Global anomalies on Lorentzian space-times Jochen Zahn Universit - PowerPoint PPT Presentation

Global anomalies on Lorentzian space-times Jochen Zahn Universit¨ at Leipzig based on 1609.06562 (Ann. H. Poincar´ e) [joint work with A. Schenkel] LQP 41, G¨ ottingen, February 2018

Global anomalies in the path integral § Chiral SU p 2 q doublet: Not anomalous w.r.t. infinitesimal (local) gauge trafos. § But: Anomalous w.r.t. large (global) gauge trafos [Witten 82] . § As π 4 p SU p 2 qq “ Z 2 , there are compactly supported gauge trafos g that can not be deformed to the identity. § However, one may deform A to A g via a path A λ of connections that are not gauge equivalent to A . Along such a path, the fermion path integral  1 „ż 2 ‰ 1 d ψ d ¯ ψ exp p ¯ ψ i { “ det i { D A λ ψ q “ D A λ 2 changes sign as A is varied to A g (mod 2 index theorem). § This implies that the full partition function ż ˆ 1 ż ˙ ‰ 1 2 exp “ det i { Z “ d A D A ´ tr F ^ ‹ F 2 g 2 YM vanishes, as the contributions from A and A g always cancel. § The theory is thus inconsistent. § Non-perturbative effect, not visible in perturbation theory around single background.

Riemannian vs. Lorentzian § The computations of global anomalies involve fermions in background fields in Riemannian signature. § No clear relation to Lorentzian signature. § What is an appropriate condition for global anomalies in Lorentzian signature (based on free fermions in non-trivial backgrounds)? § How does a global anomaly render a theory inconsistent?

The framework (I) § As in the path integral framework, we formulate a criterion for global anomalies based on free chiral fermions in generic gauge backgrounds. § Gauge backgrounds described by principal bundle connection ¯ A . A 1 differ by Lie-algebra valued one-form A “ ¯ § Two backgrounds ¯ A , ¯ A ´ ¯ A 1 . § Locally covariant field theory [Hollands, Wald 01; Brunetti, Fredenhagen, Verch 03] adapted to the gauge theory setting [Z. 14] : Local covariance also w.r.t. principal bundle morphisms. § Fields provide a consistent assignment of observables to different backgrounds. Example: The current ż � δ ¯ � ψ { j ¯ A p A q “ A S , A “ ´ A ψ vol δ ¯ defined by point-splitting w.r.t. the Hadamard parametrix. § No local anomalies, i.e., the current is conserved A p Λ q . ¯ A p ¯ “ j ¯ d Λ q “ 0 . δ j ¯ (CC) It is then unique up to charge renormalization [Z. 14] A ` λ ¯ δ ¯ A Ñ j ¯ j ¯ F .

The framework (II) A 1 differ only in a compact region, there is a § When two backgrounds ¯ A , ¯ natural isomorphism of the corresponding algebras, the retarded variation A 1 : A p ¯ A 1 q Ñ A p ¯ τ r A q . A , ¯ ¯ A 1 ´ ¯ § It acts trivially on observables localized in the past of supp p ¯ A q . § Perturbative agreement (PA) [Hollands, Wald 05] is the requirement that it should not matter whether one puts quadratic terms in the free or interaction part of the action: A q q . A p e F b e ij p ¯ A p e F ; e ij p ¯ A 1 ´ ¯ A p e ij p ¯ A 1 ´ ¯ A 1 ´ ¯ A q q ´ 1 T ¯ A q q A 1 p e F qq “ R ¯ τ r A 1 p T ¯ “ T ¯ A , ¯ ¯ § The infinitesimal retarded variation around ¯ A in the direction of A is A p A q . denoted by δ r ¯ § (PA) can be fulfilled provided that A p A 1 , A 2 q . “ δ r A p A 1 q j p A 2 q ´ δ r A p A 2 q j p A 1 q ´ i r j p A 2 q , j p A 1 qs “ 0 . E ¯ ¯ ¯ In dimension d ď 4, (CC) implies E ¯ A p A 1 , A 2 q “ 0 [Z. 15] .

The phase of the S matrix § Our criterion for the occurrence of a global anomaly will be a non-trivial phase of the S matrix for ¯ A Ñ ¯ A g . Need to fix the phase of the S matrix. A 1 “ ¯ § Formally, the S matrix for ¯ A Ñ ¯ A ` A is given by and fulfills A p e ij p A q q S ¯ A p A q “ T ¯ A p e ij p A ´ A 1 q b e ij p A 1 q q A p e ij p A 1 q q T ¯ A p e ij p A 1 q q ´ 1 T ¯ “ T ¯ A p e ij p A 1 q q R ¯ A p e ij p A ´ A 1 q ; e ij p A 1 q q “ T ¯ A p A 1 q τ r A ` A 1 p A ´ A 1 qq “ S ¯ A ` A 1 p S ¯ A , ¯ ¯ § With the further constraints A p 9 S ¯ A p 0 q “ ✶ , B λ S ¯ A p A λ q| λ “ 0 “ ij ¯ A 0 q , we may integrate S matrix for any path r 0 , 1 s Q λ ÞÑ A λ from 0 to A : ż 1 ˆ ˙ A p A q “ ¯ A ` A λ p 9 S ¯ P exp i τ r A ` A λ p j ¯ A λ qq d λ (PO) A , ¯ ¯ 0 § Path independence is equivalent to E “ 0. § Unique up to ˆ ż “ ‰˙ L YM p ¯ A ` A q ´ L YM p ¯ A p A q Ñ exp A q A p A q . S ¯ i λ S ¯

