Volume 80B, number I, 2 PHYSICS LETTERS 18 December 1978

A PHYSICAL PICTURE OF THE K3 GRAVITATIONAL INSTANTON

Don N. PAGE Department o f Applied Mathematics and Theoretical Physics, University o f Cambridge, Cambridge CB3 9EI¢, England

Received 5 September 1978

The self-dual metric on K3 (the manifold of Kummer's quartic surface) has 58 parameters. A physical interpretation of these is given in terms of an approximate construction of the K3 gravitational instanton from 16 Eguchi-Hanson metrics arranged in a lattice with certain identifications.

A particularly interesting gravitational instanton is Einstein metric on a K3 surface, the only compact simply-connected 4-manifold to admit a metric with self-dual curvature. K3 is defined to be a regular com- pact complex analytic nonsingular surface with va- nishing first Chern class. Kodaira [1 ] has proved every K3 surface is a deformation of a non-singular quartic (Kummer) surface in a projective 3-space (Cp3). Yau's proof [2] of Calabi's conjecture [3] implies that as a compact K/ihler manifold, the K3 surface admits a metric satisfying the vacuum Einstein equations. Hitchin [4] has shown that if K3 admitted an Einstein metric, it would be the only compact simply-connected 4-di- mensional Einstein manifold with Hirzebruch signature r equal to two-thirds the Euler number X. These results imply that K3 is unique in admitting a metric having

1 ~ D e f Rabcd = *Rabcd ~ 2 ~abef l" cd " (1)

(Usually K3 is taken to have self-dual Kghler form and negative r. This makes the metric anti-self dual. How- ever, to be consistent with other papers, I shall choose the opposite orientation in which the metric is self- dual.)

Next one might ask how many such self-dual metrics a K3 surface admits. More precisely, what are the num- ber of parameters of the general self-dual metric on K3? Hawking and Pope [5] answered this by spinor analysis showing there are 58 inequivalent zero-modes of the gravitational field (all self-dual), and Hitchin got the same number of parameters by arguments from algebraic geometry [6].

Can one get a physical picture of what these 58 pa- rameters represent? This letter will present such a picture in terms of an approximate construction of the self-dual metric on K3 [7]. The 58 zero modes in ref. [5 ] will be shown to arise naturally from the param- eters of this construction.

Gibbons and Pope [7] show how to construct K3 by starting with an infinite 4-dimensional lattice of flat hypercubes as unit cells. This manifold is made compact by identifying each unit cell, giving the hyper- torus T 4 , i.e. the points (x 1 + n 1 , x 2 + n 2 , x 3 + n 3 , x 4 + n 4) are identified for all sets of the four integers n a, a = 1,2, 3,4. Then points reflected through the origin are identified, i.e. the points x a and - x a are identified. This gives singularities at the 16 fixed points ( f l , f 2 , f 3 , f 4 ) , where e a c h f a is either 0 or ½. Finally, a neighbourhood of each singularity is excised and re- placed by an Eguchi-Hanson metric [8], which is a nonsingular self-dual metric that is asymptotically flat with antipodal points identified [9].

Because of the imperfect matching across the bound- ary of each excised region, the resulting metric will only be an approximation to a complete self-dual metric, but the approximation becomes arbitrarily good when each Eguchi-Hanson metric is given a sufficiently small scale parameter a. For f'mite values of the a's, some smoothing of the above construction is needed so that the curvature will be nonsingular and self-dual every- where, including the joins. There appears to be no ob- struction to this in principle, though no explicit self- dual metric on K3 has yet been found.

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Volume 80B, number 1,2 PHYSICS LETTERS 18 December 1978

Now let us see how the 58 parameters arise. The flat hypertorus T 4 has 10 parameters, the dot products of the 4 edges of the unit cell. Alternatively, if T 4 is given four coordinates x a which range from 0 to 1, then in the gauge in which the connection coefficients are zero, the 10 parameters are the 10 constant com- ponents of the metric tensor gab" Each Eguchi-Hanson metric inserted has two parameters describing its orien- tation, since only four of the six generators of the SO(4) rotation group are Killing vectors of this metric. When one adds the scale parameter a, one gets 3 param- eters for each of the 16 Eguchi-Hanson metrics, or 10 + 3 × 16 = 58 total parameters in the construction.

