Higher Derivative Supersymmetric Nonlinear Sigma Models on Hermitian Symmetric Spaces and BPS States Therein

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 . We assume that the chiral super- branch, while the on-shell Lagrangian in the noncanonical fields Φiaði ¼ 1; …; dim G; a ¼ 1; …; dim G0Þ belong to branch consists solely of the CPN Skyrme-Faddeev term the fundamental representations of the global and gauge

025001-2 HIGHER DERIVATIVE SUPERSYMMETRIC NONLINEAR SIGMA … PHYS. REV. D 103, 025001 (2021) symmetries G and G0, respectively. The four-dimensional The other examples include the Galileon inflation and the N ¼ 1 chiral superfield is expanded, in the chiral basis ghost condensation models [21,22] and a superconformal ym ¼xm þiθσmθ¯,as higher derivative nonlinear sigma model [51]. Now we make the global symmetry G0 be gauged Φia ¼ φiaðyÞþθψiaðyÞþθ2FiaðyÞ: ð2:2Þ by introducing the N ¼ 1 vector multiplet V ¼ Vaˆ Taˆ ðaˆ ¼ 0; 1; …; dim G0Þ, We stress that the pure bosonic component in the fourth derivative part 1 V ¼ −ðθσmθ¯ÞA ðxÞþiθ2θ¯ λ¯ðxÞ − iθ¯2θλðxÞþ θ2θ¯2DðxÞ: ¯ ¯ m 2 α ia kb ¯ †j ¯ α_ †l D Φ DαΦ Dα_ Φc D Φ ð2:3Þ d ð2:7Þ saturates the Grassmann coordinates, and only the bosonic Λ ˆ fields in the Kähler tensor ijk¯ l¯ contribute to this pure Here we have employed the Wess-Zumino gauge and Ta bosonic term [20]. Indeed, the component expansion of are the generators of G0 satisfying the normalization ˆ ˆ the term (2.3) can be evaluated as TrðTaˆ TbÞ¼kδaˆ b. In the derivative corrections, the vector ¯ ¯ multiplet is introduced in the gauge covariantization of the αΦiα Φkb ¯ Φ†j ¯ α_ Φ†l D Dα Dα_ c D d supercovariant derivative:, ¯ 16θ2θ¯ 2 ∂ φia∂mφkb ∂ φ¯ j ∂nφ¯ l¯ ¼ ½ð m Þð n c dÞ Φia → D Φia Φia Γ a Φib 1 ¯ ¯ ¯ ¯ Dα α ¼ Dα þð αÞ b : ð2:8Þ − ∂ φiaFkb Fia∂ φkb ∂mφ¯ j F¯ l F¯ j ∂mφ¯ l 2 ð m þ m Þð c d þ c dÞ ¯ ¯ ia ¯ j kb ¯ l The gauge superconnection Γα is defined by þ F FcF Fdþ: ð2:4Þ

Here are terms involving fermions. Due to this remarkable a −2gV 2gV ðΓαÞ ¼ e Dαe ; ð2:9Þ structure, a large class of supersymmetric models with b derivative corrections can be realized by the Lagrangian (2.1). For example, the N ¼ 1 supersymmetric Dirac-Born- where g is the gauge coupling constant. The gauge invariant Infeld model is given by the Kähler tensor (scalar) [15,16] derivative term is given by a (2,2) Kähler tensor of the following structure: 1 Λ Φ Φ† pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∂ Φ∂mΦ† ð ; Þ¼ 2 ;A¼ m ; 1 þ A þ ð1 þ AÞ − B Λ cd Λ Φ Φ† 2gV c 2gV d ijk¯ l;ab¯ ¼ ijk¯ l¯ð ; Þðe Þ aðe Þ b; ð2:10Þ m † n † B ¼ ∂mΦ∂ Φ∂nΦ ∂ Φ : ð2:5Þ Λ where ijk¯ l¯ is composed of gauge invariant quantities made A supersymmetric Skyrme-Faddeev model is given by the † 1 φ 2 −2 of Φ; Φ ;V. These include the gauge covariant derivatives Kähler metric Kφφ¯ ð þj j Þ and the Kähler scalar [27], ¯ † of the chiral superfields DαΦ, Dα_ Φ . We note that terms † 1 made of ∂mΦ, ∂mΦ , which are allowed in the ungauged ΛðΦ;Φ†Þ¼ð∂ Φ∂mΦ∂ Φ†∂nΦ†Þ−1 m n ð1þΦΦ†Þ4 case, are forbidden in the gauged models due to supersymmetry and the gauge covariance. Then, the gauged ∂ Φ†∂mΦ 2 −∂ Φ∂mΦ∂ Φ†∂nΦ† ×½ð m Þ m n : ð2:6Þ Lagrangian can be written as

Z Z 1 L 4θ Φ Φ† − 4θΛ Φ Φ† D¯ Φ†j¯ 2gV DαΦi D¯ α_ Φ†l¯ 2gVD Φk ¼ d Kð ; ;VÞ 16 d ijk¯ l¯ð ; Þð α_ e Þð e α Þ Z Z Z 2 1 2 α 4 þ d θWðΦÞþH:c: þ Tr d θW Wα þðH:c:Þ − 2κg d θTrV; ð2:11Þ 16kg2 where we have introduced the gauge kinetic and Fayet-Iliopoulos (FI) terms with the FI parameter κ. The superpotential and the FI term are necessary to impose the supersymmetric constraints. The bosonic part of the Lagrangian is given by [29]

025001-3 MUNETO NITTA and SHIN SASAKI PHYS. REV. D 103, 025001 (2021) 2 2 ∂ K ¯ ∂ K ¯ g ¯ ∂K ∂K L − φ¯ j mφib − ¯ j ib aˆ φ¯ j aˆ c aˆ c φid − 2κδaˆ boson ¼ ¯ Dm aD ¯ FaF þ D cðT Þ ¯ þ ðT Þ 0 ∂φ¯ j ∂φib ∂φ¯ j ∂φib 2 d ∂φ¯ j ∂φic d a a d ¯ 1 1 1 ∂W ∂W ¯ þ Tr − F Fmn þ D2 þ Fia þ F¯ i k 4 mn 2 ∂φia ∂φ¯ i¯ a a j¯ ¯ Λ ¯ φ φ¯ mφ¯ nφia φ¯ l φkb þ ikj¯ lð ; Þ ðD aD ÞðDm bDn Þ 1 ¯ ¯ ¯ ¯ ¯ ¯ − D φiaFkb FiaD φkb Dmφ¯ j F¯ l F¯ j Dmφ¯ l FiaF¯ j FkbF¯ l ; : 2 ð m þ m Þð a b þ a bÞþ a b ð2 12Þ

0 0 where Fmn ¼ ∂mAn − ∂mAn þ i½Am;An and T is the Uð1Þ generator in G . A notable fact about the Lagrangian (2.12) is that, due to the derivative corrections, the equation of motion for the auxiliary field F is not linear, ∂2 ∂ ¯ 1 K W j¯ ¯0¯ ¯0 ¯ ¯ − ib Λ ¯ φ φ¯ − φib ka ib φka mφ δj l δj j mφ¯ l ¯0 F þ ¯0 þ ¯ ð ; Þ ððDm F þ F Dm ÞÞðD þ D Þ ∂φ¯ j ∂φib ∂φ¯ j ikj l 2 b b a a ¯ iaδj¯0j¯ kb ¯ l¯ ib ¯ j kaδj¯0l¯ 0 þ F F Fb þ F FbF ¼ : ð2:13Þ

