# Hidden symmetries of a self-dual monopole

L Feher, P Horváthy, L O’Raifeartaigh

► To cite this version:

L Feher, P Horváthy, L O’Raifeartaigh. Hidden symmetries of a self-dual monopole. Symmetries in science III, Jul 1988, Lochau, Austria. pp.525-529. hal-05076494

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# Hidden symmetries of a self-dual monopole *

L. Feher$^{1,2}$ †, P. Horváthy$^{2,3}$ ‡ and L. O’Raifeartaigh$^{2}$ §

¹ *Bolyai Institute, Jate H-6720 - Szeged (Hungary)*

² *Dublin Institute for Advanced Studies, - Dublin (Ireland)*

³ *Laboratoire de Mathématiques et Applications de Metz (Bât. A, Ile du Saulcy), Université Paul Verlaine - Metz (France)*

*(Dated: May 21, 2025)*

## Abstract

The symmetries of a spinning particle in the field of a self-dual monopole are studied from the viewpoint of supersymmetric quantum mechanics.

<footnote>* Originally appeared in: *Symmetries in Science III*, Proc. Schloss Hofen Meeting, Gruber and Iachello (eds.), pp. 525-529. Plenum (1989). Available in inspire (code Feher:1988ym).</footnote>

<footnote>† Present address: lfeher@physx.u-szeged.hu</footnote>

<footnote>‡ Present address: horvathy@univ-tours.fr</footnote>

<footnote>§ deceased</footnote>
A self-dual (SD) SU(2) monopole is a static solution of the first order Bogomolny [1] equations

$$D_k \Phi = B_k \left( = \frac{1}{2} \epsilon_{kij} F_{ij} \right). \qquad (1)$$

The monopole field can be identified with a pure Yang-Mills configuration $A_0 = 0$, $A_k$, $A_4 = \Phi$ in (1+4)-dimensional flat space which does not depend on the coordinates $x^0$ and $x^4$. The Dirac equation,

$$i \frac{\partial \Phi}{\partial x^0} = i \gamma^0 \not D_4 \Psi, \qquad (2)$$

plays a decisive role [2–4] in describing the fluctuations around a SD monopole. Here we are interested in the symmetries of the 4-dimensional, Euclidean Dirac operator

$$\not D_4 = \gamma^k D_k + \gamma^4 D_4 = \frac{\sqrt{2}}{i} \begin{pmatrix} 0 & Q \\ Q^+ & 0 \end{pmatrix}, \qquad (3)$$

where

$$Q = \frac{1}{\sqrt{2}} (\Phi \cdot \mathbf{1}_2 - i\vec{\pi} \cdot \vec{\sigma}), \\ \vec{\pi} = -i\vec{D} = -i\vec{\nabla} - \vec{A}^a I_a, \quad D_4 = \frac{\partial}{\partial x^4} - i\Phi^a I_a. \qquad (4)$$

We use the following $\gamma$-matrices:

$$\gamma^k = \begin{pmatrix} 0 & i\sigma^k \\ -i\sigma^k & 0 \end{pmatrix}, \quad \gamma^4 = \begin{pmatrix} 0 & \mathbf{1}_2 \\ \mathbf{1}_2 & 0 \end{pmatrix}, \quad \gamma^0 = i\gamma^5 = i \begin{pmatrix} \mathbf{1}_2 & 0 \\ 0 & -\mathbf{1}_2 \end{pmatrix}. \qquad (5)$$

The $I_a$ ($a = 1, 2, 3$) are the standard isospin matrices in some representation and we suppose that nothing depends on the extra coordinate $x^4$.

Let us now consider the supersymmetric quantum mechanical system defined by $(-1)^F = \gamma^5$, and

$$H = \left( \frac{i}{\sqrt{2}} \not D_4 \right)^2 = \begin{pmatrix} QQ^+ & 0 \\ 0 & Q^+Q \end{pmatrix} = \begin{pmatrix} H_1 & 0 \\ 0 & H_0 \cdot \mathbf{1}_2 \end{pmatrix}. \qquad (6)$$

