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N=2 Sigma Model with Twisted Mass and Superpotential: Central Charges and Solitons

A. Losev, M. Shifman

TL;DR

This work shows that ${\cal N}=2$ two-dimensional sigma models with twisted mass on Kähler targets admitting a holomorphic Killing vector admit a special superpotential preserving the isometry, leading to central charges in the anticommutators ${\{Q_L,Q_L\}}$ and ${\{Q_R,Q_R\}}$ that scale with system size. These central charges enable spontaneous partial breaking of supersymmetry from ${\cal N}=2$ to ${\cal N}=1$, even though the vacuum energy is nonzero, and render standard indices like the Witten or CFIV inapplicable due to the absence of a fermion-number operator. The authors construct internal indices ${\cal I}_{2,1/2}$ and ${\cal I}_{2,1/4}$ to count short multiplets (1/2-BPS vacua and 1/4-BPS kinks) in this setting and illustrate their ideas with the ${\cal C}^*$ (CP(1) descendant) model, including explicit 1/4-BPS kink solutions. They also outline an anomaly in the anticommutator $\{\bar Q_L,Q_R\}$, connect central charges to vacua order parameters, and discuss loop corrections and their impact on central charges. The results reveal a novel, purely two-dimensional phenomenon with potential implications for soliton spectra and supersymmetric dynamics in low-dimensional field theories.

Abstract

We consider supersymmetric sigma models on the Kahler target spaces, with twisted mass. The Kahler spaces are assumed to have holomorphic Killing vectors. Introduction of a superpotential of a special type is known to be consistent with N=2 superalgebra (Alvarez-Gaume and Freedman). We show that the algebra acquires central charges in the anticommutators {Q_L, Q_L} and {Q_R, Q_R}. These central charges have no parallels, and they can exist only in two dimensions. The central extension of the N=2 superalgebra we found paves the way to a novel phenomenon -- spontaneous breaking of a part of supersymmetry. In the general case 1/2 of supersymmetry is spontaneously broken (the vacuum energy density is positive), while the remaining 1/2 is realized linearly. In the model at hand the standard fermion number is not defined, so that the Witten index as well as the Cecotti-Fendley-Intriligator-Vafa index are useless. We show how to construct an index for counting short multiplets in internal algebraic terms which is well-defined in spite of the absence of the standard fermion number. Finally, we outline derivation of the quantum anomaly in {\bar Q_L, Q_R}.

N=2 Sigma Model with Twisted Mass and Superpotential: Central Charges and Solitons

TL;DR

This work shows that two-dimensional sigma models with twisted mass on Kähler targets admitting a holomorphic Killing vector admit a special superpotential preserving the isometry, leading to central charges in the anticommutators and that scale with system size. These central charges enable spontaneous partial breaking of supersymmetry from to , even though the vacuum energy is nonzero, and render standard indices like the Witten or CFIV inapplicable due to the absence of a fermion-number operator. The authors construct internal indices and to count short multiplets (1/2-BPS vacua and 1/4-BPS kinks) in this setting and illustrate their ideas with the (CP(1) descendant) model, including explicit 1/4-BPS kink solutions. They also outline an anomaly in the anticommutator , connect central charges to vacua order parameters, and discuss loop corrections and their impact on central charges. The results reveal a novel, purely two-dimensional phenomenon with potential implications for soliton spectra and supersymmetric dynamics in low-dimensional field theories.

Abstract

We consider supersymmetric sigma models on the Kahler target spaces, with twisted mass. The Kahler spaces are assumed to have holomorphic Killing vectors. Introduction of a superpotential of a special type is known to be consistent with N=2 superalgebra (Alvarez-Gaume and Freedman). We show that the algebra acquires central charges in the anticommutators {Q_L, Q_L} and {Q_R, Q_R}. These central charges have no parallels, and they can exist only in two dimensions. The central extension of the N=2 superalgebra we found paves the way to a novel phenomenon -- spontaneous breaking of a part of supersymmetry. In the general case 1/2 of supersymmetry is spontaneously broken (the vacuum energy density is positive), while the remaining 1/2 is realized linearly. In the model at hand the standard fermion number is not defined, so that the Witten index as well as the Cecotti-Fendley-Intriligator-Vafa index are useless. We show how to construct an index for counting short multiplets in internal algebraic terms which is well-defined in spite of the absence of the standard fermion number. Finally, we outline derivation of the quantum anomaly in {\bar Q_L, Q_R}.

Paper Structure

This paper contains 14 sections, 93 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Sigma model and twisted mass
  3. Introduction of superpotential
  4. Superalgebra
  5. Implications for vacua
  6. Counting vacua and kinks, or what replaces the CFIV index
  7. A brief reminder of the ${\cal N}=1$ story
  8. Counting 1/2 BPS states in ${\cal N}=2$ SUSY
  9. Counting 1/4 BPS states in ${\cal N}=2$ SUSY
  10. Showcase example of $C^*$ model (descendant of CP(1))
  11. Kinks
  12. Loop corrections
  13. Anomaly
  14. Conclusions

Figures (1)

  • Figure 1: The vacuum manifold in the simplest model with twisted mass and superpotential, see (\ref{['Aone']}). The target space is $S_2$ with the punctured poles (the poles are marked by closed circles). The vacuum manifold consists of two disconnected $S_1$ submanifolds. The trajectory running along a meridian and interpolating between vacua belonging to the first and second $S_1$ respectively (thick solid line) is a 1/4 BPS saturated kink.