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Applications of the Ashtekar gravity to four-dimensional hyperkähler geometry andYang–Mills instantonsYoshitake Hashimoto, Yukinori Yasui, Sayuri Miyagi, and Takayoshi Ootsuka Citation: Journal of Mathematical Physics 38, 5833 (1997); doi: 10.1063/1.532169 View online: http://dx.doi.org/10.1063/1.532169 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/38/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Applications of Yang-Mills thermodynamics AIP Conf. Proc. 1479, 545 (2012); 10.1063/1.4756188 Gravity and YangMills Fields: Geometrical Approaches AIP Conf. Proc. 751, 64 (2005); 10.1063/1.1891531 Yang–Mills instantons with Lorentz violation J. Math. Phys. 45, 3228 (2004); 10.1063/1.1767624 Zero curvature formalism of the four-dimensional Yang–Mills theory in superspace J. Math. Phys. 40, 2735 (1999); 10.1063/1.532726 Reductions by isometries of the selfdual Yang–Mills equations in fourdimensional Euclidean space J. Math. Phys. 34, 3245 (1993); 10.1063/1.530074
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Applications of the Ashtekar gravity to four-dimensionalhyperka hler geometry and Yang–Mills instantons
Yoshitake Hashimotoa)
Department of Mathematics, Osaka City University, Sumiyoshiku, Osaka, Japan
Yukinori Yasui,b) Sayuri Miyagi,c) and Takayoshi Ootsukad)
Department of Physics, Osaka City University, Sumiyoshiku, Osaka, Japan
~Received 11 March 1997; accepted for publication 30 June 1997!
The Ashtekar–Jacobson–Smolin–Mason–Newman equations are used to constructthe hyperka¨hler metrics on four-dimensional manifolds. These equations are closelyrelated to anti-self-dual Yang–Mills equations of the infinite-dimensional gaugeLie algebras of all volume-preserving vector fields. Several examples of hyper-kahler metrics are presented through the reductions of anti-self-dual connections.For any gauge group anti-self-dual connections on hyperka¨hler manifolds are con-structed using the solutions of both Nahm and Laplace equations. ©1997 Ameri-can Institute of Physics.@S0022-2488~97!01811-2#
I. INTRODUCTION
Four-dimensional hyperka¨hler geometry has been studied extensively in connection with sev-eral issues: gravitational instantons, supersymmetric nonlinears models, and compactifications ofsuperstrings.1–3 A hyperkahler manifold is a Riemannian manifold equipped with three covariantlyconstant complex structuresJa (a51,2,3) which obey the conditionJaJb52dab2eabcJ
c. In fourdimensions, a Riemannian manifold is hyperka¨hler if and only if its Ricci curvature is zero and itsWeyl curvature is either self-dual or anti-self-dual, namely, the metric is the solution of~anti!self-dual Einstein equation.
It is known that all asymptotically locally Euclidean~ALE! hyperkahler manifolds are clas-sified by Dynkin diagrams of ADE types. These spaces are constructed as hyperka¨hler quotients offlat Euclidean spaces.4
On the other hand, in the Hamiltonian approach for general relativity, Ashtekar–Jacobson–Smolin and Mason–Newman~AJSMN! reduced the problem of finding hyperka¨hler metrics tothat of finding linearly independent four-vector fieldsVm ~m50,1,2,3! and a volume formv on afour-dimensional manifoldX that satisfy the following two conditions:5–7
~a! volume-preserving condition:
LVmv50, ~1!
~b! half-flat condition
12h mn
a @Vm , Vn#50, ~2!
whereh mna (a51,2,3) are the ’t Hooft matrices satisfying the relations
h mna 52h nm
a , h mna h ms
b 5dabdns1eabch nsc . ~3!
a!Electronic mail: [email protected]!Electronic mail: [email protected]!Electronic mail: [email protected]!Electronic mail: [email protected]
0022-2488/97/38(11)/5833/7/$10.005833J. Math. Phys. 38 (11), November 1997 © 1997 American Institute of Physics
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The hyperka¨hler metric onX is given by g(Vm ,Vn)5fdmn with f5v(V0 ,V1 ,V2 ,V3) ~wechoose the sign ofv such thatf is positive.!. Then three complex structuresJa are expressed by
Ja~Vm!5h nma Vn . ~4!
