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Instantons at angles
12 February 1998
Physics Letters B 419 Ž1998. 115–122
Instantons at angles G. Papadopoulos, A. Teschendorff DAMTP, UniÕersity of Cambridge, SilÕer Street, Cambridge CB3 9EW, UK Received 11 September 1997; revised 14 November 1997 Editor: P.V. Landshoff
Abstract We interpret a class of 4k-dimensional instanton solutions found by Ward, Corrigan, Goddard and Kent as four-dimensional instantons at angles. The superposition of each pair of four-dimensional instantons is associated with four angles which depend on some of the ADHM parameters. All these solutions are associated with the group SpŽ k . and are examples of Hermitian-Einstein connections on E 4 k . We show that the eight-dimensional solutions preserve 3r16 of the ten-dimensional N s 1 supersymmetry. We argue that under the correspondence between the BPS states of Yang-Mills theory and those of M-theory that arises in the context of Matrix models, the instantons at angles configuration corresponds to the longitudinal intersecting 5-branes on a string at angles configuration of M-theory. q 1998 Elsevier Science B.V.
1. Introduction Instantons are Yang-Mills configurations which are characterized by the first Pontryagin number, n , and obey the antiself-duality condition 1 wF s yF, where F is the field strength of a gauge potential A, the star is the Hodge star and ww s 1 for Euclidean signature 4-manifolds. One key property of this condition is that together with the Bianchi identities, =w M FN Lx s 0, it implies the field equations of the Yang-Mills theory. At first, a large class of instanton solutions was found using the ansatz of w1,2x. Later a
1
We have chosen the antiself-duality condition in the definition of the instanton but our results can be easily adapted to the self-duality one.
systematic classification of the solutions of the antiself-duality conditions was done in w3x, which has become known as the ADHM construction. Subsequently, many generalizations of the self-duality condition beyond four dimensions have been proposed and for some of them ADHM-like constructions have been found w4–6x. The understanding of non-perturbative aspects of various superstring theories involves the investigation of configurations which have the interpretation of intersecting branes. In the context of the effective theory, these are extreme solutions of various supergravity theories w7x. Such solutions are constructed using powerful superposition rules from the ‘elementary’ brane solutions that preserve 1r2 of the spacetime supersymmetry Žfor recent work see w8,9x and references within.. One feature of the intersecting
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G. Papadopoulos, A. Teschendorffr Physics Letters B 419 (1998) 115–122
branes is that the intersections can occur at angles in such a way that the configurations preserve a proportion of supersymmetry w10x. Solutions of various supergravity theories that have the interpretation of intersecting branes at angles have been found in w11–16x. More recently, a non-perturbative formulation of M-theory and string theory was proposed using the Matrix models w17–20x. These involve the study of certain Yang-Mills theories on a torus which is dual to the compactification torus of M-theory. The U-duality groups w21x of IIA superstring theory are naturally realised in the context of matrix models w22x. As a result, there is a correspondence between the BPS states of these Yang-Mills theories w23x and those of M-theory and superstring theories. In the low energy effective theory, the BPS states of the latter can be described by various classical solitonic solutions of various supergravity theories. This indicates that there is a correspondence between BPS states of Yang-Mills theories with the solitonic solutions of supergravity theories. Since in some cases, the solitonic BPS states of Yang-Mills theories can be described by classical solutions that carry the appropriate charges, the above argument suggests that there is a correspondence between solitonic solutions of supergravity theory with solitonic solutions of Yang-Mills theory. Although this correspondence is established for a compactification torus of finite size, we expect that the correspondence between the solutions of the Yang-Mills theory and supergravity theory will persist in the various limits where the torus becomes very large or very small. One example of such correspondence is the following: The longitudinal five-brane of the Matrix model is described either as the five-brane solution of D s 11 supergravity superposed with a pp-wave and compactified on a four-torus or as an instanton solution of the Yang-Mills theory Žsee for example w22,23x. on the dual four-torus. ŽThe pp-wave carries the longitudinal momentum of the five-brane.. Ignoring the size of the compactifying torus, one can simply say that there is a correspondence between longitudinal fivebranes and instantons. One purpose of this letter is to demonstrate the correspondence between BPS solutions of the Yang-Mills theories with BPS solutions of the supergravity theories with another example. In D s 11 supergravity theory, there is a configuration
with the interpretation of intersecting five-branes at angles w11x on a string. This configuration is associated with the group SpŽ2.; the proportion of supersymmetry preserved is directly related to the number of singlets in the decomposition of the spinor representation in eleven dimensions under SpŽ2.. Superposing this configuration with a pp-wave along the string directions corresponds to a Matrix theory configuration with the interpretation of intersecting longitudinal fiÕebranes at angles. Now using the correspondence between the longitudinal fivebranes and the Yang-Mills instantons mentioned above, one concludes that there should exist solutions of YangMills theory that have the interpretation of instantons at angles. Moreover since the instanton configuration must preserve the same proportion of supersymmetry as the supergravity one, it must also be associated with the group SpŽ2.. To superpose Yang-Mills BPS configurations, like for example Yang-Mills instantons or monopoles, at angles, we embed them as solutions of an appropriate higher-dimensional Yang-Mills theory. Such solutions will be localized on a subspace of the associated spacetime. Then given such a configuration we shall superpose it with another similar one which, however, may be localized in a different subspace. We require that the solutions of the Yang-Mills equations which have the interpretation of BPS configurations at angles to have the following properties: Ži. They solve a BPS-like equation which reduces to the standard BPS conditions of the original BPS configurations in the appropriate dimension Žii. if the solution which describes the superposition can be embedded in a supersymmetric theory, then it preserves a proportion of supersymmetry. In this letter, we shall describe the superposition of four-dimensional instantons. Since instantons are associated with four-planes, it is clear that the configuration that describes the superposition of two instantons at an angle is eight-dimensional; in general the configuration for k instantons lying on linearly independent four-planes is 4k-dimensional. It turns out that the appropriate BPS condition in 4 k dimensions necessary for the superposition of fourdimensional instantons has being found by Ward w5x. Here we shall describe how this BPS condition is associated with the group SpŽ k .. This allows us to interpret a class of the 4 k-dimensional instanton
G. Papadopoulos, A. Teschendorffr Physics Letters B 419 (1998) 115–122
solutions of Corrigan, Goddard and Kent w6x as four-dimensional instantons at angles. We shall find that there four angles associated with the superposition of each pair of two four-dimensional instantons which depend on the ADHM parameters. The eightdimensional solutions preserve 3r16 of the N s 1 ten-dimensional supersymmetry and correspond to the intersecting longitudinal fivebranes at angles configuration of the eleven-dimensional supergravity.
