Abstract
We define lowest weight polynomials (LWPs), motivated by so(d, 2) representation theory, as elements of the polynomial ring over d × n variables obeying a system of first and second order partial differential equations. LWPs invariant under Sn correspond to primary fields in free scalar field theory in d dimensions, constructed from n fields. The LWPs are in one-to-one correspondence with a quotient of the polynomial ring in d × (n − 1) variables by an ideal generated by n quadratic polynomials. The implications of this description for the counting and construction of primary fields are described: an interesting binomial identity underlies one of the construction algorithms. The product on the ring of LWPs can be described as a commutative star product. The quadratic algebra of lowest weight polynomials has a dual quadratic algebra which is non-commutative. We discuss the possible physical implications of this dual algebra.
References
R. de Mello Koch, P. Rabambi, R. Rabe and S. Ramgoolam, Free quantum fields in 4D and Calabi-Yau spaces, Phys. Rev. Lett. 119 (2017) 161602 [arXiv:1705.04039] [INSPIRE].
R. de Mello Koch, P. Rabambi, R. Rabe and S. Ramgoolam, Counting and construction of holomorphic primary fields in free CFT 4 from rings of functions on Calabi-Yau orbifolds, JHEP 08 (2017) 077 [arXiv:1705.06702] [INSPIRE].
B. Henning, X. Lu, T. Melia and H. Murayama, Operator bases, S-matrices and their partition functions, JHEP 10 (2017) 199 [arXiv:1706.08520] [INSPIRE].
R. de Mello Koch and S. Ramgoolam, CFT 4 as SO(4, 2)-invariant TFT 2, Nucl. Phys. B 890 (2014) 302 [arXiv:1403.6646] [INSPIRE].
F.A. Dolan, Character formulae and partition functions in higher dimensional conformal field theory, J. Math. Phys. 47 (2006) 062303 [hep-th/0508031] [INSPIRE].
D.A. Cox, J.B. Little and D. O’Shea, Ideals, varieties, and algorithms, fourth edition, Springer, Cham, Switzerland, (2015).
D. Eisenbud, Commutative algebra. With a view toward algebraic geometry, Grad. Texts Math. 150, Springer-Verlag, New York, U.S.A., (1995) [ISBN:0-387-94268-8].
E. Grigorescu, Hilbert series and free resolutions, Senior thesis, Bard College, Annandale-on-Hudson, NY, U.S.A., (2003).
G. Wu, Koszul algebras and Koszul duality, Masters thesis, University of Ottawa, Ottawa, ON, Canada, (2016).
R.C. King, Young tableaux, Schur functions and SU(2) plethysms, J. Phys. A 18 (1985) 2429.
W. Fulton and J. Harris, Representation theory: a first course, Springer, New York, U.S.A., (1991).
C.W. Ayoub, On constructing bases for ideals in polynomial rings over the integers, J. Num. Theor. 17 (1983) 204.
R. De Mello Koch, P. Rambambi and H.J.R. Van Zyl, From spinning primaries to permutation orbifolds, JHEP 04 (2018) 104 [arXiv:1801.10313] [INSPIRE].
A. Ram, Characters of Brauer’s centralizer algebras, Pacific J. Math. 169 (1985) 173.
S. Corley, A. Jevicki and S. Ramgoolam, Exact correlators of giant gravitons from dual N = 4 SYM theory, Adv. Theor. Math. Phys. 5 (2002) 809[hep-th/0111222] [INSPIRE].
J. Pasukonis and S. Ramgoolam, Quivers as calculators: counting, correlators and Riemann surfaces, JHEP 04 (2013) 094 [arXiv:1301.1980] [INSPIRE].
Y. Kimura, Noncommutative Frobenius algebras and open-closed duality, arXiv:1701.08382 [INSPIRE].
P. Mattioli and S. Ramgoolam, Permutation centralizer algebras and multi-matrix invariants, Phys. Rev. D 93 (2016) 065040 [arXiv:1601.06086] [INSPIRE].
R. Bhattacharyya, S. Collins and R. de Mello Koch, Exact multi-matrix correlators, JHEP 03 (2008) 044 [arXiv:0801.2061] [INSPIRE].
R. Bhattacharyya, R. de Mello Koch and M. Stephanou, Exact multi-restricted Schur polynomial correlators, JHEP 06 (2008) 101 [arXiv:0805.3025] [INSPIRE].
Y. Kimura and S. Ramgoolam, Branes, anti-branes and Brauer algebras in gauge-gravity duality, JHEP 11 (2007) 078 [arXiv:0709.2158] [INSPIRE].
T.W. Brown, P.J. Heslop and S. Ramgoolam, Diagonal multi-matrix correlators and BPS operators in N = 4 SYM, JHEP 02 (2008) 030 [arXiv:0711.0176] [INSPIRE].
T.W. Brown, P.J. Heslop and S. Ramgoolam, Diagonal free field matrix correlators, global symmetries and giant gravitons, JHEP 04 (2009) 089 [arXiv:0806.1911] [INSPIRE].
A. Polischchuk and L. Positselski, Quadratic algebras, Univ. Lect. Ser. 37, American Mathematical Society, Providence, RI, U.S.A., (2005).
Koszul duality Wikipedia article, https://en.wikipedia.org/wiki/Koszul_duality.
R.J. Szabo, Quantum field theory on noncommutative spaces, Phys. Rept. 378 (2003) 207 [hep-th/0109162] [INSPIRE].
R. Fröberg and C. Löfwal, Koszul homology and Lie algebras with application to generic forms and points, Homology Homotopy Appl. 4 (2002) 227.
S. Benvenuti, B. Feng, A. Hanany and Y.-H. He, Counting BPS operators in gauge theories: quivers, Syzygies and Plethystics, JHEP 11 (2007) 050 [hep-th/0608050] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1806.01085
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
de Mello Koch, R., Ramgoolam, S. Free field primaries in general dimensions: counting and construction with rings and modules. J. High Energ. Phys. 2018, 88 (2018). https://doi.org/10.1007/JHEP08(2018)088
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2018)088
Keywords
- AdS-CFT Correspondence
- Conformal and W Symmetry
- Differential and Algebraic Geometry