- Email: [email protected]
Non-self-dual gauge fields
Volume 78B, number 4
PHYSICS LETTERS
9 October 1978
NON-SELF-DUAL GAUGE FIELDS ~
James ISENBERG and Philip B. YASSKIN Center for Theoretical Physics, Department of Physics and Astronomy, University o]'Maryland, College Park, MD 20742, USA and Paul S. GREEN Department of Mathematics, University of Maryland, College Park, MD 20742, USA Received 25 April 1978
The Ward construction is generalized to non-self-dual gauge fields. Reality and currentless conditions are specified.
Recently, Ward [1] has developed a twistorial method for constructing self-dual Yang-Mills fields on pieces of conformally completed complex Minkowski space (CM), and by restriction, on pieces of real Minkowski space (RM), and real euclidean space (RE). The solutions on all the RE, with certain regularity conditions at infinity, are the instantons [ 2 4] which are of interest in quantum field theory. We have modified the Ward construction to produce non-self-dual as well as self-dual Yang-Mills fields on pieces of CM, RM and RE. Unlike self-dual fields, non-self-dual fields do not automatically satisfy the currentless Yang-Mills equations. We give a necessary and sufficient condition for a gauge field to be currentless. Also, unlike self-dual fields, more general fields may be "real" on RM. We present the condition for a field, constructed by our procedure, to have a real restriction to RM (or RE). Like the Ward method, our construction yields a correspondence between gauge fields on the one hand, and holomorphic, pricipal G-bundles over certain "twistorial" base spaces on the other. The base spaces used by Ward are P~ and Pt3, which are the two connected components of the set of all totally Supported in part by the National Science Foundation under Grant PHY-76-20029. 462
null complex planes (two complex dimensions) in CM. (These are parametrized by the sets of projective twistors and projective dual twistors, resp.) Our principal innovation is to replace P~ or P~ by the base space, L, which is the set of all null complex lines (one complex dimension) in CM. This modification of Ward's construction has been investigated independently by E. Witten. He obtains similar results. For any set S within the conformal completion of CM, we let P~(S), P¢(S) and L(S) denote the sets of s-planes,/3-planes and null lines which intersect S. In particular, for each point, P E S, P~(p), Pt~(p) and L(p) denote the sets of s-planes,/3-planes and null lines through P. Our basic result is stated in the following theorem.
Theorem 1 (correspondence). Let G be a complex Lie subgroup of GL(n, C). Let S be an open subset of the conformal completion of CM whose intersection with each null line is either empty or connected and simply connected. There is a 1-1 correspondence between (1) holomorphic, principal G-bundles, ES, over S with holomorphic connection, A, and curvature, F; and (2) holomorphic, principal G-bundles, E L (s), over L(S) which have trivial restriction, EL(p), to L(p) for
Volume 78B, number 4
PHYSICS LETTERS
all p ~ S. The constructions which make this correspondence explicit are outlined in the appendix. Complete proofs of this and our other theorems will appear in a separate publication. We now discuss those properties of the bundle, EL(S), which are equivalent to the gauge field, (A, F), being flat, self-dual, currentless or real. The condition for flatness is the simplest:
Theorem 2 (flatness condition). F = 0 iff E L (S) is trivial. The self-duality and currentless conditions require an understanding of the relation of L to P~ and Pp. An arbitrary c~-plane, Z E Pc~ = Cp3, and ~-plane, W ~ P/3 = ~'-p3, intersect iff Z • W = 0. (Here Z and W are each regarded as four complex numbers whose scale is irrelevant.) When Z and W intersect, their intersection is a null line. In fact, every null line is the intersection of a unique c~-plane and ~3-plane. Thus there are projections pa: L -+ Pc~ and PlY: L -+ P~ and L may be regarded as the five complex dimensional hypersurface, Z • W = 0, within P~ × P~.
