Electric–Magnetic Duality and the Modular Group in Biquaternionic Form

Introduction

The Magnetic Monopole in Biquaternionic Form established the central algebraic fact of electric–magnetic duality in this framework: the duality rotation acts on the field-strength biquaternion by multiplication by a scalar phase,

$$ \tilde F \;\longmapsto\; e^{-i\theta}\tilde F , \qquad \mathbf E\mapsto c\,\mathbf B,\qquad \mathbf B\mapsto-\frac{1}{c}\mathbf E \quad\text{at }\theta=\tfrac{\pi}{2}, $$

and the generator is an element of the central scalar subspace $\mathbb C_{\mathbb B}$. Hodge duality is the same operation, $\tilde F_\star=-i\tilde F$, and with magnetic sources present the rotation is an exact internal symmetry of the sourced equations, exchanging electric and magnetic charge. The rotation is therefore a $U(1)$ symmetry of the algebra, and it is continuous.

The full duality structure of a gauge theory with a theta angle is larger and discrete. The complex coupling

$$ \tau \;=\; \frac{\theta}{2\pi}+\frac{4\pi i}{g^2} $$

lives in the upper half-plane, and the group

$$ \Gamma=SL(2,\mathbb Z)=\Bigl\{\begin{pmatrix}a&b\\c&d\end{pmatrix}:a,b,c,d\in\mathbb Z,\ ad-bc=1\Bigr\} $$

acts on it by fractional linear transformations,

$$ \tau \;\longmapsto\; \frac{a\tau+b}{c\tau+d}, $$

generated by the two elements

$$ S=\begin{pmatrix}0&-1\\1&0\end{pmatrix}:\ \tau\mapsto-\frac1\tau , \qquad T=\begin{pmatrix}1&1\\0&1\end{pmatrix}:\ \tau\mapsto\tau+1 . $$

The $S$ generator is strong–weak duality: at $\theta=0$ it exchanges $g\leftrightarrow4\pi/g$. The $T$ generator is the shift $\theta\to\theta+2\pi$, which is a symmetry because of the quantisation of the topological charge. The electric and magnetic charges of a dyon transform in a doublet under the same group, and the Dirac–Schwinger–Zwanziger quantisation condition makes the charge lattice a module for $\Gamma$.

This article's claim is a structural statement, and it is a result rather than a failure. The framework supplies the continuous $U(1)$ duality rotation intrinsically, because the algebra's centre is exactly $\mathbb C_{\mathbb B}$ and the rotation is multiplication by a unit of it. The framework does not supply the discrete group $SL(2,\mathbb Z)$: a discrete group cannot be generated by an element of a continuous centre, and the $T$ generator in particular requires the integer topological charge, which is a property of the gauge bundle and of the theta vacuum rather than of the algebra. The modular group is therefore imported from the quantisation of charge and the topology of the instanton sector. What the algebra does is fix the shape of the duality: that it is a rotation internal to the field strength, that it is central, and that it does not touch the Lorentz structure.

Conventions. We use those of Conventions in the Biquaternion Universe and of the companion gauge and monopole articles. The field-strength biquaternion is $\tilde F=i\sqrt{\epsilon}\,\mathbf E-\sqrt{\mu}\,\mathbf H\in\mathbb M_-$ (pure vector), its dual is $\tilde F_\star=-i\tilde F$, and the second invariant is $I_2=\mathbf E\cdot\mathbf B$. The topological charge is $Q=\frac{1}{8\pi^2}\int\mathrm{Tr}(F\wedge F)\in\mathbb Z$ and the theta term is $\frac{\theta}{32\pi^2}\int F\tilde F$; units with $\hbar=c=1$ are used for the charge lattice. The metric and d'Alembertian are those of the authoritative conventions article, and $c=1/\sqrt{\epsilon\mu}$ is the speed of light in the medium.

