Charge Lattices, Duality Frames, and Local Systems

Electromagnetic charges do not form one globally trivial lattice over a Coulomb branch. They form an integral local system whose bases undergo symplectic monodromy around singular loci. A duality frame is a local polarization in which a mutually local subset is called electric; no single choice need cover the whole branch.

Required background. The abelian Coulomb-branch theory supplies the periods and central charge, while electric–magnetic lattices and global form supplies the integral quantum constraints. Helpful background. Special Kähler geometry explains the symplectic section.

The integral charge local system

Remove the discriminant $\Delta$ from the Coulomb branch and write

$$\mathcal B^\circ=\mathcal B\setminus\Delta.$$

Over each $u\in\mathcal B^\circ$, the electromagnetic charge lattice is locally

$$\Gamma_u\simeq\mathbb Z^{2r}$$

with an integral antisymmetric Dirac pairing. For a principally polarized, unimodular lattice, choose charge columns

$$\gamma=\binom{p^I}{q_I}, \qquad \langle\gamma,\gamma'\rangle =p^Iq'_I-q_Ip'^I.$$

Flat transport along a path identifies nearby lattices. Transport around a closed loop $\ell$ can return a different basis, giving a representation

$$\rho:\pi_1(\mathcal B^\circ,u_0) \longrightarrow Sp(2r,\mathbb Z).$$

The integrality encodes Dirac quantization. The symplectic condition preserves mutual locality. More generally, a polarization of type $D=\operatorname{diag}(d_1,\ldots,d_r)$ has matrix

$$J_D= \begin{pmatrix} 0&D\\ -D&0 \end{pmatrix},$$

and its monodromy group is $\operatorname{Aut}(\Gamma,J_D)$ rather than the standard $Sp(2r,\mathbb Z)$ written in a principal basis. The ambient electromagnetic lattice, the dynamical-particle sublattice, and a genuine-line lattice therefore need not carry the same primitive polarization. The flat symplectic bundle underlying rigid special Kähler geometry is described in Freed 1999, §§1 and 5.

Periods and contragredient charges

Order the period vector as

$$\Pi=\binom{a_D}{a}.$$

In the charge ordering above,

$$Z_\gamma=\gamma^T\Pi=p^Ia_{D,I}+q_Ia^I.$$

Suppose analytic continuation from patch $\alpha$ to patch $\beta$ gives

$$\Pi_\beta=M_\Pi\Pi_\alpha.$$

To describe the same transported physical charge in the compatible coordinate convention, write

$$\gamma_\beta=M_\Gamma\gamma_\alpha, \qquad M_\Gamma=M_\Pi^{-T}.$$

Then

$$\gamma_\beta^T\Pi_\beta =\gamma_\alpha^T\Pi_\alpha,$$

so the central charge and BPS mass are invariant.

An alternative convention transports cycles and keeps their integer coefficients fixed. It produces a different-looking rule but the same invariant pairing. A calculation must say which objects are actively transported and which basis is reset at the endpoint.

The period-map figure and its structured equations show $M_\Pi$ and $M_\Gamma$ on separate arrows so they cannot be mistaken for the same matrix action.

Electric frames and locality

A complete electric polarization is a primitive maximal isotropic, or Lagrangian, direct summand of the charge lattice. Particles with charges in that summand can be coupled to ordinary abelian gauge potentials. An arbitrary rank-$r$ isotropic subgroup need not be primitive and can miss global quotient data. If two light charges obey

$$\langle\gamma_1,\gamma_2\rangle\neq0,$$

no symplectic frame makes both purely electric. They cannot both appear as ordinary local elementary fields in one four-dimensional abelian Lagrangian.

This does not make the theory inconsistent. It means the electric patch is inadequate. At an isolated singularity with one primitive vanishing charge, a suitable frame usually exists. At a mutually nonlocal collision, the infrared theory can be an interacting non-Lagrangian SCFT.

Rank-one frame changes

For rank one, two standard matrices in the period ordering $(a_D,a)$ are

$$S=\begin{pmatrix}0&1\\-1&0\end{pmatrix}, \qquad T=\begin{pmatrix}1&1\\0&1\end{pmatrix}.$$

$S$ exchanges electric and magnetic periods up to a sign and sends

$$\tau\longmapsto-\frac1\tau.$$

$T$ sends $a_D\mapsto a_D+a$, corresponding to

$$\tau\longmapsto\tau+1.$$

Whether $S$ or $T$ maps a theory to itself depends on the ultraviolet global form, genuine lines, spin structure, and discrete theta data. They always act as useful changes of local polarization on the low-energy equations when their integrality conditions are met; they need not be internal symmetries of one fixed global theory. The rank-one period monodromies and duality frames are constructed in Seiberg and Witten 1994, §§3–6.

Flavor charges and affine shifts

With flavor masses $m^a$, the central charge is

$$Z_\gamma=p^Ia_{D,I}+q_Ia^I+s_am^a.$$

Monodromy around a locus where a flavored hypermultiplet becomes massless can shift electromagnetic periods by lattice-quantized linear combinations of $m^a$ in the declared mass and residue normalization. The complete structure is then an affine extension of the electromagnetic local system. Residues of the Seiberg–Witten differential encode the flavor masses; the relation is worked out explicitly in Seiberg and Witten 1994, §15.

