Fractional BPS Lumps

• Fractional BPS lumps are soliton solutions that either preserve a reduced fraction of supersymmetry or exhibit fractionalized topological charge, broadening the classical lump paradigm.
• The analysis employs higher-derivative supersymmetric chiral models, gauged vortex-lump composites, and impurity-induced deformations to derive modified BPS equations and energy bounds.
• These constructions offer practical pathways for exploring compact soliton solutions and establishing a bridge between low-dimensional sigma models and higher-dimensional gauge theories.

Searching arXiv for recent and relevant papers on fractional BPS lumps and closely related constructions. Fractional BPS lumps are Bogomol'nyi-saturated solitons for which the standard lump paradigm is altered either by a reduced supersymmetry fraction, by a fractionalization of the relevant topological charge, or by both. In the literature this label is not attached to a single universal object. It encompasses deformed $1/4$ BPS lumps in higher-derivative supersymmetric chiral models, vortex-lump composites with gauge-covariant and potentially fractional lump charge, impurity-bound lump–antilump-like BPS configurations, semilocal codimension-four instanton analogues, and twisted $\mathbb{C}P^{N-1}$ fractional instantons on a torus; closely related vortex constructions provide an important comparative background for the meaning of “fractional” in BPS soliton theory (Nitta et al., 2014, Nitta et al., 2015, Queiruga, 2021, Hayashi et al., 17 Jul 2025).

1. Terminology and baseline lump structure

In the standard $\mathbb{C}P^1$ sigma model written in stereographic coordinates $u\in\mathbb{C}$,

$$\mathcal{L}_{\mathbb{C}P^1}= \frac{1}{2}\frac{\partial_\mu u\, \partial^\mu \bar{u}}{(1+|u|^2)^2},$$

BPS lumps are holomorphic maps $u=u(z)$ or antiholomorphic maps for opposite charge. The topological charge is

$$Q=\frac{1}{\pi}\int d^2x \,\frac{u_z \bar u_{\bar z}-u_{\bar z}\bar u_z}{(1+|u|^2)^2},$$

and the static energy satisfies $E\ge \pi |Q|$ (Queiruga, 2021). In supersymmetric higher-derivative chiral models, the canonical lump equation is likewise the holomorphic condition

$\bar\partial \varphi = 0,$

which gives ordinary $1/2$ BPS lumps (Nitta et al., 2014).

Taken together, the cited works show that “fractional” is used in two non-equivalent senses. In one sense it refers to the preserved supersymmetry fraction, as in $\mathbb{C}P^{N-1}$0 BPS lumps in the non-canonical branch of higher-derivative chiral models. In another, it refers to a fractionalized topological quantity, as in gauged vortex-lumps with fractional lump charge, impurity-induced degree shifts, or toroidal $\mathbb{C}P^{N-1}$1 instantons with charge $\mathbb{C}P^{N-1}$2 (Nitta et al., 2014, Nitta et al., 2015, Hayashi et al., 17 Jul 2025).

Framework	BPS fraction	Fractional feature
Higher-derivative chiral model	$\mathbb{C}P^{N-1}$3	Deformed nonholomorphic lump equation
Gauged higher-derivative model	$\mathbb{C}P^{N-1}$4 or $\mathbb{C}P^{N-1}$5	Fractional lump or baby-Skyrme charge
Impurity $\mathbb{C}P^{N-1}$6 model	BPS	Degree accounting altered by frozen antilump
Twisted $\mathbb{C}P^{N-1}$7 on $\mathbb{C}P^{N-1}$8	BPS	$\mathbb{C}P^{N-1}$9

2. Higher-derivative supersymmetric chiral models and $\mathbb{C}P^1$0 BPS lumps

A central construction is the 4D $\mathbb{C}P^1$1 supersymmetric chiral model with Kähler potential $\mathbb{C}P^1$2, superpotential $\mathbb{C}P^1$3, and higher-derivative tensor $\mathbb{C}P^1$4. The off-shell Lagrangian remains supersymmetric, and the auxiliary fields $\mathbb{C}P^1$5 remain algebraic rather than derivative-bearing. For a single complex scalar $\mathbb{C}P^1$6, the bosonic sector contains

$\mathbb{C}P^1$7

Because the auxiliary-field equation is polynomial rather than linear, eliminating $\mathbb{C}P^1$8 produces distinct on-shell branches (Nitta et al., 2014).

