Framed BPS States, Line Defects, and Defect Moduli

A framed BPS state is a supersymmetric state in the presence of a specified defect. The defect changes the Hilbert space, selects a preserved subalgebra, and supplies a heavy core around which ordinary BPS particles can bind. Its spectrum is therefore not the bulk BPS spectrum with one extra charge appended: boundary conditions, global form, scalar couplings, and the core–halo split are part of the definition.

Required background. BPS solitons and extended charges supplies the projector logic, while disorder operators explains singular boundary data. Helpful background. Genuine line operators and charge lattices distinguishes an allowed line from a merely formal electric–magnetic label.

Defining the line before counting states

Place a straight line defect $L_\zeta$ along time at the spatial origin. A complete definition names:

• the ultraviolet operator or singular boundary condition;
• its electric, magnetic, flavor, and discrete charges, including the global gauge group;
• scalar couplings required by supersymmetry;
• the phase $\zeta\in U(1)$ selecting the preserved supercharges;
• counterterms and framing conventions localized on the line;
• the vacuum at spatial infinity.

For an abelianized Wilson–’t Hooft line, the magnetic boundary condition is schematically

$$F\sim\frac{p}{2}\sin\theta\,d\theta\wedge d\varphi,$$

so the flux through a linking sphere is $2\pi p$ in this convention. A BPS completion also imposes a $1/r$ singularity for an adjoint scalar whose sign and phase are tied to $\zeta$. The electric label $q$ enters through a Wilson coupling. Dirac quantization, Weyl equivalence, screening, and the chosen global form decide which pairs $(p,q)$ define genuine lines.

Changing the scalar singularity while keeping $(p,q)$ fixed can change the preserved subalgebra and hence the framed index. The charge label alone is not the operator.

The framed Hilbert space and its charge torsor

Radial quantization around the line, or ordinary Hamiltonian quantization with the line extending through time, gives a defect Hilbert space

$$\mathcal H_{L_\zeta,u} =\bigoplus_{\gamma\in\Gamma_L}\mathcal H_{L_\zeta,u,\gamma}.$$

Here $u$ is the vacuum and $\Gamma_L$ is generally a torsor for the bulk charge lattice $\Gamma$: differences of two framed charges lie in $\Gamma$, but a preferred zero need not exist. Choosing a reference core charge $\gamma_c$ lets one write

$$\gamma=\gamma_c+\gamma_h, \qquad \gamma_h\in\Gamma,$$

where $\gamma_h$ is the total halo charge carried by bound bulk particles. A different reference shifts $\gamma_c$ and $\gamma_h$ oppositely without changing the physical total charge. This torsor-valued charge decomposition and the framed Hilbert space are developed in Gaiotto, Moore, and Neitzke 2013, pp. 241–397, arXiv:1006.0146.

In a fixed convention for the preserved supercharge, positivity gives a framed energy bound of the form

$$E_{\mathrm{ren}} \ge -\operatorname{Re}\!\left(\zeta^{-1}Z_\gamma(u)\right).$$

The line’s divergent rest energy has been subtracted into $E_{\mathrm{ren}}$. Reversing the definition of $\zeta$ or of the preserved supercharge reverses the displayed sign; the operational statement is that a framed BPS state is annihilated by the supercharges preserved by $L_\zeta$ and saturates their positive anticommutator.

Core and halo states

Far from the line, a bulk BPS particle of charge $\gamma_h$ feels long-range electromagnetic and scalar fields sourced by the core. When its central-charge phase and the defect phase allow a supersymmetric equilibrium, it can form a halo. The Dirac pairing

$$\langle\gamma_c,\gamma_h\rangle$$

controls the electromagnetic angular momentum and the degeneracy of orbital states. The sign of a phase-dependent stability parameter determines on which side of a wall the halo has positive radius. At the wall the radius diverges, the bound state joins the continuum, and the framed spectrum jumps.

This picture is controlled when the halo radius is much larger than the core size and constituent Compton wavelengths. It predicts universal wall-crossing factors but not the microscopic core degeneracy. Near a nonabelian core or when halos overlap strongly, the simple multicenter approximation can fail.

As a minimal fixture, suppose one halo species has charge $\gamma_h$, protected degeneracy $\Omega(\gamma_h)=1$, and

$$k=\langle\gamma_c,\gamma_h\rangle>0.$$

Quantization of the relative angular degrees of freedom produces a spin $(k-1)/2$ multiplet, hence $k$ orbital states before the universal supersymmetric factors are organized into the chosen index. This is the origin of the $|k|$ factor in primitive framed wall crossing. If $k=0$, there is no electromagnetic angular-momentum multiplet and no corresponding universal halo jump.