Conversely the pair (Vm ,v) can be locally constructed from any hyperka¨hler structure (g,Ja)as follows:7,8 for a harmonic functiont and the volume formvg with respect tog, the vector fieldsVm are defined by
V05f gradt, ~5!
Va52Ja~V0!, a51,2,3, ~6!
where f5g(gradt,gradt)21. Then Vm preserve the volume formv5f21vg and satisfy thehalf-flat condition. We can choose the local coordinates such thatV05]/]t, and then~2! reducesto the Nahm equation,
]Va
]t5 1
2eabc@Vb , Vc#, ~7!
whereVa (a51,2,3) are volume-preserving vector fields on a three-ball. This form was given byAshtekar, Jacobson, and Smolin.5,6
In this paper we present an explicit construction of hyperka¨hler metrics based on~1! and~2!.We also give a new construction of anti-self-dual Yang–Mills connections on hyperka¨hler mani-folds generalizing the multi-instanton ansatz of ’t Hooft, Jackiw–Nohl–Rebbi,9 and furtherPopov.10–12
II. REDUCTIONS OF THE AJSMN EQUATIONS
We start with the construction of four vector fieldsVm corresponding to the Gibbons–Hawking metrics. We use the standard coordinates (x0,x1,x2,x3) and the volume formv5dx0
`dx1`dx2`dx3 for the underlying space–timeR4. Let f andc i ( i 51,2,3) be smooth functionsand define13
V05f]
]x0 , ~8!
Vi5]
]xi 1c i
]
]x0 . ~9!
Then the volume-preserving condition implies that the functionsf, c i are independent ofx0 andthe half-flat condition implies
* df5dc, ~10!
wherec5( i 513 c i dxi and * denotes the Hodge operator on the three-dimensional subspaceR3
5$(x1,x2,x3)%. These conditions are identical to the ansatz used by Gibbons and Hawking toconstruct hyperka¨hler metrics with a triholomorphic U~1! symmetry.14
Remark 2.1:The Gibbons–Hawking ansatz is characterized by the following Lie algebragGH.Let gGH be the Lie algebra generated by the Gibbons–Hawking vector fieldsVm . ThengGH isgiven by the extension of the two Abelian Lie algebras^] If]/]x0& and R3. ~] If denotes themultiple partial differentiation off with respect tox1, x2, andx3.! In other words the followingsequence is exact:
5834 Yasui et al.: Applications of the Ashtekar gravity to four
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0→K ] If]
]x0L→gGH→R3→0. ~11!
We note that ] If]/]x0& is a left D-module for
D5RF ]
]x1,
]
]x2,
]
]x3G Y XS ]
]x1D 2
1S ]
]x2D 2
1S ]
]x3D 2C. ~12!
We now describe an approach which allows us to obtain the vector fieldsVm satisfying theconditions~1! and~2! from ~anti! self-dual Yang–Mills connections of infinite-dimensional gaugegroups. LetS (n) (n51,2,3,4) be an-dimensional manifold equipped with a volume formv (n).Then we assume that the gauge Lie algebra is the Lie algebrasdi f f(S (n)) of all volume-preserving vector fields onS (n). Connections on Euclidean spaceR45$(x0,x1,x2,x3)% may beexplicitly expressed by one-forms onR4 valued in sdi f f(S (n)). We write these one-forms asAmdxm ~m50,1,2,3! and require the following conditions:
~a! Am areRn invariant with respect to the coordinates (x0,...,xn21).~b! Am are anti-self-dual connections, namely, the covariant differentiationsDm5]/]xm
1Am satisfy the equation
12hmn
a @Dm , Dn#50. ~13!
