166].”
• Hilbert Series for Constructing Lagrangians: expanding the phenomenologist's toolbox (2015, background): “6) is an example of a plethystic exponential [7, 9, 10, 13, 21], a generating function of all symmetric combinations of its argument (see A.”
• On instantons as Kaluza-Klein modes of M5-branes (2011, background): “Perhaps a subtlety in imposing constraints [58, 59] should be properly understood.”
• Counting gauge invariants: The Plethystic program (2007, methodology): “Take the example of G = D̂4, the binary dihedral group of order 8, which was investigated in detail in [4], the Hilbert series is the Molien series”
• Comments on twisted indices in 3d supersymmetric gauge theories (2016, background): “81) becomes the ‘Molien formula’ for the Higgs branch HS [35, 84]:”
• The Hilbert series of the one instanton moduli space (2010, background): “) instanton moduli space is therefore C2N=Z 2. One can see the last statement clearly from the Hilbert series. The Hilbert series { 19 { of C2N=Z 2 is given by the discrete Molien formula (see, e.g., [30]): g(t;x 1;:::;x N;C2N=Z 2) = 1 2 PE [1;0;:::;0] Sp(N)t + PE [1;0;:::;0] Sp(N)( t) = X1 k=0 [2k;”
• Relation between the 4d superconformal index and the S3 partition function (2011, methodology): “This quantity is exactly calculable, and is conveniently used as a tool to check Seiberg-duality[15, 16, 17, 18] and AdS5/CFT4[13, 19, 20, 21, 22].”