Framed protected indices

A framed index traces over defect BPS states after removing universal noncompact or center-of-mass factors appropriate to the line. Package the charge sectors into a generating function

$$F(L_\zeta) =\sum_{\gamma\in\Gamma_L} \underline{\Omega}(L_\zeta,\gamma;u)\,X_\gamma.$$

The underlined symbol emphasizes that this is a framed index, not the bulk index $\Omega(\gamma;u)$. The formal variables multiply with a sign or quantum-torus factor determined by the charge pairing and refinement convention. Those conventions must be fixed before comparing formulas.

Crossing a framed BPS wall conjugates or multiplies $F(L_\zeta)$ by a halo factor constructed from bulk indices. The transformation is universal because it follows from adding or removing halo Fock states. It determines the protected jump, not every unprotected defect excitation.

Framed spectra can be more informative than bulk spectra. Consistency of their jumps for many line defects leads to the Kontsevich–Soibelman wall-crossing identity. In this sense a heavy probe turns a change in the bulk particle spectrum into a computable transformation of defect observables.

Defect moduli and monopole bubbling

An ’t Hooft defect prescribes a singular magnetic charge near the line, but smooth monopoles can approach and partially screen it. This monopole bubbling changes the effective magnetic charge seen away from the core while preserving the ultraviolet defect. The associated moduli space consists of Bogomolny solutions with a prescribed singularity and asymptotic data.

Consequently, a line labeled by an ultraviolet coweight can contribute to several effective charge sectors. Quantizing the defect moduli space, including singular strata and normalizability conditions, supplies framed states. Treating the ultraviolet coweight as a fixed unscreened infrared charge would miss these sectors. The singular-monopole moduli-space analysis is given in Moore, Royston, and Van den Bleeken 2014, JHEP 10, article 142, arXiv:1404.5616.

The same lesson holds more generally: defect-localized zero modes, boundary degrees of freedom, and smooth solitons bound to the defect must be included when they are part of its definition. Their moduli are physical only when their kinetic norm is finite.

Fusion and operator products

Bring two parallel BPS lines together while preserving a common supercharge. Their operator product expands into line defects,

$$L_1L_2=\sum_a c_a L_a,$$

and the framed generating functions multiply in the corresponding charge algebra. Noncommutativity can appear in refined or compactified settings through the symplectic pairing. Fusion is sensitive to line ordering, global form, and local counterterms; it is not simply vector addition of core charges. For the Wilson–’t Hooft operator algebra, see Kapustin and Saulina 2009, pp. 327–365, arXiv:0710.2097.

This provides a valuable cross-check: the framed spectrum assigned to a line should be compatible with known line-operator products and with wall-crossing transformations.

Common pitfalls

Counting bulk particles as framed states automatically. A bulk BPS particle contributes only if it forms a normalizable supersymmetric state in the defect background. Phase alignment, charge pairing, and boundary conditions matter.

Identifying the core charge with the total charge. Halo particles and monopole bubbling shift the infrared charge sector. The total charge lies in a torsor; a core–halo decomposition requires a reference choice.

Specifying only a Wilson–’t Hooft pair. Scalar couplings, singular data, global form, and the preserved phase distinguish inequivalent supersymmetric lines with the same formal pair.

Exercises

A core of charge $\gamma_c$ binds a single halo species $\gamma_h$ with $\Omega(\gamma_h)=1$ and pairing $\langle\gamma_c,\gamma_h\rangle=-3$.

1. What is the spin of the universal orbital multiplet?
2. How many orbital states does it contain?
3. Which additional datum decides on which side of the wall the halo exists?

Solution

The spin is $(|{-3}|-1)/2=1$, so the orbital multiplet has $2j+1=3$ states. The absolute pairing fixes this multiplicity, while the sign of the phase-dependent stability parameter—equivalently the ordering of the central-charge phases relative to the preserved defect phase—decides which chamber supports a positive-radius bound state. The pairing alone does not select the stable side.

References

• Gaiotto, Davide, Gregory W. Moore, and Andrew Neitzke. “Framed BPS States.” Advances in Theoretical and Mathematical Physics 17 (2013): 241–397. arXiv:1006.0146.
• Kapustin, Anton, and Natalia Saulina. “The Algebra of Wilson–’t Hooft Operators.” Nuclear Physics B 814 (2009): 327–365. arXiv:0710.2097.
• Moore, Gregory W., Andrew B. Royston, and Dieter Van den Bleeken. “Parameter Counting for Singular Monopoles on $\mathbb R^3$.” Journal of High Energy Physics 10 (2014): 142. arXiv:1404.5616.