Ignoring these shifts can make a monodromy matrix appear nonintegral or make $Z$ fail to return correctly. Include electromagnetic and flavor charges in one declared convention.

Global form and the lattice of genuine objects

The charge lattice of possible particles, the lattice of genuine line defects, and the lattice generated by light BPS states are related but not identical. Dynamical particles need not populate every allowed charge. Genuine lines can carry charges unavailable to finite-energy particles and depend on the global gauge group.

For an $SU(2)$ ultraviolet theory, fundamental Wilson probes and adjoint dynamical fields lead to one line lattice; for an $SO(3)$ theory, magnetic sectors and discrete theta choices alter it. The same local Seiberg–Witten curve can describe their Coulomb-branch couplings while the allowed global monodromy subgroup and line spectrum differ. These line-operator and global-form choices are classified in Aharony, Seiberg, and Tachikawa 2013, §§1–2.

Therefore a theory specification should list:

Datum	Content
Particle lattice	Charges allowed by finite-energy states and flavor representations
Line lattice	Genuine Wilson–’t Hooft charges modulo screening
Pairing	Integral antisymmetric form and basis orientation
Polarization	Locally electric isotropic sublattice
Monodromy	Matrices for based, oriented loops and their ordering
Global form	Ultraviolet group, discrete theta choice, and background bundles

No single row determines the others.

Pure SU(2) normalization crosswalk

The chapter benchmark uses root-unit electric labels. The following layers must not be collapsed:

Object	Coordinates and pairing
Curve homology	$\delta_\gamma=pB+qA$ with $A\circ B=+1$ and $\delta_\gamma\circ\delta_{\gamma'}=-(pq'-qp')$
Dynamical particle labels	$\gamma=(p,q)=(n_m,n_e)$, $Z=pa_D+qa$, and $\langle\gamma,\gamma'\rangle_{\mathrm{part}}=2(pq'-qp')$
Distinguished states	$\gamma_m=(1,0)$, $\gamma_d=(1,-1)$, and $\gamma_W=(0,1)$; hence $\langle\gamma_m,\gamma_d\rangle_{\mathrm{part}}=-2$
Common GMN labels	$(n_e^{\mathrm{GMN}},n_m^{\mathrm{GMN}})=(2q,p)$, giving the strong states $(0,1)$ and $(2,-1)$ and $W=(2,0)$; this map reverses the site’s determinant orientation, so product order and pairing signs must be translated together
Genuine lines	A global-form and discrete-theta choice; not determined by the populated particle charges or by the bare curve alone

Equivalently one may use weight-unit electric labels throughout, but then $a$, the dyon and $W$ labels, and every matrix must be conjugated together. The root-unit convention is retained here because it matches $a\sim\sqrt{2u}$ and the original pure-theory periods.

Base points, paths, and matrix order

Monodromy matrices depend on a base point, branch cuts, and loop generators. If a path crosses a cut before circling a singularity, the local vanishing charge must first be transported into the base-point frame. Reversing loop orientation inverts the matrix. Changing the symplectic basis conjugates every monodromy simultaneously:

$$M_\ell\longmapsto P M_\ell P^{-1}.$$

Only conjugacy-invariant data and explicitly based products can be compared without further translation.

For loops composed as paths, matrix order follows the convention for active action on column periods. State whether the rightmost or leftmost loop acts first before checking a global product.

Common pitfalls

Calling one charge basis global. Monodromy is precisely the obstruction to doing so.

Transforming periods and charges by the same matrix. With the displayed dot-product central charge, charges transform by the inverse transpose.

Equating possible charges with an actual BPS spectrum. The local system supplies kinematics; existence and stability are chamber-dependent dynamics.

Exercises

Let

$$M=\begin{pmatrix}1&2\\0&1\end{pmatrix}, \qquad \Pi=\binom{a_D}{a}.$$

1. Find the transformed period vector.
2. Find the contragredient transformation of $\gamma=(p,q)^T$.
3. Verify $Z=pa_D+qa$ is invariant.

Solution

The periods transform as

$$a_D'=a_D+2a, \qquad a'=a.$$

Since

$$M^{-T}=\begin{pmatrix}1&0\\-2&1\end{pmatrix},$$

the charges transform as $p'=p$, $q'=q-2p$. Then

$$p'a_D'+q'a' =p(a_D+2a)+(q-2p)a =pa_D+qa.$$

References

• Aharony, Ofer, Nathan Seiberg, and Yuji Tachikawa. “Reading between the Lines of Four-Dimensional Gauge Theories.” Journal of High Energy Physics 08 (2013): 115. arXiv:1305.0318.
• Freed, Daniel S. “Special Kähler Manifolds.” Communications in Mathematical Physics 203 (1999): 31–52. arXiv:hep-th/9712042.
• Seiberg, Nathan, and Edward Witten. “Electric–Magnetic Duality, Monopole Condensation, and Confinement in $\mathcal N=2$ Supersymmetric Yang–Mills Theory.” Nuclear Physics B 426 (1994): 19–52; erratum 430 (1994): 485–486. arXiv:hep-th/9407087.
• Seiberg, Nathan, and Edward Witten. “Monopoles, Duality and Chiral Symmetry Breaking in $\mathcal N=2$ Supersymmetric QCD.” Nuclear Physics B 431 (1994): 484–550. arXiv:hep-th/9408099.