The Spin Connection of Twisted Geometry

The spin connection of twisted geometry

Hal M. Haggard,∗ Carlo Rovelli,† and Wolfgang Wieland‡ Centre de Physique Th´eoriquede Luminy,§ Case 907, 13288 Marseille, France

Francesca Vidotto¶ Institute for Mathematics, Astrophysics and Particle Physics, Radboud University Faculty of Science, Mailbox 79, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands (Dated: January 15, 2013) Twisted geometry is a piecewise-flat geometry less rigid than Regge geometry. In Loop Gravity, it provides the classical limit for each step of the truncation utilized in the definition of the quantum theory. We define the torsionless spin-connection of a twisted geometry. The difficulty given by the discontinuity of the triad is addressed by interpolating between triads. The curvature of the resulting spin connection reduces to the Regge curvature in the case of a Regge geometry.

I. INTRODUCTION and Γ = Γ(e) is the spin connection determined by the triad e via the first Cartan structure equation, namely the condition of vanishing torsion. As pointed out in Twisted geometry [1–4] is a discrete (piecewise-flat) [4], the same decomposition is not easily achieved in the geometry found in loop gravity. Here we define and com- discrete setting, because the Cartan equation does not pute the torsionless spin connection of a twisted geome- make sense on the boundary between piecewise-flat cells. try. Therefore a twisted geometry is a generalization of a 3d In loop gravity, the quantities determining the 3d ge- metric space for which a notion of spin connection has ometry of physical space are non-commuting quantum not yet been given. This has made the separation of the operators [5, 6], therefore a quantum geometry is never part of the connection that codes the extrinsic curvature a classical geometry (discrete, twisted, polymeric or oth- from the part that doesn’t problematic. In this paper, we erwise): no more than a quantum particle with spin is a fill this gap, providing a definition of Γ(e) that remains classical rotating sphere. But the notion of twisted geom- meaningful in twisted geometry. etry is nevertheless a powerful tool, because it provides the classical limit for each step in the truncation utilized in the definition of the quantum theory [7, 8]. It is there- II. DEFINITION OF THE CONNECTION fore similar to the picture of the states of a quantum field theory as configurations of n classical particles. By a twisted geometry we mean: an oriented 3d sim- The basic operators of the loop theory are the flux op- plicial complex (a triangulation) T , equipped with a flat erators, which define the 3d geometry, and the holonomy metric on each 3-simplex (which makes it a flat tetrahe- operators, which define an SU(2) connection on the same dron), along with the condition that for any two tetrahe- 3d space [9–11]. Since the two are independent, the con- dra sharing a face the area of the face is the same whether nection in general has torsion, as is the case in the con- it is computed from the metric on one side or the other tinuous (Ashtekar-Barbero) Hamiltonian theory: SU(2)- ( FIG. 1 ). If in addition we require the length of the connection degrees of freedom are independent from the edges to be the same, we have a Regge geometry. If not, 3d-metric degrees of freedom. The mismatch between we have a non-Regge twisted geometry.1 this connection and the spin connection determined by the intrinsic geometry (namely, by definition, the torsion) codes the information about the extrinsic curvature, which is the canonical variable conjugate to the intrinsic 3-geometry. In the continuum theory, the SU(2) connection A is neatly formed by two parts: A = Γ + γK, where γ is the Barbero-Immirzi parameter, K the extrinsic curvature FIG. 1: In a twisted geometry two adjacent triangles have the same area and the same normal, but the angles and the edge lengths can differ. The two triangles can be identified, at the §Unitéde recherche (UMR 6207) du CNRS et du Aix-Marseille price of a discontinuity of the metric across the triangle. Université;affiliéàla FRUMAM (FR 2291).