This apparently allows several solutions for the auxiliary A. CPN − 1 model field. The solutions provide several distinct on-shell La- We first consider the nonlinear sigma model whose 0 grangians. For example, when W ¼ , the inhomogeneous target space is CPN−1 ¼ SUðNÞ=½SUðN − 1Þ × Uð1Þ.We 0 term in (2.13) vanishes and it is obvious that F ¼ is a formulate this model as a gauged linear sigma model with solution. We call this the canonical branch. In addition, we the global symmetry G ¼ SUðNÞ and gauge symmetry have another solution F ≠ 0 and we call the corresponding 0 0 G ¼ Uð1Þ, where the latter is complexified GC ¼ Uð1ÞC. theory the noncanonical branch. Both the branches exhibit †¯ K δ ¯ΦiΦ j remarkable features [16,23,26,27]. Although this shows an We consider the flat Kähler potential ¼ ij ¼ Φ†iΦi 0 Φi Φ†i unusual situation in supersymmetric theories, we stress that and W ¼ . The chiral superfields ; ; ði ¼ 1 … N Eq. (2.13) is the algebraic equation, not the kinematical ; ;NÞ belong to the fundamental representation of one, which guarantees that F still keeps the role of the the global SUðNÞ symmetry, and their Uð1Þ charges are auxiliary field. assigned as ðþ1; −1Þ. We assume that the (2,2) Kähler In the following sections, we examine the nonlinear tensor is given by the flat metric as sigma model limit g → ∞ in the gauged linear sigma Λ δ δ Λ Φ Φ† models. In the limit, the vector multiplet carries non- ikj¯ l¯ ¼ ij¯ kl¯ ð ; Þ; ð3:1Þ propagating degrees of freedom and it will be integrated out. We in particular focus on the Hermitian symmetric where ΛðΦ; Φ¯ Þ is a gauge invariant Kähler scalar spaces of the type G=H [52]. This procedure makes us to composed of the chiral superfields and their gauge covar- ¯ † 2 write down a variety of supersymmetric nonlinear sigma iant derivatives DαΦ, Dα_ Φ . Then, the Lagrangian models with higher derivative terms. becomes [53] Z III. NONLINEAR SIGMA MODELS 4 † 2 L ¼ d θΦ ie VΦi WITH D-TERM CONSTRAINTS Z 1 In this section, we discuss supersymmetric nonlinear 4θΛ Φ Φ† δ δ 4VDαΦiD ΦkD¯ Φ†j¯D¯ α_ Φ†l¯ þ 16 d ð ; Þ ij¯ kl¯e α α_ sigma models whose target spaces are the complex pro- Z Z C N−1 jective space P and the Grassmann manifold GM;N. 1 2 α 4 þ d θW Wα þðH:c:Þ − 2κ d θV: ð3:2Þ It is known that they are obtained in the sigma model 16g2 limit g → ∞ of supersymmetric gauge theories with – D-term constraints [52 54]. The former is obtained from Here we have rescaled V → 1 V. We have introduced the FI an Abelian gauge theory, while the latter comes from g a non-Abelian gauge theory. In the following, we term which provides the D-term constraints on the com- Φi explicitly integrate out the gauge field both in the canonical ponent fields in . Note that for the Abelian gauge group, and the noncanonical branches in the sigma model limit g → ∞. 2The case of Λ ¼ const: was studied in Appendix B of Ref. [49].

025001-4 HIGHER DERIVATIVE SUPERSYMMETRIC NONLINEAR SIGMA … PHYS. REV. D 103, 025001 (2021)

¯ † ¯ α_ † i i we have DαΦ ¼ DαΦ þðDαVÞΦ, Dα_ Φ ¼ D Φ þ φ φ¯ ¼ κ: ð3:6Þ ðD¯ α_ VÞΦ†. The bosonic part of the Lagrangian (3.2) is then given by The equation of motion for the gauge field Am is

L ¼ −D φiDmφ¯ i þ FiF¯ i þ Dðφ¯ iφi − κÞ ∂L boson m 0 ¼ 1 1 ∂A 2 − mn m þ D 2 FmnF ∂Λ 2g 4g −1 ¼ −2κðiκ φi∂mφ¯ i þ AmÞþi ðφiDmφ¯ i − φ¯ iDmφiÞ i j¯ m k n l¯ ∂X þ δ ¯δ ¯Λðφ; φ¯ ÞððDmφ Dnφ¯ ÞðD φ D φ¯ Þ ij kl 2 Λ φi φ¯ i φj pφ¯ j ¯ ¯ ¯ ¯ þ i ðXÞ½ð Dp ÞðDm D Þ − 2ðD φiDmφ¯ jÞðFkF¯ lÞþFiF¯ jFkF¯ lÞ; ð3:3Þ m i i j p j − ðφ¯ Dpφ ÞðDmφ¯ D φ Þ: ð3:7Þ where Fmn ¼ ∂mAn − ∂nAm and the gauge covariant deriva- i i i Using the constraint (3.6), we find that the solution to this tive is defined by Dmφ ¼ ∂mφ þ iAmφ . In the following, we assume that the gauge invariant Kähler scalar Λðφ; φ¯ Þ is equation is given by i m i given by a function of X ¼ Dmφ D φ¯ as a typical example. κ−1φ¯ i∂ φi The most plausible reason for this assumption is that this Am ¼ i m : ð3:8Þ quantity contains only the first order time derivative of fields. This guarantees the absence of the Ostrogradski’s Substituting the solution (3.8) into (3.5) and using the ghost [19] in the theory. This assumption is easily relaxed constraint (3.6), we obtain the Lagrangian in the canonical but allowing other gauge and Lorentz invariant quantities branch, i i i n i m j n j such as X ¼ φ φ¯ ;Dmφ D φ¯ D φ D φ¯ and so on does not L − ˜ φ¯ i ˜ mφi Λ ˜ ˜ φi ˜ φ¯ i ˜ mφj ˜ nφ¯ j change the following discussions. c ¼ Dm D þ ðXÞðDm Dn ÞðD D Þ; In the sigma model limit g → ∞, the gauge kinetic and 2 ð3:9Þ D terms vanish and the gauge field Am becomes an auxiliary field. Before integrating out these fields, we first ˜ φi ∂ φi − κ−1 φ¯ j∂ φj φi integrate out the auxiliary fields in the chiral superfields. where we have defined D ¼ m ð m Þ and ¯ ˜ ˜ φi ˜ mφ¯ i The equation of motion for the auxiliary field F¯ i is X ¼ Dm D . It is convenient to solve the constraint (3.6) explicitly by ∂L using the parametrization 0 ¼ ∂ ¯ i¯ ffiffiffi F pκ i j m ¯ j i j ¯ j i φi i ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ F − 2ΛðXÞððDmφ D φ ÞF − ðF F ÞF Þ: ð3:4Þ ¼ W p : ð3:10Þ W† · W