~c! Am (0<m<n21) are linearly independent at each point onS (n).We then define the four vector fieldsVm on S (n)3R42n as follows:
Vm5H Am , 0<m<n21,
Dm , n<m<3.~14!
These vector fields evidently preserve the volume formv5v (n)`dxn`•••`dx3 and satisfy thehalf-flat condition~2! and hence induce a hyperka¨hler metric onS (n)3R42n.
We note the previous Gibbons–Hawking vector fields can be obtained by applying the aboveconstruction to the caseS (1)5R. @In the cases ofS (4) andS (3), we simply recover the equations~1!, ~2!, and~7!.#
We now concentrate our attention on the explicit construction of hyperka¨hler metrics in thecase ofX5S3R2, where (S,vS) is a two-dimensional symplectic manifold. Let us assume thatthe R2-invariant connectionsAm are the Hamiltonian vector fieldsXf m
alongS3$(x2,x3)% asso-ciated with some functionsf m ~m50,1,2,3! on S3R2. Then~14! becomes
V05Xf 0, ~15!
V15Xf 1, ~16!
V25]
]x2 1Xf 2, ~17!
V35]
]x3 1Xf 3. ~18!
Thus our problem consists in finding the four functionsf m satisfying the anti-self-dual condition~13!:
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$ f 0 , f 1%1] f 3
]x22
] f 2
]x31$ f 2 , f 3%50, ~19!
$ f 0 , f 2%2] f 0
]x21
] f 1
]x31$ f 3 , f 1%50, ~20!
$ f 0 , f 3%2] f 0
]x32
] f 1
]x21$ f 1 , f 2%50. ~21!
We will not attempt to answer this question in any generality, but consider three examplesbelow.
~i! Let f m(m50,1,2) bex2 independent andf 350. Then~19!–~21! yield the Ward equation15
] f 0
]x35$ f 1 , f 2%,
] f 1
]x35$ f 2 , f 0%,
] f 2
]x35$ f 0 , f 1%, ~22!
where $,% denotes the Poisson bracket onS induced by the symplectic structurevS . Theseequations~but with Poisson brackets replaced by commutators of matrices of some finite-dimensional Lie algebra! arose in Nahm’s construction of monopole solutions in Yang–Millstheory.16,17
The group SL~2,R! acts onS5R2, H ~the complex upper half-plane! and SU~2! acts onS5S2 preserving the standard volume form for each case. So we can construct hyperka¨hler metricson S3R3I ~I is an open interval! from the solutions of the Nahm equation valued in sl~2,R! orsu~2!. For example, we can express such solutions by Jacobi elliptic functions as follows:
~a! sl~2,R!
f 05k sn ~x3,k!h0 , ~23!
f 15k cn ~x3,k!h1 , ~24!
f 25dn ~x3,k!h2 , ~25!
~b! su~2!
f 05ns ~x3,k!h0 , ~26!
f 15ds ~x3,k!h1 , ~27!
f 25cs ~x3,k!h2 , ~28!
wherekPR\$0% and
h05$h1 ,h2%, h152$h2 ,h0%, h252$h0 ,h1%,
h052$h1 ,h2%, h152$h2 ,h0%, h252$h0 ,h1%.
Some explicit solutions of the Nahm equations valued in finite-dimensional Lie algebras wereconstructed in Refs. 15, 18, and 19.
~ii ! We consider the hyperka¨hler metric with one rotational Killing symmetry preserving onecomplex structure but not triholomorphic.20 We takeS5R2 with the coordinates (y0,y1) andintroduce ay0-independent functionc(y1,x2,x3). Let us assume the form
5836 Yasui et al.: Applications of the Ashtekar gravity to four
J. Math. Phys., Vol. 38, No. 11, November 1997
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f 0522ec/2 cosy0
2, ~29!
f 152ec/2 siny0
2, ~30!
f 252Ey1 ]c
]x3 dy1, ~31!
f 35Ey1 ]c
]x2 dy1. ~32!