∗Electronic address: [email protected] †Electronic address: [email protected] ‡Electronic address: [email protected] 1 This deﬁnition is slightly stronger than the one emerging from ¶Electronic address: [email protected] the classical limit of loop quantum gravity, since it ﬁxes the full 2

In general, the metric is discontinuous across a triangle by a foliated 3d region τ × [0, ∆] where z ∈ [0, ∆], see τ. Therefore there are two distinct flat metrics induced Fig. 2. Now, we can interpolate the triad by e(z), such on the same face: one from the left tetrahedron and one that e(0) = 1l and e(∆) = e. The (finite) holonomy of from the right tetrahedron. The twisting of the geometry the connection across the face, U(e), can be defined as measures the difference between these two metrics. Since the ∆ → 0 limit of the holonomy of the spin connection a 2d flat metric is determined by three numbers, and, by of e(z), which is calculated across the thickened triangle. definition, the two metrics define the same area, there There is a highly nontrivial condition on the interpolat- are two twisting parameters. ing triad: the resulting holonomy must transform as an Let us setup a coordinate system (x, y, z) covering the holonomy under a change of frame on either tetrahedron. two tetrahedra bounding a triangle face τ. It is con- That is, venient to choose these coordinates so that the triangle −1 −1 τ is at z = 0. Without loss of generality, we can always U(ΛseΛt ) = ΛsU(e)Λt (4) choose the coordinate system and the triad in such a way that the triad is cartesian, namely ei = dxi, on the left for any Λs, Λt ∈ SO(3). An interpolating triad that sat- tetrahedron. Then the discontinuity of the metric implies isfies this condition can be obtained starting from the that the triad on the right hand side tetrahedron can be polar decomposition of e chosen to have the constant form e = eAeS (5) e1 = e1 dx + e1dy, e2 = e2 dx + e2dy, e3 = dz, (1) x y x y where A is antisymmetric and S is symmetric, by writing The condition that the area is the same from both sides gives det e = 1. Therefore the matrix e(z) = ezAezS. (6)  1 1  ex ey 0 i 2 2 This defines a continuous triad joining the two tetra- e = {e a} =  ex ey 0  (2) 0 0 1 hedra, differentiable in (0, 1). Here and below we take ∆ = 1 because, as can be checked, the holonomy is independent of ∆, making the ∆ → 0 limit trivial. We can is in SL(3, R), or, more specifically it is in the SL(2, R) now compute the spin connection of the interpolating re- upper block diagonal subgroup of SL(3, R). The geometrical interpretation of these groups is gion. This defines a torsionless spin connection on the straightforward: e is the linear transformation that sends twisted geometry. a cartesian triangle with the dimensions given by the left metric into the cartesian triangle with the dimensions FIG. 2: We “thicken” the triangle in order to smooth-out given by the right. In other words, e is the linear trans- the discontinuity. The path γ formation that makes the two triangles of FIG. 1 match. goes from one tetrahedron to Since the triangle is two dimensional, this linear transfor- the other through the thick- mation can always be chosen in the SL(2, R) subgroup. ened region. On a Riemannian space, once we choose a triad field ei, then the torsionless Cartan spin connection is the unique Let us compute this connection explicitly. From the solution of the first Cartan structure equation last equation, we have i i j k de + jk ω ∧ e = 0. (3) i zA −zA i j de = (A + e Se ) j dz ∧ e . (7)

On a twisted geometry this definition does not make Using this in the Cartan equation (and lowering an index) sense, because of the discontinuity of the triad on the we have triangles that makes dei ill defined. To define the con- zA −zA j j k nection on the twisted geometry, we therefore extend this (A + e Se )ij dz ∧ e = −ijk ω ∧ e . (8) equation “across” the triangle, where ei is a discontinuous field. One can check that the solution of this equation is given For this purpose, let us “thicken” the triangle, in order by to smooth-out the discontinuity, replacing the triangle τ i i j ω = B j e (9)

where triangulation and not just its dual graph. Also, the deﬁnition i ikl zA −zA z 1 klm z i given here refers only to the intrinsic geometry. The full deﬁ- B j = − (A + e Se )jkel + Aklem δj, (10) nition of the twisted geometry that appears in quantum gravity 2 includes also the extrinsic curvature, which plays no role here. z Finally, for simplicity we restrict our attention to triangulations, where ei is a matrix element of the inverse triad. but the results presented extend to generic cellular decomposi- What is relevant for us is, of course, only the holonomy tions (and therefore to polyhedra other than tetrahedra). of this connection across the thickened region. Consider 3 a path crossing this region at constant x and y. The and so on cyclically. These equations can be inverted, holonomy of ωi across the region is given by giving the vectors as functions of the normals:

