Sigma models in geometry.

*(English)*Zbl 1051.53044
Slovák, Jan (ed.) et al., The proceedings of the 23th winter school “Geometry and physics”, Srní, Czech Republic, January 18–25, 2003. Palermo: Circolo Matemàtico di Palermo. Suppl. Rend. Circ. Mat. Palermo, II. Ser. 72, 79-90 (2004).

Three lectures on the geometry of sigma models are presented. The first lecture deals with supersymmetry arising in the sigma model. The second lecture deals with quotients of the sigma model. The goal of the last lecture is the quiver diagram encoding and description by closed contours on the quiver of the holomorphic moment map [U. Lindström, M. Rocek and R. von Unge, Hyper-Kähler quotients and algebraic curves, J. High Energy Phys. 4, No. 1, Paper No. 01 (2000) 022 (2000; Zbl 0989.53027)].

The model considered in these lectures is the map from \(\Sigma\), \(S(\Sigma)\), where \(S(\Sigma)\) is the spin bundle over \(\Sigma\), to \(M, T(M)\), \(T(M)\) the tangent bundle of \(M\). Denoting \(\Phi:\Sigma\to M\) and \(\psi: S(\Sigma)\to T(M)\), the Grassmann odd map, the action, for \(\dim\Sigma= 3\), is given by \[ S= {1\over 4\pi} \int_\Sigma \phi^*(| d\phi|^2+ \langle\psi,\mathbb{D}\rangle+ {1\over 4}\text{\,Riem}(\psi\otimes \psi\otimes\psi\otimes\psi)_0). \] As a symmetry of the action under the transformations \[ \delta\phi^i\propto(\varepsilon\otimes \psi^i)_0,\quad \delta\psi^i\propto (\not\partial\phi^i+ \Gamma^i_{jk}\not\partial\phi^j)\psi^k)\varepsilon, \]

where \(\varepsilon\) is a parallel spinor, the supersymmetry of the model is found. Relation of the numbers of supersymmetry and the holonomy of \(M\) is listed as table 1. In the rest of the lectures, \(\phi\) and \(\psi\), \(\Sigma\) and \(s(\Sigma)\) are combined as superfield (supermultiplet) and superspace. In lecture 2, metrics \(h\) and \(g\) of \(\Sigma\) and \(M\) are considered.

Let \(X\) be a Killig vector field on \(M\). Then the action \(\int_\Sigma | d\phi|^2_{g,h}\) of the immersion \(\phi\) is invariant under the isometry generated by \(X\). If the orbits of the action of \(X\) are compact, a \(U(1)\) action is defined for the sigma model. Introducing a \(U(1)\)-connection, the sigma model on the quotient manifold is defined. If \(M\) is a Kähler manifold with Kähler form \(\omega\), the moment map \(\mu\) is defined by \(d\mu^X= i_X\omega\). The quotient is studied as symplectic reduction.

In lecture 3, the first sigma model for \(\mathbb{C}^n\) with \(N= 2\) supersymmetry is examined. In this case, the Kähler potential is \(\sum \Phi^i\overline\Phi^i\). As symplectic quotient, one obtains \(\mathbb{C}\mathbb{P}^{n-1}\) and the Kähler potential for the Fubini-Study metric. Next, applying algebraic geometry arguments, basic supersymmetric representation of \(N= 4\) supersymmetry is given by a hypermultiplet which in \(N= 2\) language consist of two chiral superfields \(\Phi_{\pm}\) and vector multiples which consist of a chiral superfield \(S\) and an \(N =1\) vector multiplet \(V\). Then, considering variation of the action for a quotient of some flat quaternionic space described as a complex even-dimensional flat space, the constraint of the holomorphic moment map \(\sum \Phi_+\Phi_-= b\), where \(b\) is a constant contained in the action, is derived. The lecture concludes with a review of encoding and description of holomorphic moment maps by quiver diagrams and contours on the quiver.

For the entire collection see [Zbl 1034.53002].

The model considered in these lectures is the map from \(\Sigma\), \(S(\Sigma)\), where \(S(\Sigma)\) is the spin bundle over \(\Sigma\), to \(M, T(M)\), \(T(M)\) the tangent bundle of \(M\). Denoting \(\Phi:\Sigma\to M\) and \(\psi: S(\Sigma)\to T(M)\), the Grassmann odd map, the action, for \(\dim\Sigma= 3\), is given by \[ S= {1\over 4\pi} \int_\Sigma \phi^*(| d\phi|^2+ \langle\psi,\mathbb{D}\rangle+ {1\over 4}\text{\,Riem}(\psi\otimes \psi\otimes\psi\otimes\psi)_0). \] As a symmetry of the action under the transformations \[ \delta\phi^i\propto(\varepsilon\otimes \psi^i)_0,\quad \delta\psi^i\propto (\not\partial\phi^i+ \Gamma^i_{jk}\not\partial\phi^j)\psi^k)\varepsilon, \]

where \(\varepsilon\) is a parallel spinor, the supersymmetry of the model is found. Relation of the numbers of supersymmetry and the holonomy of \(M\) is listed as table 1. In the rest of the lectures, \(\phi\) and \(\psi\), \(\Sigma\) and \(s(\Sigma)\) are combined as superfield (supermultiplet) and superspace. In lecture 2, metrics \(h\) and \(g\) of \(\Sigma\) and \(M\) are considered.

Let \(X\) be a Killig vector field on \(M\). Then the action \(\int_\Sigma | d\phi|^2_{g,h}\) of the immersion \(\phi\) is invariant under the isometry generated by \(X\). If the orbits of the action of \(X\) are compact, a \(U(1)\) action is defined for the sigma model. Introducing a \(U(1)\)-connection, the sigma model on the quotient manifold is defined. If \(M\) is a Kähler manifold with Kähler form \(\omega\), the moment map \(\mu\) is defined by \(d\mu^X= i_X\omega\). The quotient is studied as symplectic reduction.

In lecture 3, the first sigma model for \(\mathbb{C}^n\) with \(N= 2\) supersymmetry is examined. In this case, the Kähler potential is \(\sum \Phi^i\overline\Phi^i\). As symplectic quotient, one obtains \(\mathbb{C}\mathbb{P}^{n-1}\) and the Kähler potential for the Fubini-Study metric. Next, applying algebraic geometry arguments, basic supersymmetric representation of \(N= 4\) supersymmetry is given by a hypermultiplet which in \(N= 2\) language consist of two chiral superfields \(\Phi_{\pm}\) and vector multiples which consist of a chiral superfield \(S\) and an \(N =1\) vector multiplet \(V\). Then, considering variation of the action for a quotient of some flat quaternionic space described as a complex even-dimensional flat space, the constraint of the holomorphic moment map \(\sum \Phi_+\Phi_-= b\), where \(b\) is a constant contained in the action, is derived. The lecture concludes with a review of encoding and description of holomorphic moment maps by quiver diagrams and contours on the quiver.

For the entire collection see [Zbl 1034.53002].

Reviewer: Akira Asada (Takarazuka)