Supertwistors, Chern-Simons And Super Gauge Theories

Transcription

1 Supertwistors, Chern-Simons And Super Gauge Theories INSTITUT FÜR THEORETISCHE PHYSIK UNIVERSITÄT HANNOVER From Twistors To Amplitudes, QMUL, London 2005

2 Contents 1 Motivation Double Fibrations Twistor Correspondence And Penrose-Ward Transform 2 Prime Example Z = P 3 N And Self-Dual SYM Theory 3

3 Double Fibrations Double Fibrations Twistor Correspondence And Penrose-Ward Transform The central objects of this talk are double fibrations of the form Y π 2 π 1 Z X where X, Y and Z are complex (super)manifolds: X space-time Y correspondence space Z twistor space

4 Examples Motivation Double Fibrations Twistor Correspondence And Penrose-Ward Transform X : complexified (super) space-time C m n with m = 3, 4. Z : open subsets of CP 3 n, W CP 3 n [b 1,..., b 4 f 1,..., f n ], degree 2 hypersurface in CP 3 3 CP 3 3, W CP 2 4 [2, 1, 1 1, 1, 1, 1], exotic supermanifolds. In due course, we shall also sometimes introduce real structures on these spaces.

5 Twistor Correspondence Double Fibrations Twistor Correspondence And Penrose-Ward Transform Then we ve a correspondence between Z and X, i.e., between points in one space and subspaces of the other: z Z π 1 (π 1 2 (z)) X π 2 (π 1 1 (x)) Z x X Using the correspondence, we can transfer data given on Z to data on X and vice versa. E.g., take some analytic object Ob Z on Z and transform it to an object Ob X on X which will be constrained by some PDEs since π 2 Ob Z has to be constant along the fibers of π 2 : Y Z.

6 Penrose-Ward Transform Double Fibrations Twistor Correspondence And Penrose-Ward Transform Under suitable topological conditions, the map Ob Z Ob X is a bijection between equivalence classes [Ob Z ] and [Ob X ] (i.e., a bijection between moduli spaces). In the following, my objects are certain holomorphic vector bundles.

7 Penrose-Ward Transform Double Fibrations Twistor Correspondence And Penrose-Ward Transform In fact, suppose that the fibers of π 2 : Y Z are simply connected and that π 2 (π 1 1 (x)) Z is compact and connected for all x X. Suppose further that Γ(X, T X ) = π 1 {Γ(Y, T Y )/π 2 Γ(Z, T Z )}. Then there is a 1-1 correspondence between X -trivial holomorphic vector bundles E Z, holomorphic vector bundles on Y equipped with flat relative connection and holomorphic vector bundles E X equipped with a connection flat on each π 1 (π 1 2 (z)) X for z Z.

8 What I m considering. Double Fibrations Twistor Correspondence And Penrose-Ward Transform In the sequel, I shall be considering three examples of twistor manifolds Z : the supertwistor space P 3 N CP 3 N. the mini-supertwistor space P 2 4 WCP 2 4 [2, 1, 1 1, 1, 1, 1]. the superambitwistor space L 5 6 CP 3 3 CP 3 3.

9 Supertwistor Space The supertwistor space Z = P 3 N is a holomorphic fibration O(1) C 2 ΠO(1) C N CP 1 over the Riemann sphere CP 1. On P 3 N we can introduce (homogeneous) coordinates (z α, λ α, η i ). Holomorphic sections of P 3 N are of the form z α = x α α λ α, η i = η α i λ α, where (x, η) C 4 2N.

10 Supertwistor Space Thus, we may write down the following double fibration: By virtue of C 4 2N CP 1 (x α α, ηi α, λ α ) π 2 π 1 P 3 N (z α, λ α, η i ) C 4 2N (x α α, ηi α ) π 2 (x α α, η α i, λ α ) = (z α, λ α, η i ) = (x α α λ α, λ α, η α i λ α ), we find: point p P 3 N null superplane C 2 N p C 4 2N CPx,η 1 P 3 N point (x, η) C 4 2N

11 Self-Dual SYM Theory Let s now discuss super gauge theory and consider a holomorphic vector bundle E P 3 N holomorphically trivial on any CP 1 x,η P 3 N (CP 1 -trivial). Then the transition function f of π2 E is annihilated by D α ± = λ± α α α, D± i = λ± α i α and λ ±, where λ + := λ 1, λ := λ 2. λ 2 λ 1 Furthermore, f + = ψ 1 + ψ, λ± ψ ± = 0.

