We construct infinite-dimensional symmetries of the two dimensional equation which results from the dimensional reduction of the self-duality condition in (2, 2) signature space-time. These are symmetries of the dimensionally reduced Chalmers-Siegel action and so hold off-shell

Mansfield, P.

Wardlow, A.

Journal of High Energy Physics

English

arXiv

Durham Research Online

arXiv:1011.0301v2  [hep-th]  8 Dec 2010Preprint typeset in JHEP style - HYPER VERSION DCPT-10/51Symmetries of Self-Dual Yang-Mills EquationsDimensionally Reduced from (2,2) Space-timePaul MansfieldDepartment of Mathematical Sciences, University of DurhamSouth Road, Durham, DH1 3LE, U.K.E-mail: p.r.w.mansfield@durham.ac.ukAdam WardlowDepartment of Mathematical Sciences, University of DurhamSouth Road, Durham, DH1 3LE, U.K.E-mail: a.b.wardlow@durham.ac.ukAbstract: We construct infinite-dimensional symmetries of the two dimensional equationwhich results from the dimensional reduction of the self-duality condition in (2, 2) signaturespace-time. These are symmetries of the dimensionally reduced Chalmers-Siegel action andso hold off-shell.Keywords: Space-Time Symmetries, Gauge Symmetry.Contents1. Introduction 12. The Hitchin system 23. The Self-Duality Equations in Complex and Light-Cone Co-ordinates 34. Symmetries of the 2D Euclidean Self-Dual Equations on the plane 45. Summary 71. IntroductionWe shall construct infinite dimensional symmetries of the actionTr∫d2xΨ((∂2u + ∂2z)Ψ−Ψ [∂zΨ, ∂uΨ])with Euler-Lagrange equation(∂2u + ∂2z )Ψ− [∂zΨ, ∂uΨ] = 0resulting from the dimensional reduction of the self duality condition in (2, 2) space. Theself-dual Yang-Mills action written in four dimensions has physically relevant solutionsand in particular in Euclidean space the solutions are referred to as instantons. Startingwith the Euclidean equations in R4 they may be dimensionally reduced by demanding thatsolutions are invariant under translations of two of the coordinates, [1]. The solutions ofthe dimensionally reduced equations, called Hitchin’s equations, have the property thatthey may be defined over a Riemann surface using analytic maps and are conformallyinvariant, and they have found applications in the field of integrability amongst others; seefor example [2, 3, 4] and [5]. We review the procedure taken by Hitchin in [1].In the paper, [6], it is shown how to construct infinite symmetries of the ChalmersSiegel action [7] describing the self-duality equations in (1, 3) space using (complex) light-cone coordinates, xo = t−x3, xō = t+x3, z = x1+ix2, z̄ = x1−ix2 for t, x1, x2 and x3 real.