2. Tri-Hermitian-Einstein connections The BPS condition of Ward w5x is naturally associated with the group SpŽ k .. For this, we first observe that the curvature FM N s E M A N y E N A M q w A M , A N x ,
Ž 1.
of a connection A on E 4 k can be thought as an element of L2 ŽE 4 k . with respect to the space indices. Now L2 ŽE 4 k . s soŽ4 k . where soŽ4 k . is the Lie algebra of SO Ž4 k .. Decomposing soŽ4 k . under SpŽ k . P SpŽ1., we find that
L2 Ž E 4 k . s sp Ž k . [ sp Ž 1 . [ l20 m s 2 ,
Ž 2.
where we have set L1 ŽE 4 k . s l m s under the decomposition of one forms under SpŽ k . P SpŽ1., l20 is the symplectic traceless two-fold anti-symmetric irreducible representation of SpŽ k . and s 2 is the symmetric two-fold irreducible representation of SpŽ1.. The integrability condition of w5x can be summarized by saying that the components of the curvature along the last two subspaces in the decomposition Ž2. vanish. This implies that the space indices of the curvature take values in spŽ k .. It is convenient to write this BPS condition in a different way. Let Ir ;r s 1,2,3.4 be three complex structures in E 4 k that obey the algebra of imaginary unit quaternions, i.e. Ir Is s ydr s q e r st It .
Ž 3.
The BPS condition then becomes
Ž Ir .
P M
Ž Ir .
R
N FP R
a
b s FM N
a b
,
Ž 4.
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Žno summation over r ., i.e. the curvature is Ž1,1. with respect to all complex structures. We shall refer to the connections that satisfy this condition as triHermitian-Einstein connections. We remark that this is precisely the condition required by Ž4,0. supersymmetry on the curvature of connection of the Yang-Mills sector in two-dimensional sigma models 2 . It is straightforward to show that Ž4. reduces to the antiself-duality condition in four dimensions 3 and that it implies the Yang-Mills field equations in 4k dimensions. Moreover any solution of Ž4. is an example of an Hermitian-Einstein Žor Hermitian Yang-Mills. connection 4 Žsee for example w28,29x.. The latter are connections for which Ži. the curvature tensor is Ž1,1. with respect to a complex structure and Žii. g ab Fab s 0 ,
Ž 5.
where g is a hermitian metric. We remark that the above two conditions for Hermitian-Einstein connections on R 2 n imply that the non-vanishing components of the curvature are in suŽ n.. ŽThe condition Ž5. together with the Bianchi identity imply the Yang-Mills field equations.. The connections that satisfy Ž4. are Hermitian-Einstein with respect to three different complex structures.
3. One-angle solutions To interpret some of the solutions of w5x and w6x as four-dimensional instantons at angles, we first use an appropriate generalization of the ’t Hooft ansatz. For
2 For the application of the instantons of w6x in sigma models see w27,24–26x. 3 In fact it reduces to either the self-duality or the antiself-duality condition depending on the choice of orientation of the fourmanifold. 4 For k )1, the Hermitian-Einstein BPS condition is weaker than that of Ž4..
G. Papadopoulos, A. Teschendorffr Physics Letters B 419 (1998) 115–122
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this, we write E 4 k s E 4 m E k , i.e. we choose the coordinates
This is a harmonic-like equation and a solution is
y M 4 s x i m ; i s 1, . . . ,k ; m s 0,1,2,3 4 ,
fs1q
Ž 6.
r2
4k
on E . Then we introduce three complex structures on E 4 k
Ž Ir . i m
jn s Ir
m n
dij ,
Ž 7.
where Ir ; r s 1,2,34 are three complex structures on E 4 associated with the Kahler forms ¨
v 1 s dx 0 n dx 1 q dx 2 n dx 3 , v 2 s ydx 0 n dx 2 q dx 1 n dx 3 , v 3 s dx 0 n dx 3 q dx 1 n dx 2 ,
Ž 8.
respectively. Choosing the volume form as e s dx 0 n dx 1 n dx 2 n dx 3, these form a basis of self-dual 2-forms in E 4 . We also write the Euclidean metric on E 4 k as ds 2 s dmn d i j dx i m dx jn .
Ž 9.
Similarly, the curvature two-form is FM N s Fi m jn .
Ž 10 .
Writing Fi m jn s Fw mn x Ž i j. q FŽ mn . w i j x ,
Ž 11 .
we find that Ž4. implies that 1 2
Fw mn x Ž i j. e mn rs s yFw rs x Ž i j. ,
FŽ mn . w i j x s dmn f i j ,
Ž 12 .
where f i j is a k = k antisymmetric matrix. Next, we shall further assume that the gauge group is SUŽ2. s SpŽ1. and write the ansatz A i m s i Sm nE i n log f Ž x . ,
Ž 13 .
where the matrix two-form Smn is the ’t Hooft tensor, i.e.
Smn s 12 v r mn s r ,
Ž 14 .
and sr ;r s 1,2,34 are the Pauli matrices. After some computation, we find that the field strength F of A in Ž13. satisfies the first equation in Ž12. provided that 1
f
Ei P Ej f s 0 .
Ž 15 .
Ž pi x i m y a m .
2
,
Ž 16 .
where p4 s Ž p 1 , p 2 , . . . , p k . are k real numbers, a m is the centre of the harmonic function and r 2 is the parameter associated with the size of the instanton. More general solutions can be obtained by a linear superposition of the above solution for different choices of p4 and a m leading to
fs1q Ý
Ý
p4 a
r a2 Ž p 4 .
Ž pi x i m y a m Ž p 4 . .
2
.
Ž 17 .
