Abstract
Counting formulae for general primary fields in free four dimensional conformal field theories of scalars, vectors and matrices are derived. These are specialised to count primaries which obey extremality conditions defined in terms of the dimensions and left or right spins (i.e. in terms of relations between the charges under the Cartan subgroup of SO(4, 2)). The construction of primary fields for scalar field theory is mapped to a problem of determining multi-variable polynomials subject to a system of symmetry and differential constraints. For the extremal primaries, we give a construction in terms of holomorphic polynomial functions on permutation orbifolds, which are shown to be Calabi-Yau spaces.
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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].
R. de Mello Koch and S. Ramgoolam, Strings from Feynman graph counting: without large-N , Phys. Rev. D 85 (2012) 026007 [arXiv:1110.4858] [INSPIRE].
J. Pasukonis and S. Ramgoolam, Quivers as calculators: counting, correlators and Riemann surfaces, JHEP 04 (2013) 094 [arXiv:1301.1980] [INSPIRE].
R. de Mello Koch and S. Ramgoolam, From matrix models and quantum fields to Hurwitz space and the absolute Galois group, arXiv:1002.1634 [INSPIRE].
J. Ben Geloun and S. Ramgoolam, Counting tensor model observables and branched covers of the 2-sphere, arXiv:1307.6490 [INSPIRE].
S. Rychkov and Z.M. Tan, The ϵ-expansion from conformal field theory, J. Phys. A 48 (2015) 29FT01 [arXiv:1505.00963] [INSPIRE].
P. Basu and C. Krishnan, ϵ-expansions near three dimensions from conformal field theory, JHEP 11 (2015) 040 [arXiv:1506.06616] [INSPIRE].
S. Ghosh, R.K. Gupta, K. Jaswin and A.A. Nizami, ϵ-expansion in the Gross-Neveu model from conformal field theory, JHEP 03 (2016) 174 [arXiv:1510.04887] [INSPIRE].
A. Raju, ϵ-expansion in the Gross-Neveu CFT, JHEP 10 (2016) 097 [arXiv:1510.05287] [INSPIRE].
K. Nii, Classical equation of motion and anomalous dimensions at leading order, JHEP 07 (2016) 107 [arXiv:1605.08868] [INSPIRE].
F. Gliozzi, A. Guerrieri, A.C. Petkou and C. Wen, Generalized Wilson-Fisher critical points from the conformal operator product expansion, Phys. Rev. Lett. 118 (2017) 061601 [arXiv:1611.10344] [INSPIRE].
F. Gliozzi, A.L. Guerrieri, A.C. Petkou and C. Wen, The analytic structure of conformal blocks and the generalized Wilson-Fisher fixed points, JHEP 04 (2017) 056 [arXiv:1702.03938] [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].
T.H. Newton and M. Spradlin, Quite a character: the spectrum of Yang-Mills on S 3, Phys. Lett. B 672 (2009) 382 [arXiv:0812.4693] [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].
I. Frenkel and M. Libine Quarternionic analysis, representation theory and physics, Adv. Math. 218 (2008) 1806.
J. Gray, A. Hanany, Y.-H. He, V. Jejjala and N. Mekareeya, SQCD: a geometric apercu, JHEP 05 (2008) 099 [arXiv:0803.4257] [INSPIRE].
A. Hanany, N. Mekareeya and G. Torri, The Hilbert series of adjoint SQCD, Nucl. Phys. B 825 (2010) 52 [arXiv:0812.2315] [INSPIRE].
R. Stanley, Hilbert functions of graded algebras, Adv. Math. 28 (1978) 57.
M.R. Gaberdiel and R. Gopakumar, Higher spins & strings, JHEP 11 (2014) 044 [arXiv:1406.6103] [INSPIRE].
R. de Mello Koch, P. Rabambi, R. Rabe and S. Ramgoolam, Free quantum fields in 4D and Calabi-Yau spaces, to appear.
V.K. Dobrev, V.B. Petkova, S.G. Petrova and I.T. Todorov, Dynamical derivation of vacuum operator product expansion in euclidean conformal quantum field theory, Phys. Rev. D 13 (1976) 887 [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning conformal correlators, JHEP 11 (2011) 071 [arXiv:1107.3554] [INSPIRE].
R. Goodman and N.R. Wallach, Representations and invariants of the classical groups, Cambridge University Press, Cambridge U.K. (1998).
S. Minwalla, Restrictions imposed by superconformal invariance on quantum field theories, Adv. Theor. Math. Phys. 2 (1998) 781 [hep-th/9712074] [INSPIRE].
K. Roumpedakis, Leading order anomalous dimensions at the Wilson-Fisher fixed point from CFT, JHEP 07 (2017) 109 [arXiv:1612.08115] [INSPIRE].
M. Hamermesh, Group theory and its application to physical problems, Addison-Wesley Publishing Company Inc., U.S.A. (1962).
S. Giombi and V. Kirilin, Anomalous dimensions in CFT with weakly broken higher spin symmetry, JHEP 11 (2016) 068 [arXiv:1601.01310] [INSPIRE].