The 58 zero modes in ref. [5] are directly related to these 58 parameters. The pure trace zero mode corresponds to a constant infinitesimal scaling of gab. The other 57 zero modes are all tracelss. In 2-spinor notation they are of the form (JAB OgA'B', a n d in tensor notat ion they are

hab = FQ')ac J(i)c b , (2)

where ®lAB or F( / )ac is one of the r + 3 = 19 (labelled b y / ) self-dual harmonic 2-forms (Maxwell fields) and OetA,B , o r J(i)c b is one of the 3 (labelled by i) anti-self- dual harmonic 2-forms (covariantly constant in the self-dual metric). 16 of the F ' s are localised in the Eguchi-Hanson regions and have nonzero flux through the respective minimal 2-surfaces at the centres of those regions. The remaining 3 F ' s and the 3 J ' s thread through the hypertorus and are constant far from the Eguchi-Hanson regions. The 9 combinations of these in eq. (2) give constant h's that correspond to traceless perturbations o f gab outside the Eguch i - Hanson regions.

The remaining 48 combinations of the 16 localised F ' s with the 3 constant J ' s in eq. (2) correspond to infinitesimal rotations and dilations of the Eguch i - Hanson metrics. This can be seen explicitly by con- sidering combinations of the F and the 3J ' s for a single Eguchi-Hanson metric.

In terms of the orthonormal frame (12) in ref. [8] with e 0 reversed to give a self-dual metric, one can easily show

F = (a4 / r4 ) (eO^ e 3 + e 1 ^ e2 ) , (3)

j ( 1 ) = e 0 ^ e 1 _ e 2 ^ e 3 , (4)

j ( 2 ) = e 0 ^ e 2 _ e 3 ^ e 1 , (5)

j(3) = e 0 ^ e 3 e I ^ e 2 . (6)

Then the zero modes are

h(1) = F . J ( 1 ) = (a4/r 4) (e0® e 2 + e2® e 0

- e 1 ® e 3 - e 3 ® e l ) = 2~lla);b ) dx a d x b , (7)

h(2) = F . J ( 2 ) = (a4 /r 4) ( - e O ® e I _ e 1 ® e 0

- e2® e 3 - e30 e 2) = 2~12a);b ) dx a d x b , (8)

h (3) = F ' J ( 3 ) = (a4/r 4) ( _ e 0 ® e 0 + e 1 ® e 1

2 ~(3) ~dx a d x b (9) + e2 ® e2 - e3 ® e3) = (gab + ~(a;b)"

where

~(1) =- ~(1) d x a = _ ½ r(1 - a4/r4) 1/2 e 2 , (10)

i r(1 - a4/r4) 1/2 e 1 (11) ~(2) _= ~a(2) d x a = + ~

~(3) = ~a(3) dx a = + ½ r(1 - a4 / r4 ) l /2e 0 . (12)

Thus h (1) and h (2) are pure gauge transformations that correspond to infinitesimal rotations of the coordinate system generated by ~(1) and ~(2). In K3 these pertur- bations are rotations of the Eguchi-Hanson metric relative to the lattice orientation and so are not merely global gauge transformations. It is interesting to note that ~(1) is the vector potential for j (2) , a n d ~(2) is the vector potential for _ j ( 1 ) . (In fact, this ansatz was used to show h (1) and h (2) are pure gauge, contrary to expectations based on an incomplete version of the spinor analysis.)

The remaining zero mode in Eguchi-Hanson, h (3), is a dilation or infinitesimal scaling of the metric plus a radial coordinate transformation. It has the effect of changing the parameter a in the metric but leaving each point at the same value o f r 4 - a 4.

All 3 of these zero modes for the Eguchi-Hanson metric die off rapidly with radius. Hence if the scale parameters are chosen small enough for the Eguch i - Hanson metrics in the construction of K3, the zero modes will have negligible effect on the matching at the boundaries of the excised regions. Of course, the approximate construction does not prove there is an exact self-dual metric on K3 with 58 parameters, but since that is known from refs. [2], [5] and [6], the approximate construction does give a good picture of what these parameters represent.

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Volume 80B, number 1,2 PHYSICS LETTERS 18 December 1978

This paper was stimulated by discussions with S.W. Hawking, G.W. Gibbons, C.N. Pope and N. Hitchin.

References

[I] K. Kodaira, Am. J. Math. 86 (1964) 751. [2] S.-T. Yau, Commun. Pure Appl. Math. 31 (1978) 339. [3} E. Calabi, Proc. Intern. Congr. Math. (Amsterdam, 1954)

Vol. 2, p.206.

[4] N. Hitchin, J. Diff. Geom. 9 (1974) 435. [5] S.W. Hawking and C.N. Pope, Symmetry breaking by

instantons in supergravity, D.A.M.T.P. preprint. [6] N. Hitchin, private communication. [7] G.W. Gibbons and C.N. Pope, The positive action conjec-

ture and asymptotically euclidean metrics in quantum gravity, D.A.M.P.T. preprint.

[8] T. Eguchi and A.J. Hanson, Phys. Lett. 74B (1978) 249. [9] V.A. Belinskii, G.W. Gibbons, D.N. Page and C.N. Pope,

Phys. Lett. 76B (1978) 433.

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