PHYSICAL REVIEW D 103, 025001 (2021) Higher derivative supersymmetric nonlinear sigma models on Hermitian symmetric spaces and BPS states therein † Muneto Nitta1,* and Shin Sasaki2, 1Department of Physics, and Research and Education Center for Natural Sciences, Keio University, Hiyoshi 4-1-1, Yokohama, Kanagawa 223-8521, Japan 2Department of Physics, Kitasato University, Sagamihara 252-0373, Japan (Received 19 November 2020; accepted 14 December 2020; published 4 January 2021) We formulate four-dimensional N ¼ 1 supersymmetric nonlinear sigma models on Hermitian symmetric spaces with higher derivative terms, free from the auxiliary field problem and the Ostrogradski’s ghosts, as gauged linear sigma models. We then study Bogomol’nyi-Prasad-Sommerfield equations preserving 1=2 or 1=4 supersymmetries. We find that there are distinct branches, that we call canonical (F ¼ 0) and noncanonical (F ≠ 0) branches, associated with solutions to auxiliary fields F in chiral multiplets. For the CPN model, we obtain a supersymmetric CPN Skyrme-Faddeev model in the canonical branch while in the noncanonical branch the Lagrangian consists of solely the CPN Skyrme-Faddeev term without a canonical kinetic term. These structures can be extended to the Grassmann manifold GM;N ¼ SUðMÞ=½SUðM − NÞ × SUðNÞ × Uð1Þ. For other Hermitian symmetric spaces such as the quadric surface QN−2 ¼ SOðNÞ=½SOðN − 2Þ × Uð1ÞÞ, we impose F-term (holomorphic) constraints for embedding them into CPN−1 or Grassmann manifold. We find that these constraints are consistent in the canonical branch but yield additional constraints on the dynamical fields, thus reducing the target spaces in the noncanonical branch. DOI: 10.1103/PhysRevD.103.025001 I. INTRODUCTION called the auxiliary field problem [4,5]. This stems from the fact that the equation of motion for the auxiliary field F in Nonlinear sigma models are typical examples of low- the chiral superfield Φ ceases to be an algebraic equation energy effective theories. When a global symmetry G is and becomes a kinematical one. As a consequence, it is spontaneously broken down to its subgroup H, there appear hard to integrate out the auxiliary field and the interactions massless Nambu-Goldstone bosons dominant at low of physical fields are not apparent. Supersymmetric exten- energy, and their low-energy dynamics can be described sions of the Wess-Zumino-Witten term [6] and the Skyrme- by nonlinear sigma model whose target space is a coset Faddeev model [7,8] are such examples with the auxiliary space G=H [1,2]. If one wants to study physics at higher field problem, while low-energy Lagrangians for super- energy, one needs higher derivative correction terms, symmetric gauge theories in Refs. [9–11] (see also typically appearing in the chiral perturbation theory of Refs. [12–14]), and supersymmetric extensions of Dirac- QCD [3]. Other examples of higher derivative terms can be Born-Infeld action [15,16] and K-field theories [17,18] are found in various contexts such as the Skyrme model, low- free from this problem. A broad class of supersymmetric energy effective theories of superstring theory, and world derivative terms free from the auxiliary field problem and volume theories of topological solitons and branes such as the Ostrogradski’s ghost [19] was found as four-dimen- Nambu-Goto and Dirac-Born-Infeld actions. sional N 1 supersymmetric higher derivative chiral In supersymmetric theories, higher derivative terms often ¼ models formulated in terms of superfields [20–32].1 bring us troubles. It is known that constructing derivative The model consists of a Kähler potential K and a super- terms in the form of ∂ Φ suffers from a technical issue m potential W together with a (2,2) Kähler tensor Λ that determines derivative corrections. It was applied to a new *[email protected] † mechanism of supersymmetry breaking in modulated vacua [email protected] [36] (see also Ref. [37]). The general formulation was extended for other superfields: the most general ghost-free Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, 1Another possibility is to gauge them away [33] if there are and DOI. Funded by SCOAP3. Ostrogradski’s higher derivative ghosts [34,35]. 