For $\mathbb{C}P^1$9, the auxiliary-field equation admits a canonical branch,

$u\in\mathbb{C}$0

and a non-canonical branch,

$u\in\mathbb{C}$1

The canonical branch yields ordinary $u\in\mathbb{C}$2 BPS lumps. For configurations depending on $u\in\mathbb{C}$3, one obtains

$u\in\mathbb{C}$4

In this branch, the higher-derivative corrections cancel out of the BPS equation and the topological bound. The Bogomol’nyi completion shows that the bound is saturated precisely on the holomorphic locus (Nitta et al., 2014).

The non-canonical branch produces the fractional-BPS phenomenon in the supersymmetry-fraction sense. The BPS equation is deformed to

$u\in\mathbb{C}$5

equivalently,

$u\in\mathbb{C}$6

These objects preserve one quarter of the supersymmetry and are identified with compact baby Skyrmions. The distinction from the canonical branch is structural rather than perturbative: the higher-derivative coupling $u\in\mathbb{C}$7 survives explicitly in the BPS condition, and the on-shell theory in the non-canonical branch has no ordinary kinetic term (Nitta et al., 2014).

The same framework also furnishes a supersymmetric extension of the Faddeev-Skyrme model without four time derivatives. In the canonical branch, it supports the usual $u\in\mathbb{C}$8 BPS lumps; in the non-canonical branch, it supports compact solitons of baby-Skyrme type. This places fractional BPS lumps within a broader class of branch-dependent supersymmetric solitons in higher-derivative theories (Nitta et al., 2014).

3. Gauged vortex-lumps, fractional lump charge, and semilocal extensions

When chiral fields are gauged, the lump sector changes qualitatively. In $u\in\mathbb{C}$9 supersymmetric gauge theories coupled to higher-derivative chiral models, the canonical branch $$\mathcal{L}_{\mathbb{C}P^1}= \frac{1}{2}\frac{\partial_\mu u\, \partial^\mu \bar{u}}{(1+|u|^2)^2},$$0 still supports familiar $$\mathcal{L}_{\mathbb{C}P^1}= \frac{1}{2}\frac{\partial_\mu u\, \partial^\mu \bar{u}}{(1+|u|^2)^2},$$1 BPS states, but now includes vortex-lumps or gauged lumps with fractional lump charge. The BPS equations take the form

$$\mathcal{L}_{\mathbb{C}P^1}= \frac{1}{2}\frac{\partial_\mu u\, \partial^\mu \bar{u}}{(1+|u|^2)^2},$$2

for the general gauge-invariant Kähler potential, with $$\mathcal{L}_{\mathbb{C}P^1}= \frac{1}{2}\frac{\partial_\mu u\, \partial^\mu \bar{u}}{(1+|u|^2)^2},$$3 and $$\mathcal{L}_{\mathbb{C}P^1}= \frac{1}{2}\frac{\partial_\mu u\, \partial^\mu \bar{u}}{(1+|u|^2)^2},$$4 (Nitta et al., 2015).

The associated BPS bound contains both a lump-like and a vortex contribution,

$$\mathcal{L}_{\mathbb{C}P^1}= \frac{1}{2}\frac{\partial_\mu u\, \partial^\mu \bar{u}}{(1+|u|^2)^2},$$5

where the first term is the gauge-covariant generalization of the lump charge density and the second is the vortex flux density. The key point is that gauging covariantizes the lump topology. In the ungauged sigma model, lump charge is an integer degree of a map $$\mathcal{L}_{\mathbb{C}P^1}= \frac{1}{2}\frac{\partial_\mu u\, \partial^\mu \bar{u}}{(1+|u|^2)^2},$$6; in the gauged model, the topological density is a covariant expression, so the lump contribution can be split between localized vortex cores and the surrounding gauge field. In this sense, the lump charge becomes fractional in vortex-lump composites (Nitta et al., 2015).

