nLab
Green-Schwarz mechanism

Recall from the discussion at quantum anomaly the following situation:

• An action functional in path integral quantization is said to be anomalous if it is only locally identified with a function on the configuration space of fields, but is globally instead a section of a line bundle (usually equipped with connection).

• Given two anomalous action functionals in this sense, it may happen that while the two corresponding line bundles on configuration space are each nontrivial, their tensor product becomes trivializable. In this case one can consider the non-anomalous action function given by the sum of the two anomalous action functionals. This is what is called anomaly cancellation of one piece of an action functional against another.

• The two main sources of examples for action functionals that are anomalous are

  • the standard action functional for chiral fermions

    • this is a section of the Pfaffian line bundle for the Dirac operator

  • the action functional for differential cocycles (higher connection, higher gauge theory) in the presence of electric and magnetic charges:

    • this is a section of the transgression of the line bundle arising as the cup product of the electric and magnetic differential cocycles

A Green–Schwarz mechanism is the addition of an action functional for higher differential cocycles with magnetic charges such that their anomaly cancels a given Pfaffian line bundle: so its a choice of by itself ill-defined action functional for higher gauge theory that cancels the ill-definedness of an action functional for chiral fermions.

More strictly speaking, the Green–Schwarz mechanism is the application of this procedure in the theory called heterotic supergravity: there it so happens that the Pfaffian line bundle of the fermionic action has as Chern class the transgression of a 12 class that factorizes as $I_8\wedge I_4$. Since heterotic supergravity contains a higher gauge field that couples to strings, this is precisely of the form $J_{\mathrm{electric}}\wedge J_{\mathrm{magnetic}}$ that the anomaly for the corresponding higher gauge theory in the presence of magnetic charges gives rise to. So the original Green–Schwarz anomaly cancellation mechanism consist of modifying the “naive” action functional for heterotic supergravity by adding the contribution that corresponds to adding a magnetic current of the form

Further details for the moment at

• Charges and Twisted Bundles, IV: Anomaly Cancellation