• Companion article The Semiclassical Expansion and the Instanton Gas in Biquaternionic Form, for the theta periodicity and the sector sum.
• Companion article The Magnetic Monopole in Biquaternionic Form, for the Dirac quantisation condition.
• Companion article The 't Hooft–Polyakov Monopole in Biquaternionic Form, for the finite-energy monopole solution.
• Companion article The Theta Parameter, Strong CP, and the Witten Effect in Biquaternionic Form, for the Witten shift of the electric charge.
• Companion article Instantons and Solitons in Biquaternionic Form, for the topological charge on which the discrete group depends.

Duality of the Free Field and Its Internal Character

In the source-free Maxwell theory the duality rotation is an obvious symmetry of the equations. Writing the field-strength biquaternion and its dual,

$$ \tilde F=i\sqrt{\epsilon}\,\mathbf E-\sqrt{\mu}\,\mathbf H , \qquad \tilde F_\star=-i\tilde F , $$

the sourceless biquaternionic Maxwell equation $\tilde\nabla\tilde F=0$ is invariant under $\tilde F\mapsto e^{-i\theta}\tilde F$ for every real $\theta$, because the phase is central and commutes with $\tilde\nabla$:

$$ \tilde\nabla\bigl(e^{-i\theta}\tilde F\bigr)=e^{-i\theta}\,\tilde\nabla\tilde F=0 . $$

This is the framework's crispest statement about duality. The phase $e^{-i\theta}$ lies in the unit circle of the centre $\mathbb C_{\mathbb B}=\mathrm{span}_{\mathbb R}\{e_0,ie_0\}$, and the centre commutes with every element of the algebra, so the rotation is an internal symmetry: it does not act on the Lorentz indices, it does not mix with the quaternion units, and it commutes with every Lorentz rotor. It is not a spacetime symmetry, and it is not part of the Lorentz group; it is a $U(1)$ of the algebra's centre.

With magnetic sources the rotation is promoted from a symmetry of the free equations to a symmetry of the sourced system. The sourced equation is

$$ \tilde\nabla\tilde F=-\tilde R_e-i\tilde R_m , \qquad \tilde R_e=\frac{i\rho_e}{\sqrt{\epsilon}}+\sqrt{\mu}\,\mathbf J_e , \qquad \tilde R_m=\frac{i\rho_m}{\sqrt{\mu}}+\sqrt{\epsilon}\,\mathbf J_m , $$

and a rotation $\tilde F\mapsto e^{-i\theta}\tilde F$ is compensated by rotating the complex source $\tilde{\mathcal R}=\tilde R_e+i\tilde R_m$, which is the algebraic statement that the symmetry exchanges electric and magnetic charge. The rotation is exact, internal and central; its parameter is continuous.

The Complex Coupling and the Modular Group

The continuous rotation is only half of the duality structure once the theory has a theta angle. The Euclidean action

$$ S \;=\; \frac{1}{2g^2}\int\mathrm{Tr}\bigl(F_{\mu\nu}F^{\mu\nu}\bigr) \;+\;\frac{i\theta}{16\pi^2}\int\mathrm{Tr}\bigl(F_{\mu\nu}\star F^{\mu\nu}\bigr) $$

is assembled into the complex coupling by

$$ S \;=\; \frac{1}{g^2}\int\mathrm{Tr}(F\wedge\star F)\;+\;\frac{i\theta}{8\pi^2}\int\mathrm{Tr}(F\wedge F) , $$

which is the real and imaginary part of a single complex number multiplying the (anti-)self-dual parts of the curvature. The two forms agree, using $\int\mathrm{Tr}(F\wedge\star F)=\tfrac12\int d^4x\,\mathrm{Tr}(F_{\mu\nu}F^{\mu\nu})$ and $\int\mathrm{Tr}(F\wedge F)=\tfrac12\int d^4x\,\mathrm{Tr}(F_{\mu\nu}\tilde F^{\mu\nu})$: the theta term is $\frac{i\theta}{16\pi^2}\int d^4x\,\mathrm{Tr}(F\tilde F)$ in the first display and $\frac{i\theta}{8\pi^2}\int\mathrm{Tr}(F\wedge F)$ in the second, the two differing by the factor two that relates $F\wedge F$ to $F\tilde F$, and both equal $i\theta Q$ after the normalisation $\int\mathrm{Tr}(F\wedge F)=8\pi^2Q$. The natural variable is therefore

$$ \tau=\frac{\theta}{2\pi}+\frac{4\pi i}{g^2}, $$

and the action's dependence on $\tau$ and $\bar\tau$ organises the theory. The group that preserves the form of the action and the charge lattice is $SL(2,\mathbb Z)$ acting by $\tau\mapsto(a\tau+b)/(c\tau+d)$.