9 October 1978
Theorem 4 (currentless condition with holonomic communitivity). D • F = 0 and [.2q)a, .t2q~¢] = 0 iff E L (S) has an extension from L(S) to a neighborhood of L(S) within P~(S) X P¢(S). For G = SL(2, C), the condition, [~oo/,, .67~] = 0, means that F is self-dual, anti-self-dual, or abelian. However, at least locally, there exist [5] currentless SL(2, C) Yang-Mills fields which do not satisfy this condition. Thus we see that the extensibility of EL(S) to P~(S) X P~(S) is sufficient to guarantee that the gauge field is currentless, but is stronger than necessary. We therefore consider extensions to various orders. Recall that L is the surface Z • W = 0 within P~ × P~. We say that E L (S) has an nth order extension from L(S) to P~(S) × P~(S) if there exist patches, Ui, on Pa(S) × P~(S) and functions, gi/: UinU/-~ G, which (a) satisfy the cocycle condition, g i f g / k = gi]c, to nth order in a parameter proportional to Z • W; and (b) become the transition functions of E L (s) when restricted to L(S)nUinU j.
Theorem 5 (currentless condition). D* F = 0 iff EL(S) has a third order extension from L(S) to
p (s3 x ion(s). Theorem 3 (self-duality condition), i • F = F iff EL (S) is the pullback along p~ : L(S) -+ Pa(S) of a bundle Epa(s ) over P~(S). This theorem relates our construction to Ward's. The gauge field he constructs from Epc~(S) is the same as the one we produce from E L (S) in the situation described in theorem 3. A somewhat broader class of fields may be described in terms of the holonomy Lie algebra, Z?~, defined as the subalgebra of the gauge Lie algebra, Z?G, spanned by F and all of its covariant derivatives. From F, we can extract a self-dual part, F~ = ½(F + i * F), and an anti-self-dual part, Ft3 = ~(F - i • F). Although Fa and F~ are not curvatures, we define the self-dual (respectively anti-self-dual) part of the holonomy Lie algebra, . C ~ , (respectively Z?d/,~) as the subspace of Z?G spanned by Fa (respectively F¢3) and all of its covariant' derivatives. It is easy to show that Z?q~ = o ~ a + ~q)t3 where the sum is not necessarily direct. We can characterize those currentless Yang-Mills fields for which Z?~c~ commutes with Z?~5~.
On RiM (or RE), we may be interested in a real gauge group H (such as U(n), GL(n, R), SO(n, R), etc.). We then let G be the complexification of H and ask whether the restricted bundle, ESnRM (or ESnRE) and the restricted connection, A, are reducible from G to H. Our criteria for this reducibility involves several conjugations. There are two conjugations, •, defined on CM: one leaves RM invariant; the other leaves RE invariant. Either conjugation takes a null line into another null line, and hence defines a conjugation, k-: L(Sf~KS) -+ L(snKs). A conjugation on EL(SnKS ) is any map, ~': EL(SnKS ) -+ EL(SnKS), such that k"2 = id, 7r o ~ = ~- o n, and for all g E G, ~ o pg = p~go ~, where p is the right action of G on E L (S) and X is the conjugation on G for which H is the set of fixed points.
Theorem 6 (reality condition). The restricted bundle, ESnRM (or ESnRE), and the restricted gauge field, (A, F), are reducible from G to H iff there exists a conjugation on the restricted bundle EL(Sn KS).
463
Volume 78B, number 4
PHYSICS LETTERS
We have translated the problem of finding real solutions of the currentless Yang-Mills equations on S into the problem of finding a bundle E L(S) over L(S) which has a conjugation and has a third order extinction to P~(S) × P~(S). So far we have not said anything about the finiteness of the action. If, however, we choose S to contain the conformal completion of either RM (S 3 × S 1) or RE ($4), then the total action is automatically finite. For the particular group SU(2), one would like to know [6, p. 699] whether every finite action, globally defined, currentless solution on S4 is self-dual or anti-self-dual. Since the complexification of SU(2) is SL(2, C), this translates into the question of whether there are any SL(2, C)-bundles, EL (S), with a conjugation which have a third order extension to Pc~(S) X P~(S) but are not fully extensible. Finally, we point out that our results may generalize from conformally flat background spacetimes to more general background spacetimes, since unlike P~ and P~, the space L, has an analog in any complex spacetime.