Finally, it is straightforward to verify that the curvature of the connection Ž13. with f given as in Ž17., satisfies Ž4.. To investigate the properties of the above solutions, let us suppose that the harmonic function f involves only one p4 as in Ž16.. In this case it is always possible, after a coordinate transformation, to set p4 s 1,0, . . . ,04 . Then f becomes a harmonic function on the ‘first’ copy of E 4 in E 4 k . The resulting configuration is that derived from the ansatz of w1,2x on E 4 . In particular, the instanton number is equal to the number of centres a m4 of the harmonic function. In this case the singularities at the centres of the harmonic function can be removed with a suitable choice of a gauge transformation w30x. Next, let us suppose that our solution involves two linearly independent vectors p4 , say p4 s p4 and p4 s q4 . If we take < pi x i < ™ `, then the configuration reduces to the solution associated with a harmonic function that involves only q4 . As we have explained above, this is the instanton solution of w1,2x on an appropriate 4-plane in E 4 k . Similarly if we take < qi x i < ™ `, then the configuration reduces to the solution associated with the harmonic function that involves only p4 . We can therefore interpret this solution as the superposition of two four-dimensional instantons at angles in E 4 k . The angle is cos u s
pPq < p< < q<
.
Ž 18 .
As it may have been expected from the non-linearity of the gauge potentials in Ž4., this superposition is
G. Papadopoulos, A. Teschendorffr Physics Letters B 419 (1998) 115–122
non-linear even in the case for which p4 and q4 are orthogonal. The solution appears to be singular at the positions of the harmonic function. For simplicity let us consider two instantons at angles with instanton number one. In this case we have two centres. The singular set is defined by the two codimension-fourplanes im
m
pi x y a s 0 , qi x i m y b m s 0 ,
Ž 19 .
8
in E . We have investigated the singularity structure of the solution when p4 and q4 are orthogonal using a method similar to w30x. We have found that these singularities can be removed everywhere apart from the intersection of the two singular sets. A similar observation has been made in w6x. To conclude, let us briefly consider the general case. The number of four dimensional instantons involved in the configuration is equal to the number of linearly independent choices for p4 . Their instanton number is equal to the number of centres associated with each choice of p4 . The general solution describes the superposition of these four-dimensional instantons at angles.
4. Four-angle solutions The ADHM ansatz of w6x describes solutions with more parameters than those found in the previous section using the ’t Hooft ansatz. Here we shall interpret some of these parameters as angles. In fact we shall find that there are four angles between each pair of planes associated with these solutions all determined in terms of ADHM parameters. For simplicity, we shall take SpŽ1. as the gauge group. The ADHM ansatz is A s Õ†dÕ ,
Ž 20 .
where Õ is an Ž l q 1. = 1 matrix of quaternions normalized as Õ† Õ s 1, and l is an integer which is identified with the ‘total’ instanton number Žsee w6x.. The matrix Õ satisfies the condition Õ†D ' Õ† Ž a q bi x i . s 0 ,
Ž 21 .
where a,bi are Ž l q 1. = l matrices of quaternions; we have arranged the 4k coordinates into quaternions x i ;i s 1, PPP ,k 4 ŽE 4 k s H k .. The connection Ž20.
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satisfies the condition Ž4. provided that the matrices a†a,b†i bj ,a† bi are symmetric as matrices of quaternions. A convenient choice of matrices a,bi leads to
Ž D . 0 n s ya0 l n , Ž D . n m s Ž pni† x i y a n . dn m ,
Ž 22 .
where l n ;n s 1, . . . , l 4 are real numbers, and pn i ;n s 1, . . . , l ;i s 1, . . . ,k 4 and a0 ,a n ;n s 1, . . . , l 4 are quaternions. The solutions discussed in the previous section using the ’t Hooft ansatz correspond to choosing pn i ;n s 1, . . . , l ,i s 1, . . . ,k 4 to be real numbers. For Õ, we find a 0 y1 f , Ž Õ.0 sy < a0 < y1
†
Ž Õ . n s y< a0 < l n Ž x i . pn i y a†n
fy1 ,
Ž 23 .
where f is the normalization factor f 2 s 1 q < a0 < 2
l2n
l
Ý ns1
< pn† i x i y a n < 2
.
Ž 24 .
We remark that the sum in the normalization factor is over the centres a n ;n s 1, . . . , l 4 . It is clear from the form of the normalization factor that this solution is associated with l codimension-four-planes in E 4 k . The equations of these planes are pn† i x i y a n s 0 ,
Ž 25 .
for n s 1, . . . , l . The four normalized normal vectors to these planes are 1 Nn s
< pn <
Ž pni .
†
E E xi
,
Ž 26 .
in quaternionic notation, where < pn < 2 s d i j Ž pni . † pnj. Next let us consider two such codimension-four-plane determined say by p s p 1 and q s p 2 and with associated normal vectors N and N˜ , respectively. The angles associated with these planes are given by the inner product of their normal vectors, so †
†
cos Ž u . s Ž N˜ . N s
di j Ž q i . p j , < p< < q<
Ž 27 .
where cosŽ u . is a quaternion in the obvious notation. There are four angles because cosŽ u . has four components. These four angles are all different for a
G. Papadopoulos, A. Teschendorffr Physics Letters B 419 (1998) 115–122
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generic choice of p,q and independent from the choice of gauge fixing for the residual symmetries of the ADHM construction w6x. It is worth mentioning that each pair of four-dimensional instantons are superposed at SpŽ2. angles. For this we observe that the normal vectors Ž26. of the two codimensionfour-planes span an eight-dimensional subspace in E 4 k and their coefficients are quaternions. So, they can be related with an SpŽ2. rotation Žafter choosing a suitable basis in E 4 k .. As a result the two fourplanes spanned by the normal vectors in E 8 are at SpŽ2. angles. Such four-planes are parameterized by the four-dimensional coset space SpŽ2.rSO Ž4.; this explains the presence of four angles in Ž27.. It also suggests that there may be a generalization of the solutions of w11x which describe the superposition of five-branes and KK-monopoles depending on four angles. A naive counting of the dimension of the moduli of the 4k-dimensional solutions which takes account of the dimension of moduli of 4-dimensional instantons involved in the superposition and their associated angles does not reproduce the dimension of the moduli space in w6x. However, all their solutions have a decaying behaviour at large distances consistent with the interpretation that they are four-dimensional instantons at angles.