R R 1 − ω − ω(∂z )dz 2 U = P e γ = P e 0 . (11) v1 = n2 × n3, (19) 3V Observe now that since e(z) ∈ SL(2, R) ⊂ SL(3, R) it where V is the volume of the tetrahedron, given by follows that (A + ezASe−zA) is upper block diagonal and so is B, and therefore ω is determined just by the second 1 r 2 z V = (v × v ) · v = (n × n ) · n . (20) term in (10). Explicitly, 3! 1 2 3 9 1 2 3

k 1 kij Say we are interested in the face f deﬁned by the vec- ω (∂z) = Aij (12) 2 tors v1 and v2, or equivalently by the normal n3. It is convenient2 to use the linear but non-orthogonal coordi- So that nates adapted to the face, determined by the triple ua = (v1, v2, nˆ3), where nˆ3 = n3/|n3|. That is, we use coordi- U = exp A (13) a nates x = (x, y, z) deﬁned by x = x v1 + y v2 + z nˆ3. It that is, the holonomy is precisely the orthogonal matrix is immediate to see that in these coordinates the metric in the polar decomposition of e. For the explicit form of of the tetrahedron is given by the polar decomposition, we have then  2    |v1| v1 ·v2 0 a b 0 T −1/2 2 U(e) = e(e e) (14) g =  v1 ·v2 |v2| 0  ≡  c d 0  . (21) 0 0 1 0 0 1 where eT is the transpose of e. Since U(e) is indepen- 2 2 2 2 dent from the size of the interpolating region, taking the Notice that |v1| |v2| − (v1 · v2) = (2A) (so that limit ∆ → 0 is immediate. The resulting distributional det g = 4A2). Without loss of generality, we can orient torsionless spin connection is concentrated on the face the cartesian frame (in both the left and right tetrahedra) τ :(σ1, σ2) 7→ xa(σ) and is given by so that

Γ = −A dτ (15) v1 = (a, 0, 0) (22) v = (b, c, 0) (23) where the distributional one-form of the triangle is de- 2 ﬁned by nˆ3 = (0, 0, 1) (24)

Z b c where 2 ∂x ∂x dτa(x) ≡ d σ 1 2 abc δ(x − x(σ)). (16) v1 ·v2 τ ∂σ ∂σ a = |v1|, b = , (25) |v1| The gauge invariance (4) can be veriﬁed by using the fact p|v |2|v |2 − (v ·v )2 that both the left and the right polar decomposition give c = 1 2 1 2 . (26) rise to the same rotation matrix. |v1| Now, observe that a triad for this metric is precisely

i i i i III. CONNECTION AS A FUNCTION OF THE e = v1dx + v2dy +n ˆ3dz, (27) NORMALS that is, In this section we compute the connection U in terms  a 0 0  of the normals to the faces of the tetrahedra, which are e = {ei } = b c 0 . (28) the basic variables defining the twisted geometry in loop a   0 0 1 gravity. Let e be a triad in the right tetrahedron ande ˜ the one in the left tetrahedron, in the same coordinate The left triade ˜ is given by by the same expression for the system. The interpolating map is given by the SL(2, R) left tetrahedron, which we indicate here by tilded quan- block diagonal matrix tities. Therefore the SL(3, R) matrix s that transforms the left triangle into the right one is s = ee˜−1 (17)  a/a˜ 0 0  and the holonomy is U(s). Consider a tetrahedron de- s = {ei (˜e−1)a } = (bc˜ − c˜b)/a˜c˜ c/c˜ 0 (29) fined by the triple of vectors v ∈ R3, a = 1, 2, 3. The a j   a 0 0 1 normals to the faces defined by two of these vectors, nor- malized to the area of the face, is given by 1 n = v × v , (18) 2 1 2 2 3 Notwithstanding the dimension mismatch. 4