Hilbert space representation § A representation ¯ π : A p ¯ A q Ñ End p ¯ H q naturally induces representations π A . A ` A : A p ¯ A ` A q Ñ End p ¯ π ˝ τ r “ ¯ H q . A , ¯ ¯ § In the representation, (PO) reads ż 1 ˆ ˙ U p A , A 1 q . A ` A p A 1 ´ A qq “ ¯ A ` A λ p 9 “ π A p S ¯ P exp π A λ p j ¯ A λ qq d λ i , 0 with A λ a path from A to A 1 . § Q: Is π p j q self-adjoint? Is U well-defined and unitary? § Assuming it is, U p A , A 1 q ´ 1 “ U p A 1 , A q . U p A , A 1 q U p A 1 , A 2 q “ U p A , A 2 q , § Furthermore, V p g q . A ` A q g ´ ¯ “ U pp ¯ A , A q “ e i φ g id is independent of A , and thus provides a representation of the gauge group Γ 8 c p M , P ˆ Ad G q . § If g is deformable to the identity, then, by (PO) and (CC), V p g q “ id . § If V p g q ‰ id for some g , then no gauge invariant vector, a global anomaly. § Same topological obstructions as in the path integral formalism and similar computation via gauge non-equivalent connections.

Global anomalies in a Hamiltonian framework § Following [Witten 82] , assume that the Hilbert space is given by sections over the space of 3 d gauge fields in temporal gauge. The gauge group is then G “ C 8 c p R 3 , G q with homotopy group π 1 p G q “ π 4 p G q . § Physical states are annihilated by the generators Q p Λ q of G . § The non-trivial element of π 1 p G q must be represented by the identity, otherwise there are no physical states. § The matter contribution to the generators is Q matter p Λ q “ j ¯ A p B q with B a µ p x q “ δ 0 µ Λ a p � x q δ p x 0 q . § E “ 0 ensures r Q p Λ q , Q p Λ 1 qs “ iQ pr Λ , Λ 1 sq . § In the case of a global anomaly, there are no physical states, as integrating up Q p Λ q along a non-trivial cycle does not yield the identity.

Perturbative agreement and the Wess-Zumino consistency condition § Assume there is a local anomaly, i.e., (CC) does not hold. Can we still A p A , A 1 q “ 0 by giving up the requirement that j is a field? obtain E ¯ § We fix a flat reference connection ¯ A 0 and specify any other background A “ ¯ ¯ A 0 ` ¯ A by a vector potential ¯ A to depend on ¯ A . Allow j ¯ A . We have � δ � δ A Λ 1 q “ A ¯ A p Λ 1 q , d ¯ A ¯ A Λ 1 � ´ ¯ A pr Λ , Λ 1 sq ! � A p d ¯ A Λ , d ¯ A Λ ´ A p Λ q , d ¯ “ 0 . E ¯ δ j ¯ δ j ¯ δ j ¯ δ ¯ δ ¯ (WZ) This is the Wess-Zumino consistency condition. § For d “ 4 and flat space-time [Z. 14] , ż i ¯ tr Λ ¯ F ^ ¯ A p Λ q “ δ j ¯ F . 8 π 2 § With [Bardeen & Zumino 84] i ż A ^ p ¯ A ^ ¯ F ` ¯ F ^ ¯ 2 ¯ A ^ ¯ A ^ ¯ A ´ 1 “ ‰ j ¯ A p A q ÞÑ j ¯ A p A q ` tr A q 24 π 2 A p A 1 , A 2 q “ 0 and the consistent anomaly one obtains E ¯ i ż ¯ Λ p d ¯ A ^ d ¯ 2 d p ¯ A ^ ¯ A ^ ¯ “ A ` 1 ‰ A p Λ q “ tr A qq δ j ¯ . 24 π 2 § For G “ U p 1 q and flat space-time, one can obtain (CC) and (WZ), but then E ¯ A p A 1 , A 2 q ‰ 0. Hence, (PA) is stronger than (WZ).

Computation of the SU p 2 q anomaly § Following [Witten 83; Elitzur & Nair 84] , compute SU p 2 q anomaly by embedding G “ SU p 2 q Ă SU p 3 q “ H with π 4 p H q “ 0. May connect the nontrivial g P π 4 p G q by a path in C 8 c p R 4 , H q to the identity. With (PO), the global anomaly of G is computed by integrating the consistent anomaly of H : ż 1 ˆ ¯˙ 1 ż A g ´ ¯ ´ h ´ 1 9 A p ¯ A q “ exp tr h ^ A ^ A ^ A ^ A S ¯ d λ 48 π 2 0 ˆ 1 ż ˙ r 0 , 1 sˆ R 4 h ˚ p µ 5 “ exp H q 240 π 2 where h p 0 q “ id , h p 1 q “ g , A “ h ´ 1 d h , and ¯ A is flat. § h defines an element of π 5 p H { G q and r h s ÞÑ 1 ş S 5 h ˚ p µ 5 H q is a group 240 π 2 homomorphism, which for the generator h 1 of π 5 p H q is normalized to 1 ż S 5 h ˚ 1 p µ 5 H q “ 2 π i . 240 π 2 § We have the exact sequence π 5 p H q “ Z Ñ π 5 p H { G q “ Z Ñ π 4 p G q “ Z 2 Ñ π 4 p H q “ 0 . A g ´ ¯ A p ¯ 1 S 5 h ˚ p µ 5 ş Hence H q is odd multiple of i π , so that S ¯ A q “ ´ id . 240 π 2

Summary & Outlook Summary: § Interpreted global anomalies in a Lorentzian setting. § Phase of the S matrix. § Pivotal role of perturbative agreement ( E “ 0). § Relation of perturbative agreement and WZ consistency. Open issues: § Unitarity of implementers in representation. § Effect of non-trivial topologies.