Appendix. We outline the constructions needed for the correspondence in theorem 1. The crucial fact in our construction, as well as Ward's, comes from complex an~alysis: Any holomorphic function from a compact, complex manifold into C must be a constant. We assume G is a complex Lie subgroup of GL(n, C). Then any holomorphic cross section of a trivial, principal G-bundle over a compact, complex manifold is determined by its value at a single point. Further, the set of such cross sections may be identified with the fibre at that point. Given the bundle, E L (S), the point set ofE S is constructed as follows: For each p E S, le1 the fibre over p be the set, 7r-1 (p), of holomorphic cross sections Of EL(p),and let E S = Up~sTr-l(p ). Since L(p)_'Z Cp1 × ~--p1 is compact and EL(p) is trivial, 7r-l(p)
464
9 October 1978
may be identified with G. The connection is constructed by defining a transport, T, along null directions: Assume p and q ~ S lie on a null line, l. Each ff E 7r-1 (q) is a holomorphic cross section of EL(q). Since every holomorphic cross section of EL(p) is determined by its value at a point, we define the transport of from q to p along l as the unique holomorphic cross section, Tp~_qt~, of/Z'L(p) whose value at l E L(p) N L(q) is (Tp+_q~)(1) = if(l). The transport, T, defines a connection, Au, which may be expressed in terms of the transition functions of EL (S) by a formula similar to Ward's. We omit the details previously omitted by Ward. Conversely, given the bundle, E S, with connection, A, we specify the point set of E l (S): For each l C L(S), let the fibre over l be the set, 7r-l(/), of covariantly constant cross sections of E l, the restriction of E S to l, and let EL(S) = Ul~L(S)Tr-l(1). Since such a cross section is determined by its value at a point, 7r-1(/) may be identified with G. The transition functions of E L(S) may be expressed as path ordered exponential integrals of A. It may be checked that the two constructions are inverses. We would like to thank M. Atiyah, G. Sparling and R. Ward for introducing us to the Ward construction. We also thank J. Swank, P. Yang, E. Witten and T. Spencer for helpful discussions.
References [1] R.S. Ward, Phys. Lett. 61A (1977) 81. [2] A.A. Belavin, A.M. Polyakov, A.S. Schwartz and Yu.S. Tyupkin, Phys. Lett. 59B (1975) 85. [3] R. Jackiw and C. Rebbi, Phys. Rev. D16 (1977) 1052. [4] M.F. Atiyah, N.J. Hitchin, V.G. Drinfeld and Yu.I. Manin, Phys. Lett. 65A (1978) 185. [5] S. Coleman, Phys. Lett. 70B (1977) 59. [6] R. Jackiw, Rev. Mod. Phys. 49 (1977) 681.
PHYSICS LETTERS
9 October 1978
NON-SELF-DUAL GAUGE FIELDS ~
James ISENBERG and Philip B. YASSKIN Center for Theoretical Physics, Department of Physics and Astronomy, University o]'Maryland, College Park, MD 20742, USA and Paul S. GREEN Department of Mathematics, University of Maryland, College Park, MD 20742, USA Received 25 April 1978
The Ward construction is generalized to non-self-dual gauge fields. Reality and currentless conditions are specified.
Recently, Ward [1] has developed a twistorial method for constructing self-dual Yang-Mills fields on pieces of conformally completed complex Minkowski space (CM), and by restriction, on pieces of real Minkowski space (RM), and real euclidean space (RE). The solutions on all the RE, with certain regularity conditions at infinity, are the instantons [ 2 4] which are of interest in quantum field theory. We have modified the Ward construction to produce non-self-dual as well as self-dual Yang-Mills fields on pieces of CM, RM and RE. Unlike self-dual fields, non-self-dual fields do not automatically satisfy the currentless Yang-Mills equations. We give a necessary and sufficient condition for a gauge field to be currentless. Also, unlike self-dual fields, more general fields may be "real" on RM. We present the condition for a field, constructed by our procedure, to have a real restriction to RM (or RE). Like the Ward method, our construction yields a correspondence between gauge fields on the one hand, and holomorphic, pricipal G-bundles over certain "twistorial" base spaces on the other. The base spaces used by Ward are P~ and Pt3, which are the two connected components of the set of all totally Supported in part by the National Science Foundation under Grant PHY-76-20029. 462
null complex planes (two complex dimensions) in CM. (These are parametrized by the sets of projective twistors and projective dual twistors, resp.) Our principal innovation is to replace P~ or P~ by the base space, L, which is the set of all null complex lines (one complex dimension) in CM. This modification of Ward's construction has been investigated independently by E. Witten. He obtains similar results. For any set S within the conformal completion of CM, we let P~(S), P¢(S) and L(S) denote the sets of s-planes,/3-planes and null lines which intersect S. In particular, for each point, P E S, P~(p), Pt~(p) and L(p) denote the sets of s-planes,/3-planes and null lines through P. Our basic result is stated in the following theorem.