(See also [8, 9, 10, 11, 12] for previous discussions on symmetries of Yang-Mills theories)This work was based on the derivation of a Lagrangian formalism to the MHV rules,[13, 14, 15, 16], in the papers [17, 18] and was extended to the N = 4 supersymmetric actionin the papers [19] and [20]. The procedure involves defining a canonical transformationthat maps the Chalmers Siegel action to a free theory. By writing symmetries of the freetheory and using the transformation and its inverse the authors construct symmetries of– 1 –the self-dual action. In particular, it was mentioned in [6] that by making x2 imaginary,the light-cone system became a set of real coordinates and the arguments in that paperextended to (2, 2) space-time where Az and Az̄ are not related by complex conjugation.By writing the Euclidean Cartesian coordinates (x1, x2, x3, x4) in terms of two complexcoordinates, z = x1 + ix2 and u = x3 + ix4 we write the Hitchin equations by assumingthat the fields do not depend on the imaginary parts of the new coordinates, x2 and x4where we assume an anti-hermitian representation of the Lie algebra valued fields withA∗ū = −Au = and A∗z̄ = −Az. In the case where x2 and x4 are pure imaginary then wearrive at the same dimensionally reduced equations but now the fields are no longer relatedby complex conjugation and on the plane we may impose the gauge Aū = 0. Since thesymmetry δAz, (written down in the introduction to [6]) with real momenta is a symmetryof the 4d action in (2, 2) space (using the light-cone gauge), where fields are not relatedby conjugation, it is necessarily a symmetry of the equation of motion. By dimensionallyreducing the expression, we write down an expression for the symmetry of the 2d equationof motion.2. The Hitchin systemThe Hitchin equations result from the dimensional reduction of the self-duality equationsin Euclidean space, [1]. We shall review their derivation using real space-time coordinates(x1, x2, x3, x4), before introducing the complex co-ordinates that allow immediate general-isation to (2,2) signature. So the metric is ds2 = dx21 + dx22 + dx23 + dx24. Hitchin considersthe Lie-algebra valued curvature two formF (A) = dA+A2∑µ<νFµνdxµ ∧ dxνwhere A is the connection over the G-bundleA = A1dx1 +A2dx2 +A3dx3 +A4dx4,and we choose an anti-hermitian representation of the Lie algebra generators, T a, so thatif A = Aa Ta, then A∗a = Aa and A∗ = −A. When written out explicitly the self-dualitycondition,Fµν =12√gεµνλρFλρ,becomes the setF12 = F34, F13 = F42, F14 = F23. (2.1)Hitchin, [1], then assumes that the functions Aµ are independent of two of the coordinates,x2 and x4 say, (note, for later convenience, this is a different choice form that of [1]). Sothat the connection becomesA = A1dx1 +A3dx3.Re-naming the fields A2 and A4 asA2 = ϕ, A4 = ψ– 2 –which are reminiscent of Higgs fields, [1], puts the equations of motion into the formF12 = [D1, ϕ] = [D3, ψ] = F34F13 = [D1,D3] = [ψ,ϕ] = F42F14 = [D1, ψ] = [ϕ,D3] = F23.3. The Self-Duality Equations in Complex and Light-Cone Co-ordinatesWe have seen the derivation of the Hitchin equations in Cartesian coordinates. The ap-proach in the papers [18], [17] and [6] is to write the self-dual action and its symmetries in(1, 3) light-cone momentum space coordinates, p̌ = pt − p3, p̂ = pt + p3, p̃ = p1 + ip2 andp̄ = p1 − ip2 where p̃ and p̄ are related by complex conjugation. However, it was discussedin [6] that we can make p2 pure imaginary thereby making all (p̌, p̂, p̃, p̄) real. Then thearguments written down in [6] extend to (2, 2) space. In fact the authors used this to derivetheir results to begin with.With that in mind, we write the self-duality equations that are dimensionally reducedfrom Euclidean space-time in a complex coordinate system. We define complex coordinates(u, ū, z, z̄) for Euclidean space byu = x3 + ix4, z = x1 + ix2ū = x3 − ix4, z̄ = x1 − ix2 (3.1)so nowds2 = dx21 + dx22 + dx23 + dx24 = dudū+ dzdz̄The self-duality equationsFµν =12√gεµνλρFλρ,with εuūzz̄ = 1, written out in components are nowFuū = −Fzz̄, Fuz = 0, Fūz̄ = 0, (3.2)or more explicitly,∂uAz − ∂zAu + [Au, Az ] = 0∂ūAz̄ − ∂z̄Aū + [Aū, Az̄ ] = 0∂uAū − ∂ūAu + [Au, Aū] = −∂zAz̄ + ∂z̄Az − [Az, Az̄ ] (3.3)where the fields are related by conjugation with A∗ū = −Au and A∗z̄ = −Az because of theanti-hermitian generators. Following Hitchin’s procedure, we assume that the fields areindependent of the coordinates x2 and x4. Then we have∂x4Aµ = ∂uAµ − ∂ūAµ = 0∂x2Aµ = ∂z̄Aµ − ∂zAµ = 0 (3.4)– 3 –for all the field components, Aµ. The Hitchin equations (3.3) in this coordinate system,are then∂uAz − ∂zAu + [Au, Az ] = 0 (3.5)∂uAz̄ − ∂zAū + [Aū, Az̄ ] = 0 (3.6)∂uAū − ∂uAu + [Au, Aū] = −∂zAz̄ + ∂zAz − [Az, Az̄ ] (3.7)and the Higgs fields are,ϕ =i2(Az̄ −Az)ψ =i2(Aū −Au) .In the papers [17, 18] and [6] we had a time-like co-ordinate and were able to use alight-cone gauge condition. This would be analogous to Aū = 0 which we are not free tochoose in Euclidean space because the components are related by A∗ū = −Au. So now weconsider the problem in (2, 2) space by writing x2 = iy and x4 = it, with t and y real.Then the coordinates (u, ū, z, z̄) are all real and independent. Furthermore, the fields Auand Aū are no longer related by complex conjugation, and neither are Az and Az̄.In an appropriate domain we can now make the gauge choice Aū = 0 and then by (3.7)we can set Az̄ = 0 (without constraining Az). Then using (3.5) we can writeAu = −∂z∂uAz (3.8)and substitute into (3.6) to arrive at(∂u +∂2z∂u)Az −[∂z∂uAz, Az]= 0 (3.9)or, to make a simplification by defining Az = ∂uΨ, we can write(∂2u + ∂2z)Ψ− [∂zΨ, ∂uΨ] = 0. (3.10)4. Symmetries of the 2D Euclidean Self-Dual Equations on the planeIn [6], infinite dimensional symmetries of the self-dual action in (1, 3) space were constructedbut by assuming real momenta then the results are valid in (2, 2) space. The procedurewas extended to the N = 4 supersymmetric theory in [20]. The Chalmers-Siegel actionthat gives rise to the self-duality equation for A = Az as its Euler-Lagrange equation isS =4g2Tr∫d4xA (∂u∂u + ∂z∂z)A+4g2Tr∫d4x(∂z∂−1u A) [A, ∂uA](4.1)We define the notation, (pnu, pnū, pnz , pnz̄ ) = (ň, n̂, ñ, n̄) and ζp = p̄/p̂, and∫1···n=∫d4p1(2π)4· · · d4pn(2π)4.