As we have noticed, there are two branches associated with 1 By using the Uð1ÞC gauge symmetry, we can fix W ¼ 1. the solutions Fi ¼ 0 and Fi ≠ 0. We examine each branch Then, we have the conventional parametrization of the separately. CPN−1 model, 1. Canonical branch ffiffiffi pκ 1 φi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; For the canonical branch corresponding to the solution ¼ 2 s i 1 þju⃗j u F ¼ 0, the Lagrangian (3.3) becomes ffiffiffi pκ i¯ s¯ L − φi mφ¯ i Λ φi φ¯ i mφj nφ¯ j φ¯ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1; u¯ Þ; ðs; s¯ ¼ 2; …;NÞ; ð3:11Þ c ¼ Dm D þ ðDm Dn ÞðD D Þ 1 þju⃗j2 þ Dðφ¯ iφi − κÞ: ð3:5Þ where us; u¯ s¯ are inhomogeneous coordinates of the CPN−1 The equation of motion for the auxiliary field D gives the manifold. By using this form, the Lagrangian in the following constraint for the scalar fields φi: canonical branch can be rewritten as

κ L ¼ − ½ð1 þju⃗j2Þð∂ u⃗ · ∂mu⃗Þ − ðu⃗· ∂ u⃗Þðu⃗ · ∂mu⃗Þ c ð1 þju⃗j2Þ2 m m κ2 þ Λðu; u¯Þ½ð1 þju⃗j2Þ2ð∂ u⃗ · ∂ u⃗Þð∂mu⃗ · ∂nu⃗Þ ð1 þju⃗j2Þ4 m n 2 m n 2 2 −2ð1 þju⃗j Þð∂mu⃗ · ∂nu⃗Þðu⃗· ∂ u⃗Þðu⃗ · ∂ u⃗Þþðu⃗· ∂mu⃗Þ ðu⃗ · ∂nu⃗Þ : ð3:12Þ

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By using the Fubini-Study metric The solution Fi ≠ 0 is found to be

1 ⃗2 δ − s ¯ t¯ κ ð þjuj Þ st¯ u u 1 gst¯ ¼ 2 2 ð3:13Þ i ¯ i − φi mφ¯ i ð1 þju⃗j Þ F F ¼ 2Λ þ Dm D : ð3:16Þ for the CPN−1 manifold, the Lagrangian (3.12) can be simply written as Substituting this solution into the Lagrangian (3.3),it becomes L − ∂ s∂m ¯ t¯ c ¼ gst¯ mu u 2 s t¯ m q n r¯ L ¼ ΛðXÞ½ðD φ¯ iD φiÞðDmφ¯ jDnφjÞ − ðD φjDmφ¯ jÞ þ gst¯gqr¯Λðu; u;¯ ∂mu; ∂mu¯Þð∂mu ∂nu¯ Þð∂ u ∂ u¯ Þ: nc m n m 1 ð3:14Þ − þ Dðφ¯ iφi − κÞ; ð3:17Þ 4ΛðXÞ The first term is the ordinary kinetic term of the CPN−1 nonlinear sigma model and the second is the derivative where we have assumed the sigma model limit g → ∞. The corrections determined by the arbitrary scalar function D-term condition leads to the constraint (3.6), while the Λ ¯ ∂ ∂ ¯ ðu; u; mu; muÞ. equation of motion for the gauge field is Before going to the discussion on the noncanonical branch, two comments in this Lagrangian are in order. This ∂Λ Λ k m k k m k i i m j n j Lagrangian with ¼ const: was obtained in Ref. [49] as 0 ¼ i ðφ D φ¯ −φ¯ D φ Þ½ðDmφ¯ Dnφ ÞðD φ¯ D φ Þ the low-energy effective theory of a BPS non-Abelian ∂X j m j 2 vortex in N ¼ 2 supersymmetric UðNÞ gauge theory [55]; −ðDmφ D φ¯ Þ a non-Abelian vortex in this case allows the supersym- þ2iΛðXÞ½−φ¯ jDqφjðDmφ¯ iD φiÞþφjDpφ¯ jðD φ¯ iDmφiÞ metric CPN−1 model on the vortex world sheet, on which q p − φi pφ¯ i φj mφ¯ j −φ¯ j mφj 1=2 BPS condition preserves four supercharges among ðDp D Þð D D Þ eight supercharges that the N ¼ 2 supersymmetric theory ∂Λ þ4iΛ−2ðXÞ ðφiDmφ¯ i −φ¯ iDmφiÞ: ð3:18Þ in the bulk has. The bosonic part of the four-derivative ∂X correction term is precisely in this form [49]. Without supersymmetry, the CPN−1 Skyrme-Faddeev −1 By using the constraint (3.6), we find that Eq. (3.18) is model was proposed in Refs. [46–48] as the CPN model again solved by with four-derivative terms. In this case, there are three kinds of four-derivative terms, and the one in Eq. (3.14) (with Λ −1 ¯ i i ¼ const:) corresponds to a particular choice among Am ¼ iκ φ ∂mφ : ð3:19Þ them. Thus, we call the Lagrangian in Eq. (3.14) (the bosonic part of) the supersymmetric CPN−1 Skyrme- Substituting this solution into the Lagrangian (3.17),we Faddeev model. find the new sigma model given by 2. Noncanonical branch L Λ ˜ ˜ φ¯ i ˜ φi ˜ mφ¯ j ˜ nφj − ˜ φj ˜ mφ¯ j 2 We next study the noncanonical branch corresponding to nc ¼ ðXÞ½ðDm Dn ÞðD D Þ ðDm D Þ solutions F ≠ 0. Again, the equation of motion for F¯ is 1 i − : ð3:20Þ 4Λ X˜ ∂L ð Þ 0 ¼ ¯ ∂F¯ i The expression (3.11) enables us to rewrite the Lagrangian i − 2Λ φj mφ¯ j i − j ¯ j i ¼ F ðXÞððDm D ÞF ðF F ÞF Þ: ð3:15Þ (3.20) as

κ2 L ¼ Λðu; u;¯ ∂ u; ∂ u¯Þ½ð1 þju⃗j2Þ2fð∂ u⃗· ∂ u⃗Þð∂mu⃗· ∂nu⃗Þ − ð∂ u⃗ · ∂mu⃗Þ2g nc ð1 þju⃗j2Þ4 m m m n m 2 m n m n − 2ð1 þju⃗j Þfð∂mu⃗ · ∂nu⃗Þðu⃗· ∂ u⃗Þðu⃗ · ∂ u⃗Þ − ð∂mu⃗ · ∂ u⃗Þðu⃗· ∂nu⃗Þðu⃗ · ∂ u⃗Þg 1 ⃗ ∂ ⃗ 2 ⃗ ∂ ⃗2 − ⃗ ∂ ⃗ ⃗ ∂m ⃗ 2 − Λ−1 ¯ ∂ ∂ ¯ þðu · mu Þ ðu · nuÞ fðu · mu Þðu · uÞg 4 ðu; u; mu; muÞ: ð3:21Þ

