Abstract
The sigma model on projective superspaces \( \mathbb{C}{\mathbb{P}^{S - 1\left| S \right.}} \) gives rise to a continuous family of interacting 2D conformal field theories which are parametrized by the curvature radius R and the theta angle θ. Our main goal is to determine the spectrum of the model, non-perturbatively as a function of both parameters. We succeed to do so for all open boundary conditions preserving the full global symmetry of the model. In string theory parlor, these correspond to volume filling branes that are equipped with a monopole line bundle and connection. The paper consists of two parts. In the first part, we approach the problem within the continuum formulation. Combining combinatorial arguments with perturbative studies and some simple free field calculations, we determine a closed formula for the partition function of the theory. This is then tested numerically in the second part. There we extend the proposal of [arXiv:0908.1081] for a spin chain regularization of the \( \mathbb{C}{\mathbb{P}^{S - 1\left| S \right.}} \) model with open boundary conditions and use it to determine the spectrum at the conformal fixed point. The numerical results are in remarkable agreement with the continuum analysis.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
N. Read and H. Saleur, Exact spectra of conformal supersymmetric nonlinear σ-models in two dimensions, Nucl. Phys. B 613 (2001) 409 [hep-th/0106124] [SPIRES].
C. Candu and H. Saleur, A lattice approach to the conformal osp(2S + 2|2S) supercoset σ-model. Part I: algebraic structures in the spin chain. The Brauer algebra, Nucl. Phys. B 808 (2009) 441 [arXiv:0801.0430] [SPIRES].
C. Candu and H. Saleur, A lattice approach to the conformal osp(2S + 2|2S) supercoset σ-model. Part II: the boundary spectrum, Nucl. Phys. B 808 (2009) 487 [arXiv:0801.0444] [SPIRES].
V. Mitev, T. Quella and V. Schomerus, Principal chiral model on superspheres, JHEP 11 (2008) 086 [arXiv:0809.1046] [SPIRES].
S. Sethi, Supermanifolds, rigid manifolds and mirror symmetry, Nucl. Phys. B 430 (1994) 31 [hep-th/9404186] [SPIRES].
M. Aganagic and C. Vafa, Mirror symmetry and supermanifolds, hep-th/0403192 [SPIRES].
S.P. Kumar and G. Policastro, Strings in twistor superspace and mirror symmetry, Phys. Lett. B 619 (2005) 163 [hep-th/0405236] [SPIRES].
C.-H. Ahn, Mirror symmetry of Calabi-Yau supermanifolds, Mod. Phys. Lett. A 20 (2005) 407 [hep-th/0407009] [SPIRES].
A. Belhaj, L.B. Drissi, J. Rasmussen, E.H. Saidi and A. Sebbar, Toric Calabi-Yau supermanifolds and mirror symmetry, J. Phys. A 38 (2005) 6405 [hep-th/0410291] [SPIRES].
R. Ricci, Super Calabi-Yau’s and special Lagrangians, JHEP 03 (2007) 048 [hep-th/0511284] [SPIRES].
S. Seki, K. Sugiyama and T. Tokunaga, Superconformal symmetry in linear σ-model on supermanifolds, Nucl. Phys. B 753 (2006) 295 [hep-th/0605021] [SPIRES].
E. Witten, Perturbative gauge theory as a string theory in twistor space, Commun. Math. Phys. 252 (2004) 189 [hep-th/0312171] [SPIRES].
O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, N = 6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [SPIRES].
H.A. Weidenmüller and M.R. Zirnbauer, Instanton approximation to the graded nonlinear σ-model for the integer quantum Hall effect, Nucl. Phys. B 305 (1988) 339 [SPIRES].
M.R. Zirnbauer, Conformal field theory of the integer quantum Hall plateau transition, hep-th/9905054 [SPIRES].
C. Candu, J.L. Jacobsen, N. Read and H. Saleur, Universality classes of dense polymers and conformal σ-models, arXiv:0908.1081 [SPIRES].
H.G. Kausch, Curiosities at c = −2, hep-th/9510149 [SPIRES].
H.G. Kausch, Symplectic fermions, Nucl. Phys. B 583 (2000) 513 [hep-th/0003029] [SPIRES].
S. Xiong, N. Read and A.D. Stone, Mesoscopic conductance and its fluctuations at a nonzero Hall angle, Phys. Rev. B 56 (1997) 3982.
R. Kuwabara, Spectrum of the Schrödinger operator on a line bundle over the complex projective spaces, Tôhoku Math. J. 40 (1988) 199.
W. Greub and H.-R. Petry, Minimal coupling and complex line bundles, J. Math. Phys. 16 (1975) 1347.
Y.-Z. Zhang and M.D. Gould, A unified and complete construction of all finite dimensional irreducible representations of u(2|2), J. Math. Phys. 46 (2005) 013505 [math.QA/0405043] [SPIRES].