Theorem 1 (correspondence). Let G be a complex Lie subgroup of GL(n, C). Let S be an open subset of the conformal completion of CM whose intersection with each null line is either empty or connected and simply connected. There is a 1-1 correspondence between (1) holomorphic, principal G-bundles, ES, over S with holomorphic connection, A, and curvature, F; and (2) holomorphic, principal G-bundles, E L (s), over L(S) which have trivial restriction, EL(p), to L(p) for
Volume 78B, number 4
PHYSICS LETTERS
all p ~ S. The constructions which make this correspondence explicit are outlined in the appendix. Complete proofs of this and our other theorems will appear in a separate publication. We now discuss those properties of the bundle, EL(S), which are equivalent to the gauge field, (A, F), being flat, self-dual, currentless or real. The condition for flatness is the simplest:
Theorem 2 (flatness condition). F = 0 iff E L (S) is trivial. The self-duality and currentless conditions require an understanding of the relation of L to P~ and Pp. An arbitrary c~-plane, Z E Pc~ = Cp3, and ~-plane, W ~ P/3 = ~'-p3, intersect iff Z • W = 0. (Here Z and W are each regarded as four complex numbers whose scale is irrelevant.) When Z and W intersect, their intersection is a null line. In fact, every null line is the intersection of a unique c~-plane and ~3-plane. Thus there are projections pa: L -+ Pc~ and PlY: L -+ P~ and L may be regarded as the five complex dimensional hypersurface, Z • W = 0, within P~ × P~.
9 October 1978
Theorem 4 (currentless condition with holonomic communitivity). D • F = 0 and [.2q)a, .t2q~¢] = 0 iff E L (S) has an extension from L(S) to a neighborhood of L(S) within P~(S) X P¢(S). For G = SL(2, C), the condition, [~oo/,, .67~] = 0, means that F is self-dual, anti-self-dual, or abelian. However, at least locally, there exist [5] currentless SL(2, C) Yang-Mills fields which do not satisfy this condition. Thus we see that the extensibility of EL(S) to P~(S) X P~(S) is sufficient to guarantee that the gauge field is currentless, but is stronger than necessary. We therefore consider extensions to various orders. Recall that L is the surface Z • W = 0 within P~ × P~. We say that E L (S) has an nth order extension from L(S) to P~(S) × P~(S) if there exist patches, Ui, on Pa(S) × P~(S) and functions, gi/: UinU/-~ G, which (a) satisfy the cocycle condition, g i f g / k = gi]c, to nth order in a parameter proportional to Z • W; and (b) become the transition functions of E L (s) when restricted to L(S)nUinU j.
Theorem 5 (currentless condition). D* F = 0 iff EL(S) has a third order extension from L(S) to
p (s3 x ion(s). Theorem 3 (self-duality condition), i • F = F iff EL (S) is the pullback along p~ : L(S) -+ Pa(S) of a bundle Epa(s ) over P~(S). This theorem relates our construction to Ward's. The gauge field he constructs from Epc~(S) is the same as the one we produce from E L (S) in the situation described in theorem 3. A somewhat broader class of fields may be described in terms of the holonomy Lie algebra, Z?~, defined as the subalgebra of the gauge Lie algebra, Z?G, spanned by F and all of its covariant derivatives. From F, we can extract a self-dual part, F~ = ½(F + i * F), and an anti-self-dual part, Ft3 = ~(F - i • F). Although Fa and F~ are not curvatures, we define the self-dual (respectively anti-self-dual) part of the holonomy Lie algebra, . C ~ , (respectively Z?d/,~) as the subspace of Z?G spanned by Fa (respectively F¢3) and all of its covariant' derivatives. It is easy to show that Z?q~ = o ~ a + ~q)t3 where the sum is not necessarily direct. We can characterize those currentless Yang-Mills fields for which Z?~c~ commutes with Z?~5~.
