Quantum Fields in Curved Spacetime Lecture 3 Finn Larsen Michigan Center for Theoretical Physics Yerevan, August 22, 2016.
Recap AdS 3 is an instructive application of quantum fields in curved space. The heat kernel for a (dimension, spin s) field is known exactly: K(t) = 1 e ( 1)2t (1 + 2s 2 t) (4πt) 3 2 UV divergences (small t) can be subtracted and absorbed into IR parameters without introducing IR ambiguities at large t. This gives the quantum effective potential in closed form V ren = ( ) 2s 1 12πl 3 A ( 1 3 3s 2 1 ) It represents a contribution to the cosmological constant due to a fluctuating quantum field. 2
Cosmological Constant Problem Our Universe exhibits accelerating expansion. The phenomenon can be interpreted as a cosmological constant encoded in the constant effective potential V eff = 0.8 (10 3 ev ) 4. The cosmological constant puzzle: this is tiny, given that physics at scale M contributes M 4 to V eff. Example: quantum fluctuations of a particle of mass m contributes m 4 to V eff and there are many (known and unknown) particles. The techniques in these lectures sharpens conceptual issues (and identifies precise functional dependence on m). So far: a single field (in 3D where V M 3 ). 3
AdS 3 /CFT 2 AdS/CFT correspondence identifies AdS 3 quantum gravity with a 2D CFT without gravity. The map identifies the obvious IR volume divergence of AdS 3 with the UV divergences of 2D CFT. Holographic renormalization: regularize the volume of spacetime and subtract local boundary terms. In odd bulk (even boundary) dimension a logarithmic divergence remains unsubtracted in the boundary theory. In the boundary theory it gives a trace anomaly, a special contribution to the energy momentum tensor T µ µ = c 12 R 4
The bulk quantum effective potential in AdS 3 is equivalent to the 2D central charge c c = 6πl 3 AV eff The central charge c is a pure number that counts the number of degrees of freedom (in the boundary theory without gravity) There is a (large) classical contribution from the AdS 3 background and we determine the (small) quantum correction. For AdS 3 sourced by bound states of N 1 D1 and N 5 D5 the central charge for large N 1 and N 5 is c = 6N 1 N 5 in both AdS 3 (determined from geometry of these objects) and CFT 2 (determined from quantum bound states of these objects). 5
SUSY Result The contribution to the central charge from a single degree of freedom: c = ( ) 2s1 2 ( 1 3 3s 2 1 ) The contribution from a full SUSY multiplet: c = 3( ) 2s So: there is a lot of cancellation for large, but not complete cancellation. A heavy SUSY multiplet with large gives a substantial correction to the central charge. 6
The KK tower In string theory, there is usually more structure. AdS 3 typically emerges from more dimensions through Kaluza-Klein compactification. For example, the D1/D5 system has a 6D origin as AdS 3 S 3, so that is AdS 3 with a partial wave expansion on S 3. In this context AdS 3 has an infinite KK tower of matter fields due to ever higher angular momentum. Their masses will be ever larger: 0, 0 + 1,.... The quantum corrections due to the KK tower diverge: there are infinitely many fields, some arbitrarily heavy. 7
Renormalization We have described all the matter as spectator fields, no matter how high their mass. This is wrong: the heavy ones deform spacetime. Filosophy: the heavy matter affects only the UV which we do not trust anyway. Standard strategy: regularize the divergence (ie, deform the UV theory in some arbitrary way). Then absorb the UV divergences into the original spacetime parameters (subtract them). The true quantum correction to the central charge c = 3( ) 2s ( ) ren 8
Results The string theory context specifies the precise values of. The correction becomes c = +6 after summing over the entire KK tower of modes. It corrects the classical result c = 6N 1 N 5 to c = 6(N 1 N 5 + 1) In string theory black holes are bound states of microscopic strings, roughly N 1 N 5 of them. The +6 is due to the overall center of mass motion of the bound state. No details here: the main point is that often there are two (or more) totally different ways to get the same result. 9
Four Dimensions Quantum corrections are more complicated in higher dimensions, at least technically. Some perspectives: - Loop integrals over more dimensions. - More local operators needed to subtract UV divergences. In D = 4 the constant term in the heat kernel is of order R 2 : K(t) = 1 (4πt) 2 ( 1 + c1 Rt + c 2 R 2 t 2 +... ) Upon integration (with measure dt 2t ) this gives a UV divergence that must be subtracted. But, in order to do so without changing the IR an arbitrary distance L cutoff must be introduced trace anomaly. 10
4D Trace Anomaly The 4D trace anomaly is much studied, especially in the context of 4D Conformal Field Theory IN curved space. Complication: the curvature symbol has four indices R αβγδ so there are multiple contraction giving scalars of order R 2. Standard notation: T µ µ = 2π µν δs g g δg = 1 µν 16π (cw µνρσw µνρσ ae 2 4 ), Euler density E 4 = R µνρσ R µνρσ 4R µν R µν + R 2. Square of Weyl tensor W µνρσ W µνρσ = R µνρσ R µνρσ 2R µν R µν + 1 3 R2. 11