2470-0010=2021=103(2)=025001(16) 025001-1 Published by the American Physical Society MUNETO NITTA and SHIN SASAKI PHYS. REV. D 103, 025001 (2021) (and tachyon-free) higher derivative N ¼ 1 supersymmet- (the term without fourth order time derivatives). We then ric Lagrangian for vector multiplets (1-form gauge fields) study BPS states in these models and find that lumps (sigma [38] and 3-form gauge field [39]. As for nonlinear sigma model instantons) identical to those without higher deriva- models of chiral superfields, only known examples are the tive terms remain 1=2 BPS states in the canonical branch, supersymmetric CP1 model (or baby Skyrme model) while compact baby skyrmions are 1=4 BPS states in the [24,25,27,40–42] and the supersymmetric Skyrme model noncanonical branch. [30]. When one solves the auxiliary field equations of The organization of this paper is as follows. In the next motion for the auxiliary field F, it in general allows more section, we introduce the supersymmetric higher derivative than one solution F ¼ 0 and F ≠ 0. The former is called term in the gauged chiral models. We show that a derivative the canonical branch and the latter the noncanonical term determined by a (2,2) Kähler tensor provides super- branch. The bosonic part of the Lagrangian of the canonical symmetric higher derivative terms free from the auxiliary branch consists of a canonical kinetic term and a four- field problem. We will see that there are two on-shell derivative term for the CP1 model [43]. This admits lump branches associated with the solutions to the auxiliary field (sigma model instanton) solutions identical to those in the in the chiral multiplets. In Sec. III, we study the sigma model CP1 model without a higher derivative term [44],as limit of the gauged chiral models. We first focus on the sigma Bogomol’nyi-Prasad-Sommerfield (BPS) states preserving models whose target spaces are the Hermitian symmetric C N−1 1=2 supersymmetric charges among the original supersym- spaces P and GM;N. They are defined only by the metry [27]. On the other hand, in the noncanonical branch, D-term constraints. We examine the limit in the two distinct the bosonic part of the Lagrangian consists of only the four- branches and write down the higher derivative terms in derivative term of the Skyrme-Faddeev type [43] (that is the nonlinear sigma models. In Sec. IV, we discuss the without fourth order time derivatives) without any canoni- nonlinear sigma models defined by the F-term constraints in addition to D-term constraints, which include the cal kinetic term. This admits a compact baby Skyrmion [45] −2 1 4 target spaces QN , SOð2NÞ=UðNÞ, SpðNÞ=UðNÞ, which is a BPS state preserving = supersymmetry [27]. 10 1 1 While the general Lagrangian in superspace [20] can be E6=½SOð Þ × Uð Þ, E7=½E6 × Uð Þ. We will show that defined on any Kähler target spaces, solving auxiliary field the F-term constraints give additional conditions on equations consisting of more than one chiral multiplet is the target spaces in the nocanonical branch thus reduc- still an open problem, except for a single matrix chiral ing the target spaces, while they do not in the canonical C N−1 superfield [30]. branch. This distinguishes the situation from the P and The purpose of this paper is to present higher derivative GM;N cases. supersymmetric nonlinear sigma models with a wider class In Sec. V, we discuss the BPS states in the nonlinear of target spaces—Hermitian symmetric spaces, sigma models, and in Sec. VI, we mention fermionic terms in the models. Section VII is devoted to conclusion and CPN−1 ¼ SUðNÞ=½SUðN − 1Þ × Uð1Þ; discussions. As for a superfield notation, we follow the Wess-Bagger convention [50]. GM;N ¼ UðMÞ=½UðM − NÞ × UðNÞ; QN−2 ¼ SOðNÞ=½SOðN − 2Þ × Uð1Þ; II. SUPERSYMMETRIC HIGHER DERIVATIVE TERMS IN THE CHIRAL MODEL SOð2NÞ=UðNÞ;SpðNÞ=UðNÞ; In this section, we briefly introduce the supersymmetric E6=½SOð10Þ × Uð1Þ;E7=½E6 × Uð1Þ; ð1:1Þ higher derivative term to the chiral model that is free from the auxiliary field problem. The Lagrangian is given by [20,29] for which we solve auxiliary field equations. In the case Z without higher derivative terms, supersymmetric nonlinear L 4θ Φ Φ† sigma models on Hermitian symmetric spaces can be ¼ d Kð ; Þ Z constructed by imposing supersymmetric constraints on 1 ¯ ¯ 4 cd † α ia kb ¯ †j ¯ α_ †l d θΛ ¯ ¯ Φ;Φ D Φ DαΦ Dα_ Φ D Φ gauged linear sigma models [27]. This formulation is found þ 16 ijkl;ab ð Þ c d to help us to solve auxiliary field equations even with higher Z derivative terms. We employ the gauged linear sigma models þ d2θWðΦÞþH:c: ; ð2:1Þ with higher derivative terms for chiral multiplets [29] and then take the strong gauge coupling (nonlinear sigma model) † limit. Solving auxiliary field equations yields the canonical where KðΦ; Φ Þ is a Kähler potential, WðΦÞ is a super- Λ Φ Φ† and noncanonical branches for the on-shell actions. For the potential, and ijk¯ l¯ð ; Þ is a (2,2) Kähler tensor defined by N † CP model, we obtain a supersymmetric extension of the the chiral superfields Φ; Φ and their derivatives DαΦ, C N – ¯ † † P Skyrme-Faddeev model [46 49] in the canonical Dα_ Φ , ∂mΦ, and ∂mΦ .