Then the functionsf m satisfy~19!–~21!, if c is a solution of the three-dimensional continual Todaequation:
S ]
]x2D 2
c1S ]
]x3D 2
c1S ]
]y1D 2
ec50. ~33!
This solution leads to a hyperka¨hler metric with the Killing vector field]/]y0, which is known asthe real heaven solution in the Pleban´ski formalism.21
~iii ! Using the solutions of both Nahm and Laplace equations, Popov presented a constructionof self-dual Yang–Mills connections onR4.10 We may apply the same method to our case, i.e.,R2-invariant Yang–Mills connections of the gauge Lie algebrasdi f f(S). Let Ta(t) (a51,2,3)andu(x2,x3) be solutions of the Nahm equation associated with the Lie algebrasdi f f(S) and theLaplace equation onR2, respectively:
]Ta
]t5
1
2eabc$T
b,Tc%, a,b,c51,2,3, ~34!
(m52
3 S ]
]xmD 2
u50. ~35!
Then we obtain the solutionsf m ~m50,1,2,3!:
f m5 (n52
3
hmna ]u
]xn Ta~u!. ~36!
Remark 2.2: We note that the ansatz for the vector fields~15!–~18! is similar to that in Refs.22–24, which should be regarded as the ansatz for the vector fields on a complex four-dimensionalmanifold, so that it is not so obvious whether the corresponding metrics satisfy the reality condi-tion. However, our construction manifestly satisfies such a condition.
III. YANG–MILLS INSTANTONS ON HYPERKA HLER MANIFOLDS
Now we present a construction of anti-self-dual Yang–Mills connections of arbitrary gaugeLie algebrag on a hyperka¨hler manifoldX. As mentioned above, Popov has obtained a formulafor self-dual Yang–Mills connections onR4. We generalize his construction to the case of four-dimensional hyperka¨hler manifolds by using Ashtekar variables.
SupposeX is a hyperka¨hler manifold expressed by linearly independent four-vector fieldsVm
and a volume formv as mentioned in~1! and ~2!.Proposition 3.1:The g-valued connectionA on X given by
5837Yasui et al.: Applications of the Ashtekar gravity to four
J. Math. Phys., Vol. 38, No. 11, November 1997
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A~Vm!5hmna ~Vnu!Ta~u! ~37!
satisfies the anti-self-dual condition if and only if the set ofg-valued functionsTa (a51,2,3) onR is a solution of the Nahm equation:
dTa
dt5
1
2eabc@Tb, Tc#, ~38!
and the functionu on X is a solution of the equation:
(m50
3
~VmVm!u50. ~39!
Proof: Using the identities~3!, we calculate the anti-self-dual part of the curvatureFmn
5F(Vm ,Vn) as follows:
1
2hmn
a Fmn51
2hmn
a $Vm„A~Vn!…2Vn„A~Vm!…2A~@Vm , Vn#!1@A~Vm!, A~Vn!#%
52(s
~VsVsu!Ta2(s
~Vsu!~Vsu!H dTa
dt2
1
2eabc@Tb, Tc#J
21
2A~ hmn
a @Vm , Vn#!.
It follows from ~2! and our assumptions~38! and ~39! that the above quantity vanishes.hRemark 3.2: The equation~39! is identical to the Laplace equation associated with the hyper-
kahler metric. So we can choose the harmonic functions asu in ~39!. For example, if the solutionsof ~2! are the Hamiltonian vector fields for functionsf m on a symplectic four-dimensional mani-fold, then f m are harmonic.
Remark 3.3: In the above proof we do not use the volume-preserving condition for the vectorfields Vm . Hence it is not necessary thatX is a four-dimensional hyperka¨hler manifold. Therestriction for ann-dimensional manifoldX we require is simply the existence of the linearlyindependentn vector fieldsVm such that12hmn
a @Vm , Vn#50 with the relations~3! for the matriceshmn
a .25
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5838 Yasui et al.: Applications of the Ashtekar gravity to four
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