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By using the Fubini-Study metric (3.13), this can be simply The derivative corrections are given in the double trace expressed as form, and we have included the FI term which provides the D-term conditions on the chiral multiplets. L Λ ¯ ∂ ∂ ¯ nc ¼ ðu; u; mu; muÞ In the nonlinear sigma model limit g → ∞, the bosonic s t¯ m q n q¯ s m t¯ 2 part of the Lagrangian becomes × ½ðgst¯∂mu ∂nu¯ Þðgqr¯∂ u ∂ u¯ Þ − ðgst¯∂mu ∂ u¯ Þ 1 − Λ−1 ¯ ∂ ∂ ¯ L ¼ −Tr½D φDmφ¯ þTr½φ¯ Dφ − κDþTr½FF¯ 4 ðu; u; mu; muÞ: ð3:22Þ boson m m n þ Λðφ;φ¯ ÞfTr½DmφDnφ¯ Tr½D φD φ¯ This four-derivative term does not contain fourth order time − D φDmφ¯ FF¯ derivatives, unlike the one in Eq. (3.14) in the canonical Tr½ m Tr½ m ¯ ¯ ¯ branch. It is obvious that the target space of the scalar − Tr½FDmφ¯ Tr½D φFþTr½FFTr½FFg: ð3:26Þ fields is the CPN−1 space, but the canonical kinetic term ∂ s∂m t¯ gst¯ mu u is absent. The gauge covariant derivative is D φ ¼ ∂ φ þ iφA .In 1 m m m In particular, for the N ¼ 2 case of CP , the Lagrangian the following, we assume Λ is given by the gauge invariant m reduces to quantity X ¼ Tr½DmφD φ¯ . As clarified before, this condition is easily relaxed allowing for any other gauge ð∂ u∂mu¯Þ2 − j∂ u∂muj2 1 L ¼ −κ2Λ m m − ; ð3:23Þ invariant quantities. The equation of motion for the nc ð1 þjuj2Þ4 4Λ auxiliary field D gives the constraint which is known as the supersymmetric baby Skyrme model φφ¯ ¼ κ1 : ð3:27Þ [24,25,27,40–42]. One notices that the first term (with N Λ ¼ 1) is nothing but the Skyrme-Faddeev fourth deriva- On the other hand, the equation of motion for the auxiliary tive term (without the fourth order time derivatives), while F¯ the second term provides higher derivative corrections field is given by determined by Λðu; u;¯ ∂mu; ∂mu¯Þ. The Lagrangian in −1 Λ − φ mφ¯ Eq. (3.22) may be called the supersymmetric CPN baby F þ ðXÞf Tr½Dm D F m ¯ Skyrme model. − Tr½FDmφ¯ D φ þ 2Tr½FFFg¼0: ð3:28Þ

G B. M;N model 1. Canonical branch We next consider the nonlinear sigma model whose It is obvious that F ¼ 0 is a solution. In this case, the target space is the Grassmann manifold G M;N ¼ Lagrangian becomes UðMÞ=½UðM − NÞ × UðNÞ. We consider a gauged linear sigma model with the global symmetry G ¼ SUðMÞ and L − φ mφ¯ Λ φ φ¯ mφ nφ¯ i c ¼ Tr½Dm D þ ðXÞTr½Dm Dn Tr½D D ; gauge symmetry UðNÞ, of which the chiral superfield Φa (i ¼ 1; …M; a ¼ 1; …N) belong to the ðM; N¯ Þ represen- ð3:29Þ tation. The global symmetry G ¼ SUðMÞ and gauge symmetry UðNÞC ¼ GLðN;CÞ act on it as where the scalar fields satisfy the constraint (3.27). The

† equation of motion for Am then becomes Φ → gΦeiΘ0 ;e2V → e−iΘ0 e2VeiΘ0 ; ð3:24Þ ∂Λ ∈ Θ0 θ θ¯ φ¯ ∂ φ þ iκA þ i ðφDmφ¯ − Dmφφ¯ Þ where g SUðMÞ, ðx; ; Þ is a chiral superfield of a m m ∂X UðNÞC gauge parameter, and we have introduced the − ΛðXÞTr½D φD φ¯ þ D φD φ¯ ðφ¯ ∂nφ þ iκAnÞ¼0; associated UðNÞ vector superfield Vðx; θ; θ¯Þ. The G × G0 m n n m invariant Lagrangian is therefore [52,54] ð3:30Þ Z L¼ d4θTr½Φe2VΦ† where we have used the constraint (3.27). We again find Z that the exact solution to this equation is given by 1 4 † α 2V ¯ † 2V ¯ α_ † − d θΛðΦ;Φ ÞTr½D Φe Dα_ Φ Tr½DαΦe D Φ −1 16 A ¼ iκ φ¯ ∂ φ: ð3:31Þ Z Z m m 1 2 α 4 þ d θTr½W WαþðH:c:Þ −2κ d θTrV: 16g2 Plugging this solution back into the Lagrangian, we obtain the nonlinear sigma model supplemented by the derivative ð3:25Þ corrections