3r32 of spacetime supersymmetry. This is in agreement with the proportion of supersymmetry preserved by the intersecting five-branes on a string at angles and superposed with a pp-wave solution of D s 11 supergravity. However there is a difference. In the case of two orthogonal intersecting five-branes on a string with a pp-wave superposed the proportion of the supersymmetry preserved is 1r8 but this is not the case for the configuration of two orthogonal instantons. To see this, first observe that the solutions of the previous two sections will preserve 1r4 of the N s 1 ten-dimensional supersymmetry if the components of the curvature are in the spŽ1. [ spŽ1. subalgebra of spŽ2.. However this is not so because a direct calculation reveals that the mixed components Fm1, n 2 of the curvature tensor do not vanish. Further we remark that one might have thought that it is possible to superpose two four-dimensional instantons Bm and Ca with gauge group G localised at two orthogonal four-planes in E 8 by simply setting A s Ž B,C . and requiring that the gauge group of the new connection is again G. However, this is not a solution of the BPS condition Ž4. for a non-abelian gauge group in eight-dimensions. This is because Ž4. is non-linear in the connection. However the above linear superposition is a solution if the gauge groups of B and C are treated independently, i.e. if the gauge group of A is G = G.
5. Supersymmetry 6. Concluding remarks Among the above Yang-Mills configurations on E 4 k only the instantons at angles in E 8 and the instantons on E 4 are solutions of ten-dimensional supersymmetric Yang-Mills theory preserving a proportion of supersymmetry. The supersymmetry condition is FM N G
MN
es0 ,
Ž 28 .
where e is the supersymmetry parameter. In the four-dimensional case the instantons preserve 1r2 of the supersymmetry. In the eight-dimensional case condition Ž4. implies that FM N is in spŽ2., an argument similar to that in w11x can be used to show that the instantons at angles in E 8 will preserve 3r16 of supersymmetry. As a solution of the Matrix theory, the eight-dimensional instantons at angles preserve
We have interpreted some of the 4k-dimensional instantons of w5x and w6x as superposition of four-dimensional instantons at angles. We have shown that there four angles associated with the superposition of two four-dimensional instantons which depend on the ADHM parameters. These solutions are examples of Hermitian-Einstein connections in 4k dimensions and the eight-dimensional ones preserve 3r16 of the N s 1 ten-dimensional supersymmetry. Because of the close relation between the self-duality condition and the BPS equations for magnetic monopoles we expect that one can generalize the above construction to find solutions of Yang-Mills equations on E 3 k which have the interpretation of monopoles at angles. In addition, the Yang-Mills configurations of
G. Papadopoulos, A. Teschendorffr Physics Letters B 419 (1998) 115–122
w6x and the HKT geometries described in w31x combined give new examples of two-dimensional supersymmetric sigma models with Ž4,0. supersymmetry. These provide consistent backgrounds for the propagation of the heterotic string w32x. Another application of the instantons of w6x is in the context of D-branes. In IIA theory, they are associated with the D-brane bound state of a 0-brane within a 4-brane within an 8-brane, and in the IIB theory they are associated with the D-brane bound state of a D-string within a D-5-brane within a D-9-brane. There are many other ways to superpose two or more four-dimensional instantons. Here we have described the superposition using the condition that the components of the curvature F of the Yang-Mills connection are in spŽ k .. Another option is to use the Hermitian-Einstein condition associated with suŽ2 k . which we have already mentioned in section two. In eight dimensions this will lead to configurations preserving 1r8 of the N s 1 ten-dimensional supersymmetry. In addition in eight dimensions one can allow the components of F to lie in spinŽ7.. This will result in superpositions of instantons preserving 1r16 of the N s 1 ten-dimensional supersymmetry. In fact, some spinŽ7. instanton solutions have already been found w33x. It will be of interest to see whether they can be interpreted as four-dimensional instantons at angles. Since SpŽ2. is a subgroup of SpinŽ7., the solutions of w6x are also examples of spinŽ7. instantons, albeit of a particular type. There are more possibilities in lower dimensions, for example the G 2 instantons in seven dimensions w34,35x which may have a similar interpretation. 7. Note added While we were revising our work, the paper w36x by N. Ohta and J-G. Zhou appeared in which the relation between supergravity configurations and Yang-Mills ones is also discussed. However, they use constant Yang-Mills configurations on the eight torus. Acknowledgements We would like to thank P. Goddard and A. Kent for helpful discussions. A.T. thanks PPARC for a
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studentship. G.P. is supported by a University Research Fellowship from the Royal Society. References w1x A.A. Belavin, A.M. Polyakov, A.S. Schwartz, Yu.S. Tyupkin, Phys. Lett. B 59 Ž1975. 85. w2x G. ’t Hooft, Phys. Rev. D 14 Ž1976. 3432. w3x M.F. Atiyah, V.G. Drinfeld, N.J. Hitchin, Yu.I. Manin, Phys. Lett. A 65 Ž1978. 185. w4x E. Corrigan, C. Devchand, D.B. Fairlie, J. Nuyts, Nucl. Phys. B 214 Ž1983. 452. w5x R.S. Ward, Nucl. Phys. B 236 Ž1984. 381. w6x E. Corrigan, P. Goddard, A. Kent, Commun. Math. Phys. 100 Ž1985. 1. w7x G. Papadopoulos, P.K. Townsend, Phys. Lett. B 380 Ž1996. 273. w8x E. Bergshoeff, M. de Roo, E. Eyras, B. Janssen, J.P. van der Schaar, Multiple Intersections of D-branes and M-branes, hep-thr9612095. w9x H. Lu, C.N. Pope, T.A. Tran, K.W. Xu, Classification of p-branes, Nuts, Waves and Intersections, hep-thr9708055. w10x M. Berkooz, M.R. Douglas, R.G. Leigh, Nucl. Phys. B 480 Ž1996. 265. w11x J.P. Gauntlett, G.W. Gibbons, G. Papadopoulos, P.K. Townsend, Hyper-Kahler manifolds and multiply intersecting ¨ branes, Nucl. Phys. Ž1997., to appear: hep-thr9702202. w12x K. Behrndt, M. Cvetic, ˇ Phys. Rev. D 56 Ž1997. 