Abstract. We study a family of BPS Wilson loops in N = 4 SYM theory that is defined on the hyperbolic H3 subspace of Minkowski space R1,3. More precisely we ...
BPS Wilson loops on Hyperbolic Space Fabrizio Pucci Department of BioModeling, BioInformatics & BioProcesses, Universit´e Libre de Bruxelles, Roosevelt Avenue 50, 1050 Brussels, Belgium Abstract We study a family of BPS Wilson loops in N = 4 SYM theory that is defined on the hyperbolic H3 subspace of Minkowski space R1,3 . More precisely we compute perturbatively the expectation value (VEV) of certain 1/8 BPS operators supported on loops with compact shape that lie on H2 ⊂ H3 . An agreement with the VEV of the analogous observables in the zero instanton sector of YM2 on H2 has been found.
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Introduction
Supersymmetric Wilson loops are very interesting non-local operators that give us important information about the AdS/CFT correspondence [1, 2, 3]. In order to probe this duality many examples of these operators have been considered in N = 4 Supersymmetric Yang Mills Theory (SYM) and their systematic classification has been given by Pestun and Dymarsky in [4]. One of the most famous example that has been analyzed in great detail is the 1/2 BPS circular Wilson loop. In [5, 6], from the crucial observations that the gauge-scalar propagator between two points on the loop is a constant and that at g 4 order interactions cancel, the expectation value of this operator is conjectured to be captured by a Gaussian Matrix Model. This matrix model can be exactly solved and in the large λ limit the result nicely agrees with the solution to the minimal area problem [7, 8]. Successively this conjecture has been rigorously proven in an interesting paper [9] where the author shows that the path integral for the observable localizes on a finite dimensional space and reduce to the Gaussian Matrix Model. In order to generalize this operator, a new class of BPS Wilson loops has been introduced in [10, 11]. These loops are supported on a three-sphere and differently from the circular operator they couple to three of the six scalars of the theory. They can be written as I ¡ ¢ 1 i W = Tr P exp dxµ iAµ − σµν xν M i I ΦI , (1) N where the 3 × 6 dimensional matrix M i I satisfies M M T =Id while σ’s are basically the same as ’t Hooft’s η symbols used in writing down instanton solutions. The requirement that the supersymmetry variation of these loops vanishes for arbitrary curves on S 3 leads to some equations that can be consistently solved showing that a generic curve preserve 1/16 of the original supersymmetries.
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When the loop lies on a great S 2 inside S 3 , the situation becomes more interesting [10, 11]. An enhancement of preserved supersymmetries occurs and the Wilson loop operators that belong to this class are generically 1/8 BPS object. Moreover there are strong evidences that their expectation values are captured by the Wilson loop computation in the zero instanton sector of the ordinary bosonic 2d Yang Mills theory on a two-dimensional sphere. This relation is suggested in [10, 11] where, considering an operator supported on a generic curve inside S 2 , the authors performed a perturbative first order computation finding hWN =4 i = 1 + g 2 N
A1 A2 + O(g 4 ) , 2 (A1 + A1 )2
(2)
where A1 and A2 are the two areas determined by the loop. After identifying the 2 four-dimensional coupling constant g with the two-dimensional one g2d through the relation 2 g2d =−
g2 , (A2 + A1 )
(3)
one can see that the result (2) coincides with the perturbative expansion of the matrix model that describes the VEV of the Wilson loop supported on the same curve in the zero instanton sector of YM2 on S 2 [12] : µ ¶ · 2 ¸ 1 A1 A2 g A1 A2 2 hWYM2 i = L1N −1 g2d exp − 2d , (4) N (A1 + A2 ) 2 (A1 + A2 ) where L1N −1 (x) is a Laguerre polynomial. In [13, 14] a two loops analysis of this conjecture has been performed, finding an exact agreement between the N = 4 SYM calculation and the YM2 prediction. From these results one could suppose that a localization procedure like the one presented in [9] for the Maldacena-Wilson loop could also apply to this more general class of operators. Indeed in [15] the author has shown that the 4d path integral localizes to the semi-topological Hitchin/Higgs-Yang-Mills (HYM) theory and that at least perturbatively the computation of the operators in HYM agrees with the computation in the zero instanton sector of YM2 . The relation between the N = 4 SYM in four dimension and the ordinary bosonic YM2 has been studied intensively