Monopole Dynamics and BPS Dyons N=2 Super-Yang-Mills Theories

Abstract

We determine the low energy dynamics of monopoles in pure N=2 Yang-Mills theories for points in the vacuum moduli space where the two Higgs fields are not aligned. The dynamics is governed by a supersymmetric quantum mechanics with potential terms and four real supercharges. The corresponding superalgebra contains a central charge but nevertheless supersymmetric states preserve all four supercharges. The central charge depends on the sign of the electric charges and consequently so does the BPS spectrum. We focus on the SU(3) case where certain BPS states are realised as zero-modes of a Dirac operator on Taub-NUT space twisted by the tri-holomorphic Killing vector field. We show that the BPS spectrum includes hypermultiplets that are consistent with the strong- and weak-coupling behaviour of the Seiberg-Witten theory. Comment: LaTeX, 18 pages, 1 figure, references added

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arXiv:hep-th/9912082v2 17 Dec 1999
KIAS-P99103
QMW-PH-99-15
IASSNS-HEP-99/110
hep-th/9912082
Monopole Dynamics and BPS Dyons
in N = 2 Super-Yang-Mills Theories
Jerome P. Gauntlett∗,1, Nakwoo Kim∗,2, Jaemo Park+,3, and Piljin Yi#,4
∗Department of Physics, Queen Mary and Westfield College
Mile End Rd, London E1 4NS, UK
+School of Natural Sciences, Institute for Advanced Study
Princeton, NJ 08540, USA
#School of Physics, Korea Institute for Advanced Study
207-43, Cheongryangri-Dong, Dongdaemun-Gu, Seoul 130-012, Korea
ABSTRACT
We determine the low energy dynamics of monopoles in pure N = 2 Yang-
Mills theories for points in the vacuum moduli space where the two Higgs
fields are not aligned. The dynamics is governed by a supersymmetric
quantum mechanics with potential terms and four real supercharges. The
corresponding superalgebra contains a central charge but nevertheless su-
persymmetric states preserve all four supercharges. The central charge
depends on the sign of the electric charges and consequently so does the
BPS spectrum. We focus on the SU(3) case where certain BPS states are
realised as zero-modes of a Dirac operator on Taub-NUT space twisted
by the tri-holomorphic Killing vector field. We show that the BPS spec-
trum includes hypermultiplets that are consistent with the strong- and
weak-coupling behaviour of the Seiberg-Witten theory.
1E-mail: j.p.gauntlett@qmw.ac.uk
2E-mail: n.kim@qmw.ac.uk
3E-mail: jaemo@sns.ias.edu
4E-mail: piljin@kias.re.kr

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1Introduction
The BPS spectrum of monopoles and dyons are an important non-perturbative feature
of supersymmetric Yang-Mills theories. At weak coupling one can determine the
BPS spectrum using semi-classical techniques. Following [1, 2], the BPS spectrum of
N = 2 and N = 4 theories was studied in a number of papers [3, 4, 5, 6] at points
in the moduli space of vacua where only a single Higgs field was involved, or more
precisely where all of the Higgs fields were aligned. In these cases one studies certain
supersymmetric quantum mechanics models with the target manifold given by the
moduli space of classical BPS monopole solutions.
New features arise when one studies the spectrum at points in the moduli space
where the Higgs fields are not aligned [7, 8, 9, 10, 11, 12]. For theories with N = 4
supersymmetry, the BPS bound is determined by two complex central charges that
appear in the supersymmetry algebra. For aligned Higgs fields the two charges are
necessarily equal and a BPS state preserves 1/2 of the supersymmetry. When the
six Higgs fields are not aligned the central charges can be different and then the BPS
states preserve 1/4 of the supersymmetry.
The low energy dynamics of monopoles for non-aligned Higgs fields in N = 4
theories were recently studied by Bak. et. al [13]. The supersymmetric quantum
mechanics is still based on the same BPS monopole moduli space, but is now sup-
plemented by a supersymmetric potential term which is constructed from a set of
tri-holomorphic Killing vector fields that generate unbroken U(1) gauge symmetries.
It was noticed in Ref. [12] that this potential naturally appears in the expression
for the energy of BPS states, while Bak et.al. later showed how the same potential
occurs in the low energy dynamics, albeit with an important multiplicative factor of
1/2, and used the resulting Lagrangian to study the spectrum of N = 4 Yang-Mills,
including 1/4 BPS states.
