nLab CP^N sigma-model



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By “P N\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 P N\mathbb{C}P^N such as the Riemann sphere P 1\mathbb{C}P^1.

NFJKK22: “The CP(N1)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 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(N1)CP(N-1) models and QCD contain many nonperturbative phenomena, which cannot be addressed with conventional lattice techniques.”


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

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

The lattice field theory-formulation:

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 P N\mathbb{C}P^N-model:

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

See also:

  • G. Sumbatian, E. Ievlev, A. Yung, Large-N Solution and Effective Action of “Twisted-Mass” Deformed P(N1)\mathbb{C}P(N-1) Model [arXiv:2312.12569]

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

Flag manifold sigma-models

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

See also:

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 P 1 \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 KK is essentially the quantum cohomology ring of the flag variety of its complexification by its Borel subgroup is attributed (“Peterson isomorphism”) to

see also

and proven in

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:

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

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

See also:

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

Last revised on December 29, 2023 at 07:58:34. See the history of this page for a list of all contributions to it.