Here

$$H_0 = \frac{1}{2} (\pi^2 + \Phi^2), \quad H_1 = H_0 \cdot \mathbf{1}_2 - \vec{\sigma} \cdot \vec{B}. \qquad (7)$$
As a consequence of self-duality, the spin drops out of $Q^+Q$, while $QQ^+$ contains the term $\vec{\sigma} \cdot \vec{B}$ characteristic of a particle of “apparent” gyromagnetic ratio 4. Note that $H_0$ is the Hamiltonian of a nonrelativistic, spinless particle moving in the field of the monopole. The Pauli Hamiltonian $H_1$ describes a nonrelativistic particle of spin $\frac{1}{2}$. For $E \neq 0$ the relation

$$H_1 = U^+ (H_0 \cdot \mathbf{1}_2) U, \quad U = \frac{1}{\sqrt{H_0}} Q^+ \qquad (8)$$

is satisfied. Thus $H_1$ and $H_0 \cdot \mathbf{1}_2$ have identical spectra apart from zero modes which are counted by the index theorem [5]. Let us now consider particles of definite momenta, helicity, and electric charge,

$$q = \frac{I_a \Phi^a}{\|\Phi\|}, \qquad (9)$$

scattering on the monopole. At large distances $A_k$ tends to the potential of a Dirac monopole, while the Higgs field behaves as $\Phi \sim q(1 - \frac{1}{r})$. This implies that the transformation $U$ preserves the quantum numbers of the scattering states. Therefore $H_1$ and $H_0 \cdot \mathbf{1}_2$ have identical $S$-matrices with a factorized, purely kinematical spin dependence. (The spin-charge factorization of the $S$-matrix is inherited by the vector fluctuations scattering off the monopole [6].) One can understand the factorization of the $S$-matrix also as a consequence of the conservation of $\vec{S}_0 = \frac{1}{2} \begin{pmatrix} 0 & 0 \\ 0 & \vec{\sigma} \end{pmatrix}$ for $H_0 \cdot \mathbf{1}_2$ and

$$\vec{S}_1 = U^+ \vec{S}_0 U = \frac{-1}{2H_1} \vec{\Omega}, \quad \vec{\Omega} = \frac{1}{2} (\pi^2 - \Phi^2) \vec{\sigma} + \Phi(\vec{\pi} \times \vec{\sigma}) - (\vec{\sigma} \cdot \vec{\pi})\vec{\pi} \qquad (10)$$

for $H_1$. Now we introduce the following vector supercharges commuting with $H$:

$$Q_1 = \frac{i}{\sqrt{2}} \not D_4 = \begin{pmatrix} 0 & Q \\ Q^+ & 0 \end{pmatrix}, \quad Q_2 = -i\gamma^5 Q_1, \quad \vec{Q}_\alpha = 2i[\vec{S}_0, Q_\alpha], \quad (\alpha = 1, 2). \qquad (11)$$

The even generators $\gamma^5, \vec{S}_0, \vec{S}_1$ and the odd ones $Q_\alpha, \vec{Q}_\alpha$ yield, for fixed $E \neq 0$, a closed superalgebra determined by the commutation relations:

$$[S_{\alpha}^{i}, S_{\beta}^{j}] = i\delta_{\alpha\beta}\epsilon_{ijk} \quad (\alpha, \beta = 0, 1), \qquad (12a)$$

$$\left. \begin{array}{l} [\gamma^5, Q_\alpha] = 2i\epsilon_{\alpha\beta}Q_\beta, \quad [\gamma^5, Q_\alpha^k] = 2i\epsilon_{\alpha\beta}Q_\alpha^k \quad (\alpha, \beta = 1, 2), \\ [S_0^k, Q_\alpha] = -\frac{1}{2}iQ_\alpha^k, \quad [S_0^k, Q_\alpha^j] = \frac{1}{2}i(\delta_{jk}Q_\alpha + \epsilon_{kjn}Q_\alpha^n), \\ [S_1^k, Q_\alpha] = \frac{1}{2}iQ_\alpha^k, \quad [S_1^k, Q_\alpha^j] = -\frac{1}{2}i(\delta_{jk}Q_\alpha - \epsilon_{kjn}Q_\alpha^n), \end{array} \right\} \qquad (12b)$$