M. Hochster and J. Roberts, Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay, Adv. Math. 13 (1974) 115.
I.R. Klebanov and A.M. Polyakov, AdS dual of the critical O(N ) vector model, Phys. Lett. B 550 (2002) 213 [hep-th/0210114] [INSPIRE].
M.A. Vasiliev, Consistent equation for interacting gauge fields of all spins in (3 + 1)-dimensions, Phys. Lett. B 243 (1990) 378 [INSPIRE].
F.A. Dolan, Counting BPS operators in N = 4 SYM, Nucl. Phys. B 790 (2008) 432 [arXiv:0704.1038] [INSPIRE].
R. de Mello Koch, P. Diaz and H. Soltanpanahi, Non-planar anomalous dimensions in the sl(2) sector, Phys. Lett. B 713 (2012) 509 [arXiv:1111.6385] [INSPIRE].
O. Aharony, J. Marsano, S. Minwalla, K. Papadodimas and M. Van Raamsdonk, The Hagedorn-deconfinement phase transition in weakly coupled large-N gauge theories, Adv. Theor. Math. Phys. 8 (2004) 603 [hep-th/0310285] [INSPIRE].
P.J. Cameron, Combinatorics: topics, techniques, algorithms, Cambridge University Press, Cambridge U.K. (1994).
M.A. Vasiliev, Multiparticle extension of the higher-spin algebra, Class. Quant. Grav. 30 (2013) 104006 [arXiv:1212.6071] [INSPIRE].
O.A. Gelfond and M.A. Vasiliev, Operator algebra of free conformal currents via twistors, Nucl. Phys. B 876 (2013) 871 [arXiv:1301.3123] [INSPIRE].
G. Benkart, T. Halverson and N. Harman, Dimensions of irreducible modules for partition algebras and tensor power multiplicities for symmetric and alternating groups, arXiv:1605.06543.
V.M. Braun, G.P. Korchemsky and D. Mueller, The uses of conformal symmetry in QCD, Prog. Part. Nucl. Phys. 51 (2003) 311 [hep-ph/0306057] [INSPIRE].
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].
D. Berenstein, A Toy model for the AdS/CFT correspondence, JHEP 07 (2004) 018 [hep-th/0403110] [INSPIRE].
H. Lin, O. Lunin and J.M. Maldacena, Bubbling AdS space and 1/2 BPS geometries, JHEP 10 (2004) 025 [hep-th/0409174] [INSPIRE].
L. Grant, P.A. Grassi, S. Kim and S. Minwalla, Comments on 1/16 BPS quantum states and classical configurations, JHEP 05 (2008) 049 [arXiv:0803.4183] [INSPIRE].
J. Pasukonis and S. Ramgoolam, From counting to construction of BPS states in N = 4 SYM, JHEP 02 (2011) 078 [arXiv:1010.1683] [INSPIRE].
J. Pasukonis and S. Ramgoolam, Quantum states to brane geometries via fuzzy moduli spaces of giant gravitons, JHEP 04 (2012) 077 [arXiv:1201.5588] [INSPIRE].
A. Mikhailov, Giant gravitons from holomorphic surfaces, JHEP 11 (2000) 027 [hep-th/0010206] [INSPIRE].
I. Biswas, D. Gaiotto, S. Lahiri and S. Minwalla, Supersymmetric states of N = 4 Yang-Mills from giant gravitons, JHEP 12 (2007) 006 [hep-th/0606087] [INSPIRE].
R. de Mello Koch, M. Dessein, D. Giataganas and C. Mathwin, Giant graviton oscillators, JHEP 10 (2011) 009 [arXiv:1108.2761] [INSPIRE].
R. de Mello Koch and S. Ramgoolam, A double coset ansatz for integrability in AdS/CFT, JHEP 06 (2012) 083 [arXiv:1204.2153] [INSPIRE].
D. Berenstein, Giant gravitons: a collective coordinate approach, Phys. Rev. D 87 (2013) 126009 [arXiv:1301.3519] [INSPIRE].
D. Berenstein, Sketches of emergent geometry in the gauge/gravity duality, Fortsch. Phys. 62 (2014) 776 [arXiv:1404.7052] [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].
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].
J. Pasukonis and S. Ramgoolam, Quivers as calculators: counting, correlators and Riemann surfaces, JHEP 04 (2013) 094 [arXiv:1301.1980] [INSPIRE].
R. de Mello Koch, R. Kreyfelt and N. Nokwara, Finite N quiver gauge theory, Phys. Rev. D 89 (2014) 126004 [arXiv:1403.7592] [INSPIRE].
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de Mello Koch, R., Rabambi, P., Rabe, R. et al. Counting and construction of holomorphic primary fields in free CFT4 from rings of functions on Calabi-Yau orbifolds. J. High Energ. Phys. 2017, 77 (2017). https://doi.org/10.1007/JHEP08(2017)077
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DOI: https://doi.org/10.1007/JHEP08(2017)077