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L − ˜ φ ˜ mφ¯ Λ ˜ ˜ φ ˜ φ¯ ˜ mφ ˜ nφ¯ c ¼ Tr½Dm D þ ðXÞTr½Dm Dn Tr½D D : Therefore, even in the noncanonical branch, we find that the solution to the equation of motion for Am is given by ð3:32Þ −1 Am ¼ iκ φ∂mφ¯ . For the other gauge invariant combina- tions Y such as (3.34), the result is the same. The constraint Here D˜ φ ∂ φ − κ−1φ φ¯ ∂ φ and X˜ Tr D˜ φD˜ mφ¯ . m ¼ m ð m Þ ¼ ½ m (3.27) is solved by the parametrization (3.33) and the The constraint (3.27) is solved by the parametrization Lagrangian becomes again the Grassmanian sigma model ffiffiffi pκ with the derivative corrections. φ ffiffiffiffiffiffiffiffiffiffiffi We comment on the symmetric nature of the Grassmanian ¼ W p † ; ð3:33Þ W W manifold GM;N ¼ GM;M−N. We have constructed the Grassmanian sigma models by gauging the symmetry where W is an M × N matrix. Substituting this into the G0 ¼ UðNÞ, but there is an alternative way to define the Lagrangian, the first term in (3.32) becomes the kinetic same target space by gauging G0 ¼ UðM − NÞ. It is obvious term of the sigma model whose target space is the that the latter construction is essentially the same with the Grassmann manifold G , while the second term gives M;N former one except the N ¼ 1 case. In this case, one notices derivative corrections determined by the arbitrary gauge N−1 that G 1 ¼ G −1 ¼ CP . This equality implies that we invariant function Λ W;W†; ∂ W;∂ W† . N; N;N ð m m Þ can construct supersymmetric sigma model whose target C N−1 2. Noncanonical branch space is P , either by gauging the non-Abelian UðN − 1Þ symmetry or the Abelian Uð1Þ symmetry. The We next examine the noncanonical branch. It is obvious former is nothing but the construction discussed in this that there are nonzero solutions F ≠ 0 to Eq. (3.28). section, while the latter is the formulation discussed in the Compared with the case of the Abelian gauge symmetry, previous section. We find apparently different Lagrangians it is not straightforward to write down explicit solutions for (3.35) and (3.20) in the derivative corrections. This means F. However, the gauge invariant solution Tr½FF¯ should be that the equivalence of the target spaces in different given by the gauge and the Lorentz invariant quantities Y formalism are not taken over to the derivative corrections such as in the superfield language. In the next section, we will see a similar situation in the canonical branch of the nonlinear m m n Y ¼ Tr½DmφD φ¯ ; Tr½DmφDnφ¯ D φD φ¯ ; sigma models defined by the F-term constraints. We will, among other things, encounter qualitatively different struc- Tr½D φD φ¯ DnφDmφ¯ ; ð3:34Þ m n tures of the target spaces in the noncanonical branch. and so on. Using the equation of motion for F, the noncanonical Lagrangian becomes IV. NONLINEAR SIGMA MODELS WITH D- AND F-TERM CONSTRAINTS L − φ mφ¯ nc ¼ Tr½Dm D In this section, we discuss the nonlinear sigma m n ¯ 2 models defined by F-term constraints in addition to the þ ΛðXÞfTr½DmφDnφ¯ Tr½D φD φ¯ − Tr½FFðYÞ g: D-term constraints [52,56]. Such Hermitian symmetric ð3:35Þ target spaces include the quadric surface QN−2¼ SOðNÞ=½SOðN−2Þ×Uð1Þ, the quotient spaces of the types ¯ ≠ 0 Here Tr½FFðYÞ is a solution F given by Y. The SOð2NÞ=UðNÞ, Spð2NÞ=UðNÞ, E6=½SOð10Þ × Uð1Þ, and variation of the Lagrangian (3.35) by A is therefore N−2 m E7=½E6 × Uð1Þ. We begin with the target space Q ¼ SOðNÞ=½SOðN − 2Þ × Uð1Þ as a typical example. δL φ¯ ∂mφ κ m δ nc ¼ iTr½ð þ i A Þ Am ∂Λ QN − 2 m m A. Quadric surface þ i Tr½ðφD φ¯ − D φφ¯ ÞδAm ∂X We consider a gauged linear sigma model with the global − iΛðXÞTr½DmφDnφ¯ þ DnφDmφ¯ symmetry G ¼ SOðNÞ and the gauge symmetry Uð1Þ. The Φi 1 … ×Tr½ðφ¯ ∂ φ þ iκA ÞδA chiral superfield (i ¼ ; ;N) belongs to the funda- n n m mental representation N of SOðNÞ and has the Uð1Þ charge ¯ ¯ − 2Λðφ; φ¯ ÞTr½FFTr½ðFFÞ0δY: ð3:36Þ þ1. The Lagrangian in the nonlinear sigma model limit is Z Z Here ðFF¯ Þ0 stands for the differentiation with respect to Y. L 4θ Φi 2VΦ†i − 2κ 2θΦ Φ⃗t Φ⃗ In the last term in (3.36), one finds that the variation of the ¼ d ð e VÞþ d 0 J þ H:c: φ¯ ∂mφ κ m Z gauge invariant quantity Y is proportional to þ i A . 1 Indeed, for Y ¼ Tr½D φDmφ¯ ,wehave 4θΛ Φ Φ† 4VDαΦiD ΦjD¯ Φ†iD¯ α_ Φ†j m þ 16 d ð ; Þe α α_ ; m m δY ¼ 2iTr½ðφ¯ ∂ φ þ iκA ÞδAm: ð3:37Þ ð4:1Þ

025001-8 HIGHER DERIVATIVE SUPERSYMMETRIC NONLINEAR SIGMA … PHYS. REV. D 103, 025001 (2021) where we have introduced the gauge invariant superpoten- The gauge transformations are given by ⃗t ⃗ tial W ¼ Φ0Φ JΦ. This will provide the F-term constraints on the chiral multiplet. Here Φ0 is the SOðNÞ singlet and whose Uð1Þ charge is assigned to −2, V is the Uð1Þ vector † Φ → −iΦ0 Φ Φ → 2iΘ0 Φ 2V → −iΘ0 2V iΘ0 multiplet, and the matrix J is given by e ; 0 e 0;e e e e : 0 1 ð4:3Þ 001 B C J ¼ @ 0 1N−2 0 A: ð4:2Þ 100 Then the bosonic part of the Lagrangian is evaluated as

L − φi mφ¯ i i ¯ i φ⃗t φ⃗ ¯ φ⃗¯ t φ⃗¯ φ¯ iφi − κ boson ¼ Dm D þ F F þ F0ð J ÞþF0ð J ÞþDð Þ i ¯ i m j n ¯ j i m ¯ i j ¯ j i i 2 þ ΛðXÞfðDmφ Dnφ ÞðD φ D φ Þ − 2ðDmφ D φ ÞF F þðF F Þ g XN−1 XN−1 1 N N 1 iˆ iˆ 1 ¯ N N ¯ 1 iˆ ¯ iˆ þ 2φ0 φ F þ φ F þ φ F þ 2φ¯ 0 φ¯ F þ φ¯ F þ φ¯ F ; ð4:4Þ iˆ¼2 iˆ¼2 where we have again assumed that Λ is a function of the 1. Canonical branch φi mφ¯ i gauge invariant quantity X ¼ Dm D . The equation of One finds that a solution to Eq. (4.8) is given by motion for D gives the constraint i φ0 ¼ 0;F¼ 0: ð4:9Þ φ¯ iφi ¼ κ: ð4:5Þ This corresponds to the canonical branch. We note that in this branch, the constraint (4.7) becomes trivial. Then the Since Φ0 does not propagate, it is also integrated out. The Lagrangian becomes F-term constraints are given by the equation of motions for F0; φ0, L − φi mφ¯ i Λ φi φ¯ i mφj nφ¯ j c ¼ Dm D þ ðXÞðDm Dn ÞðD D Þ: ð4:10Þ t δF0∶φ⃗Jφ⃗¼ 0; ð4:6Þ The equation of motion for Am is therefore

XN−1 ∂Λ ˆ ˆ − κ κ−1φi∂ φ¯ i φi mφ¯ i − φ¯ i mφi δφ ∶φ1 N φN 1 φi i 0 ði m þ AmÞþi ð D D Þ 0 F þ F þ F ¼ : ð4:7Þ ∂X ˆ i¼2 i i j n j þ iΛðφ; φ¯ Þfðφ Dnφ¯ ÞðDmφ D φ¯ Þ − ðφ¯ iDnφiÞðD φ¯ jDnφjÞg ¼ 0; ð4:11Þ These give the constraints for the fields φi; φ¯ i. The m equations of motion for F¯ i are where we have used the constraint (4.5). We find that the solution is given by ¯ 1 1 j m j 1 j ¯ j 1 δF ∶F − 2ΛðXÞfðDmφ D φ¯ ÞF − ðF F ÞF g κ−1φ¯ i∂ φi N Am ¼ i m : ð4:12Þ þ 2φ¯ 0φ¯ ¼ 0; ¯ iˆ iˆ j m ¯ j iˆ j ¯ j iˆ δF ∶F − 2ΛðXÞfðDmφ D φ ÞF − ðF F ÞF g Substituting this solution to the Lagrangian, the covariant derivative D φi in the Lagrangian (4.10) is replaced by iˆ ˆ m þ 2φ¯ 0φ¯ ¼ 0; ði ¼ 2; …;N− 1Þ; ˜ i Dmφ . Now, we solve the remaining constraints (4.5) and ¯ N N j m j N j ¯ j N δF ∶F − 2ΛðXÞfðDmφ D φ¯ ÞF − ðF F ÞF g (4.6). The constraint (4.5) is solved by the following 1 parametrization: þ 2φ¯ 0φ¯ ¼ 0: ð4:8Þ ffiffiffi pκ φi i ffiffiffiffiffiffiffiffiffiffiffiffiffiffi – ¼ W p † : ð4:13Þ Equations (4.5) (4.8) should be solved simultaneously. In W · W the following, we solve these equations in the canonical and the noncanonical branches separately. Since we have