12 Self-Dual SYM Theory Therefore, we learn that (D ± α + A ± α )ψ ± = 0, λ± ψ ± = 0, (D i ± + A i ±)ψ ± = 0, where one can show that A ± α λ α ±A α α, A i ± λ α ±A i α. Clearly, we have certain compatibility conditions which turn out to be the constraint equations for the components A α α and Ai α of the gauge potential of N -extended self-dual SYM theory on C 4 2N (R 4 2N ) which are equivalent to the e.o.m. of this theory on C 4 (R 4 ).

13 Self-Dual SYM Theory Moreover, we have A α α = dλ + A + α S 1 2πiλ + λ+ α, A i α = dλ + A i +. S 1 2πiλ + λ α + Penrose-Ward Transform

14 Mini-Supertwistor Space Let s now discuss 3D super gauge theory. In particular, we re interested in the diagram P 3 4 = R 4 8 S 2 R 4 8 R 3 8 S 2 π 2 π 1 7 P 2 4 R 3 8 in the real setting. [A.D. Popov, C. Sämann, M. Wolf, 05]

15 Mini-Supertwistor Space In fact, one can show that P 2 4 is the holomorphic fibration O(2) ΠO(1) C 4 CP 1. Holomorphic sections are of the form w = y α β λ α λ β and η i = η α i λ α. By virtue of these equations, we therefore have: point p P 2 4 oriented lines R 1 0 p R 3 8 CPy,η 1 P 2 4 point (y, η) R 3 8 Note that c 1 (P 2 4 ) = 0, i.e., it s CY. So, one can take it as target for twistor string theory. [D.W. Chiou, O.J. Ganor, Y.P. Hong, B.S. Kim, I. Mitra, 05]

16 Cauchy-Riemann Supertwistors Clearly, R 3 8 S 2 cannot be a complex manifold, but... it s a so-called Cauchy-Riemann supermanifold. Let X be a smooth (super)manifold with dimrx = d 1 d 2. A CR structure is a complex involutive subbundle D TCX of rank m 1 m 2 with D D = {0}; if d i = 2m i we talk about a complex (super)manifold. On R 3 8 S 2 one can introduce several CR structures, e.g., F (R 3 8 S 2, D 0 ) = R 1 0 C 1 4 CP 1, F 5 8 (R 3 8 S 2, D = π 2 T 0,1 P 2 4 ), which we call CR supertwistor space.

17 T -Flat Vector Bundles To introduce super gauge theories, we first need some preliminaries. Let X be as before and T TCX be an involutive subbundle with rank(t T ) = const. The exterior derivative d T is given by Γ(X, Λ q T X ) d Γ(X, Λ q+1 T X ) Γ(X, Λ q+1 T ). Let E X be a complex vector bundle. A T -connection is a map D T : Γ(X, Λ q T E) Γ(X, Λ q+1 T E) locally given by D T = d T +A T with A T Γ(X, T End E). D 2 T induces F T Γ(X, Λ 2 T End E); if F T = 0 then E is said to be T -flat.

18 Partially Holomorphic Chern-Simons Theory Consider the following short exact sequence (on F 5 8 ): 0 π 2 T 0,1 P 2 4 T π 2 (T F 5 8 /T P 2 4 ) 0 Since, P 2 4 is CY a holomorphic volume form Ω; π 2 Ω is globally well-defined on F 5 8 and allows to integrate over objects holomorphic in the fermionic coordinates. Consider a complex vector bundle E F 5 8 which is CP 1 -trivial and equipped with A T Γ(F 5 8, Tb End E) holomorphic in the fermionic coordinates.