$$\left. \begin{array}{l} \{Q_\alpha, Q_\beta\} = 2\delta_{\alpha\beta}H, \quad \{Q_\alpha, Q_\alpha^k\} = -4H\epsilon_{\alpha\beta}(S_0^k + S_1^k), \\ \{Q_\alpha^k, Q_\beta^j\} = 2H\delta_{\alpha\beta}\delta_{jk} - 4H\epsilon_{\alpha\beta}\epsilon_{kjn}(S_1^n - S_0^n). \end{array} \right\} \qquad (12c)$$
For fixed $E > 0$, this algebra is $SU(2/2) \oplus U(1)$. The $U(1)$ factor is generated by $\gamma^5$. Observe that $\vec{S} = \vec{S}_0 + \vec{S}_1$ generates an $SO(3)$ invariance algebra of the Dirac operator $\not{D}_4$. The same algebraic structure is present (at least locally) in an instanton and in a $SD$ gravitational instanton background. This could be useful for analysing the Dirac equation.

Our investigations presented so far are valid in a general $SD$ background in the whole space. From now on we shall restrict ourselves to the behaviour at large distances. We show, in fact, that for the asymptotic, long range field of the $BPS$ 1-monopole there exists also a large dynamical symmetry [7]. As $r \to \infty$, the fields tend to those of a singular monopole

$$A_i^a = \epsilon_{\alpha ij} \frac{x_j}{r^2}, \quad \Phi^a = -\frac{x^a}{r} \left(1 - \frac{1}{r}\right). \qquad (13)$$

For the asymptotic system, the electric charge is conserved and the angular momentum (without spin) becomes

$$\vec{L}_0 = \vec{J} - \frac{\vec{\sigma}}{2} = \vec{r} \times \vec{\pi} - q\hat{r}, \qquad (14)$$

since $A_i^a$ describes an embedded Dirac monopole. The Hamiltonian $H_0$ is separated as

$$H_0 = -\frac{1}{2}\Delta_r + \frac{L_0^2 - q^2}{2r^2} + \frac{q^2}{2}\left(1 - \frac{2}{r} + \frac{1}{r^2}\right) = -\frac{1}{2}\Delta_r + \frac{L_0^2}{2r^2} - \frac{q^2}{r} + \frac{q^2}{2}. \qquad (15)$$

The $\frac{q^2}{r^2}$ terms coming from the centrifugal potential and from $\Phi^2$ cancel and we are left with a Coulomb type system. This is the “MIC-Zwanziger” Hamiltonian [8, 9] introduced originally just for the sake of its symmetries 20 years ago. It commutes with the Runge-Lenz vector

$$\vec{K}_0 = \frac{1}{2} \left( \vec{\pi} \times \vec{L}_0 - \vec{L}_0 \times \vec{\pi} \right) - q^2 \hat{r} \qquad (16)$$

Correspondingly, $\vec{L}_1 = U^+\vec{L}_0U = \vec{J} - \vec{S}_1$ and $\vec{K}_1 = U^+\vec{K}_0U$ are constants of motion for $H_1$. The Hamiltonian $H_1$ has been investigated recently by D'Hoker and Vinet [10] but without its present interpretation [11]. The spin $\frac{1}{2}$ “Runge-Lenz vector” $\vec{\Lambda}_1$ they have found is

$$\vec{\Lambda}_1 = \vec{K}_1 - q\vec{S}_1 = K_0 \cdot \mathbf{1}_2 + \vec{\pi} \times \vec{\sigma} + \left(\frac{q}{r} - \frac{q}{2}\right) \vec{\sigma} - \left(\vec{\sigma} \cdot \vec{B}\right) \vec{r}. \qquad (17)$$

For bound states $0 < E < \frac{q^2}{2}$, and from (12), the angular momentum and the Runge-Lenz vector combine into an $[SU(2/2) \oplus U(1)] \oplus O(4)$ dynamical symmetry algebra of $H$. For
$E = \frac{q^2}{2}$ and $E > \frac{q^2}{2}$ $O(4)$ is replaced by $E(3)$ and $O(3, 1)$, respectively. The bound state spectrum (without zero modes)

$$E = \frac{1}{2} \left( q^2 - \frac{1}{n^2} \right), \quad n = |q| + 1, |q| + 2, \dots \qquad (18)$$