A notable structural result is that the higher-derivative term vanishes identically on the canonical BPS locus once $$\mathcal{L}_{\mathbb{C}P^1}= \frac{1}{2}\frac{\partial_\mu u\, \partial^\mu \bar{u}}{(1+|u|^2)^2},$$7 is imposed. Thus the vortex-lump equations are unchanged from the ordinary gauged sigma model. By contrast, the non-canonical branch does not support the ordinary $$\mathcal{L}_{\mathbb{C}P^1}= \frac{1}{2}\frac{\partial_\mu u\, \partial^\mu \bar{u}}{(1+|u|^2)^2},$$8 BPS vortex-lump state. Instead it supports higher-derivative generalizations of vortices and $$\mathcal{L}_{\mathbb{C}P^1}= \frac{1}{2}\frac{\partial_\mu u\, \partial^\mu \bar{u}}{(1+|u|^2)^2},$$9 BPS gauged baby-Skyrmion configurations, which may carry fractional baby-Skyrme charges (Nitta et al., 2015).

A higher-dimensional analogue is furnished by semilocal fractional instantons in supersymmetric gauge theories with eight supercharges. In 4+1-dimensional $u=u(z)$0 gauge theory with $u=u(z)$1 Higgs fields, the strong-coupling limit yields $u=u(z)$2 or Grassmann sigma models, and the relevant solitons become codimension-four, $u=u(z)$3 BPS, and classified by $u=u(z)$4. Their instanton charge is fractional: $u=u(z)$5 in the Abelian theory and $u=u(z)$6 in the non-Abelian theory. The energy diverges in infinite volume because of the vortex-sheet background, but the instanton charge itself is finite. In this setting the fractional object is a semilocal instanton or four-dimensional lump analogue, stabilized by combined vortex and instanton topology rather than by an isolated localized instanton sector (Eto et al., 2015).

4. Holomorphic impurities and lump–antilump BPS sectors

A different mechanism for fractionalization arises in the $u=u(z)$7 model with a BPS-preserving impurity. The impurity is a fixed nondynamical complex background $u=u(z)$8, holomorphic or antiholomorphic in the spatial coordinates, introduced in a supersymmetric formulation. After eliminating the auxiliary field, the static energy can be written as

$u=u(z)$9

so the topological bound remains $$Q=\frac{1}{\pi}\int d^2x \,\frac{u_z \bar u_{\bar z}-u_{\bar z}\bar u_z}{(1+|u|^2)^2},$$0, but the BPS equation is shifted to the inhomogeneous Cauchy–Riemann form

$$Q=\frac{1}{\pi}\int d^2x \,\frac{u_z \bar u_{\bar z}-u_{\bar z}\bar u_z}{(1+|u|^2)^2},$$1

(Queiruga, 2021).

The general finite-energy BPS solution is

$$Q=\frac{1}{\pi}\int d^2x \,\frac{u_z \bar u_{\bar z}-u_{\bar z}\bar u_z}{(1+|u|^2)^2},$$2

or, for the rational impurity profiles emphasized in the analysis,

$$Q=\frac{1}{\pi}\int d^2x \,\frac{u_z \bar u_{\bar z}-u_{\bar z}\bar u_z}{(1+|u|^2)^2},$$3

This decomposition has a direct interpretation: $$Q=\frac{1}{\pi}\int d^2x \,\frac{u_z \bar u_{\bar z}-u_{\bar z}\bar u_z}{(1+|u|^2)^2},$$4 is the usual holomorphic lump contribution, while $$Q=\frac{1}{\pi}\int d^2x \,\frac{u_z \bar u_{\bar z}-u_{\bar z}\bar u_z}{(1+|u|^2)^2},$$5 is a fixed antiholomorphic impurity contribution. The resulting configuration is lump–antilump-like but nevertheless BPS because the impurity deforms the first-order equation. The impurity can therefore be viewed as a frozen lump with opposite charge (Queiruga, 2021).

