nLab CP^N sigma-model

Idea

By “$\mathbb{C}P^N$-models” one refers to (topological) sigma-models — often (topological) string theories if the worldvolume is 2-dimensional — with target space a complex projective space $\mathbb{C}P^N$ such as the Riemann sphere $\mathbb{C}P^1$.

NFJKK22: “The $CP(N-1)$ models in 1+1 dimensions share many properties with QCD in 3+1 dimensions, among them confinement, asymptotic freedom, instantons, a $1/N$ expansion, a topological charge, and a θ-term. Thus, they serve as benchmark models for developing and testing both new numerical techniques and new proposed solutions to open questions of QCD. Both the $CP(N-1)$ models and QCD contain many nonperturbative phenomena, which cannot be addressed with conventional lattice techniques.”

References

$\mathbb{C}P^N$-models

The general notion of sigma models with target space a sphere or a complex projective space $\mathbb{C}P^n$:

• Golo, Perelomov, Solution of the duality equations for the two-dimensional $SU(N)$-invariant chiral model, Physics Letters B 79 1–2 (1978) 112-113 [doi:10.1016/0370-2693(78)90447-1]

• Harald Eichenherr, $SU(N)$ invariant non-linear σ models, Nuclear Physics B 146 1 (1978) 215-223 [doi:10.1016/0550-3213(78)90439-X, errata:doi:10.1016/0550-3213(79)90287-6]

• Harald Eichenherr, Michael Forger, Higher local conservation laws for nonlinear sigma models on symmetric spaces, Commun. Math. Phys. 82 (1981) 227–255 [doi:10.1007/BF02099918]

• Edward Witten, §3 of: Instatons, the quark model, and the $1/N$ expansion, Nuclear Physics B 149 2 (1979) 285-320 [doi:10.1016/0550-3213(79)90243-8

The lattice field theory-formulation:

• Paolo Di Vecchia, A. Holtkamp, R. Musto, F. Nicodemi, R. Pettorini, Lattice $CP^{N-1}$ models and their large-$N$ behaviour, Nuclear Physics B 190 4 (1981) 719-733 [doi:10.1016/0550-3213(81)90047-X]

and its supersymmetric version:

• Christian Wozar, §6 of: Low-dimensional supersymmetric field theories on the lattice (1981) [pdf]

On the quantum cohomology inducd by the $\mathbb{C}P^N$-model:

• Sergio Cecotti, Cumrun Vafa, Exact Results for Supersymmetric Sigma Models, Phys. Rev. Lett. 68 (1992) 903-906 [arXiv:hep-th/9111016, doi:10.1103/PhysRevLett.68.903]

• Sergio Cecotti, Cumrun Vafa, On Classification of $N=2$ Supersymmetric Theories, Comm. Math. Phys. 158 (1993) 569-644 [arXiv:hep-th/9211097, doi:10.1007/BF02096804]

• M. F. Bourdeau, Michael R. Douglas, Topological-Antitopological Fusion and the Large $N$ $\mathbb{C}P^N$ Model, Nucl. Phys. B 420 (1994) 243-267 [doi:hep-th/9312095, doi:10.1016/0550-3213(94)90380-8]

On the Nicolai map for the $\mathbb{C}P^n$ sigma-model:

• Lorenzo Casarin, Olaf Lechtenfeld, Maximilian Rupprecht, Nicolai maps with four-fermion interactions [arXiv:2310.19946]

See also:

• G. Sumbatian, E. Ievlev, A. Yung, Large-N Solution and Effective Action of “Twisted-Mass” Deformed $\mathbb{C}P(N-1)$ Model [arXiv:2312.12569]

The $\mathbb{C}P^1$-model

• Edward Witten, On the structure of the topological phase of two-dimensional gravity, Nuclear Physics B

  340 2–3 (1990) 281-332 [doi:10.1016/0550-3213(90)90449-N]

• Tohru Eguchi, Sung-Kil Yang, The Topological $C P^1$ Model and the Large-$N$ Matrix Integral, Mod. Phys. Lett. A 9 (1994) 2893-2902 [arXiv:hep-th/9407134, doi:10.1142/S0217732394002732]

• Shmuel Elitzur, A. Forge, Eliezer Rabinovici, On effective theories of topological strings, Nuclear Physics B 388 1 (1992) 131-155 [doi:10.1016/0550-3213(92)90548-P]

• Shmuel Elitzur, Yaron Oz, Eliezer Rabinovici, Johannes Walcher, Open/Closed Topological $\mathbb{C}P^1$ Sigma Model Revisited, J. High Energ. Phys. 2012 101 (2012) [arXiv:1106.2967]

• Rajesh Gopakumar, What is the Simplest Gauge-String Duality? [arXiv:1104.2386]

• Rajesh Gopakumar, Roji Pius, Correlators in the Simplest Gauge-String Duality [arXiv:1212.1236]

• Lwazi Nkumane, The Simplest Gauge-String Duality (2015) [pdf]