On RiM (or RE), we may be interested in a real gauge group H (such as U(n), GL(n, R), SO(n, R), etc.). We then let G be the complexification of H and ask whether the restricted bundle, ESnRM (or ESnRE) and the restricted connection, A, are reducible from G to H. Our criteria for this reducibility involves several conjugations. There are two conjugations, •, defined on CM: one leaves RM invariant; the other leaves RE invariant. Either conjugation takes a null line into another null line, and hence defines a conjugation, k-: L(Sf~KS) -+ L(snKs). A conjugation on EL(SnKS ) is any map, ~': EL(SnKS ) -+ EL(SnKS), such that k"2 = id, 7r o ~ = ~- o n, and for all g E G, ~ o pg = p~go ~, where p is the right action of G on E L (S) and X is the conjugation on G for which H is the set of fixed points.
Theorem 6 (reality condition). The restricted bundle, ESnRM (or ESnRE), and the restricted gauge field, (A, F), are reducible from G to H iff there exists a conjugation on the restricted bundle EL(Sn KS).
463
Volume 78B, number 4
PHYSICS LETTERS
We have translated the problem of finding real solutions of the currentless Yang-Mills equations on S into the problem of finding a bundle E L(S) over L(S) which has a conjugation and has a third order extinction to P~(S) × P~(S). So far we have not said anything about the finiteness of the action. If, however, we choose S to contain the conformal completion of either RM (S 3 × S 1) or RE ($4), then the total action is automatically finite. For the particular group SU(2), one would like to know [6, p. 699] whether every finite action, globally defined, currentless solution on S4 is self-dual or anti-self-dual. Since the complexification of SU(2) is SL(2, C), this translates into the question of whether there are any SL(2, C)-bundles, EL (S), with a conjugation which have a third order extension to Pc~(S) X P~(S) but are not fully extensible. Finally, we point out that our results may generalize from conformally flat background spacetimes to more general background spacetimes, since unlike P~ and P~, the space L, has an analog in any complex spacetime.
Appendix. We outline the constructions needed for the correspondence in theorem 1. The crucial fact in our construction, as well as Ward's, comes from complex an~alysis: Any holomorphic function from a compact, complex manifold into C must be a constant. We assume G is a complex Lie subgroup of GL(n, C). Then any holomorphic cross section of a trivial, principal G-bundle over a compact, complex manifold is determined by its value at a single point. Further, the set of such cross sections may be identified with the fibre at that point. Given the bundle, E L (S), the point set ofE S is constructed as follows: For each p E S, le1 the fibre over p be the set, 7r-1 (p), of holomorphic cross sections Of EL(p),and let E S = Up~sTr-l(p ). Since L(p)_'Z Cp1 × ~--p1 is compact and EL(p) is trivial, 7r-l(p)
464
9 October 1978
may be identified with G. The connection is constructed by defining a transport, T, along null directions: Assume p and q ~ S lie on a null line, l. Each ff E 7r-1 (q) is a holomorphic cross section of EL(q). Since every holomorphic cross section of EL(p) is determined by its value at a point, we define the transport of from q to p along l as the unique holomorphic cross section, Tp~_qt~, of/Z'L(p) whose value at l E L(p) N L(q) is (Tp+_q~)(1) = if(l). The transport, T, defines a connection, Au, which may be expressed in terms of the transition functions of EL (S) by a formula similar to Ward's. We omit the details previously omitted by Ward. Conversely, given the bundle, E S, with connection, A, we specify the point set of E l (S): For each l C L(S), let the fibre over l be the set, 7r-l(/), of covariantly constant cross sections of E l, the restriction of E S to l, and let EL(S) = Ul~L(S)Tr-l(1). Since such a cross section is determined by its value at a point, 7r-1(/) may be identified with G. The transition functions of E L(S) may be expressed as path ordered exponential integrals of A. It may be checked that the two constructions are inverses. We would like to thank M. Atiyah, G. Sparling and R. Ward for introducing us to the Ward construction. We also thank J. Swank, P. Yang, E. Witten and T. Spencer for helpful discussions.
References [1] R.S. Ward, Phys. Lett. 61A (1977) 81. [2] A.A. Belavin, A.M. Polyakov, A.S. Schwartz and Yu.S. Tyupkin, Phys. Lett. 59B (1975) 85. [3] R. Jackiw and C. Rebbi, Phys. Rev. D16 (1977) 1052. [4] M.F. Atiyah, N.J. Hitchin, V.G. Drinfeld and Yu.I. Manin, Phys. Lett. 65A (1978) 185. [5] S. Coleman, Phys. Lett. 70B (1977) 59. [6] R. Jackiw, Rev. Mod. Phys. 49 (1977) 681.