In the case of an ungauged symmetry, the product of the exponents of the action functional corresponding of all sectors of the theory needs not to be a scalar, i.e., there can be still a net anomaly as far as its symmetry is not gauged.

In this case, the exponent of the action functional will be a section of a line bundle, and the wave function(als) themselves will be sections of the dual line bundle, thus we obtain a scalar only when the action functional is sandwiched between two wave functions (giving a transition amplitude). The most famous action functional is obtained in this manner is the case chiral anomaly. The integration over the fermionic fields produces the Dirac determinant line bundle, please see the following lecture by:Juoko Mickelsson. Please see also the following work by: Orlando Alvarez giving examples that in this case the coefficients of certain terms in the Lagrangian need to be quantized.

Positive energy unitary highest weight representations of anomalous current algebras exist in $1+1$ dimensions, and in this case the anomaly is essential to obtain nontrivial representations. Representations of this kind were obtained in Pressley and Segal: Loop groups. No generalizations to realistic theories in $3+1$ dimensions are known. Juoko Mickelsson was able to construct a representation on a Hilbert bundle.

The connection with the Borel-Weil theorem is that in the finite dimensional case, the tautological line bundle of the complex Grassmannian $Gr(k,n)$ (Please see: Stefan Berceanu’s article) is a determinant line bundle of the k dimensional subspaces of an n-dimensional complex vector space, associated with a antisymmetric representation corresponding (according to the Borel-Weil theorem) to the $k$-box column Young tableaux. This connection extends to the case of loop groups, but again, no generalization for groups of mapping to higher dimensional manifolds (such as three dimensional) are known.

In the case of an ungauged symmetry, the product of the exponents of the action functional corresponding of all sectors of the theory needs not to be a scalar, i.e., there can be still a net anomaly as far as its symmetry is not gauged.

In this case, the exponent of the action functional will be a section of a line bundle, and the wave function(als) themselves will be sections of the dual line bundle, thus we obtain a scalar only when the action functional is sandwiched between two wave functions (giving a transition amplitude). The most famous action functional is obtained in this manner is the case chiral anomaly. The integration over the fermionic fields produces the Dirac determinant line bundle, please see the following lecture by:Juoko Mickelsson. Please see also the following work by: Orlando Alvarez giving examples that in this case the coefficients of certain terms in the Lagrangian need to be quantized.

Positive energy unitary highest weight representations of anomalous current algebras exist in $1+1$ dimensions, and in this case the anomaly is essential to obtain nontrivial representations. Representations of this kind were obtained in Pressley and Segal: Loop groups. No generalizations to realistic theories in $3+1$ dimensions are known. Juoko Mickelsson was able to construct a representation on a Hilbert bundle.

The connection with the Borel-Weil theorem is that in the finite dimensional case, the tautological line bundle of the complex Grassmannian $Gr(k,n)$ (Please see: Stefan Berceanu’s article) is a determinant line bundle of the k dimensional subspaces of an n-dimensional complex vector space, associated with a antisymmetric representation corresponding (according to the Borel-Weil theorem) to the $k$-box column Young tableaux. This connection extends to the case of loop groups, but again, no generalization for groups of mapping to higher dimensional manifolds (such as three dimensional) are known.