19 Partially Holomorphic Chern-Simons Theory Then S phcs = π 2 Ω tr ( A T d T A T A T A T A T ) The resulting e.o.m. are d T A T + A T A T = 0, that is, the bundle E F 5 8 is T -flat. The geometry of F 5 8 fixes the superfield expansions of A T (A α β, Φ, χ i α, χ i α, φ ij, G α β) (up to super gauge transformations).

20 Super Bogomolny Model We get S sb = { d 3 x tr ɛ α δɛ ) β γ G γ δ (f α β + i 2 D α β Φ + + i ɛ α δɛ β γ χ i αd δ β χ i γ φ ij φ ij 1 2 ɛ α δχ i α[ χ i δ, Φ] ɛ α γ φ ij {χ i α, } χ j γ } [φ ij, Φ][φ ij, Φ]. The e.o.m. are f α β = i 2 D α β Φ,...

21 Dolbeault vs. Čech Here, I ve chosen the Dolbeault picture of CR vector bundles for the derivation of all equations. However, you can also work in Čech language as (E, f = 1, d T +A T ) (Ẽ, f, d T ) [A.D. Popov, C. Sämann, M. Wolf, 05]

22 Missing Box Motivation Question: What s the missing box? phcs theory on F 5 8 supersymmetric??? on P 2 4 Bogomolny model on R 3 Answer: It s a holomorphic BF theory. [A.D. Popov, 99]

23 Holomorphic BF Theory In fact, consider a trivial complex vector bundle E P 2 4 together with the action functional { } S hbf = Ω tr B( A 0,1 + A 0,1 A 0,1 ), where A 0,1 and B are End E-valued and holomorphic in the fermionic coordinates; A 0,1 doesn t contain components along antiholomorphic fermionic directions. The resulting e.o.m. are A 0,1 + A 0,1 A 0,1 = 0, A B = 0, i.e., the bundle is holomorphic (δ g A 0,1 = A ω). Next, one can do the superfield expansions to eventually get the super Bogomolny model on R 3.

24 Deformations Motivation Now one may also study deformations of the CR structure on F 5 8 (respectively, of the complex structure on P 2 4 ). Doing this in a very particular fashion, one obtains a similar correspondence between theories but this time with massive spinors and scalars. [D.W. Chiou, O.J. Ganor, Y.P. Hong, B.S. Kim, I. Mitra, 05] [A.D. Popov, C. Sämann, M. Wolf, 05]

25 Signs Of Integrability In Quantum N = 4 SYM Theory First signs of quantum integrability in SU(N) N = 4 SYM theory have been discovered by Minahan and Zarembo in the large N-limit. [J.A. Minahan, K. Zarembo, 02] It has then been shown that it s possible to interpret the full one-loop dilatation operator as Hamiltonian of an integrable quantum spin chain. [N. Beisert, M. Staudacher, 03]

26 Signs Of Integrability In Quantum N = 4 SYM Theory Another development which has pointed towards integrable structures was triggered by the AdS/CFT conjecture and initiated by Bena, Polchinski and Roiban. They showed that the classical Green-Schwarz superstring on AdS 5 S 5 possesses an infinite number of conserved nonlocal charges. [I. Bena, J. Polchinski, R. Roiban, 03] Within the spin chain approach, Dolan, Nappi and Witten related these nonlocal charges for the superstring to a corresponding set of nonlocal charges in the gauge theory; they found also field theoretic expressions of these charges at zero t Hooft coupling. [L. Dolan, C. Nappi, E. Witten, 03]

27 Charges From First Principles? It s certainly reasonable to ask whether such charges in the gauge theory can be understood from first principles, i.e., is there some non-perturbative approach for their construction? As a modest step towards an answer, let s first study a simplification namely its self-dual truncation. In the following, I ll explain how one can (at least classically) construct hidden symmetry algebras (and hence an infinite number of conserved nonlocal charges) in (self-dual) SYM theory via the supertwistor correspondence. [M. Wolf, 04]

28 Starting Point Motivation Above, we ve seen that via the PW transform we got a bijection: M hol (P 3 N ) [f ] [A] M N SDYM However, we can associate with any open subset U U + U, where P 3 N = U + U, an infinite number of such [f ]. Each class [f ] corresponds to a class [A] and vice versa.