The generators. $S$ and $T$ generate $\Gamma$ up to the centre. Their matrix properties were verified explicitly:

$$ S^2=\begin{pmatrix}-1&0\\0&-1\end{pmatrix}, \qquad T^3=\begin{pmatrix}1&3\\0&1\end{pmatrix}, \qquad \det S=\det T=1 , $$

so $S$ has order four modulo $\pm1$ and $T$ has infinite order; the group generated is $SL(2,\mathbb Z)/\{\pm1\}=PSL(2,\mathbb Z)$, the modular group of the torus. The combinations $ST$, $(ST)^3$ and $S$ generate it, and every element has integer entries and unit determinant, verified on four independent products.

Strong–weak duality. At $\theta=0$ the coupling is purely imaginary, $\tau=4\pi i/g^2$, and

$$ S:\ \tau\mapsto-\frac1\tau = -\,\frac{1}{4\pi i/g^2}=\frac{i g^2}{4\pi}=\frac{4\pi i}{g'^2}, \qquad g'=\frac{4\pi}{g}. $$

This was verified numerically at three couplings: for $g=1$ the image $\tau'=0.079577i$ equals $4\pi i/g'^2$ with $g'=4\pi$; for $g=2$ it is $0.318310i$ with $g'=2\pi$; and for $g=\sqrt{4\pi}$ the exchange is the self-dual point $g'=g$ at $\tau=i$. The map $g\mapsto4\pi/g$ is the strong–weak duality of the electric and magnetic couplings, and the self-dual coupling is $g^2=4\pi$. For general $\theta$ the transformation mixes the two parameters, $g'^2=g^2\bigl((\theta/2\pi)^2+(4\pi/g^2)^2\bigr)$ and $\theta'=-\theta/\bigl((\theta/2\pi)^2+(4\pi/g^2)^2\bigr)$, so the angle is not preserved by the exchange; this is the standard statement that the theta angle rotates under the duality.

The $T$ generator and the theta periodicity. $T:\tau\mapsto\tau+1$ leaves $g$ fixed and shifts $\theta\to\theta+2\pi$. It is a symmetry because the partition function is periodic in $\theta$ with period $2\pi$, as The Semiclassical Expansion and the Instanton Gas in Biquaternionic Form established from the integrality of the topological charge. The generator $T$ therefore exists because the topological charge is an integer; it is a statement about the sum over instanton sectors, not a statement about the algebra.

The Charge Lattice and the Quantisation Condition

A dyon carries electric charge $q_e$ and magnetic charge $q_m$. In units $\hbar=c=1$ the Dirac–Schwinger–Zwanziger quantisation condition for two dyons labelled $i=1,2$ is

$$ q_e^{(1)}q_m^{(2)}-q_e^{(2)}q_m^{(1)} \;=\; 2\pi\,\nu , \qquad \nu\in\mathbb Z , $$

which reduces for a purely electric and a purely magnetic particle to the Dirac condition $e\,g_m=2\pi\nu$ of The Magnetic Monopole in Biquaternionic Form, with $\nu$ the Dirac index. The index is written $\nu$ so that it does not collide with the lattice label $n$ below: the two differ by the factor of two that the next display exhibits. Writing the charges of the fundamental objects as

$$ (q_e,q_m)=\bigl(e\,m,\;\tfrac{4\pi}{e}\,n\bigr), \qquad m,n\in\mathbb Z , $$

the condition for a single species is $e\cdot\frac{4\pi}{e}mn=4\pi mn=2\pi\,(2mn)$, so $\nu=2mn$ and the lattice spacing is fixed by the same factor of two that the 't Hooft–Polyakov monopole exhibited. The pair $(m,n)$ labels the lattice of charges, and the modular group acts on it. The action is most transparently written on the complex charge

$$ q \;=\; q_e+\frac{i\,4\pi}{e^2}\,q_m , $$

for which the transformation $g=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ acts by

$$ q \;\longmapsto\; \frac{q}{c\tau+d}, $$

accompanied by the $\tau$ transformation; the lattice of integer electric and magnetic charges is mapped to itself because the entries are integers and the determinant is one. The Dirac condition is invariant under this action, which is the precise sense in which duality is a symmetry of the charge spectrum: it permutes the charged states rather than changing the quantisation.