1188. w13x J.C. Breckenridge, G. Michaud, R.C. Myers, New Angles on Branes, hep-thr9703041. w14x M. Costa, M. Cvetic, ˇ Non-Threshold D-Brane Bound States and Black Holes with Non-Zero Entropy, hep-thr9703204. w15x V. Balasubramanian, F. Larsen, R.G. Leigh, Branes at Angles and Black Holes, hep-thr9704143. w16x G. Michaud, R.C. Myers, Hermitian D-Brane Solutions, hepthr9705079. w17x T. Banks, W. Fischer, S.H. Shenker, L. Susskind, M Theory as a Matrix Model, hep-thr9610043. w18x L. Motl, Proposals on Nonperturbative Superstring Interactions, hep-thr9701025. w19x T. Banks, N. Seiberg, Strings from Matrices, hepthr9702187. w20x R. Dijkgraaf, E. Verlinde, H. Verlinde, Matrix String Theory, hep-thr9703030. w21x C.M. Hull, P.K. Townsend, Nucl. Phys. B 438 Ž1995. 109. w22x O.J. Ganor, S. Ramgoolam, W. Taylor IV, Branes, Fluxes and Duality in M Žatrix.-Theory, hep-thr9611202. w23x T. Banks, N. Seiberg, S. Shenker, Branes from Matrices, hep-thr9612157. w24x S.J. Gates, C.M. Hull, M. Rocek, ˇ Nucl. Phys. B 248 Ž1984. 157. w25x P.S. Howe, G. Papadopoulos, Nucl. Phys. B 289 Ž1987. 264; Class. Quantum Grav. 5 Ž1988. 1647. w26x P.S. Howe, G. Papadopoulos, Phys. Lett. B 379 Ž1996. 80. w27x N.D. Lambert, Nucl. Phys. B 477 Ž1996. 141, hepthr9605010.
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Physics Letters B 419 Ž1998. 115–122
Instantons at angles G. Papadopoulos, A. Teschendorff DAMTP, UniÕersity of Cambridge, SilÕer Street, Cambridge CB3 9EW, UK Received 11 September 1997; revised 14 November 1997 Editor: P.V. Landshoff
Abstract We interpret a class of 4k-dimensional instanton solutions found by Ward, Corrigan, Goddard and Kent as four-dimensional instantons at angles. The superposition of each pair of four-dimensional instantons is associated with four angles which depend on some of the ADHM parameters. All these solutions are associated with the group SpŽ k . and are examples of Hermitian-Einstein connections on E 4 k . We show that the eight-dimensional solutions preserve 3r16 of the ten-dimensional N s 1 supersymmetry. We argue that under the correspondence between the BPS states of Yang-Mills theory and those of M-theory that arises in the context of Matrix models, the instantons at angles configuration corresponds to the longitudinal intersecting 5-branes on a string at angles configuration of M-theory. q 1998 Elsevier Science B.V.
1. Introduction Instantons are Yang-Mills configurations which are characterized by the first Pontryagin number, n , and obey the antiself-duality condition 1 wF s yF, where F is the field strength of a gauge potential A, the star is the Hodge star and ww s 1 for Euclidean signature 4-manifolds. One key property of this condition is that together with the Bianchi identities, =w M FN Lx s 0, it implies the field equations of the Yang-Mills theory. At first, a large class of instanton solutions was found using the ansatz of w1,2x. Later a
1
We have chosen the antiself-duality condition in the definition of the instanton but our results can be easily adapted to the self-duality one.
systematic classification of the solutions of the antiself-duality conditions was done in w3x, which has become known as the ADHM construction. Subsequently, many generalizations of the self-duality condition beyond four dimensions have been proposed and for some of them ADHM-like constructions have been found w4–6x. The understanding of non-perturbative aspects of various superstring theories involves the investigation of configurations which have the interpretation of intersecting branes. In the context of the effective theory, these are extreme solutions of various supergravity theories w7x. Such solutions are constructed using powerful superposition rules from the ‘elementary’ brane solutions that preserve 1r2 of the spacetime supersymmetry Žfor recent work see w8,9x and references within.. One feature of the intersecting
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branes is that the intersections can occur at angles in such a way that the configurations preserve a proportion of supersymmetry w10x. Solutions of various supergravity theories that have the interpretation of intersecting branes at angles have been found in w11–16x. More recently, a non-perturbative formulation of M-theory and string theory was proposed using the Matrix models w17–20x. These involve the study of certain Yang-Mills theories on a torus which is dual to the compactification torus of M-theory. The U-duality groups w21x of IIA superstring theory are naturally realised in the context of matrix models w22x. As a result, there is a correspondence between the BPS states of these Yang-Mills theories w23x and those of M-theory and superstring theories. In the low energy effective theory, the BPS states of the latter can be described by various classical solitonic solutions of various supergravity theories. This indicates that there is a correspondence between BPS states of Yang-Mills theories with the solitonic solutions of supergravity theories. Since in some cases, the solitonic BPS states of Yang-Mills theories can be described by classical solutions that carry the appropriate charges, the above argument suggests that there is a correspondence between solitonic solutions of supergravity theory with solitonic solutions of Yang-Mills theory. Although this correspondence is established for a compactification torus of finite size, we expect that the correspondence between the solutions of the Yang-Mills theory and supergravity theory will persist in the various limits where the torus becomes very large or very small. One example of such correspondence is the following: The longitudinal five-brane of the Matrix model is described either as the five-brane solution of D s 11 supergravity superposed with a pp-wave and compactified on a four-torus or as an instanton solution of the Yang-Mills theory Žsee for example w22,23x. on the dual four-torus. ŽThe pp-wave carries the longitudinal momentum of the five-brane.. Ignoring the size of the compactifying torus, one can simply say that there is a correspondence between longitudinal fivebranes and instantons. One purpose of this letter is to demonstrate the correspondence between BPS solutions of the Yang-Mills theories with BPS solutions of the supergravity theories with another example. In D s 11 supergravity theory, there is a configuration