G. Götz, T. Quella and V. Schomerus, Tensor products of psl(2|2) representations, hep-th/0506072 [SPIRES].
V. Schomerus and H. Saleur, The GL(1|1) WZW model: from supergeometry to logarithmic CFT, Nucl. Phys. B 734 (2006) 221 [hep-th/0510032] [SPIRES].
T. Creutzig, T. Quella and V. Schomerus, New boundary conditions for the c = −2 ghost system, Phys. Rev. D 77 (2008) 026003 [hep-th/0612040] [SPIRES].
D.G. Boulware and L.S. Brown, Symmetric space scalar field theory, Ann. Phys. 138 (1982) 392 [SPIRES].
V. Schomerus, D-branes and deformation quantization, JHEP 06 (1999) 030 [hep-th/9903205] [SPIRES].
A. Abouelsaood, C.G. Callan Jr., C.R. Nappi and S.A. Yost, Open strings in background gauge fields, Nucl. Phys. B 280 (1987) 599 [SPIRES].
M. Bershadsky, S. Zhukov and A. Vaintrob, PSL(n|n) σ-model as a conformal field theory, Nucl. Phys. B 559 (1999) 205 [hep-th/9902180] [SPIRES].
T. Creutzig and P.B. Rønne, The GL(1|1)-symplectic fermion correspondence, Nucl. Phys. B 815 (2009) 95 [arXiv:0812.2835] [SPIRES].
B. Berg and M. Lüscher, Definition and statistical distributions of a topological number in the lattice O(3) σ-model, Nucl. Phys. B 190 (1981) 412 [SPIRES].
N. Seiberg, Topology in strong coupling, Phys. Rev. Lett. 53 (1984) 637 [SPIRES].
F.D.M. Haldane, Nonlinear field theory of large spin Heisenberg antiferromagnets. Semiclassically quantized solitons of the one-dimensional easy Axis Néel state, Phys. Rev. Lett. 50 (1983) 1153 [SPIRES].
I. Affleck, The quantum Hall effect, σ-modelS at θ = π and quantum spin chains, Nucl. Phys. B 257 (1985) 397 [SPIRES].
N. Read and S. Sachdev, Some features of the phase diagram of the square lattice SU(N) antiferromagnet, Nucl. Phys. B 316 (1989) 609 [SPIRES].
P.B. Wiegmann, Superconductivity in strongly correlated electronic systems and confinement versus deconfinement phenomenon, Phys. Rev. Lett. 60 (1988) 821 [SPIRES].
H. Saleur and V. Schomerus, On the SU(2|1) WZNW model and its statistical mechanics applications, Nucl. Phys. B 775 (2007) 312 [hep-th/0611147] [SPIRES].
N. Read and H. Saleur, Enlarged symmetry algebras of spin chains, loop models and S-matrices, Nucl. Phys. B 777 (2007) 263 [cond-mat/0701259] [SPIRES].
N. Read and H. Saleur, Associative-algebraic approach to logarithmic conformal field theories, Nucl. Phys. B 777 (2007) 316 [hep-th/0701117] [SPIRES].
G. Benkart, C.L. Shader and A. Ram, Tensor product representations for orthosymplectic Lie superalgebras, math.RA/9607232.
A.N. Sergeev, An analog of the classical invariant theory for Lie superalgebras. I, Michigan Math. J. 49 (2001) 113.
A.N. Sergeev, An analog of the classical invariant theory for Lie superalgebras. II, Michigan Math. J. 49 (2001) 147.
H. Saleur, Polymers and percolation in two-dimensions and twisted N = 2 supersymmetry, Nucl. Phys. B 382 (1992) 486 [hep-th/9111007] [SPIRES].
J.L. Jacobsen and H. Saleur, The arboreal gas and the supersphere σ-model, Nucl. Phys. B 716 (2005) 439 [cond-mat/0502052] [SPIRES].
R.B. Zhang and Y.M. Zou, Spherical functions on homogeneous superspaces, J. Math. Phys. 46 (2005) 043513.
T. Quella, V. Schomerus and T. Creutzig, Boundary spectra in superspace σ-models, JHEP 10 (2008) 024 [arXiv:0712.3549] [SPIRES].
J.L. Jacobsen and H. Saleur, Conformal boundary loop models, Nucl. Phys. B 788 (2008) 137 [math-ph/0611078] [SPIRES].
B. Konstant, Quantization and unitary representations, Lect. Notes Math. 170 (1970) 87, Springer-Verlag, Berlin, Heidelberg Germany and New York U.S.A. (1970).
S. Helgason, Differential geometry, Lie groups and symmetric spaces, Academic Press, New York U.S.A. (1978).
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 0908.0878
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Candu, C., Mitev, V., Quella, T. et al. The sigma model on complex projective superspaces. J. High Energ. Phys. 2010, 15 (2010). https://doi.org/10.1007/JHEP02(2010)015
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2010)015