Montonen–Olive duality. The conjecture of Montonen and Olive is that the spectrum of the theory is invariant under the exchange of electric and magnetic charges with $g\leftrightarrow4\pi/g$: the magnetic monopoles are the dual gauge bosons of a dual gauge theory. In the maximally supersymmetric $\mathcal N=4$ theory the conjecture is established at the level of the spectrum and the coupling; in theories with less supersymmetry it is modified or fails. The framework's role in the conjecture is limited but exact: it supplies the internal rotation that exchanges the two charges and the asymptotic Hopf bundle that carries the magnetic charge, and it supplies the topological integer on which the $T$ generator depends. It does not supply the dynamical statement that the magnetic objects form a gauge multiplet, which is a statement about the spectrum and the action.

The Modular Action on the Charge Lattice

The discrete group acts on the charges, and the action makes the arithmetic of the modular group and the physics of the lattice the same statement. Write a state of definite electric and magnetic charge as a column

$$ v=\begin{pmatrix} m\\ n\end{pmatrix}\in\mathbb Z^2 , \qquad q_e=e\,m , \qquad g_m=\frac{4\pi}{e}\,n , $$

with $m$ the electric and $n$ the magnetic quantum number, the magnetic charge being measured in units of $\frac{4\pi}{e}$, twice the minimal Dirac charge $\frac{2\pi}{e}$. The Dirac pairing of two charge vectors,

$$ \langle v,v'\rangle=mn'-nm'=\det\begin{pmatrix}m&m'\\ n&n'\end{pmatrix}, $$

is the symplectic form of the lattice, and the quantisation condition is the statement that it takes integer values on integer vectors: the physical pairing $q_e q_m'-q_e'q_m$ equals $4\pi\langle v,v'\rangle$, which lies in $2\pi\mathbb Z$ precisely when $\langle v,v'\rangle$ is an integer. For the minimal dyons $v=(1,0)$ and $v'=(0,1)$ the pairing is $1$, corresponding to the physical product $e\cdot\frac{4\pi}{e}=4\pi$ of a unit electric and a unit magnetic charge — two Dirac quanta, the factor of two being the same one that fixed $g_m=4\pi/e$ in the monopole article. A matrix

$$ \gamma=\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in SL(2,\mathbb Z) $$

acts on the lattice by $v\mapsto\gamma v$, and because $\det\gamma=1$ it preserves the determinant and hence the pairing:

$$ \langle \gamma v,\gamma v'\rangle=\det\bigl(\gamma\begin{pmatrix}m&m'\\ n&n'\end{pmatrix}\bigr)=\det\gamma\cdot\langle v,v'\rangle=\langle v,v'\rangle . $$

The group that preserves the Dirac pairing of integer charges is therefore exactly $SL(2,\mathbb Z)$ (up to the overall sign that is irrelevant on the lattice), and this is the precise reason the modular group and not some other arithmetic group governs the duality.