– 4 –Writing the action in momentum space givesTr∫1{p̄1p̃1 + p̂1p̌1}A1̄A1 − iTr∫123p̂1 (ζ3 − ζ2)A1̄A2̄A3̄(2π)4δ(p1 + p2 + p3)where we use the shorthand Ap = A(p) and Ap̄ = A(−p), and we perform the transforma-tion A→ A′ = A+ εδA and Ā→ Ā′ + εδĀ withδA1 = −ε∞∑n=2n∑i=2n∑j=i∫2···n1̂q̂Γ(qG, iG, · · · , jG)Γ(q, j + 1, · · · , n, 1 · · · , i− 1)××A2̄ · · ·AīG · · ·Aj̄G · · ·An̄(4.2)andδA1 = −ε∞∑n=2n∑k=2k−1∑i=2k−1∑j=i∫2···nk̂21̂q̂Γ(qG, iG, · · · , jG)Γ(q, j + 1, · · · , n, 1, · · · , i− 1)××A2̄ · · ·AīG · · ·Aj̄G · · ·Ak̄ · · ·An̄+ε∞∑n=2n∑k=2k∑i=2n∑j=k∫2···nq̂1̂(k̂G−1)2(q̂G−1)2Γ(qG−1, iG−1, · · · , jG−1)Γ(q, j + 1, · · · , n, 1, · · · , i− 1)××A2̄ · · ·AīG−1 · · ·Ak̄G−1 · · ·Aj̄G−1 · · ·An̄−ε∞∑n=2n∑k=2n∑i=k+1n∑j=i∫2···nk̂21̂q̂Γ(qG, iG, · · · , jG)Γ(q, j + 1, · · · , n, 1, · · · , i− 1)××A2̄ · · ·Ak̄ · · ·AīG · · ·Aj̄G · · ·An̄.where Γ(12 · · · n) was first written in [17] asΓ(1, · · · , n) = −(i)n 1̂(1, 2)1̂(1, 2 + 3)· · · 1̂(1, 2 + · · · (n− 1)) .and p→ pG is an isometry. The bracket (i, j) is defined as (i, j) = (̂ij̃− ĩĵ). The invarianceof the action under these transformations is proven in [6]. Given that they are symmetriesof the action, that is sufficient for us to be able to infer that they are indeed symmetriesof the equation of motion which we derived earlier (3.9). We could have also derived thisequation using the action (4.1).Here, we are concerned with finding symmetries of the dimensionally reduced equation(3.10) where, as before, we have defined Az = A = ∂uΨ,(∂2u + ∂2z)Ψ− [∂zΨ, ∂uΨ] = 0.which in momentum space is(1̌2 + 1̃2)Ψ1 −∫d2p1d2p2(2, 3)Ψ2̄Ψ3̄δ2(p1 + p2 + p3).where d2p = dp̌dp̃– 5 –By writing Ap = −ip̂Ψp the expression for δA, (written in the introduction to [6]),becomesδΨ1 = −ε∞∑n=2n∑i=2n∑j=i∫2···nq̂1̂D(qG, iG, · · · , jG)D(q, j + 1, · · · , n, 1, · · · , i− 1)××Ψ2̄ · · ·ΨīG · · ·Ψj̄G · · ·Ψn̄δ4(∑mpm)(4.3)where D(1, 2, · · · , n) is given byD(12 · · · n) = −(−1)n 1̂n−32̂3̂ · · · n̂(1, 2) (1, 2 + 3) · · · (1, 2 + 3 + · · · + (n− 1)) (4.4)in momentum space. The Ψ(p1, p2, p3, p4) are the Fourier transform of Ψ(x1, x2, x3, x4),Ψ(p1, p2, p3, p4) =∫d4x(2π)4eipµxµΨ(x1, x2, x3, x4).Since we assume that Ψ depends only on x1 and x3, we haveΨ(p1, p2, p3, p4) =∫dx22πeip2x2∫dx42πeip4x4∫d2x(2π)2eip1x1+ip3x3Ψ(x1, x3)= δ(p2)δ(p4)Ψ′(p1, p3)= δ(p̂ − p̌)δ(p̃ − p̄)Ψ′(p̌, p̃).Substitute this into (4.3) and evaluate the integrals over p̂i and p̄i for i = 2, · · · , n and wehaveδΨ1 = −ε∞∑n=2n∑i=2n∑j=i∫2···nq̂1̂D(qG, iG, · · · , jG)D(q, j + 1, · · · , n, 1, · · · , i− 1)××Ψ′2̄· · ·Ψ′īG · · ·Ψ′j̄G · · ·Ψ′n̄δ2(∑mpm)δ(p̂1 − p̌1)δ(p̃1 − p̄1)where now∫1···n=∫d2p1(2π)2· · · d2pn(2π)2and the kernels, D, (4.4) are written in terms of p̌ and p̃ using p̌ = p̂ and p̃ = p̄. Thenwrite the inverse Fourier transform, δΨ(x1, x2, x3, x4), of δΨ(p1),δΨ(x) =∫d4p1e−ip1µxµ1 δΨ(p1)and evaluating the integrals over p̂1 and p̄1 the final expression isδΨ′1 = −ε∞∑n=2n∑i=2n∑j=i∫2···nq̂1̂D(qG, iG, · · · , jG)D(q, j + 1, · · · , n, 1, · · · , i− 1)××Ψ′2̄· · ·Ψ′īG · · ·Ψ′j̄G · · ·Ψ′n̄δ2(∑mpm)– 6 –and the isometry x→ xG is simply a rotation of the plane about angle t, vizuGt = cos(t)u+ sin(t)zzGt = − sin(t)u+ cos(t)z.Given two transformations, δt1 and δt2 and the argument in [6], the