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κ φ⃗t φ⃗ 2 1 N iˆ 2 0 simultaneous equations (4.18) and (4.19) admit nontrivial J ¼ † ð W W þðW Þ Þ¼ ; W · W solutions or not. Instead, the most plausible way to work ðiˆ ¼ 2; …;N− 1Þ; ð4:14Þ with these equations is to restrict the dynamics to a subspace in QN−2. This subspace highly depends on the last constraint (4.6) is solved by explicit structures of the solution Fi. For example, if we 0 1 consider the N ¼ 4 case, one finds that a solution to the 1 equation of motion for Fi in the noncanonical branch is B ˆ C Wi ¼ @ ui A; ð4:15Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − 1 iˆ 2 1 2 ðu Þ F2 ¼ F3 ¼ F4 ¼ 0;F1 ¼ − þ D˜ φiD˜ mφ¯ i: 2ΛðX˜ Þ m 1 where we have fixed W ¼ 1 by the Uð1ÞC gauge trans- ð4:20Þ formation. Plugging this back into the Lagrangian, we obtain the QN−2 sigma model whose derivative corrections Λ † ∂ ∂ † Substituting this solution to the constraint (4.18) gives a are completely determined by ðW;W ; mW; mW Þ. condition 2. Noncanonical branch sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 φ4 − ˜ φi ˜ mφ¯ i 0 In the noncanonical branch, we find that a solution to the þ Dm D ¼ : ð4:21Þ 2ΛðX˜ Þ equations (4.8) is given by 1 In order that this is compatible with equation of motion, we φ 0 i ¯ i − φi mφ¯ i 4 2 0 ¼ ;FF ¼ þ Dm D : ð4:16Þ first choose the subspace φ ¼ 0 in Q and then solve 2ΛðXÞ Eq. (4.19) in that subspace. A comment is in order. As for the Grassmann case, we Then, the Lagrangian becomes 4 have an equality Q ¼ G4;2. The constraint (4.18) is neces- 4 L Λ φ¯ i φi mφ¯ j nφj − φj mφ¯ j 2 sary in the construction of the Q sigma model discussed in nc ¼ ðXÞ½ðDm Dn ÞðD D Þ ðDm D Þ 1 this section, but such a restriction to a subspace was absent in − the construction of the G4 2 model in the previous section. 4Λ : ð4:17Þ ; ðXÞ Apparently, they have different Lagrangians even though there is a target space isomorphism between them. The same The equation of motion for Am is the same with (3.18) 1 1 is true for the relation Q ¼ G2 1 ¼ CP . In the noncanoni- and it is again solved by (3.19). Substituting this into ; cal branch, the different constructions of sigma models the Lagrangian, the gauge covariant derivative D φi is m result in the different derivative corrections and the field ˜ φi replaced by Dm . The remaining constraints (4.5) and space that the dynamics occurs, although they give identical (4.6) are solved by the parametrizations (4.13) and (4.15) models in the canonical branches. and the target space is QN−2. However, we should keep in mind that there is an extra constraint (4.7) given by B. SOð2NÞ=UðNÞ and SpðNÞ=UðNÞ models −2 XN−1 Observed that the QN model has remarkable structures ˆ ˆ φ1FNðX˜ ÞþφNF1ðX˜ Þþ φiFiðX˜ Þ¼0; ð4:18Þ in the noncanonical branch due to the F-term constraint, iˆ¼2 we next consider nonlinear sigma models with the target spaces SOð2NÞ=UðNÞ or SpðNÞ=UðNÞ. The gauged linear ˜ where FiðXÞ ≠ 0 is the solution to the auxiliary field in the sigma model has the global symmetry G ¼ SOð2NÞ or noncanonical branch. The constraint (4.18) generically SpðNÞ½¼ USpð2NÞ and gauge symmetry G0 ¼ UðNÞ. The ¯ contains space-time derivatives of the fields φi; φ¯ i and this chiral superfield Φ belongs to the ð2N; N¯ Þ representation of has to be solved together with the following equation of G × G0 and it is expressed as an 2N × N matrix. The motion: constraint is given by the F-term superpotential

0 ˜ ˜ m i ˜ j ˜ j ˜ m k ˜ n k ˜ j ˜ m j 2 t 0 Λ ðXÞD φ ½ðDmφ¯ Dnφ ÞðD φ¯ D φ Þ − ðDmφ D φ¯ Þ W ¼ Tr½Φ0Φ J Φ; ð4:22Þ ˜ ˜ i ˜ m j ˜ n j ˜ i ˜ j ˜ n j − 2ΛðXÞ½Dnφ ðD φ¯ D φ Þ − Dmφ ðDnφ D φ¯ Þ where Φ0 is an N × N symmetric (antisymmetric) matrix 1 t Λ0 ˜ ˜ mφi 0 superfield satisfying Φ0 ¼ ϵΦ0 for G ¼ SOð2NÞ½SpðNÞ. þ 2 ˜ ðXÞD ¼ ; ð4:19Þ 4Λ ðXÞ Here ϵ ¼ 1 for G ¼ SOð2NÞ and ϵ ¼ −1 for G ¼ SpðNÞ. The Uð1Þ ⊂ UðNÞ charge of Φ0 is assigned to −2 to ˜ D where the prime in Λ0ðXÞ denotes the differentiation cancel the Uð1Þ charge þ1 of Φ. The 2N × 2N matrix J0 is with respect to X˜ . In general, it is uncertain whether the given by