with the interpretation of intersecting five-branes at angles w11x on a string. This configuration is associated with the group SpŽ2.; the proportion of supersymmetry preserved is directly related to the number of singlets in the decomposition of the spinor representation in eleven dimensions under SpŽ2.. Superposing this configuration with a pp-wave along the string directions corresponds to a Matrix theory configuration with the interpretation of intersecting longitudinal fiÕebranes at angles. Now using the correspondence between the longitudinal fivebranes and the Yang-Mills instantons mentioned above, one concludes that there should exist solutions of YangMills theory that have the interpretation of instantons at angles. Moreover since the instanton configuration must preserve the same proportion of supersymmetry as the supergravity one, it must also be associated with the group SpŽ2.. To superpose Yang-Mills BPS configurations, like for example Yang-Mills instantons or monopoles, at angles, we embed them as solutions of an appropriate higher-dimensional Yang-Mills theory. Such solutions will be localized on a subspace of the associated spacetime. Then given such a configuration we shall superpose it with another similar one which, however, may be localized in a different subspace. We require that the solutions of the Yang-Mills equations which have the interpretation of BPS configurations at angles to have the following properties: Ži. They solve a BPS-like equation which reduces to the standard BPS conditions of the original BPS configurations in the appropriate dimension Žii. if the solution which describes the superposition can be embedded in a supersymmetric theory, then it preserves a proportion of supersymmetry. In this letter, we shall describe the superposition of four-dimensional instantons. Since instantons are associated with four-planes, it is clear that the configuration that describes the superposition of two instantons at an angle is eight-dimensional; in general the configuration for k instantons lying on linearly independent four-planes is 4k-dimensional. It turns out that the appropriate BPS condition in 4 k dimensions necessary for the superposition of fourdimensional instantons has being found by Ward w5x. Here we shall describe how this BPS condition is associated with the group SpŽ k .. This allows us to interpret a class of the 4 k-dimensional instanton
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solutions of Corrigan, Goddard and Kent w6x as four-dimensional instantons at angles. We shall find that there four angles associated with the superposition of each pair of two four-dimensional instantons which depend on the ADHM parameters. The eightdimensional solutions preserve 3r16 of the N s 1 ten-dimensional supersymmetry and correspond to the intersecting longitudinal fivebranes at angles configuration of the eleven-dimensional supergravity.
2. Tri-Hermitian-Einstein connections The BPS condition of Ward w5x is naturally associated with the group SpŽ k .. For this, we first observe that the curvature FM N s E M A N y E N A M q w A M , A N x ,
Ž 1.
of a connection A on E 4 k can be thought as an element of L2 ŽE 4 k . with respect to the space indices. Now L2 ŽE 4 k . s soŽ4 k . where soŽ4 k . is the Lie algebra of SO Ž4 k .. Decomposing soŽ4 k . under SpŽ k . P SpŽ1., we find that
L2 Ž E 4 k . s sp Ž k . [ sp Ž 1 . [ l20 m s 2 ,
Ž 2.
where we have set L1 ŽE 4 k . s l m s under the decomposition of one forms under SpŽ k . P SpŽ1., l20 is the symplectic traceless two-fold anti-symmetric irreducible representation of SpŽ k . and s 2 is the symmetric two-fold irreducible representation of SpŽ1.. The integrability condition of w5x can be summarized by saying that the components of the curvature along the last two subspaces in the decomposition Ž2. vanish. This implies that the space indices of the curvature take values in spŽ k .. It is convenient to write this BPS condition in a different way. Let Ir ;r s 1,2,3.4 be three complex structures in E 4 k that obey the algebra of imaginary unit quaternions, i.e. Ir Is s ydr s q e r st It .
Ž 3.
The BPS condition then becomes
Ž Ir .
P M
Ž Ir .
R
N FP R
a
b s FM N
a b
,
Ž 4.
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Žno summation over r ., i.e. the curvature is Ž1,1. with respect to all complex structures. We shall refer to the connections that satisfy this condition as triHermitian-Einstein connections. We remark that this is precisely the condition required by Ž4,0. supersymmetry on the curvature of connection of the Yang-Mills sector in two-dimensional sigma models 2 . It is straightforward to show that Ž4. reduces to the antiself-duality condition in four dimensions 3 and that it implies the Yang-Mills field equations in 4k dimensions. Moreover any solution of Ž4. is an example of an Hermitian-Einstein Žor Hermitian Yang-Mills. connection 4 Žsee for example w28,29x.. The latter are connections for which Ži. the curvature tensor is Ž1,1. with respect to a complex structure and Žii. g ab Fab s 0 ,
Ž 5.
where g is a hermitian metric. We remark that the above two conditions for Hermitian-Einstein connections on R 2 n imply that the non-vanishing components of the curvature are in suŽ n.. ŽThe condition Ž5. together with the Bianchi identity imply the Yang-Mills field equations.. The connections that satisfy Ž4. are Hermitian-Einstein with respect to three different complex structures.
3. One-angle solutions To interpret some of the solutions of w5x and w6x as four-dimensional instantons at angles, we first use an appropriate generalization of the ’t Hooft ansatz. For
2 For the application of the instantons of w6x in sigma models see w27,24–26x. 3 In fact it reduces to either the self-duality or the antiself-duality condition depending on the choice of orientation of the fourmanifold. 4 For k )1, the Hermitian-Einstein BPS condition is weaker than that of Ž4..
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this, we write E 4 k s E 4 m E k , i.e. we choose the coordinates
This is a harmonic-like equation and a solution is
y M 4 s x i m ; i s 1, . . . ,k ; m s 0,1,2,3 4 ,
fs1q
Ž 6.
r2
4k
on E . Then we introduce three complex structures on E 4 k
Ž Ir . i m
jn s Ir
m n
dij ,
Ž 7.
where Ir ; r s 1,2,34 are three complex structures on E 4 associated with the Kahler forms ¨
v 1 s dx 0 n dx 1 q dx 2 n dx 3 , v 2 s ydx 0 n dx 2 q dx 1 n dx 3 , v 3 s dx 0 n dx 3 q dx 1 n dx 2 ,
Ž 8.
respectively. Choosing the volume form as e s dx 0 n dx 1 n dx 2 n dx 3, these form a basis of self-dual 2-forms in E 4 . We also write the Euclidean metric on E 4 k as ds 2 s dmn d i j dx i m dx jn .
