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John Baez (Nov 08 2020 at 22:13):Joachim Kock writes:
Problems with symmetries often arise from dividing out by them. You can kill symmetries by dividing out by them, but their ghosts will haunt you forever.
:ghost:
How true!
Matteo Capucci (he/him) (Nov 10 2020 at 13:46):Those ghosts literally live in a higher dimension
John Baez (Nov 10 2020 at 15:30):I like that!
In physics people deal with gauge symmetries by adding extra particles called "ghosts", but these then spawn "ghosts-on-ghosts", and so on - secretly morphisms in an -groupoid.
Cole Comfort (Nov 10 2020 at 15:34):How spooky. Like you can only see ghosts through the shadows which they leave, projected onto the wall or something...
Matteo Capucci (he/him) (Nov 10 2020 at 15:44):John Baez said:
I like that!
In physics people deal with gauge symmetries by adding extra particles called "ghosts", but these then spawn "ghosts-on-ghosts", and so on - secretly morphisms in an -groupoid.
Is this true or just speculative? Are those ghosts really morphisms in a higher groupid? :surprise:
John Baez (Nov 10 2020 at 15:45):Something is true; I'm not giving a very precise account of it.
John Baez (Nov 10 2020 at 15:46):But yes, there's a known relation between higher groupoids and the use of ghosts, ghosts-on-ghosts etc. in BRST quantization.
Matteo Capucci (he/him) (Nov 10 2020 at 15:47):Very cool
John Baez (Nov 10 2020 at 15:47):The nLab says it more precisely:
In physics and specifically in field theory, the BV-BRST formalism is a tool in homological algebra, higher differential geometry and derived geometry to handle the intersection- and quotient-constructions that appear
in the construction of reduced phase spaces of Lagrangian field theories, in particular including gauge theories; (“Lagrangian BV”)
in symplectic reduction of phase spaces (“Hamiltonian BV”)
In either case the BRST-BV complex is a model in dg-geometry of a joint homotopy intersection and homotopy quotient, hence of an (∞,1)-colimit and (∞,1)-limit, of a space in higher differential geometry/derived geometry.
Accordingly, the BRST-BV complex is built from two main pieces:
it contains in positive degree a BRST-complex: the Chevalley-Eilenberg algebra of the ∞-Lie algebroid which is the homotopy quotient (action Lie algebroid) of the gauge group (in Lagrangian BV) or of the group of flows generated by the constraints (in Hamiltonian BFV) – which is in general an ∞-group in either case – acting on configuration space CC;
it contains in negative degree a Koszul-Tate resolution of the critical locus of the action functional (for Lagrangian BV) or of the constraint surface (in Hamiltonian BFV).
John Baez (Nov 10 2020 at 15:49):I was alluding to the BRST stuff. That's where "ghosts" live.
Note a " ∞-Lie algebroid" is an infinitesimal, i.e. linearized, description of an ∞-groupoid.