4D Trace Anomaly These combinations are very special. The Euler invariant is topological (Gauss-Bonnet theorem) so it does not depend on metric at all: χ = 1 32π 2 d 4 xe 4 The square of the Weyl tensor does not depend on scale so the trace variation g µν δ δg µν vanishes. Therefore: subtractions of this form do not contribute to the trace of the EM tensor except through the cutoff L cutoff. The a and c coefficients are known as central charges. 12
Heat Kernel Computations So far: minimally coupled scalars. It is interesting to consider particles of various spins. Then the kinetic operator has spin indices Λ n m = I n m(d µ D µ ) (2ω µ D µ ) n m P n m The analysis of its spectrum (using the heat kernel equation) is more laborious but straightforward in principle. For fermions: take the square of the (generalized) Dirac operator and apply an overall minus sign. 13
Special Cases It is common to analyze the problem in spacetimes satisfying the vacuum Einstein equations. Then R µν = 0 so such considerations are sensitive to c a. A simple complementary spacetime: analyze AdS 4. The geometry is conformally flat so Weyl =0 but E 4 does not vanish. AdS 4 computations are sensitive to a but not c. 14
Central charges c and a Field c a c a Real Scalar Weyl Fermion Vector Antisymmetric Tensor 1 120 1 40 1 10 1 360 11 720 1 120 179 360 1 180 7 720 31 180-13 180 91 180 Gravitino 411 360 589 720 233 720 Graviton 783 180 571 180 53 45 These numerical coefficients contain interesting data about the quantum field theory. 15
Extended SUSY The most supersymmetric theory in 4D: N = 8 Supergravity. Conventional wisdom: this theory is finite to very high order (and perhaps no divergences at all). Let us check at one loop. The spectrum: 1 s = 2, 8 s = 3/2, 28 s = 1, 56 s = 1/2, 70 s = 0. Add anomaly coefficients for such fields c(n = 8) = 0 a(n = 8) = 5 2 This can only happen if there is a UV divergence: conventional wisdom is false. 16
The Gauss-Bonnet Invariant The UV divergence is a-type: proportional to the Euler density E 4 This term is topological upon integration: the Gauss-Bonnet invariant. Locally it is a total derivative so contributions to scattering amplitudes vanish due to a momentum conserving delta-function. Conventional wisdom: this term has no physical consequences. 17
The Gauss-Bonnet Invariant in AdS Let us compute it in AdS 4 : E 4 d 4 x = 24 Vol AdS4 So we regulate the volume but cutting off AdS 4. Physics: include local boundary terms, subtract on boundary, volume becomes finite. Mathematics: full Gauss-Bonnet invariant also has boundary terms that must be included. Either way χ = 1 32π 2 ( ) E 4 d 4 x + boundary = 1 In words: global AdS 4 (cut-off in any way) has nontrivial topology. 18
Quantum Inequivalence In 4D a massless antisymmetric tensor is classically equivalent to a scalar: H µνρ = 3 [µ B νρ] = ɛ µνρλ λ φ Is this true in the quantum theory? One test: check the table with anomaly coefficients: c as c scalar = 0 a as a scalar = 1 2 In the quantum theory these fields are inquivalent. They do not describe the same theory. 19
More Quantum Inequivalence In 4D a massless 3 tensor has no dynamics (it is classically equivalent to a constant field): F µνρλ = 4 [µ C νρλ] = 0 Is this true in the quantum theory? Anomaly analysis: a 3 tensor = 1 In the quantum theory gauge fixing of a massless 3 tensor gives 2 massless antisymmetric tensor ghosts. 20
Two Form Gauge Symmetry The massless antisymmetric tensor theory has a gauge symmetry B µν B µν + [µ A ν] The gauge symmetry leaves the field strength H µνρ = 3 [µ B νρ] invariant. So the dual scalar does not distinguish configurations that are related by gauge symmetry For a two form in AdS 4, configurations related by a gauge transformation are not always gauge equivalent. 21
Boundary Modes AdS 4 metric: ( ds 2 4 = l 2 A dρ 2 + sinh 2 ) ρdω 2 3 A gauge field with asymptotic conditions at large ρ: A ρ = 0, A i Θ(Ω 3 ) is not normalizable A 2 sinh 3 ρ 1 sinh 2 ρ Θ(Ω 3) 2 The pure gauge antisymmetric tensor [µ A ν] is normalizable (one more index, so one more inverse power of sinh 2 ρ) It is physical because A µ is not an allowed gauge transformation. 22
Counting Boundary Modes So: there are physical field configurations of B µν that do not correspond to any scalar φ. They are called boundary modes because they are locally pure gauge but not globally due their behavior at the boundary. Such configurations must be included in the heat kernel of the antisymmetric tensor but not the scalar field. We can compute their heat kernel and show it is equivalent to a boundary = 1 2. 23
Zero Modes Boundary modes have vanishing action (they have no field strength) so they are zero-modes. The original path integral was evaluated as a Gaussian: e W [g µν] = Dφ e φλφ = 1 detλ. When there are zero modes this must be corrected. Contributions with λ = 0 should not be included in the heat kernel D(t) = Tr e tλ = λ i 0 e λt Instead one must evaluate the volume of zero modes ( Dφ 0 ). 24
Back to N=8 SUGRA We included 70 scalars when finding a(n = 8) = 5 2 But there are quantum inequivalent versions of N=8 SUGRA. An appealing version has 63 scalars, 7 antisymmetric tensors, 1 antisymmetric 3-form - Classically this is equivalent to 70 scalars. - The quantum theory has no anomalies: a(n = 8) = c(n = 8) = 0 - N=8 SUGRA in D=4 is KK reduction of N=1 SUGRA in D=11. An 11D 3-form C IJK reduces to 1 antisymmetric 3-form and 7 antisymmetric tensors (and additional scalars). 25