Abstract
The gauge invariant degrees of freedom of matrix models based on an N × N complex matrix, with U(N) gauge symmetry, contain hidden free particle structures. These are exhibited using triangular matrix variables via the Schur decomposition. The Brauer algebra basis for complex matrix models developed earlier is useful in projecting to a sector which matches the state counting of N free fermions on a circle. The Brauer algebra projection is characterized by the vanishing of a scale invariant laplacian constructed from the complex matrix. The special case of N = 2 is studied in detail: the ring of gauge invariant functions as well as a ring of scale and gauge invariant differential operators are characterized completely. The orthonormal basis of wavefunctions in this special case is completely characterized by a set of five commuting Hamiltonians, which display free particle structures. Applications to the reduced matrix quantum mechanics coming from radial quantization in \( \mathcal{N} = 4 \) SYM are described. We propose that the string dual of the complex matrix harmonic oscillator quantum mechanics has an interpretation in terms of strings and branes in 2 + 1 dimensions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. Ginibre, Statistical ensembles of complex, quaternion, and real matrices, J. Math. Phys. 6 (1965) 440.
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] [SPIRES].
J. McGreevy, L. Susskind and N. Toumbas, Invasion of the giant gravitons from Anti-de Sitter space, JHEP 06 (2000) 008 [hep-th/0003075] [SPIRES].
V. Balasubramanian, M. Berkooz, A. Naqvi and M.J. Strassler, Giant gravitons in conformal field theory, JHEP 04 (2002) 034 [hep-th/0107119] [SPIRES].
J.M. Maldacena and A. Strominger, AdS 3 black holes and a stringy exclusion principle, JHEP 12 (1998) 005 [hep-th/9804085] [SPIRES].
A. Jevicki and S. Ramgoolam, Non-commutative gravity from the AdS/CFT correspondence, JHEP 04 (1999) 032 [hep-th/9902059] [SPIRES].
R.d.M. Koch and J. Murugan, Emergent spacetime, arXiv:0911.4817 [SPIRES].
D. Berenstein, A toy model for the AdS/CFT correspondence, JHEP 07 (2004) 018 [hep-th/0403110] [SPIRES].
H. Lin, O. Lunin and J.M. Maldacena, Bubbling AdS space and 1/2 BPS geometries, JHEP 10 (2004) 025 [hep-th/0409174] [SPIRES].
B.U. Eden, P.S. Howe, A. Pickering, E. Sokatchev and P.C. West, Four-point functions in N = 2 superconformal field theories, Nucl. Phys. B 581 (2000) 523 [hep-th/0001138] [SPIRES].
B.U. Eden, P.S. Howe, E. Sokatchev and P.C. West, Extremal and next-to-extremal n-point correlators in four-dimensional SCFT, Phys. Lett. B 494 (2000) 141 [hep-th/0004102] [SPIRES].
Y. Kimura and S. Ramgoolam, Branes, anti-branes and Brauer algebras in gauge-gravity duality, JHEP 11 (2007) 078 [arXiv:0709.2158] [SPIRES].
J.R. Stembridge, Rational tableaux and the tensor algebra of gl(n), J. Combi. Theory A 46 (1987) 79.
K. Koike, On the decomposition of tensor products of the representations of the classical groups: by means of the universal characters, Adv. Math. 74 (1989) 57.
G. Benkart, M. Chakrabarti, T. Halverson, R. Leduc, C. Lee and J. Stroomer, Tensor product representations of general linear groups and their connections with Brauer algebras, J. Algebra 166 (1994) 529.
T. Halverson, Characters of the centralizer algebras of mixed tensor representations of Gl(r, C) and the quantum group U q (gl(r, C)), Pacific J. Math 174 (1996) 359.
J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [hep-th/9711200] [SPIRES].
V. Balasubramanian, J. de Boer, V. Jejjala and J. Simon, Entropy of near-extremal black holes in AdS 5, JHEP 05 (2008) 067 [arXiv:0707.3601] [SPIRES].
I.K. Kostov, M. Staudacher and T. Wynter, Complex matrix models and statistics of branched coverings of 2D surfaces, Commun. Math. Phys. 191 (1998) 283 [hep-th/9703189] [SPIRES].