The two generators act on the quantum numbers as

$$ S:\begin{pmatrix}m\\ n\end{pmatrix}\mapsto\begin{pmatrix}-n\\ m\end{pmatrix}, \qquad T:\begin{pmatrix}m\\ n\end{pmatrix}\mapsto\begin{pmatrix}m+n\\ n\end{pmatrix}, $$

so that $S$ exchanges electric and magnetic charge up to a sign — the strong–weak duality of the earlier section — while $T$ shifts the electric charge by one unit of magnetic charge, which is the Witten effect of The Theta Parameter, Strong CP, and the Witten Effect in Biquaternionic Form written as a lattice translation. The relation $S^2=-I$, already verified as a matrix identity, is the statement that two duality rotations by a quarter turn produce the charge conjugation of the lattice: the electric and magnetic charges change sign while the pairing is untouched. The central charge of a state is linear in the charge vector,

$$ Z=a\,m+a_D\,n , $$

with $a$ the vacuum expectation value and $a_D$ its dual, and the BPS bound is $M\ge|Z|$ with equality for the BPS states. Since $Z$ is a linear functional on the lattice, an $SL(2,\mathbb Z)$ transformation of the charges can be compensated by the induced transformation of $(a,a_D)$, which is the modular action of the earlier section; the lattice picture and the complex-coupling picture are the same representation read on states and on the theory. The spectrum is not invariant term by term — states can appear and disappear as the coupling crosses a wall of marginal stability, the phenomenon of wall-crossing — and this non-trivial dynamics is standard in the Seiberg–Witten solution and is imported rather than derived.

Two consequences for the framework follow. First, the discreteness of the lattice is the integer structure imposed by quantisation; the algebra supplies the continuous charges, and the lattice is what quantisation makes of them, exactly as the discrete modular group is what quantisation makes of the continuous duality rotation. Second, the Dirac pairing is a symplectic form, so the charge lattice is a symplectic lattice and the modular group is its symplectic automorphism group; the algebra's role is that the abelian charges whose pairing this is are the coefficients of the central direction $\mathbb C_{\mathbb B}$, and their continuity is the algebra's continuous $U(1)$.

The Biquaternion Reading: Continuous Duality, Discrete Group

The separation between what the algebra gives and what is imported is unusually clean in this article.

• The algebra gives the continuous rotation. The duality rotation $\tilde F\mapsto e^{-i\theta}\tilde F$ is multiplication by an element of the unit circle of the centre $\mathbb C_{\mathbb B}$, which is a one-parameter continuous subgroup of the algebra's units. It is internal, central, and exact for the sourced equations. The framework's duality group, on its own, is $U(1)$.
• The algebra gives the shape of the action on the field. That the rotation exchanges the real and imaginary halves of $\tilde F$, that its generator is $ie_0$, that Hodge duality is the special case $\theta=\pi/2$ with $\tilde F_\star=-i\tilde F$, and that the rotation is Lorentz-trivial are all algebraic statements read off from $\tilde F=i\sqrt{\epsilon}\mathbf E-\sqrt{\mu}\mathbf H$ and the centrality of $i$. These are genuine results of the framework, and they fix the form of the duality.
• The algebra does not give the discrete group. A continuous centre cannot generate a discrete group, and no element of $\mathbb C_{\mathbb B}$ has the effect of $T:\theta\to\theta+2\pi$ on the spectrum except through the integer topological charge. The group $SL(2,\mathbb Z)$ arises from two imports: the quantisation of electric and magnetic charge, which is the Dirac condition and hence quantum mechanics, and the integrality of the topological charge, which is the instanton sector of The Semiclassical Expansion and the Instanton Gas in Biquaternionic Form. The modular group is thus a symmetry of the quantum theory with magnetic charges, not of the classical algebra.
• The algebra gives the self-dual point. The exchange $g\leftrightarrow4\pi/g$ has its fixed point at $g^2=4\pi$, which is a statement about the coupling and not about the algebra; the algebra fixes only that the duality is a rotation by $\pi/2$ in the internal phase and that the rotation is of order four, since $e^{-i\pi/2}=-i$ and $(-i)^2=-1$. The order-four character of the duality is the algebraic shadow of the order of $S$ modulo the centre.

The structural statement of the article is therefore: the biquaternion algebra reaches the continuous duality $U(1)$ and does not reach the discrete modular group. The former is the algebra's; the latter is the quantum theory's. This is the same kind of ceiling that The Gauge Group Ceiling: Why the Biquaternion Algebra Reaches SU(2) but Not SU(3) records for the gauge group, and it has the same character: a sharp statement of what the algebra cannot accommodate, which is a result and not a defect.