commutator of thetransformations is[δt1 , δt2 ] Ψ = 0and the infinite set of transformations clearly forms an Abelian algebra. Similarly, bydefining Ψ̄ = ∂uĀ, we can write the transformation δΨ̄ asδΨ1 = −ε∞∑n=2n∑k=2k−1∑i=2k−1∑j=i∫2···nq̂1̂D(qG, iG, · · · , jG)D(q, j + 1, · · · , n, 1, · · · , i− 1)××Ψ2̄ · · ·ΨīG · · ·Ψj̄G · · ·Ψk̄ · · ·Ψn̄+ε∞∑n=2n∑k=2k∑i=2n∑j=k∫2···nq̂1̂D(qG−1, iG−1, · · · , jG−1)D(q, j + 1, · · · , n, 1, · · · , i− 1)××Ψ2̄ · · ·ΨīG−1 · · ·Ψk̄G−1 · · ·Ψj̄G−1 · · ·Ψn̄−ε∞∑n=2n∑k=2n∑i=k+1n∑j=i∫2···nq̂1̂D(qG, iG, · · · , jG)D(q, j + 1, · · · , n, 1, · · · , i− 1)××Ψ2̄ · · ·Ψk̄ · · ·ΨīG · · ·Ψj̄G · · ·Ψn̄.5. SummaryWe reviewed the derivation of Hitchin’s equations on Euclidean space using complex co-ordinates. These give an immediate generalisation to (2, 2) space-time, resulting in thetwo-dimensional equation(∂2u + ∂2z)Ψ− [∂zΨ, ∂uΨ] = 0.because in this signature we are free to impose a light-cone gauge condition, as in [6].By dimensionally reducing the results of that paper we obtained symmetries of this two-dimensional equation δΨ (and δΨ) and, importantly, the action whose Euler-Lagrangeequation this is,S =4g2Tr∫d2xΨ((∂2u + ∂2z)Ψ−Ψ [∂zΨ, ∂uΨ])where Ψ̄ = ∂uĀ.E. Brézin (et-al), [21] consider two dimensional non-linear sigma models and constructinfinite, on-shell, conserved currents of the motion iteratively, subject to the equations ofmotion being∂µAµ = 0, [∂µ +Aµ, ∂ν +Aν ] = 0,– 7 –and the authors give a number of examples of such systems. The system we have studiedis an example of such a model1 since, from (3.8) and Az = ∂uΨ, we have Au = −∂zΨ and∂zAz + ∂uAu = 0[∂z +Az, ∂u +Au] = 0using (3.10). Our symmetries of the action extend the program of [21] and also [10, 8, 9]off-shell.In the introduction to this paper we alluded to the fact that the solutions to theHitchin equations may be defined over Riemann surfaces using analytic maps. It wouldbe interesting to consider how we might extend our approach to find symmetries of theself-dual equations on such surfaces, for example Riemann spheres or tori.AcknowledgmentsA.W. wishes to thank the STFC for a studentship. P.M. thanks STFC for support under therolling grant ST/G000433/1. We also thank Niall MacKay for pointing out the connectionto earlier work on conserved currents in sigma-models.References[1] N.J.Hitchin, The Self-Duality Equations on a Riemann Surface, Proc. London Math. Soc 55(1986) 59–126.[2] M. A. C. Kneipp, Hitchin’s equations and integrability of BPS Z(N) strings in Yang-Millstheories, JHEP 11 (2008) 049, [arXiv:0801.0720].[3] M. 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Symmetries of Self-Dual Yang-Mills Equations Dimensionally Reduced From (2,2) Space-time

Paul Mansfield

Adam Wardlow

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Symmetries of self-dual Yang-Mills equations dimensionally reduced from (2, 2) space-time

Symmetries of Self-Dual Yang-Mills Equations Dimensionally Reduced From (2,2) Space-time

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