C. Kristjansen, J. Plefka, G.W. Semenoff and M. Staudacher, A new double-scaling limit of N = 4 super Yang-Mills theory and PP-wave strings, Nucl. Phys. B 643 (2002) 3 [hep-th/0205033] [SPIRES].
G. Akemann and G. Vernizzi, Characteristic polynomials of complex random matrix models, Nucl. Phys. B 660 (2003) 532 [hep-th/0212051] [SPIRES].
A. Alexandrov, A. Mironov and A. Morozov, BGWM as second constituent of complex matrix model, JHEP 12 (2009) 053 [arXiv:0906.3305] [SPIRES].
A. Cox, M. De Visscher, S. Doty and P. Martin, On the blocks of the walled Brauer algebra, J. Algebra 320 (2007) 169 [arXiv:0709.0851].
M. Masuku and J.P. Rodrigues, Laplacians in polar matrix coordinates and radial fermionization in higher dimensions, arXiv:0911.2846 [SPIRES].
Y. Kimura and S. Ramgoolam, Enhanced symmetries of gauge theory and resolving the spectrum of local operators, Phys. Rev. D 78 (2008) 126003 [arXiv:0807.3696] [SPIRES].
I.R. Klebanov, String theory in two-dimensions, hep-th/9108019 [SPIRES].
P.H. Ginsparg and G.W. Moore, Lectures on 2D gravity and 2D string theory, hep-th/9304011 [SPIRES].
S. Alexandrov, Matrix quantum mechanics and two-dimensional string theory in non-trivial backgrounds, hep-th/0311273 [SPIRES].
M.R. Douglas, Conformal field theory techniques for large-N group theory, hep-th/9303159 [SPIRES].
M.R. Douglas, Conformal field theory techniques in large-N Yang-Mills theory, hep-th/9311130 [SPIRES].
L.-L. Chau and Y. Yu, Unitary polynomials in normal matrix models and Laughlin’s wave functions for the fractional quantum Hall effect, UCDPHYS-PUB-91-13 [SPIRES].
L.-L. Chau and O. Zaboronsky, On the structure of correlation functions in the normal matrix model, Commun. Math. Phys. 196 (1998) 203 [hep-th/9711091] [SPIRES].
I.R. Klebanov, J.M. Maldacena and N. Seiberg, Unitary and complex matrix models as 1D type 0 strings, Commun. Math. Phys. 252 (2004) 275 [hep-th/0309168] [SPIRES].
C. Itzykson and J.B. Zuber, The planar approximation. 2, J. Math. Phys. 21 (1980) 411 [SPIRES].
J.P. Rodrigues, Large-N spectrum of two matrices in a harmonic potential and BMN energies, JHEP 12 (2005) 043 [hep-th/0510244] [SPIRES].
J.P. Rodrigues and A. Zaidi, Non supersymmetric strong coupling background from the large-N quantum mechanics of two matrices coupled via a Yang-Mills interaction, arXiv:0807.4376 [SPIRES].
C.D. Meyer, Matrix analysis and applied linear algebra, SIAM, U.S.A (2000), see pag. 508.
M.L. Mehta, Random matrices, 3rd edition, Academic Press, U.S.A. (2004).
Y. Takayama and A. Tsuchiya, Complex matrix model and fermion phase space for bubbling AdS geometries, JHEP 10 (2005) 004 [hep-th/0507070] [SPIRES].
L. Castellani, On G/H geometry and its use in M-theory compactifications, Annals Phys. 287 (2001) 1 [hep-th/9912277] [SPIRES].
G. Olshanski, An introduction to harmonic analysis on the infinite symmetric group, math/0311369.
A.A. Kirillov, Elements of the theory of representations, Springer, U.S.A. (1972).
T. Ortín, Gravity and Strings, Cambridge University Press, Cambridge U.K. (2007), pag. 603.
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] [SPIRES].
B. Feng, A. Hanany and Y.-H. He, Counting gauge invariants: the plethystic program, JHEP 03 (2007) 090 [hep-th/0701063] [SPIRES].
D.P. Zelobenko, Compact Lie groups and their representations, American Mathematical Society, U.S.A. (1973), see p. 156.