Ž 9.
Similarly, the curvature two-form is FM N s Fi m jn .
Ž 10 .
Writing Fi m jn s Fw mn x Ž i j. q FŽ mn . w i j x ,
Ž 11 .
we find that Ž4. implies that 1 2
Fw mn x Ž i j. e mn rs s yFw rs x Ž i j. ,
FŽ mn . w i j x s dmn f i j ,
Ž 12 .
where f i j is a k = k antisymmetric matrix. Next, we shall further assume that the gauge group is SUŽ2. s SpŽ1. and write the ansatz A i m s i Sm nE i n log f Ž x . ,
Ž 13 .
where the matrix two-form Smn is the ’t Hooft tensor, i.e.
Smn s 12 v r mn s r ,
Ž 14 .
and sr ;r s 1,2,34 are the Pauli matrices. After some computation, we find that the field strength F of A in Ž13. satisfies the first equation in Ž12. provided that 1
f
Ei P Ej f s 0 .
Ž 15 .
Ž pi x i m y a m .
2
,
Ž 16 .
where p4 s Ž p 1 , p 2 , . . . , p k . are k real numbers, a m is the centre of the harmonic function and r 2 is the parameter associated with the size of the instanton. More general solutions can be obtained by a linear superposition of the above solution for different choices of p4 and a m leading to
fs1q Ý
Ý
p4 a
r a2 Ž p 4 .
Ž pi x i m y a m Ž p 4 . .
2
.
Ž 17 .
Finally, it is straightforward to verify that the curvature of the connection Ž13. with f given as in Ž17., satisfies Ž4.. To investigate the properties of the above solutions, let us suppose that the harmonic function f involves only one p4 as in Ž16.. In this case it is always possible, after a coordinate transformation, to set p4 s 1,0, . . . ,04 . Then f becomes a harmonic function on the ‘first’ copy of E 4 in E 4 k . The resulting configuration is that derived from the ansatz of w1,2x on E 4 . In particular, the instanton number is equal to the number of centres a m4 of the harmonic function. In this case the singularities at the centres of the harmonic function can be removed with a suitable choice of a gauge transformation w30x. Next, let us suppose that our solution involves two linearly independent vectors p4 , say p4 s p4 and p4 s q4 . If we take < pi x i < ™ `, then the configuration reduces to the solution associated with a harmonic function that involves only q4 . As we have explained above, this is the instanton solution of w1,2x on an appropriate 4-plane in E 4 k . Similarly if we take < qi x i < ™ `, then the configuration reduces to the solution associated with the harmonic function that involves only p4 . We can therefore interpret this solution as the superposition of two four-dimensional instantons at angles in E 4 k . The angle is cos u s
pPq < p< < q<
.
Ž 18 .
As it may have been expected from the non-linearity of the gauge potentials in Ž4., this superposition is
G. Papadopoulos, A. Teschendorffr Physics Letters B 419 (1998) 115–122
non-linear even in the case for which p4 and q4 are orthogonal. The solution appears to be singular at the positions of the harmonic function. For simplicity let us consider two instantons at angles with instanton number one. In this case we have two centres. The singular set is defined by the two codimension-fourplanes im
m
pi x y a s 0 , qi x i m y b m s 0 ,
Ž 19 .
8
in E . We have investigated the singularity structure of the solution when p4 and q4 are orthogonal using a method similar to w30x. We have found that these singularities can be removed everywhere apart from the intersection of the two singular sets. A similar observation has been made in w6x. To conclude, let us briefly consider the general case. The number of four dimensional instantons involved in the configuration is equal to the number of linearly independent choices for p4 . Their instanton number is equal to the number of centres associated with each choice of p4 . The general solution describes the superposition of these four-dimensional instantons at angles.
4. Four-angle solutions The ADHM ansatz of w6x describes solutions with more parameters than those found in the previous section using the ’t Hooft ansatz. Here we shall interpret some of these parameters as angles. In fact we shall find that there are four angles between each pair of planes associated with these solutions all determined in terms of ADHM parameters. For simplicity, we shall take SpŽ1. as the gauge group. The ADHM ansatz is A s Õ†dÕ ,
Ž 20 .
where Õ is an Ž l q 1. = 1 matrix of quaternions normalized as Õ† Õ s 1, and l is an integer which is identified with the ‘total’ instanton number Žsee w6x.. The matrix Õ satisfies the condition Õ†D ' Õ† Ž a q bi x i . s 0 ,
Ž 21 .
where a,bi are Ž l q 1. = l matrices of quaternions; we have arranged the 4k coordinates into quaternions x i ;i s 1, PPP ,k 4 ŽE 4 k s H k .. The connection Ž20.
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satisfies the condition Ž4. provided that the matrices a†a,b†i bj ,a† bi are symmetric as matrices of quaternions. A convenient choice of matrices a,bi leads to
Ž D . 0 n s ya0 l n , Ž D . n m s Ž pni† x i y a n . dn m ,
Ž 22 .
where l n ;n s 1, . . . , l 4 are real numbers, and pn i ;n s 1, . . . , l ;i s 1, . . . ,k 4 and a0 ,a n ;n s 1, . . . , l 4 are quaternions. The solutions discussed in the previous section using the ’t Hooft ansatz correspond to choosing pn i ;n s 1, . . . , l ,i s 1, . . . ,k 4 to be real numbers. For Õ, we find a 0 y1 f , Ž Õ.0 sy < a0 < y1
†
Ž Õ . n s y< a0 < l n Ž x i . pn i y a†n
fy1 ,
Ž 23 .
where f is the normalization factor f 2 s 1 q < a0 < 2
l2n
l
Ý ns1
< pn† i x i y a n < 2
.
Ž 24 .
We remark that the sum in the normalization factor is over the centres a n ;n s 1, . . . , l 4 . It is clear from the form of the normalization factor that this solution is associated with l codimension-four-planes in E 4 k . The equations of these planes are pn† i x i y a n s 0 ,
Ž 25 .
for n s 1, . . . , l . The four normalized normal vectors to these planes are 1 Nn s
< pn <
Ž pni .