is calculated from the $O(4)$ symmetry by Pauli's method [12]. The multiplicity (for fix $q \neq 0$) is $2(n^2 - q^2)$. The S-matrix is also easy to derive from the bosonic symmetry $O(3) \oplus O(3, 1)$ (the $O(3)$ factor comes from $\vec{S}$). In fact, Zwanziger's method [8] yields

$$S(\underline{k}', s' | \underline{k}, s) = \delta(E_{\underline{k}'} - E_{\underline{k}}) \cdot D_{s',s}^{\frac{1}{2}} (R_{\underline{k}'}^{-1} R_{\underline{k}}) \cdot S_0(\underline{k}' | \underline{k}), \\ S_0(\underline{k}' | \underline{k}) = \sum_{l \ge |q|} (2l + 1) \left[ \frac{(l - \frac{i}{k})!}{(l + \frac{i}{k})!} \right] D_{-q,q}^{l} (R_{\underline{k}'}^{-1} R_{\underline{k}}). \qquad (19)$$

Here $\underline{k}$ is the velocity and $s$ is the helicity of the incoming particles of charge $q$, $k = \sqrt{2E_k - q^2}$. The primed quantities refer to the outgoing particles. $R_{\underline{k}}$ is some rotation, chosen by convention, which brings the $\hat{z}$-direction into the direction of $\underline{k}$. The poles of $S$ are at $l - \frac{i}{k} = -T$, $T = 1, 2, \dots$ yielding the $E > 0$ bound state spectrum (18) with $n = T + l = |q| + 1, |q| + 2, \dots$.

Finally we show that, for the singular monopole (13), $H_0 \cdot \mathbf{1}_2$ and $H_1$ are essentially identical. In terms of the quantities

$$x = \vec{\sigma} \cdot \vec{J} - \frac{1}{2}, \quad H_r = -\frac{1}{2}\Delta_r - \frac{q^2}{r} + \frac{q^2}{2}, \qquad (20)$$

$H_0 \cdot \mathbf{1}_2$ is rewritten as

$$H_0 \cdot \mathbf{1}_2 = H_r + \frac{J^2 + \frac{1}{4} - x}{2r^2}. \qquad (21)$$

The operator $x$ commutes with $H_0 \cdot \mathbf{1}_2$ and with $\vec{J}$ and has eigenvalues $\pm(j + \frac{1}{2})$ because of the identity

$$x^2 = J^2 + \frac{1}{4}. \qquad (22)$$

Thus, by separating according to $J^2$, $J_3$ and $x$ (21) becomes

$$H_0 \cdot \mathbf{1}_2 = H_r + \begin{cases} \frac{(j-\frac{1}{2})(j+\frac{1}{2})}{2r^2}, & j = |q| + \frac{1}{2}, \dots \\ \frac{(j+\frac{1}{2})(j+\frac{3}{2})}{2r^2}, & j = |q| - \frac{1}{2}, \dots \end{cases}. \qquad (23)$$
On the other hand, the operator

$$y = \vec{\sigma} (\vec{J} + 2q\hat{r}) - \frac{1}{2} \qquad (24)$$

allows us to rewrite $H_1 = H_0 \cdot \mathbf{1}_2 - \vec{\sigma} \cdot \vec{B}$ as

$$H_1 = H_r + \frac{J^2 + \frac{1}{4} - y}{2r^2}. \qquad (25)$$

Moreover, $y$ commutes with $H_r$ and $\vec{J}$ and has eigenvalues $\pm (j + \frac{1}{2})$. Therefore, by diagonalizing the commuting set $J^2$, $J_3$, $y$ (25) is converted into

$$H_1 = H_r + \begin{cases} \frac{(j-\frac{1}{2})(j+\frac{1}{2})}{2r^2}, & j = |q| - \frac{1}{2}, |q| + \frac{1}{2}, \dots \\ \frac{(j+\frac{1}{2})(j+\frac{3}{2})}{2r^2}, & j = |q| - \frac{1}{2}, \dots \end{cases} \qquad (26)$$

This is the same equation as (23) describing $H_0 \cdot \mathbf{1}_2!$ The associated eigenvalue problem can be solved in terms of hypergeometric functions. The result is consistent with (18). The only difference between $H_0 \cdot \mathbf{1}_2$ and $H_1$ is that for $H_1$, in the $y = (j + \frac{1}{2})$ sector $j$ starts from $|q| - \frac{1}{2}$, while for $H_0 \cdot \mathbf{1}_2$, in the $x = (j + \frac{1}{2})$ sector $j$ starts from $|q| + \frac{1}{2}$. This is the only effect produced by the term $\vec{\sigma} \cdot \vec{B}$. In fact, this leads to the spectral asymmetry required by the index theorem [5]. The zero modes of $H_1$ carry the “unpaired” quantum numbers $j = |q| - \frac{1}{2}$, $y = |q|$. For fixed $q \neq 0$, the multiplicity of the 0-energy state is $(2j_{\min} + 1) = 2|q|$.