With the boundary condition $$Q=\frac{1}{\pi}\int d^2x \,\frac{u_z \bar u_{\bar z}-u_{\bar z}\bar u_z}{(1+|u|^2)^2},$$6, the degree constraints are

$$Q=\frac{1}{\pi}\int d^2x \,\frac{u_z \bar u_{\bar z}-u_{\bar z}\bar u_z}{(1+|u|^2)^2},$$7

and the net topological degree becomes

$$Q=\frac{1}{\pi}\int d^2x \,\frac{u_z \bar u_{\bar z}-u_{\bar z}\bar u_z}{(1+|u|^2)^2},$$8

The topological charge density is positive near roots of $$Q=\frac{1}{\pi}\int d^2x \,\frac{u_z \bar u_{\bar z}-u_{\bar z}\bar u_z}{(1+|u|^2)^2},$$9 and negative near roots of $E\ge \pi |Q|$0, so the degree is naturally understood as the difference between dynamical lumps and impurity-induced antilumps. The paper’s terminology treats this as a fractional or impurity-induced modification of the usual degree accounting (Queiruga, 2021).

The BPS solution space at fixed charge is denoted $E\ge \pi |Q|$1. Under the geodesic approximation, the finite-inertia submanifold remains Kähler. For the parametrization

$E\ge \pi |Q|$2

the kinetic term is generated by the Kähler potential

$E\ge \pi |Q|$3

This shows that impurity deformations preserve a substantial part of the geometric structure familiar from ordinary lump moduli spaces. For constant impurity,

$E\ge \pi |Q|$4

the solution is related by inversion to the critical magnetic skyrmion profile, and the topological charge density is invariant under $E\ge \pi |Q|$5 in the standard $E\ge \pi |Q|$6 model (Queiruga, 2021).

5. Twisted $E\ge \pi |Q|$7 models on the torus and exact fractional instantons

A fully explicit topological fractionalization mechanism appears in the $E\ge \pi |Q|$8 model on a two-torus

$E\ge \pi |Q|$9

with complex coordinate

$\bar\partial \varphi = 0,$0

The model is regularized by an $\bar\partial \varphi = 0,$1-component Abelian-Higgs theory with action

$\bar\partial \varphi = 0,$2

which reduces to the $\bar\partial \varphi = 0,$3 model in the strong-coupling limit $\bar\partial \varphi = 0,$4 (Hayashi et al., 17 Jul 2025).

Shift-clock twisted boundary conditions are imposed using the $\bar\partial \varphi = 0,$5 shift and clock matrices $\bar\partial \varphi = 0,$6 and $\bar\partial \varphi = 0,$7, satisfying

$\bar\partial \varphi = 0,$8

Because the twists do not commute, the cocycle condition induces a fractional $\bar\partial \varphi = 0,$9 Hooft flux,

$1/2$0

with $1/2$1. The topological charge is therefore fractionalized as

$1/2$2

In the BPS sector the equations are

$1/2$3

with the standard ansatz

$1/2$4

reducing the remaining problem to a Taubes equation for $1/2$5 (Hayashi et al., 17 Jul 2025).

The explicit solutions are written in terms of Jacobi theta functions. In the regularized theory they are exact BPS vortex-like solutions; in the $1/2$6 limit they become fractional instantons or fractional BPS lumps. Two complementary parametrizations are emphasized. The zero-locus parametrization organizes the solution by the zeros of the holomorphic components $1/2$7 and makes the small-lump singularity transparent: singular behavior occurs when $1/2$8. The theta-basis parametrization instead makes the global quotient structure and modular invariance manifest (Hayashi et al., 17 Jul 2025).