†
E E xi
,
Ž 26 .
in quaternionic notation, where < pn < 2 s d i j Ž pni . † pnj. Next let us consider two such codimension-four-plane determined say by p s p 1 and q s p 2 and with associated normal vectors N and N˜ , respectively. The angles associated with these planes are given by the inner product of their normal vectors, so †
†
cos Ž u . s Ž N˜ . N s
di j Ž q i . p j , < p< < q<
Ž 27 .
where cosŽ u . is a quaternion in the obvious notation. There are four angles because cosŽ u . has four components. These four angles are all different for a
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generic choice of p,q and independent from the choice of gauge fixing for the residual symmetries of the ADHM construction w6x. It is worth mentioning that each pair of four-dimensional instantons are superposed at SpŽ2. angles. For this we observe that the normal vectors Ž26. of the two codimensionfour-planes span an eight-dimensional subspace in E 4 k and their coefficients are quaternions. So, they can be related with an SpŽ2. rotation Žafter choosing a suitable basis in E 4 k .. As a result the two fourplanes spanned by the normal vectors in E 8 are at SpŽ2. angles. Such four-planes are parameterized by the four-dimensional coset space SpŽ2.rSO Ž4.; this explains the presence of four angles in Ž27.. It also suggests that there may be a generalization of the solutions of w11x which describe the superposition of five-branes and KK-monopoles depending on four angles. A naive counting of the dimension of the moduli of the 4k-dimensional solutions which takes account of the dimension of moduli of 4-dimensional instantons involved in the superposition and their associated angles does not reproduce the dimension of the moduli space in w6x. However, all their solutions have a decaying behaviour at large distances consistent with the interpretation that they are four-dimensional instantons at angles.
3r32 of spacetime supersymmetry. This is in agreement with the proportion of supersymmetry preserved by the intersecting five-branes on a string at angles and superposed with a pp-wave solution of D s 11 supergravity. However there is a difference. In the case of two orthogonal intersecting five-branes on a string with a pp-wave superposed the proportion of the supersymmetry preserved is 1r8 but this is not the case for the configuration of two orthogonal instantons. To see this, first observe that the solutions of the previous two sections will preserve 1r4 of the N s 1 ten-dimensional supersymmetry if the components of the curvature are in the spŽ1. [ spŽ1. subalgebra of spŽ2.. However this is not so because a direct calculation reveals that the mixed components Fm1, n 2 of the curvature tensor do not vanish. Further we remark that one might have thought that it is possible to superpose two four-dimensional instantons Bm and Ca with gauge group G localised at two orthogonal four-planes in E 8 by simply setting A s Ž B,C . and requiring that the gauge group of the new connection is again G. However, this is not a solution of the BPS condition Ž4. for a non-abelian gauge group in eight-dimensions. This is because Ž4. is non-linear in the connection. However the above linear superposition is a solution if the gauge groups of B and C are treated independently, i.e. if the gauge group of A is G = G.
5. Supersymmetry 6. Concluding remarks Among the above Yang-Mills configurations on E 4 k only the instantons at angles in E 8 and the instantons on E 4 are solutions of ten-dimensional supersymmetric Yang-Mills theory preserving a proportion of supersymmetry. The supersymmetry condition is FM N G
MN
es0 ,
Ž 28 .
where e is the supersymmetry parameter. In the four-dimensional case the instantons preserve 1r2 of the supersymmetry. In the eight-dimensional case condition Ž4. implies that FM N is in spŽ2., an argument similar to that in w11x can be used to show that the instantons at angles in E 8 will preserve 3r16 of supersymmetry. As a solution of the Matrix theory, the eight-dimensional instantons at angles preserve
We have interpreted some of the 4k-dimensional instantons of w5x and w6x as superposition of four-dimensional instantons at angles. We have shown that there four angles associated with the superposition of two four-dimensional instantons which depend on the ADHM parameters. These solutions are examples of Hermitian-Einstein connections in 4k dimensions and the eight-dimensional ones preserve 3r16 of the N s 1 ten-dimensional supersymmetry. Because of the close relation between the self-duality condition and the BPS equations for magnetic monopoles we expect that one can generalize the above construction to find solutions of Yang-Mills equations on E 3 k which have the interpretation of monopoles at angles. In addition, the Yang-Mills configurations of
G. Papadopoulos, A. Teschendorffr Physics Letters B 419 (1998) 115–122
w6x and the HKT geometries described in w31x combined give new examples of two-dimensional supersymmetric sigma models with Ž4,0. supersymmetry. These provide consistent backgrounds for the propagation of the heterotic string w32x. Another application of the instantons of w6x is in the context of D-branes. In IIA theory, they are associated with the D-brane bound state of a 0-brane within a 4-brane within an 8-brane, and in the IIB theory they are associated with the D-brane bound state of a D-string within a D-5-brane within a D-9-brane. There are many other ways to superpose two or more four-dimensional instantons. Here we have described the superposition using the condition that the components of the curvature F of the Yang-Mills connection are in spŽ k .. Another option is to use the Hermitian-Einstein condition associated with suŽ2 k . which we have already mentioned in section two. In eight dimensions this will lead to configurations preserving 1r8 of the N s 1 ten-dimensional supersymmetry. In addition in eight dimensions one can allow the components of F to lie in spinŽ7.. This will result in superpositions of instantons preserving 1r16 of the N s 1 ten-dimensional supersymmetry. In fact, some spinŽ7. instanton solutions have already been found w33x. It will be of interest to see whether they can be interpreted as four-dimensional instantons at angles. Since SpŽ2. is a subgroup of SpinŽ7., the solutions of w6x are also examples of spinŽ7. instantons, albeit of a particular type. There are more possibilities in lower dimensions, for example the G 2 instantons in seven dimensions w34,35x which may have a similar interpretation. 7. Note added While we were revising our work, the paper w36x by N. Ohta and J-G. Zhou appeared in which the relation between supergravity configurations and Yang-Mills ones is also discussed. However, they use constant Yang-Mills configurations on the eight torus. Acknowledgements We would like to thank P. Goddard and A. Kent for helpful discussions. A.T. thanks PPARC for a
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