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Sigma models as Gross–Neveu models. II

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Abstract

We summarize some (mostly geometric) facts underlying the relation between \(2\)D integrable sigma models and generalized Gross–Neveu models, emphasizing connections to the theory of nilpotent orbits, Springer resolutions, and quiver varieties. This is meant to shed light on the general setup when this correspondence holds.

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Notes

  1. Classical integrability of Gross–Neveu models was discovered in [3], [4]; see also [5]–[7] for more recent developments.

  2. For the definition, see Part VIII in [11]. Intuitively, for any generator \(T\in\mathsf g\), the function \(\mu(T)\) is the Hamiltonian for the action of the one-parametric subgroup \((e^{sT},\,s\in\mathbb R)\) on the manifold.

  3. The Peter–Weyl theorem provides an explicit way to decompose \(L^2(G/H)\) into irreducible representations of \(G\) (generalized spherical harmonics), \(L^2(G/H)=\bigoplus_{\mathrm{irreps of}\,G}V_i\otimes W_i^\ast\), where the sum is over all unitary irreducible representations of \(G\), and \(W_i^\ast\subset V_i^\ast\) is the subset of \(V_i^\ast\) on which \(H\) acts trivially. For more on this, see [13].

  4. We note that classically the sigma model action depends only on the conformal class of the worldsheet metric due to the so-called Weyl invariance. If one brings the metric to conformal coordinates, \(ds^2=e^\Lambda\,dz\,d\bar z\), the factor \(e^\Lambda\) drops out of the action, and the remaining degrees of freedom are encoded in the complex coordinates \(z\), \(\bar z\). By varying the complex structure on \(\Sigma\), one effectively varies the metric, and hence (2.1) is valid in an arbitrary metric.

  5. By a neighborhood, we mean an \(\epsilon\)-disc \(\|z-z_0\|<\epsilon\), where distance is measured with respect to the induced metric \((ds^2)_\Sigma=\operatorname{Tr}(j_z\overline{j_z})\,dz\,\overline{dz}\).

  6. Let \(x\in \mathsf{g}_C\) be a nilpotent element (i.e., \(ad_x^m=0\) for some \(m\)). By definition, the nilpotent orbit \(N_x\) is the adjoint orbit: \(N_x=\{g x g^{-1}, g\in G_C\}\). It can be shown that in this case \(x\) is a nilpotent matrix in any representation of \(\mathsf g_{\mathbb C}\), cf. [15]. In practice, when speaking of Jordan forms, we always assume that we are dealing with \(x\) in the standard (defining) representation of the corresponding Lie algebra.

  7. We can find \(\overline{\mathcal A}\), for example, by taking the scalar product of the first equation in (2.6) with \(\overline{U}\) and using the constraint \(VU=0\).

  8. For an introductory exposition of some of the material in this section, see [17]. More advanced topics on symplectic singularities and their resolutions are covered in [16], [18].

  9. In the case \(G_{\mathbb C}=SL(n,\mathbb C)\), the Borel subgroup consists of invertible upper-triangular matrices, whereas \(n(\mathsf b)\) comprises strictly upper-triangular matrices.

  10. Indeed, we have a surjective map \((G\times \mathsf{g})/H \to T(G/H)\) constructed as follows. Let \((g, a) \in (G \times \mathsf{g})/H\) and \(f(gH) = f(g)\) be an arbitrary function on \(G/H\). We can then define a vector field \(v\) on \(G/H\) as \(v f(gH):=\frac{d}{d\epsilon}\,f(g\,e^{\epsilon a}H)\big|_{\epsilon=0}\). Because \(G/H\) is a homogeneous space, any vector field can be constructed in this way, and therefore the above map is surjerctive. In addition, two elements \((g, a_1)\) and \((g, a_2)\) define the same vector field only if \(a_2 - a_1 \in\mathsf h\). Hence, replacing \(\mathsf g\) in the above map with the quotient \(\mathsf g/\mathsf h\) gives a one-to-one map, which proves (3.3).

  11. For such orbits, there is also a canonical choice of \((U,V)\)-type Darboux coordinates, i.e., a polarization, as shown in [19].

  12. As discussed in [18], the relevant analogue of the Springer map (3.7) for an arbitrary parabolic subgroup is a surjective map of degree \(d\ge 1\), and it only defines a resolution of singularities if \(d=1\). At present, it is unclear what the implications of \(d>1\) are for the relation to sigma models. In the foregoing, we therefore restrict to those cases where \(d=1\).

  13. We have shown in [1] that these maps are well defined in the case of \(\mathsf{sl}_n\) flags, where the relevant flag is the flag of kernels \(\mathrm{Ker}(j_z^\ell)\), \(\ell=1,2,\dots\).

  14. In the \(\mathsf{sl}\) case, this was shown in [20], and generalized to \(\mathsf o\) and \(\mathsf{sp}\) orbits in [21].

  15. On \(\Sigma=\mathbb{CP}^1\), nontrivial line bundles \(L\) correspond to “instanton” solutions of the sigma model, the degree of \(L\) being related to the instanton number.

  16. The orbits that do lead to sigma models with Grassmannian target spaces (in the \(\mathsf u\), \(\mathsf o\) and \(\mathsf{sp}\) cases) are briefly discussed in [22].

  17. The hyper-Kähler metric on it is the Eguchi–Hanson metric [23], [24].

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Acknowledgments

This paper is dedicated to the memory of my scientific supervisor A. A. Slavnov. I will remember him as a profound yet cheerful person, of immense scientific integrity and dedication to science, and I will always be grateful for his benevolence and support. I would like to thank E. Ivanov, V. Krivorol, A. Nersessian, A. Smilga and members of the I. R. Shafarevich’s seminar, where part of this work was presented, for discussions, useful remarks and suggestions.

Funding

This work was supported by the Russian Science Foundation under grant No. 22-72-10122, https://rscf.ru/ en/ project/ 22-72-10122/.

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  1. Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia

    D. V. Bykov

  2. Institute for Theoretical and Mathematical Physics, Lomonosov Moscow State University, Moscow, Russia

    D. V. Bykov

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Translated from Teoreticheskaya i Matematicheskaya Fizika, 2023, Vol. 217, pp. 499–514 https://doi.org/10.4213/tmf10528.

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Bykov, D.V. Sigma models as Gross–Neveu models. II. Theor Math Phys 217, 1842–1854 (2023). https://doi.org/10.1134/S0040577923120048

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