[Paper Review] SO and USp Kahler and Hyper-Kahler Quotients and Lumps

Minoru Eto, Toshiaki Fujimori+3 more|arXiv (Cornell University)|Sep 11, 2008

Algebraic Geometry and Number Theory46 references6 citations

TL;DR

This paper constructs explicit Kähler potentials and metrics for the Higgs branches of $\mathcal{N}=1$ and $\mathcal{N}=2$ supersymmetric $SO(N_{\rm C})$ and $USp(2M_{\rm C})$ gauge theories using a superfield formalism with Lagrange multipliers to enforce orthogonal and symplectic gauge algebras. It derives curvature expressions and identifies a new type of lump singularity due to singular submanifolds in the target space, extending the understanding of vortex-lump moduli space relations.

ABSTRACT

We study non-linear sigma models whose target spaces are the Higgs phases of supersymmetric SO and USp gauge theories by using the Kahler and hyper-Kahler quotient constructions. We obtain the explicit Kahler potentials and develop an expansion formula to make use of the obtained potentials from which we also calculate the curvatures of the manifolds. The 1/2 BPS lumps in the U(1) x SO and U(1) x USp Kahler quotients and their effective descriptions are also studied. In this connection, a general relation between moduli spaces of vortices and lumps is discussed. We find a new singular limit of the lumps with non-vanishing sizes in addition to the ordinary small lump singularity. The former is due to the existence of singular submanifolds in the target spaces.

Motivation & Objective

• To explicitly construct the Kähler potential and metric for the Higgs branch of $\mathcal{N}=1$ and $\mathcal{N}=2$ supersymmetric $SO(N_{\rm C})$ and $USp(2M_{\rm C})$ gauge theories, which had not been previously derived in full generality.
• To overcome the difficulty of solving $D$-term constraints for $SO$ and $USp$ groups by introducing a superfield trick: relaxing the gauge algebra to $U(N_{\rm C})$ and imposing constraints via Lagrange multipliers.
• To provide a direct method for computing geometric quantities like curvature by expanding the derived Kähler potentials, avoiding the algebraic constraints inherent in geometric invariant theory.
• To study 1/2 BPS lumps in $U(1)\times SO(N_{\rm C})$ and $U(1)\times USp(2M_{\rm C})$ models and identify a new singular limit of lumps with non-vanishing size due to singular submanifolds in the target space.
• To clarify the general relation between moduli spaces of vortices and lumps in these gauge theories, particularly in the context of hyper-Kähler quotient constructions.

Proposed method

• Use of the superfield formalism to solve $D$-term constraints for $SO(N_{\rm C})$ and $USp(2M_{\rm C})$ gauge groups by complexifying the gauge group and introducing Lagrange multiplier superfields to restrict the algebra to $\mathfrak{so}(N_{\rm C})$ and $\mathfrak{usp}(2M_{\rm C})$.
• Derivation of explicit Kähler potentials in terms of chiral superfields and holomorphic gauge invariants, enabling direct computation of the metric and curvature via expansion.
• Application of the hyper-Kähler quotient construction to obtain the target space geometry of the low-energy $\mathcal{N}=2$ non-linear $\sigma$-model, with the Higgs branch as a hyper-Kähler manifold.
• Expansion of the Kähler potential in powers of the scalar fields $\phi$ to compute the Riemann curvature tensor at $\phi = 0$, yielding explicit curvature expressions in terms of $\mu_i'$ and $\varepsilon$.
• Identification of singularities in the moduli space when gauge symmetry is partially restored, confirmed by curvature divergences.
• Analysis of 1/2 BPS lumps in $U(1)\times SO(N_{\rm C})$ and $U(1)\times USp(2M_{\rm C})$ models, revealing a new type of singularity distinct from the standard small-lump limit.

Experimental results

Research questions

• RQ1 How can the $D$-term constraints for $SO(N_{\rm C})$ and $USp(2M_{\rm C})$ gauge groups be solved explicitly in the superfield formalism to obtain the Kähler potential of the Higgs branch?
• RQ2 What is the explicit form of the Kähler potential and metric for the Higgs branch of $U(1)\times SO(N_{\rm C})$ and $U(1)\times USp(2M_{\rm C})$ gauge theories?
• RQ3 What is the curvature of the target space manifold at the origin of the moduli space, and how does it depend on the mass parameters and the deformation parameter $\varepsilon$?
• RQ4 What is the nature of the singularity in the lump moduli space, and how does it differ from the standard small-lump limit?
• RQ5 What is the general relation between the moduli spaces of vortices and lumps in these $SO$ and $USp$ gauge theories?

Key findings

• The Kähler potential for the Higgs branch of $SO(N_{\rm C})$ and $USp(2M_{\rm C})$ gauge theories is derived explicitly using a superfield method with Lagrange multipliers, enabling direct access to geometric data.
• The curvature of the target space at $\phi = 0$ is computed and expressed as a sum of rational functions in $\mu_i'$ and $\varepsilon$, with explicit divergences signaling symmetry restoration.
• A new type of lump singularity is identified: it arises not from shrinking size but from the presence of singular submanifolds in the target space geometry, distinct from the standard small-lump limit.
• The curvature expression includes terms up to $\mathcal{O}(\varepsilon^8)$ and involves sums over $i,j,k$ indices, reflecting the non-trivial structure of the hyper-Kähler manifold.
• The moduli space of vortices and lumps is shown to be related through the geometry of the target space, with the new singularity arising from the intrinsic singularities of the quotient manifold.
• The method successfully reproduces known results such as the Lindström-Roček metric in special cases and confirms the appearance of singularities when gauge symmetry is partially restored.