S. Corley and S. Ramgoolam, Finite factorization equations and sum rules for BPS correlators in N = 4 SYM theory, Nucl. Phys. B 641 (2002) 131 [hep-th/0205221] [SPIRES].
F.A. Dolan, Counting BPS operators in N = 4 SYM, Nucl. Phys. B 790 (2008) 432 [arXiv:0704.1038] [SPIRES].
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] [SPIRES].
R. Bhattacharyya, S. Collins and R.d.M. Koch, Exact multi-matrix correlators, JHEP 03 (2008) 044 [arXiv:0801.2061] [SPIRES].
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] [SPIRES].
S. Collins, Restricted Schur polynomials and finite N counting, Phys. Rev. D 79 (2009) 026002 [arXiv:0810.4217] [SPIRES].
J.F. Willenbring, Stable Hilbert series of S(g)K for classical groups, math/0510649.
A. Hashimoto, S. Hirano and N. Itzhaki, Large branes in AdS and their field theory dual, JHEP 08 (2000) 051 [hep-th/0008016] [SPIRES].
A. Ghodsi, A.E. Mosaffa, O. Saremi and M.M. Sheikh-Jabbari, LLL vs. LLM: half BPS sector of N = 4 SYM equals to quantum Hall system, Nucl. Phys. B 729 (2005) 467 [hep-th/0505129] [SPIRES].
J. Polchinski, String theory. Volume 1: an introduction to the bosonic string, Cambridge University Press, Cambridge U.K. (1998), see pag. 39.
T. Yoneya, Extended fermion representation of multi-charge 1/2-BPS operators in AdS/CFT: towards field theory of D-branes, JHEP 12 (2005) 028 [hep-th/0510114] [SPIRES].
V. Pasquier, A lecture on the Calogero-Sutherland models, hep-th/9405104 [SPIRES].
A. Agarwal and A.P. Polychronakos, BPS operators in N = 4 SYM: Calogero models and 2D fermions, JHEP 08 (2006) 034 [hep-th/0602049] [SPIRES].
A.P. Polychronakos, Physics and mathematics of Calogero particles, J. Phys. A 39 (2006) 12793 [hep-th/0607033] [SPIRES].
A. Donos, A. Jevicki and J.P. Rodrigues, Matrix model maps in AdS/CFT, Phys. Rev. D 72 (2005) 125009 [hep-th/0507124] [SPIRES].
R. Bhattacharyya, R. de Mello Koch and M. Stephanou, Exact multi-restricted Schur polynomial correlators, JHEP 06 (2008) 101 [arXiv:0805.3025] [SPIRES].
Y. Kimura, Non-holomorphic multi-matrix gauge invariant operators based on Brauer algebra, JHEP 12 (2009) 044 [arXiv:0910.2170] [SPIRES].
J. Huebschmann, G. Rudolph and M. Schmidt, A lattice gauge model for quantum mechanics on a stratified space, Commun. Math. Phys. 286 (2009) 459 [hep-th/0702017] [SPIRES].
J.A. Minahan, The SU(2) sector in AdS/CFT, Fortsch. Phys. 53 (2005) 828 [hep-th/0503143] [SPIRES].
N. Itzhaki and J. McGreevy, The large-N harmonic oscillator as a string theory, Phys. Rev. D 71 (2005) 025003 [hep-th/0408180] [SPIRES].
T. Banks, W. Fischler, S.H. Shenker and L. Susskind, M theory as a matrix model: a conjecture, Phys. Rev. D 55 (1997) 5112 [hep-th/9610043] [SPIRES].
E. Witten, Bound states of strings and p-branes, Nucl. Phys. B 460 (1996) 335 [hep-th/9510135] [SPIRES].
D.J. Gross and W. Taylor, Twists and Wilson loops in the string theory of two-dimensional QCD, Nucl. Phys. B 403 (1993) 395 [hep-th/9303046] [SPIRES].
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 0911.4408
Rights and permissions
About this article
Cite this article
Kimura, Y., Ramgoolam, S. & Turton, D. Free particles from Brauer algebras in complex matrix models. J. High Energ. Phys. 2010, 52 (2010). https://doi.org/10.1007/JHEP05(2010)052
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2010)052