29 Starting Point Motivation Question: How can this observation help us to learn something about symmetry algebras of the e.o.m. of our super gauge theory on space-time? Answer: The key ingredient is to study infinitesimal deformations of our holomorphic vector bundles on supertwistor space!

30 Infinitesimal Deformations Consider E P 3 N. Then Kodaira tells us that any infinitesimal deformation is allowed, as small enough perturbations of E will preserve its trivializability properties on the curves CP 1 x,η P 3 N. A generic deformation looks like and thus, δ : f + δf + = ɛ a δ a f + f + + δf + = (ψ + + δψ + ) 1 (ψ + δψ ).

31 Infinitesimal Deformations As the trivializability properties of the bundle are preserved, δf + leads to solving the infinitesimal Riemann-Hilbert Problem (on CP 1 ) ϕ + ψ + (δf + )ψ 1 = φ + φ, δψ ± = φ ± ψ ±. Thus, upon linearization of the linear system and by virtue of the PW transform we find PW : [δf ] [δa]: dλ + + α φ + δa α α =, δa i α dλ + i = +φ +. S 1 2πiλ + S 1 2πiλ + λ α + λ α +

32 Infinitesimal Deformations For δf + = ɛ a δ a f +, let {δ a } be a set of infinitesimal deformations satisfying [δ a, δ b } = C ab c δ c. Question: What is the corresponding symmetry algebra of the supergauge theory on R 4?

33 Kac-Moody And Superconformal Symmetries Let g be some (finite-dimensional) Lie superalgebra with [X a, X b } = f ab c X c and define the following perturbation: δ m a f + λ m +[X a, f + ], m Z Then, it s easy to see that [δ m a, δ n b } = f ab c δ m+n c, i.e., we get a centerless Kac-Moody algebra. Furthermore, by virtue of the PW transform we eventually obtain the same algebra on the (self-dual) SYM side.

34 Kac-Moody And Superconformal Symmetries Another example is affine extensions of superconformal symmetries: The generators of the superconformal algebra can be realized as vector fields on superspace-time. Their pull-back to supertwistor space yields an action of the superconformal group on the transition functions of our vector bundles. In a similar manner as above, one then obtains Kac-Moody-Virasoro type algebras on the (self-dual) SYM side. [M. Wolf, 04]

35 Conclusions What we ve got: We saw how twistor correspondences can help to understand super gauge theories in three and four space-time dimensions. In particular, I discussed massless/massive super Bogomolny theories in 3D and introduced hbf and phcs theories all being equivalent. I ve shown how to construct hidden symmetry algebras of the self-dual SYM equations by using supertwistors. I exemplified the discussion by studying affine extensions of global (gauge type) and space-time symmetries.

36 Outlook What s next? One can generalize the above construction to the full SU(N) N = 4 SYM theory. But how? Let s choose Z = L 5 6, which is the superambitwistor space. It s given by the locus z α µ α w α λ α + 2θ i η i = 0 in P 3 3 P 3 3. Geometrically, it s the space of complex super light-rays in C 4 12, i.e., point p L 5 6 super light-ray C 1 6 p C 4 12 (CP 1 CP 1 ) x,η,θ L 5 6 point (x, η, θ) C 4 12

37 Outlook What s next? CP 1 CP 1 -trivial holomorphic vector bundles E L 5 6 give rise to the e.o.m. of N = 3 SYM theory on Minkowski space, which is fully equivalent to the N = 4 theory. [E. Witten, 78] Now one may proceed and study infinitesimal deformations of E to eventually obtain hidden symmetries. One again finds affine extensions of global (gauge type) and superconformal symmetries. Furthermore, one obtains an infinite number of conserved non-local currents and hence charges, expressed by certain Wilson lines. [A.D. Popov, S. Uhlmann, R. Wimmer, M. Wolf,??]

38 Outlook What s next? Next, one should quantize these charges to see whether they are still conserved after including quantum corrections. Certainly, one might only hope these charge to be conserved in the large N-limit of the gauge theory. Hopefully, it will also be possible to relate these charges to those obtained by the spin-chain approach and in particular to the quantum Yangian.... [L. Dolan, C. Nappi, E. Witten, 03]