025001-10 HIGHER DERIVATIVE SUPERSYMMETRIC NONLINEAR SIGMA … PHYS. REV. D 103, 025001 (2021) 0 1N derived by the superpotential, the target space of the sigma J0 ¼ ; ð4:23Þ ϵ1 0 model is restricted to a subspace of E6=½SOð10Þ × Uð1Þ in N the noncanonical branch. 1 which is an invariant tensor of SOð2NÞ (ϵ ¼ 1)or For the target space E7=½E6 × Uð Þ, the global sym- 0 1 SpðNÞ (ϵ ¼ −1). metry is G ¼ E7 and the G ¼ Uð Þ symmetry is gauged in The equation of motion for D gives the constraint (3.27), the gauge linear sigma model. The F-term constraint is [52] while that for F0 gives the constraint α β γ δ W ¼ dαβγδΦ0Φ Φ Φ ; ð4:28Þ φtJ0φ ¼ 0 ð4:24Þ where dαβγδ is a rank-4 symmetric invariant tensor of E7. Since the gauge symmetry is Abelian, the structure of the and that for φ0 gives a constraint sigma model is essentially the same with the QN−2 case. Again, the constraint φtJ0F ¼ 0: ð4:25Þ α β γ dαβγδφ φ F ¼ 0 ð4:29Þ The equation for F is (4.8), but the gauge symmetry is non- Abelian. One finds that φ0 ¼ 0 and F ¼ 0 are a solution to defines a subspace in E7=½E6 × Uð1Þ in the noncanonical the equation of motion for F. Therefore, the constraint branch. (4.25) becomes trivial in the canonical branch. We find that the gauge field is integrated out by the solution Am ¼ V. BOGOMOL’NYI-PRASAD-SOMMERFIELD κ−1φ¯ ∂ φ i m in the sigma model limit which is the same with STATES the case without the derivative corrections. Then the target space of the sigma model is nothing but SOð2NÞ=UðNÞ or In this section, we study the BPS conditions in the SpðNÞ=UðNÞ and the derivative corrections appear as in nonlinear sigma models discussed in the previous section. the Grassmann case in Eq. (3.35). BPS properties of Skyrme type and higher derivative −2 – Compared with the QN model, it is not straightforward models have been studied in various contexts [57 60].It to write down solutions F ≠ 0 explicitly due to the has been discussed that the gauged chiral models admit the 1 2 1 4 non-Abelian gauge symmetry. However, similar to the = BPS vortex state and the = BPS states [29]. In the 1 2 1 4 FF¯ following, we derive the = and = BPS conditions in Grassmann case, the solution Tr½ is given by the gauge → ∞ invariant quantities such as Tr½D φDmφ¯ . Therefore, we the nonlinear sigma models in the limit g of those in m the gauged chiral models. again find that the gauge field Am is integrated out by A ¼ iκ−1φ¯ ∂ φ. The subsequent discussion is parallel to m m A. BPS states in canonical branch the QN−2 case. Due to the extra constraint (4.25), we find that the target spaces SOð2NÞ=UðNÞ or SpðNÞ=UðNÞ In the canonical branch, the 1=2 BPS vortex equation for should be restricted to their subspaces in the noncanonical the finite g for gauged linear sigma model is branches. 1 ¯ i D φ ¼ 0; F12 − ðφφ¯ − κ1Þ¼0; ð5:1Þ z g C. E6=½SOð10Þ × Uð1Þ and E7=½E6 × Uð1Þ models 1 2 For the case of the target space E6=½SOð10Þ × Uð1Þ, the where z ¼ x þ ix is the complex coordinate in the 0 1 2 global symmetry is G ¼ E6 and the G ¼ Uð1Þ symmetry ðx ;x Þ-plane. These are UðMÞ BPS semilocal vortex is gauged in the gauged linear sigma model [52]. The chiral equations with N-flavors in the case of Grassmann case superfield Φ belongs to the fundamental representation 27 [55,61], for which we have found that higher derivative of E6. This is decomposed into the maximal subgroup corrections do not appear. They reduce to the ordinary BPS SOð10Þ × Uð1Þ. The F-term constraint is given by the vortex equations in the Abelian-Higgs model for the Uð1Þ superpotential gauge theory. The energy bound for this configuration is [29]3 W ¼ Γ Φi ΦjΦk; ð4:26Þ ijk 0 0 E ¼ −κF12: ð5:2Þ where Γijk is a rank-3 symmetric invariant tensor of E6. Since the gauge symmetry is Abelian, the structure of the 3 −2 We comment on the positive semidefiniteness of the energy in sigma model is essentially the same with the QN case. the higher derivative chiral models. It is not always true that the Due to the constraint energy density derived from Eq. (2.1) is positive semidefinite for arbitrary Λ and the Kähler potential K. In order to define the BPS Λ Γ φi k 0 states as minima of energy, we have to choose appropriate and ijk F ¼ ð4:27Þ K that makes the energy be positive semidefinite [27].

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¯ ¯ RThe configuration is classified by the vortex number D˜ φ¯ D˜ φ − D˜ φiD˜ φ¯ i ¼ Λ−1ðφ; φ¯ Þ: ð5:9Þ 2 0 z z z z d xF12. In the nonlinear sigma model limit g → ∞, the second This is nothing but a compacton type equation. To see this, condition in (5.1) gives the D-term constraint, while the we choose an appropriate function Λ. For example, we first one becomes consider the following function in the CP1 model: ∂¯ φi − κ−1 φ¯ j∂¯ φj φi 0 2 −s z ð z Þ ¼ : ð5:3Þ juj 2 Λ ¼ 2 ; ð5:10Þ 1 þjuj2 This implies the following 1=2 BPS lump equation in the sigma models: where s is a constant. Then, the BPS equation (5.9) results in the following compacton equation [45]: ¯ i ∂zφ ¼ 0: ð5:4Þ s 2 ngy ¼ −g : ð5:11Þ The bound (5.2) survives in the limit g → ∞ and the result is Here we have assumed the ansatz

1 1 2 E ∂ φ 2 inθ r ¼ j z j : ð5:5Þ u ¼ e fðrÞ; 1 − g ¼ ;y¼ ; ð5:12Þ 2 1 þ f2 2

This provides the lump charge density in the nonlinear where r and θ are the polar coordinates in the ðx1;x2Þ plane. sigma model. We note that the derivative corrections never show up in Eq. (5.4) and the BPS bound (5.5) in the canonical branch.4 VI. COMMENT ON FERMIONIC INTERACTIONS B. BPS states in noncanonical branch Finally, we comment on the fermionic interactions of the In the noncanonical branch, we have only 1=4 BPS models, while in the previous sections, we have focused on state [29]. The BPS equations in the gauged linear sigma the bosonic sector of the supersymmetric models and have models are shown that the vector field Am is integrated out exactly. However, the integration of the gaugino is rather awkward 1 from the reason described below. We illustrate the problem ¯ −1 D φ ¼ −iF; F12 ¼ φφ¯ − κ1: ð5:6Þ C N z g in the P model. Assuming that Λ does not contain V for simplicity, the In the nonlinear sigma model limit g → ∞, the BPS equation of motion for the vector superfield V in the sigma → ∞ equations become model limit g is given by 1 D¯ φ −iF; φφ¯ κ1: 5:7 2Φi 2VΦ†i − κ Λ Φ Φ† 4VDαΦiD ΦjD¯ Φ†iD¯ α_ Φ†j z ¼ ¼ ð Þ e þ 4 ð ; Þe α α_ 1 † 4V i α j ¯ †i ¯ α_ †j The latter equation gives the D-term constraint. From the þ Dα½ΛðΦ; Φ Þe Φ D Φ Dα_ Φ D Φ first equation, we obtain 4 1 ¯ Λ Φ Φ† 4VD ΦiDαΦjΦ†iD¯ α_ Φ†j 0 ¯ ¯ þ 4 Dα_ ½ ð ; Þe α ¼ : ð6:1Þ DzφDzφ¯ ¼ FF: ð5:8Þ If the term in the second line is ignored, we find that the For the CPN−1, QN−2 cases, the gauge group is Abelian and vector superfield is solved by we find an explicit solution for F. Then the BPS condition is rewritten as 1 − 2κ−1ΦiΦ†i V ¼ 2 logð Þ: ð6:2Þ 4As denoted below Eq. (3.14), for the case of the CPN−1 model, the Lagrangian in Eq. (3.14) (with Λ ¼ const: can be This is easily confirmed if one notices that the following obtained [49] as the low-energy effective theory of a BPS non- relation holds for Eq. (6.2): Abelian vortex in N ¼ 2 supersymmetric UðNÞ gauge theory [55]. In this context, CPN−1 lumps (sigma model instantons) on Φ†iD Φi Φ†i Φi 2 Φi the vortex world sheet represent Yang-Mills instantons in the bulk α ¼ ½Dα þ ðDαVÞ theory, and such instanton-vortex composites are 1=4 BPS states 1 Φ†i Φi − Φ†j Φj ΦiΦ†i 0 [62]. It was discussed in Ref. [49] that this fact implies that lump ¼ Dα 2 Dα ð Þ¼ : ð6:3Þ solutions should not have derivative corrections. jΦj