also in [16, 17, 18, 19, 20, 21] where the equivalence has been extended to the cases of correlators of Wilson loops and of correlators between Wilson loops and local operators. Furthermore in [22, 23] the authors have conjectured that in N = 4 SYM the insertion of ’t Hooft operators on a maximal circle on S 2 is captured by the non-zero (unstable) instanton contributions to the partition function of the 2d Yang-Mills theory. Finally in [24] the magnetic dual operators of this class of observables have been introduced. In this paper we discuss a family of Wilson loops in Minkowski space-time that has been introduced in [25]. These loops are supported on curves that lie on a three dimensional hyperbolic space H3 (Euclidean AdS3 ) and generically 2
preserve two supercharges that are combinations of super-Poincar´e and superconformal charges (1/16 BPS operators). Here we will analyze a sub-class of these operators considering only loops that are restricted on a two-dimensional hyperbolic space H2 ⊂ H3 (1/8 BPS operators). A first order computation of the VEV of these observables suggests the equivalence with the VEV of the ordinary Wilson loop in the bosonic YM2 on a H2 whose expectation value is captured by the expression [25][26]
hWYM2 i =
µ ¶ · 2 ¸ 1 1 g A(A + 4π) 2 A(A + 4π) LN −1 g2d exp − 2d , N 4π 2 4π
(5)
after the identification of the 2d coupling constant with the four dimensional one trough the relation g2 , (6) 4π and where A is the finite area delimited by the loops. In this paper we perform a perturbative analysis of the expectation value of these operators in order to test the conjecture at least up to the g 4 perturbative order. The plan of the paper is the following : in section 2 we briefly review the construction and the supersymmetry properties of this class of Wilson loops. In section number 3 we compute numerically up to the g 4 order the VEV of a 1/8 BPS Wilson operator that belongs to this family and we show the agreement with the YM2 prediction. The last section is dedicated to the summary. There are two technical appendices, in the first one we show how the divergences cancel in the intermediate steps of the computation while in the second one all the O(g 4 ) interaction diagrams are written explicitly. 2 g2d =
2
Supersymmetric Wilson loops on H2
We consider a class of Wilson loops defined on hyperbolic sub-manifold of the Minkowski space [25] identified by the constraint −x20 + x21 + x22 + x23 = 1.
(7) 3
Essentially these operators are constructed from those that live on S by an appropriate analytic continuation in the scalar couplings. Indeed in order to define the three scalar fields that couple to the loop, the invariant one-form i σ = σµν xµ dxν introduced in [10] can be rotated in ω as ω1 ω2
= x0 dx1 − x1 dx0 + i(x2 dx3 − x3 dx2 ) = x0 dx2 − x2 dx0 + i(x3 dx1 − x1 dx3 )
ω3
= x0 dx3 − x3 dx0 + i(x1 dx2 − x2 dx1 ).
Then in terms of this one-form one can construct the modified connection as 3
A˜ = Aµ dxµ + i ωi MIi φI with
(8)
MIi
a 3×6 norm-preserving matrix and define the Wilson loop as I 1 ˜ W = Tr P exp i A. (9) N Being ²(x) = ²0 + xµ γµ ²1 the conformal Killing spinor that generates the superconformal transformations, with ²0 and ²1 two constant sixteen-component Majorana-Weyl spinors, the variation of a generic loop on H3 can be recast in the following form i I + iτ i ²+ 1 = MI ρ ²0
(10)
− ²− 1 = ²0 = 0
(11)
where τ i are the Pauli matrices, ρI ’s are the gamma matrices belonging to the ± 1 1 1 Clifford Algebra of SO(6), ²± 1 = 2 (1 ± γ5 ) ²1 and ²0 = 2 (1 ± γ5 ) ²0 . From (10) and exploiting a little bit of Diracology one can show that there are only two + + independent constraints on ²+ 1 while ²0 is fully determined by ²1 . This make us conclude that an operator supported on a generic circuits is 1/16 BPS. An enhancement of the supersymmetries can occur if the shape of the loop satisfies certain geometrical conditions. For example if the loop is restricted to the twodimensional hyperbolic space H2 ⊂ H3 the supersymmetries preserved are four (1/8 BPS) while the hyperbolic line and the circle with arbitrary radius in H2 are 1/2 and 1/4 BPS objects respectively. In order to approach the two-loop analysis of the conjecture, the generic circuit on H2 is quite complicated. Thus we decide to study numerically a Wilson loop defined on a circuit C made by two finite rays with a cusp in the origin plus an arc that close the contour (see fig.1). C is composed by three different edges that can be parameterized as: xµ (t) = (cosh t, sinh t, 0, 0) yµ (t) = (cosh t, − sinh t cos δ, − sinh t sin δ, 0) zµ (t) = (cosh θ, sinh θ cos t, sinh θ sin t, 0)
−θ