The orthogonal part of the polar decomposition of this between the curvature of ω(e) and the Riemann curva- matrix is, with some algebra, a rotation in the xy plane ture: if the connection satisfies the Cartan equation, then ij ij i kj with angle determined by its curvature F = dω +ω k ∧ω is related to the Rie- √ √ mann tensor of the Riemannian manifold defined by the ˜ cos(θ) = (ca˜ + ac˜)/ D, sin(θ) = (bc˜ − cb)/ D, metric g = e ei by (30) ab ai b D =c ˜2(a2 + b2) + c2(ã2 + ˜b2) + 2cc˜(aa˜ − b˜b). 1 F ij[ω(e)] = ei ejd Rc [g(e)] dxa ∧ dxb. (33) The holonomy U is a rotation in the plane of the face 2 c dab by this angle, where a, b, c, a,˜ ˜b andc ˜ are given explicitly above in terms of the normals. Finally, the torsionless In the general twisted case, the curvature may not be of spin connection is the characteristic Regge form e δabel f Γ = θ e(nˆ3) dτ. (31) Rabcd ∼ e cdf l . (34)

This gives the torsionless connection explicitly in terms where ~l is the bone on which the curvature is concen- of the normals ni, which are the independent variables trated. In fact, investigating the general form of the in the loop-gravity twisted-geometry framework. curvature tensor arising from the connection presented here may give insights into the type of generalization that twisted geometries provide. For example, it may be pos- IV. CURVATURE sible to characterize what Petrov classes are possible in a twisted geometry and to see if they are more general Let Ul be the holonomy of the connection Γ around than the single class that Regge geometry captures. a circle that surrounds a bone l. Recall that the Regge P deficit angle δl of a bone l is defined as δl = 2π − i θi where θi are the dihedral angles at l of the (d − 1)- V. CLOSING CONSIDERATIONS simplices in the link of l. The following holds: Proposition: If the twisted geometry is Regge, then Ul We have defined a connection Γ in the context of i is a rotation around the axis e (l), by an angle equal to twisted geometry. This is determined by the normals to the Regge deficit angle. the triangles of the tetrahedra. It reduces to the standard To show this, note that the holonomy Ul can always spin-connection in the Regge case, where its curvature be decomposed into a product of contributions from each gives the Regge deficit angle. tetrahedron meeting at l. In turn, the tetrahedral con- The result reinforces the twisted geometry construc- tributions can be further decomposed into a product of tion, and its interpretation as a classical limit of a trun- two pieces: the holonomy coming from crossing the ini- cation of quantum gravity. tial triangle τ upon entering the tetrahedron, U , and i τi The construction should also contribute to dispelling the holonomy arising from changing frames within the two possible sources of confusion. The first is the idea tetrahedron σi in order to adapt to the triangle through that the twisting might code torsion. It does not, since a which the path leaves the tetrahedron, U , thus, σi torsionless connection can be defined in the presence of twisting. The key point is that twisting is a purely metric Ul = Uσ Uτ ··· Uσ Uτ . (32) n n−1 1 1 notion: it refers to discontinuities in the metric, and it is When the geometry is Regge the triangles all have match- determined by the property of the metric space defined ing shapes and each of the contributions Uτi are the iden- by the discrete geometry. Torsion, on the other hand, is tity. Meanwhile, the changes of frame within each tetra- not a purely metric notion: a metric does not define tor- hedron bring the initial triangle’s inward normal into the sion. It is only the existence of a connection independent final triangle’s outward normal and this is just a rota- from the metric that can determine a torsion. There- tion about the bone by the dihedral angle, θi. Thus the fore twisting cannot define torsion. The idea of relating transport around the loop, Ul, amounts to rotating the twisting and torsion, although intuitively attractive, is orginal frame by δl just as in Regge calculus. misled. Put more simply, the point is that for a Regge geom- The second confusion is the idea that twisting needs etry the spin connection defined here simply agrees with to be suppressed in order to recover the classical limit of the spin connection which is defined directly by the fact general relativity. A twisted geometry is a generalization that there is a flat metric without discontinuities around of a Regge geometry. It is a discretization of a metric the bone. This characterization of a Regge geometry is space that is distinct and no less honorable than Regge explicit when that geometry is viewed as arising by re- geometry. moving the (d − 2)-skeleton of a triangulation from a The conditions under which a twisted geometry re- d-dimensional manifold M [12]. duces to the Regge case have been studied [13, 14]. At- The proposition shows that in the Regge case the con- tempts to relate these to the vanishing of the torsion of nection defined agrees with the standard torsionless Car- the four-dimensional spin connection, and therefore to tan connection. It is the discrete analog of the relation the simplicity constraints of general relativity have been 5 explored [14, 15]. But twisting appears in the classi- ——– cal limit of the standard time-gauge Hamiltonian theory, where there are no residual simplicity constraints to deal We thank Simone Speziale for extensive discussions with. Therefore there is no reason for the simplicity con- on twisted geometries. We thank Luca Fabbri for hav- straints to suppress twisting. Of course, one can assume ing questioned the role of torsion in twisted geome- that the classical limit of discrete general relativity must tries. HMH acknowledges support from the National Sci- be Regge geometry, but the results presented here put ence Foundation (NSF) International Research Fellow- into question the need for this assumption. In particu- ship Program (IRFP) under grant OISE-1159218. FV lar, there is no clash between the existence of twisting acknowledges support from the Netherlands Organisa- and the possibility of defining the discrete version of the tion for Scientific Research (NWO) Rubicon Fellowship first Cartan equation. Program. Twisted geometry is a bona fide discretization of 3d geometry.