Summary

Electric–magnetic duality in the biquaternionic framework is the internal rotation $\tilde F\mapsto e^{-i\theta}\tilde F$ generated by an element of the central scalar subspace $\mathbb C_{\mathbb B}$. It is continuous, central, Lorentz-trivial, and exact for the sourced Maxwell equations; Hodge duality is its special case $\theta=\pi/2$, $\tilde F_\star=-i\tilde F$. With the theta angle, the theory's coupling is the complex parameter

$$ \tau=\frac{\theta}{2\pi}+\frac{4\pi i}{g^2}, $$

and the symmetry group of the charge lattice and the action is the modular group $SL(2,\mathbb Z)$, generated by

$$ S:\tau\mapsto-\frac1\tau \quad(\text{strong–weak duality},\ g\mapsto\tfrac{4\pi}{g}), \qquad T:\tau\mapsto\tau+1 \quad(\theta\mapsto\theta+2\pi). $$

The matrix properties $\det S=\det T=1$, $S^2=-I$, $T^3=\begin{pmatrix}1&3\\0&1\end{pmatrix}$, and the exchange $g\leftrightarrow4\pi/g$ with self-dual point $g^2=4\pi$ were verified numerically. The charge lattice $(q_e,q_m)=(em,4\pi n/e)$ is a module for the group, and the Dirac–Schwinger–Zwanziger condition is invariant under it.

The framework supplies the continuous $U(1)$ duality intrinsically, because the centre of the algebra is a continuous field whose unit circle is the rotation group; it does not supply the discrete modular group, whose $T$ generator requires the integer topological charge and whose $S$ generator requires the quantised electric and magnetic charges. The modular group is a symmetry of the quantum theory with magnetic charges and is imported from the Dirac condition and the instanton sector, while the algebra fixes the internal, central and Lorentz-trivial shape of the duality. This is a structural ceiling of the same character as the gauge-group ceiling of the companion articles.

Summary of Notation

Further Reading

• Peter Goddard, Jean Nuyts and David Olive, "Gauge theories and magnetic charge", Nuclear Physics B 125 (1977) 1–28, for the electric–magnetic duality conjecture and the charge lattice.
• Claus Montonen and David Olive, "Magnetic monopoles as gauge particles?", Physics Letters B 72 (1977) 117–120, for the original duality conjecture.
• Edward Witten and David Olive, "Supersymmetry algebras that include topological charges", Physics Letters B 78 (1978) 97–101, for the central charges and the BPS spectrum that underlie the duality.
• Ashoke Sen, "Electric magnetic duality in string theory", Nuclear Physics B 404 (1993) 109–126, and Nathan Seiberg and Edward Witten, "Electric–magnetic duality, monopole condensation, and confinement in $N=2$ supersymmetric Yang–Mills theory", Nuclear Physics B 426 (1994) 19–52, for the modern non-perturbative formulation of the duality.
• David Tong, Gauge Theory (lecture notes, University of Cambridge, 2018), for the modular group, the charge lattice and the Montonen–Olive conjecture.
• Sidney Coleman, "The magnetic monopole fifty years later", in Aspects of Symmetry (Cambridge University Press, 1985), for the duality rotation and the theta dependence of the dyon spectrum.
• Kentaro Hori, Sheldon Katz, Albrecht Klemm, Rahul Pandharipande, Richard Thomas, Cumrun Vafa, Ravi Vakil and Eric Zaslow, Mirror Symmetry (American Mathematical Society, 2003), for the modular group, its action on the complex coupling and the charge lattice, and the wall-crossing of the BPS spectrum.
• Yakov M. Shnir, Magnetic Monopoles (Springer, 2005), for the Dirac–Schwinger–Zwanziger quantisation condition and its duality invariance.
• Edward Witten, "Dyons of charge $e\theta/2\pi$", Physics Letters B 86 (1979) 283–287, for the theta dependence of the dyon charges that the modular action organises.
• Luis Álvarez-Gaumé and Miguel A. Vázquez-Mozo, An Introduction to Quantum Field Theory (Springer, 2012), for the complex coupling, instanton sectors and the theta dependence.