The moduli space is denoted

$1/2$9

and its complex dimension is

$\mathbb{C}P^{N-1}$00

Globally it is a $\mathbb{C}P^{N-1}$01-fiber bundle over a torus, equivalently a quotient of $\mathbb{C}P^{N-1}$02, and it is Kähler. The Abelian-Higgs regularization smooths the small-lump singularity: for finite $\mathbb{C}P^{N-1}$03, a common zero of all components becomes a local vortex of size $\mathbb{C}P^{N-1}$04 rather than a singular point of the sigma model (Hayashi et al., 17 Jul 2025).

The toroidal construction also provides a direct bridge to four-dimensional $\mathbb{C}P^{N-1}$05 Yang-Mills theory on $\mathbb{C}P^{N-1}$06 with $\mathbb{C}P^{N-1}$07 Hooft twists. After dimensional reduction on a small $\mathbb{C}P^{N-1}$08, the twisted four-dimensional problem yields a two-dimensional effective theory whose fractional instantons have the same moduli-space topology, as complex manifolds, as the twisted $\mathbb{C}P^{N-1}$09 model. This places fractional BPS lumps within a precise 2D/4D correspondence rather than treating them as purely sigma-model artifacts (Hayashi et al., 17 Jul 2025).

6. Related vortex constructions, nonrelativistic analogues, and conceptual boundaries

Several nearby constructions clarify what fractional BPS lumps are not. In a CPT-even Lorentz-violating Abelian Maxwell-Higgs model, gauge-sector Lorentz violation produces compactlike uncharged BPS vortices, while including Higgs-sector Lorentz violation leads to fractional BPS vortices with modified vacuum and magnetic flux

$\mathbb{C}P^{N-1}$10

The stable fractional sector requires $\mathbb{C}P^{N-1}$11, giving the modified vacuum

$\mathbb{C}P^{N-1}$12

These are BPS and fractional, but they are vortices rather than lumps; the fractionalization occurs in magnetic flux and winding data rather than in a sigma-model lump charge (Miller et al., 2011).

A nonrelativistic analogue appears in the 2D Gross-Pitaevskii equation, where the energy can be rewritten as a sum of squares plus two boundary terms $\mathbb{C}P^{N-1}$13 and $\mathbb{C}P^{N-1}$14. The BPS system implies the full second-order Gross-Pitaevskii equations and admits an exact fractional $\mathbb{C}P^{N-1}$15-vorticity solution on an annulus,

$\mathbb{C}P^{N-1}$16

Here the first boundary term is a vorticity-like charge dressed by the condensate profile, while the second is a “skewness” term. This again yields a fractional BPS soliton, but in a condensate system rather than a relativistic sigma model or supersymmetric gauge theory (Canfora et al., 7 Jan 2025).

An opposite comparison is provided by thick oriented and nonoriented center-vortex $\mathbb{C}P^{N-1}$17 configurations with fractional topological charge lumps. These smooth gauge fields exhibit localized fractional charge contributions such as $\mathbb{C}P^{N-1}$18 and $\mathbb{C}P^{N-1}$19, and explicit thick mixed configurations can have total charge $\mathbb{C}P^{N-1}$20. However, no BPS equations or self-duality conditions are derived. The relevance of these configurations is therefore structural: they show that fractional topological lumps need not be BPS, and that non-Abelian branch changes mediated by monopole junctions can localize fractional charge even outside the Bogomol'nyi framework (Junior et al., 14 Nov 2025).

Taken together, these works indicate that fractional BPS lumps arise by several distinct mechanisms: auxiliary-field branching in higher-derivative supersymmetry, gauge-covariantization of lump topology, frozen impurity sectors, twisted toroidal monodromy, and semilocal embedding into higher-dimensional gauge theories. A plausible implication is that “fractionality” should always be specified in context—whether it refers to preserved supersymmetry, to a fractionalized topological number, or to both—because the mathematical origin and physical interpretation depend strongly on the model class.