[1] L. Freidel and S. Speziale, “From twistors to twisted arXiv:1102.3660. geometries,” arXiv:1006.0199. [9] C. Rovelli, Quantum Gravity. Cambridge University [2] L. Freidel and S. Speziale, “Twisted geometries: A Press, Cambridge, U.K., 2004. geometric parametrisation of SU(2) phase space,” [10] T. Thiemann, Modern Canonical Quantum General Phys.Rev. D82 (2010) 084040, arXiv:1001.2748. Relativity. Cambridge University Press, Cambridge, [3] M. Dupuis, J. P. Ryan, and S. Speziale, “Discrete U.K., 2007. gravity models and Loop Quantum Gravity: a short [11] A. Ashtekar and J. Lewandowski, “Background review,” arXiv:1204.5394. independent quantum gravity: A status report,” Class. [4] S. Speziale and W. M. Wieland, “The twistorial Quant. Grav. 21 (2004) R53, arXiv:gr-qc/0404018. structure of loop-gravity transition amplitudes,” [12] E. Bianchi, “Loop Quantum Gravity `ala arXiv:1207.6348. Aharonov-Bohm,” arXiv:0907.4388. [5] C. Rovelli and L. Smolin, “Discreteness of area and [13] B. Dittrich and S. Speziale, “Area-angle variables for volume in quantum gravity,” Nucl. Phys. B442 (1995) general relativity,” New J. Phys. 10 (2008) 083006, 593–622, arXiv:gr-qc/9411005. arXiv:0802.0864. [6] A. Ashtekar, A. Corichi, and J. A. Zapata, “Quantum [14] B. Dittrich and J. P. Ryan, “Simplicity in simplicial theory of geometry III: Noncommutativity of phase space,” Phys. Rev. D82 (2010) 064026, Riemannian structures,” Class.Quant.Grav. 15 (1998) arXiv:1006.4295. 2955–2972, arXiv:gr-qc/9806041. [15] B. Dittrich and J. P. Ryan, “On the role of the [7] C. Rovelli and S. Speziale, “On the geometry of loop Barbero-Immirzi parameter in discrete quantum quantum gravity on a graph,” Phys. Rev. D82 (2010) gravity,” arXiv:1209.4892. 044018, arXiv:1005.2927. [8] C. Rovelli, “Zakopane lectures on loop gravity,”