In this paper we will analyse analogous issues for pure N = 2 supersymmetric
Yang-Mills theories. Since the N = 2 supersymmetry algebra has one complex central
charge there can only be BPS states preserving 1/2 of the supersymmetry. Since the
pure N = 2 Yang-Mills theory can be embedded in the N = 4 theory it is not
surprising that the central charge is one of the central charges that appear in the
N = 4 theory. It is interesting that the other N = 4 central charge also appears as
a bound on the classical mass of dyons, but it is no longer related to preservation of
supersymmetry. If this latter bound is stronger than the BPS bound for a given set
of charges, then no BPS state can exist with those charges.
At points in the vacuum moduli space of pure N = 2 Yang-Mills theories where
the Higgs fields are aligned, the low energy dynamics is a supersymmetric quantum
mechanics on the moduli space of BPS monopoles with four real supersymmetries [1].
The BPS states correspond to harmonic spinors on the monopole moduli space (or
equivalently, on a hyperK¨ ahler manifold, harmonic holomorphic forms). This follows
from the simple fact that one of the low energy supercharges Q is proportional to a
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Dirac operator on the moduli space,
D = −iγµ∇µ,(1)
with covariant derivative on the moduli space ∇, and its square gives the supersym-
metric sigma-model Hamiltonian.
Q2=1
2D2= H0. (2)
With two Higgs fields active, we will argue that the low energy dynamics includes
a supersymmetric potential term as in the N = 4 theories [13]. The only difference
from the N = 4 case is that the number of fermions and the number of supercharges is
reduced by half. We will write down this low energy Lagrangian explicitly in section
3. The supercharge is now given by the Dirac operator twisted by a tri-holomorphic
vector field G. The superalgebra then has the general form,
Q2=1
2(D − γµGµ)2= H − Z,(3)
where H is the modified Hamiltonian, and Z is a real central charge defined by the
Lie derivative along G,
Z = −iLG,
and measures a linear combination of electric charges.
The BPS states with H = Z preserve not only the supercharge Q but, as we
will show, all four supercharges. This is consistent with preservation of 1/2 of the
spacetime supersymmetry. It is interesting to note that if we flip the signs of the
electric charges so that Z → −Z, the state will no longer be BPS. This should be
contrasted with the N = 4 theory, where BPS states with electric charges of both
signs may occur and break a further half of the supersymmetries in general.
We will analyze in some detail the simplest case of SU(3) broken to U(1) × U(1)
by two adjoint Higgs. In particular we will focus on BPS states with a (1,1) magnetic
charge. The BPS monopole moduli space for this case is given by R3× (R × M)/Z
where M is Taub-NUT space [5, 14]. The BPS spectrum is then determined by solving
the Dirac equation on the Taub-NUT manifold twisted by the tri-holomorphic Killing
vector field and we will be able to utilise the results of Pope who studied precisely
the same operator in [16].
An early analysis of the BPS spectrum of N = 2 SU(3) Yang-Mills theories in the
weak coupling regime was carried out in the context of Seiberg-Witten theories [17] by
Fraser and Hollowood [18]. Acting with semi-classical monodromy transformations on
purely magnetic states, they found new states in a certain part of the vacuum moduli
space whose magnetic charge is (1,1) and whose electric charge is (n,n − 1) with
arbitrary integer n. Since the monodromy cannot alter the supermultiplet structures,
all of these dyons fill out hypermultiplets. By solving the low energy dynamics of
two distinct monopoles, we will find that these are a particular case of more general
(4)
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states with electric charges, (m,l) where integers m and l are such that m > l. The
size of supermultiplet of the BPS state grows linearly with the positive integer m−l.
This paper is organised as follows. Section 2 will briefly summarise the classical
energy bound of the pure N = 2 Yang-Mills theory. We will show that there are two
bounds on the classical energy and only one of them corresponds to a BPS bound.
In section 3 we will present the supersymmetric quantum mechanics with potential
that should describe the low-energy dynamics of monopoles and dyons. We analyze
the conditions for preserved supersymmetry and use this in section 4 to analyze
the BPS spectrum for the case of SU(3). In Section 5, we summarise some of the
previously known results on the spectrum of pure SU(3) Seiberg-Witten theory from
monodromies as well as strong coupling singularities in the vacuum moduli space,
and show that the results are consistent with those in Section 4. We conclude in
section 6.
2BPS bound
The N = 2 super-Yang-Mills Lagrangian is given by
L =
1
2tr
?