• R. Banerjee, Quantum equivalence of $O(3)$ nonlinear σ model and the $CP^1$ model: A gauge-independent Hamiltonian approach, Phys. Rev. D 49 2133 (1994) [doi:10.1103/PhysRevD.49.2133]

• Katsumasa Nakayama, Lena Funcke, Karl Jansen, Ying-Jer Kao, Stefan Kühn, Phase structure of the $CP(1)$ model in the presence of a topological $\theta$-term, Phys. Rev. D 105 054507 (2022) [arXiv:2107.14220, doi:10.1103/PhysRevD.105.054507]

Flag manifold sigma-models

More generally, on sigma models with flag manifold target spaces and relation to Gross-Neveu models:

• Dmitri Bykov, Quantum flag manifold σ-models and Hermitian Ricci flow, Commun. Math. Phys. 401 1-32 (2023) [arXiv:2006.14124, doi:10.1007/s00220-022-04532-5]

• Ian Affleck, Dmitri Bykov, Kyle Wamer, Flag manifold sigma models: spin chains and integrable theories, Phys. Rept. 953 (2022) 1-93 [arXiv:2101.11638, doi:10.1016/j.physrep.2021.09.004]

See also:

• Dmitri Bykov, Anton Pribytok, Supersymmetric deformation of the $\mathbb{C}P^1$ model and its conformal limits [arXiv:2312.16396]

Quantum cohomology as Pontrjagin rings

On the relation between quantum cohomology rings, hence of Gromov-Witten invariants in topological string theory, for flag manifold target spaces (such as the $\mathbb{C}P^1$-sigma model) and Pontrjagin rings (homology-Hopf algebras of based loop spaces):

That the Pontryagin ring-structure on the ordinary homology of the based loop space of a simply-connected compact Lie group $K$ is essentially the quantum cohomology ring of the flag variety of its complexification by its Borel subgroup is attributed (“Peterson isomorphism”) to

• Dale H. Peterson (notes by Arun Ram), Quantum Cohomology of $G/H$, MIT (1997) $[$web, pdf, pdf$]$

see also

• Yasha Savelyev, Quantum characteristic classes and the Hofer metric, Geom. Topol. 12 (2008) 2277-2326 $[$arXiv:0709.4510, doi:10.2140/gt.2008.12.2277$]$

and proven in

• Thomas Lam, Mark Shimozono, §6.2 in: Quantum cohomology of $G/P$ and homology of affine Grassmannian, Acta Mathematica 204 (2010) 49–90 $[$arXiv:0705.1386, doi:10.1007/s11511-010-0045-8$]$

• Chi Hong Chow, Peterson-Lam-Shimozono’s theorem is an affine analogue of quantum Chevalley formula $[$arXiv:2110.09985$]$

• Chi Hong Chow, On D. Peterson’s presentation of quantum cohomology of $G/P$ $[$arXiv:2210.17382$]$

reviewed in

• Jimmy Chow, Homology of based loop groups and quantum cohomology of flag varieties, talk at Western Hemisphere Virtual Symplectic Seminar (2021) $[$pdf, pdf, video:YT$]$

with further discussion in:

• Naichung Conan Leung, Changzheng Li, Gromov-Witten invariants for $G/B$ and Pontryagin product for $\Omega K$, Transactions of the American Mathematical Society 364 5 (2012) 2567-2599 $[$web, arXiv:0810.4859, jstor:41524936$]$

On the variant for Pontryagin products not on ordinary homology but in topological K-homology:

• Thomas Lam, Changzheng Li, Leonardo C. Mihalcea, Mark Shimozono, A conjectural Peterson isomorphism in K-theory, Journal of Algebra 513 (2018) 326-343 $[$doi:10.1016/j.jalgebra.2018.07.029, arXiv:1705.03435$]$

• Takeshi Ikeda, Shinsuke Iwao, Toshiaki Maeno, Peterson Isomorphism in K-theory and Relativistic Toda Lattice, International Mathematics Research Notices 2020 19 (2020) 6421–6462 $[$arXiv:1703.08664, doi:10.1093/imrn/rny051$]$

• Syu Kato, Loop structure on equivariant K-theory of semi-infinite flag manifolds $[$arXiv:1805.01718$]$

• Syu Kato, On quantum $K$-groups of partial flag manifolds $[$arXiv:1906.09343$]$

• Syu Kato, Darboux coordinates on the BFM spaces $[$arXiv:2008.01310$]$

• Syu Kato, Quantum $K$-groups on flag manifolds, talk at IMPANGA 20 (2021) $[$pdf$]$

On the example of the CP^1 sigma-model: LLMS18, §4.1, Kato21 p. 17, Chow22 Exp. 1.4.

See also:

• Constantin Teleman, Gauge theory and mirror symmetry $[$arXiv:1404.6305$]$

Relation to chiral rings of D=3 N=4 super Yang-Mills theory:

• Constantin Teleman, Coulomb branches for quaternionic representations $[$arXiv:2209.01088$]$