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The would-be solution (6.2), whose θσmθ¯ component Lagrangians of the CPN−1 nonlinear sigma model. The −1 ¯ i i gives Am ¼ iκ φ ∂mφ , precisely gives the correct canonical quadratic kinetic term is absent in the nonca- Kähler potential K ¼ − logð2κ−1ΦΦ†Þ for the CPN−1 nonical branch, and there is only the fourth derivative term model. However, when the term in the second line in as a generalization of the Skyrme-Faddeev term of the CP1 Eq. (6.1) is included, Eq. (6.2) fails to be a solution. This model. We thus have obtained the supersymmetric CPN−1 implies that the expression in Eq. (6.2) is correct only in the Skyrme-Faddeev model. bosonic sector. Including the second line in Eq. (6.1) For the formulation of the GM;N model, a non-Abelian modifies the equality in Eq. (6.2) in the fermionic sector. UðNÞ symmetry is gauged in the gauged linear sigma We note that in solving the auxiliary field F, the model. Compared with the CPN−1 case for which the gauge fermionic fields ψ in the chiral multiplets are introduced symmetry is Abelian in the gauged linear sigma model, the φ φ¯ perturbatively around the bosonic solution F ¼ Fð ; Þ equation of motion for the auxiliary field in the GM;N model [20]. The same is true for the vector multiplet. Since the is rather complicated. Although it is hard to find explicit interaction terms are all expressed by the superfields F ≠ 0 solutions, we have been able to integrate out the explicitly, it is in principle possible to write down the gauge field with the help of the gauge invariance of the equations for the fermions. We can integrate out the vector higher derivative terms. multiplet starting from the bosonic part of the solution (6.2) For other Hermitian symmetric spaces, in addition to and introduce the fermions perturbatively. This procedure D-term constraints, we further impose F-term constraints determines the fermionic interactions in the sigma model yielding holomorphic embedding of the target spaces into limit g → ∞. Even though the fermionic interactions are CPN−1 or Grassmann manifold, as was done for the case rather cumbersome, we stress that the BPS conditions in the without higher derivative terms. In the canonical branches, nonlinear sigma models are not affected by details of the these constraints are consistent in the presence of higher fermionic interactions. As we have shown, this is obtained derivative terms, but we find that in the noncanonical by those in the gauged chiral models. branch these constraints yield further additional constraints reducing the target spaces to their submanifolds. N−2 VII. CONCLUSION AND DISCUSSIONS For the formulation of the Q model, an Abelian symmetry is gauged in the gauged linear sigma model, but N 1 In this paper, we have constructed ¼ supersym- we have an extra constraint coming from the F-term metric higher derivative terms in nonlinear sigma models embedding QN−2 to CPN−1. The nonzero solution F ≠ 0 whose target spaces are Hermitian symmetric spaces. We yields an extra constraint on the scalar fields. We have have considered the sigma model limit of the supersym- found that in order that these constraints are compatible metric gauged linear sigma models involving the derivative with the equation of motion in the noncanonical branches, terms for chiral multiplets that are free from the auxiliary the dynamics of fields should be restricted to a subspace ’ field problem and the Ostrogradski s ghost. In the sigma in the QN−2 manifold, which is a peculiar property of model limit, the vector multiplet does not propagate any- this model. We have also discussed the sigma models more and can be integrated out. Due to the supersymmetric whose target spaces are SOð2NÞ=UðNÞ, SpðNÞ=UðNÞ, derivative corrections, there are two distinct on-shell E6=½SOð10Þ × Uð1Þ, and E7=½E6 × Uð1Þ. The first two branches called the canonical and noncanonical branches. cases are essentially similar to the cases of the Grassmann This structure is carried over to the nonlinear sigma models. −2 and the QN cases. For the latter two cases, although the We have shown that the gauge field A is explicitly m superpotential is complicated compared with the QN−2 integrated out, both in the canonical and the noncanonical case, the structure of the higher derivative terms is similar branches, even in the presence of higher derivative terms. N−2 In the canonical branch, where the auxiliary field is given to that of the Q case. We also have found that these constructions provide the different sigma models even by F ¼ 0, we explicitly have written down the Lagrangians of the nonlinear sigma models. They consist of the though there are several isomorphisms among the target ≃ ≃ C N−1 1 ≃ ≃ C 1 canonical kinetic terms and the derivative corrections spaces such as GN;1 GN;N−1 P , Q G2;1 P , 4 ≃ characterized by the arbitrary function Λ. This is a natural Q G4;2, and so on. generalization of the sigma model construction discussed Finally, we provide a comment on the BPS equations and in Ref. [52]. the integration of the fermionic terms in the vector For the CPN−1 model, we have obtained the super- multiplet. In the canonical branches, BPS equations and symmetric CPN−1 Skyrme-Faddeev model. In the nonca- their solutions are the same with those without higher nonical branch, where the auxiliary field is given by F ≠ 0, derivative terms, while in the noncanonical branches they the situation changes drastically. For the CPN−1 model, an give compacton type configurations. Abelian symmetry is gauged and we have found the explicit It would be interesting to study explicit solutions to the solutions of the nonzero auxiliary field F. We have solved BPS equations in the noncanonical branch. As we have 1 4 all the constraints and have written down the explicit noted, the = BPS equation in the noncanonical branch of

025001-13 MUNETO NITTA and SHIN SASAKI PHYS. REV. D 103, 025001 (2021) the CPN−1 model reduces to that of the compactons. walls, their junctions, and so on [27]. Introducing potential Finding solutions in the other sigma models is interesting. terms and studying associated BPS states remain as a We will come back to these issues in future studies. future work. In this paper, we have considered the case that there are no potential terms in the nonlinear sigma models (although ACKNOWLEDGMENTS we have introduced superpotentials for the F-term constraints in the gauged linear sigma models). If we also This work was supported in part by Grant-in-Aid introduce superpotentials or twisted masses (from dimen- for Scientific Research, JSPS KAKENHI Grants sional reductions), there are more BPS states such as domain No. JP18H01217 (M. N.) and No. JP20K03952 (S. S.).

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