−1
2FµνFµν+ DµφIDµφI+ e2[φ1,φ2]2
+i¯ χΓµDµχ − e¯ χ[φ1,χ] − ie¯ χγ5[φ2,χ]
?
,(5)
where φI, I = 1,2 denote the two real Higgs fields, DµφI= ∂µφI−ie[Aµ,φI] and χ is
a Dirac spinor and all fields are in the adjoint representation of the gauge group G.
The vacuum moduli space demands that [φ1,φ2] = 0; we may choose the asymptotic
values of the Higgs fields along the positive z-axis, say, to be in the Cartan sub-
algebra, φI= φI· H, where φIare vectors of dimension r=rank(G). This does
not completely fix the gauge transformations as one has the freedom to perform
discrete gauge transformations by elements of the Weyl group. These can be fixed
by demanding, for example, that φ1·βa≥ 0 for a given set of simple roots βaof the
Lie algebra G of G. We will only consider points in the moduli space of vacua where
the symmetry is maximally broken to U(1)r.
For a given vacuum we can define electric and magnetic charge two-vectors
?Qe= tr
?
dSiEi?φ,
?Qm= tr
?
dSiBi?φ,
(6)
with i = 1,2,3 and?φ = (φ1,φ2). These can be written as
QI
e= φI· q,QI
m= φI· g,(7)
where we have introduced the electric and magnetic charge vectors given by
q = ena
eβa,
g =
4π
ena
mβ∗
a,(8)
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respectively, where βaare the simple roots and β∗
na
electric quantum numbers.
By determining the central charges that appear in the supersymmetry algebra as
in [15] we can determine the BPS bound:
aare the simple co-roots of G, and
eare, in the quantum theory, the
mare the topological winding numbers and na
M ≥ |Z−= (Q1
e− Q2
m) + i(Q1
m+ Q2
e)|.(9)
Note that if we introduce a complex rescaled Higgs vector A = e(φ1+ iφ2) and
rescale the charge vectors via ˆ q = q/e and ˆ g = (e/4π)g then the BPS condition
becomes M = |A· ˆ q+ADˆ ·g| where AD= (i4π/e2)A which is the form familiar from
Seiberg-Witten theory (for vanishing θ) [17].
It is illuminating to rederive the BPS bound using Bogomol’nyi’s method of rewrit-
ing the energy as a sum of squares plus conserved charges. Indeed we will see that
this gives rise to two bounds on the classical energy. Since the bosonic part of the
N = 2 Lagrangian differs from the N = 4 theory only in the fact that there are two
Higgs fields instead of six Higgs, one can immediately adapt the derivation of general
BPS bound for the N = 4 theory [7, 10] to the N = 2 case. One finds that the most
stringent bound on the mass is given by
M≥
= Max (
?
|?Qe|2+ |?Qm|2+ 2|?Qe||?Qm|sinξ
?
|?Qe|2+ |?Qm|2± 2(Q2
mQ1
e− Q1
mQ2
e)), (10)
where 0 ≤ ξ ≤ π is the angle between the two 2-vectors?Qeand?Qm. This is equivalent
to
M ≥ Max |Z±= (Q1
In N = 4 theories, Z±appear as central charges in the supersymmetry algebra.
If a state saturates the BPS bound (11) it will preserve 1/4 of the supersymmetry.
In cases where Z+= Z−, which occurs when the angle between?Qeand?Qmvanishes,
the state will preserve 1/2 of the supersymmetry. By contrast, in N = 2 theories
there is only one complex central charge that appears in the supersymmetry algebra1,
Z−, giving rise to the BPS bound (9). A state saturating this bound will preserve
1/2 of the supersymmetry. A classical soliton can only saturate the larger of the two
bounds, |Z±|. Thus, if it so happens that |Z−| < |Z+| then there can be no classical
BPS soliton with such charges in such a vacuum. In particular, suppose that a state
of charge (g,q) saturates the BPS bound |Z−| > |Z+|. Then, for a state of charge
(g,−q), the BPS bound |Z−| will be smaller than the classical energy bound |Z+|.
In the asymptotic region of vacuum moduli space where a semi-classical analysis is
suitable the quantum corrections to the classical soliton mass will be small and we
conclude that that the latter state cannot be BPS saturated. This asymmetry with
respect to the sign of the electric charge is a generic feature of the N = 2 dyon
e± Q2
m) + i(Q1
m∓ Q2
e)|.(11)
1If φ2→ −φ2in (5) the central charge would be Z+.
4