**Note added in 2024.** This paper was originally presented by L. Feher at the Symmetries in Science III meeting at Schloss Hofen (July 25-28, 1988) [13].

[1] For a review, see P. Goddard and D. I. Olive, “New Developments in the Theory of Magnetic Monopoles,” Rept. Prog. Phys. **41** (1978), 1357 doi:10.1088/0034-4885/41/9/001

[2] L. S. Brown, R. D. Carlitz and C. k. Lee, “Massless Excitations in Instanton Fields,” Phys. Rev. D **16** (1977), 417-422 doi:10.1103/PhysRevD.16.417

[3] R. Jackiw and C. Rebbi, “Spin from Isospin in a Gauge Theory,” Phys. Rev. Lett. **36** (1976), 1116 doi:10.1103/PhysRevLett.36.1116

[4] F. A. Bais and W. Troost, “Zero Modes and Bound States of the Supersymmetric Monopole,” Nucl. Phys. B **178** (1981), 125-140 doi:10.1016/0550-3213(81)90499-5
[5] E. J. Weinberg, “Parameter Counting for Multi-Monopole Solutions,” Phys. Rev. D **20** (1979), 936-944 doi:10.1103/PhysRevD.20.936

[6] K. J. Biebl and J. Wolf, “Scattering of gauge and Higgs fields by selfdual monopoles,” Nucl. Phys. B **279** (1987), 571-592 doi:10.1016/0550-3213(87)90011-3

[7] J. F. Schönfeld, “Dynamical Symmetry and Magnetic Charge,” J. Math. Phys. **21** (1980), 2528 doi:10.1063/1.524360, L. G. Feher, “Dynamical O(4) Symmetry in the Asymptotic Field of the Prasad-sommerfield Monopole,” J. Phys. A **19** (1986), 1259-1270 doi:10.1088/0305-4470/19/7/026, G. W. Gibbons and P. J. Ruback, “The Hidden Symmetries of Multicenter Metrics,” Commun. Math. Phys. **115** (1988), 267 doi:10.1007/BF01466773

[8] D. Zwanziger, “Exactly soluble nonrelativistic model of particles with both electric and magnetic charges,” Phys. Rev. **176** (1968), 1480-1488 doi:10.1103/PhysRev.176.1480

[9] H. V. Mcintosh and A. Cisneros, “Degeneracy in the presence of a magnetic monopole,” J. Math. Phys. **11** (1970), 896-916 doi:10.1063/1.1665227

[10] E. D'Hoker and L. Vinet, “Constants of motion for a spin 1/2 particle in the field of a dyon,” Phys. Rev. Lett. **55** (1985), 1043-1046 doi:10.1103/PhysRevLett.55.1043, E. D'Hoker and L. Vinet, “Hidden Symmetries and Accidental Degeneracy for a Spin 1/2 Particle in the Field of a Dyon,” Lett. Math. Phys. **12** (1986), 71 doi:10.1007/BF00400305

[11] L. G. Feher and P. A. Horvathy, “Non-relativistic scattering of a spin-1/2 particle off a self-dual monopole,” Mod. Phys. Lett. A **3** (1988), 1451-1460 doi:10.1142/S0217732388001744 [arXiv:0903.0249 [hep-th]].

[12] W. Pauli, “Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik,” Z. Phys. **36** (1926) no.5, 336-363 doi:10.1007/BF01450175

[13] L. Feher, P. Horvathy and L. O’Raifeartaigh, “Hidden symmetries of a self-dual monopole,” In: *Symmetries in Science III*, Proc. Schloss Hofen Meeting, Gruber and Iachello (eds.), pp.525-529. Plenum (1989).