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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMay 28th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded to [[quantum anomaly]] * an uncommented link to [[Liouville cocycle]] * a paragraph with the basic idea of fermioninc anomalies * the missing reference to Witten's old article on spin structures and fermioninc anomalies. The entry is still way, way, stubby. But now a little bit less than a minute ago ;-

   added to quantum anomaly

   • an uncommented link to Liouville cocycle

   • a paragraph with the basic idea of fermioninc anomalies

   • the missing reference to Witten’s old article on spin structures and fermioninc anomalies.

   The entry is still way, way, stubby. But now a little bit less than a minute ago ;-

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMar 4th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded the story of gauge anomalies via first BV-BRST cohomology to [[quantum anomaly]]

   added the story of gauge anomalies via first BV-BRST cohomology to quantum anomaly

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeApr 19th 2019
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexAdded the reference * {#1903.02828} [[Samuel Monnier]], _A Modern Point of View on Anomalies_, ([arxiv/1903.02828](http://arxiv.org/abs/1903.02828)) <a href="https://ncatlab.org/nlab/revision/diff/quantum+anomaly/54">diff</a>, <a href="https://ncatlab.org/nlab/revision/quantum+anomaly/54">v54</a>, <a href="https://ncatlab.org/nlab/show/quantum+anomaly">current</a>

   Added the reference

   • {#1903.02828} Samuel Monnier, A Modern Point of View on Anomalies, (arxiv/1903.02828)

   diff, v54, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJan 21st 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointers to electronic repositories (`doi:`, `euclid:`, `jstor:`) for all of the following items: * [[Luis Alvarez-Gaumé]], [[Edward Witten]], _Gravitational Anomalies_ Nucl. Phys. B234 (1984) 269 (<a href="https://doi.org/10.1016/0550-3213(84)90066-X">doi:10.1016/0550-3213(84)90066-X</a>, [spire:192309](http://inspirehep.net/record/192309)) * [[Luis Alvarez-Gaumé]], [[Paul Ginsparg]], _The structure of gauge and gravitational anomalies_, Ann. Phys. 161 (1985) 423. (<a href="https://doi.org/10.1016/0003-4916(85)90087-9">doi:10.1016/0003-4916(85)90087-9</a>, [spire:202565](http://inspirehep.net/record/202565)) * [[Edward Witten]], _Global gravitational anomalies_, Commun. Math. Phys. 100 (1985) 197. ([doi:10.1007/BF01212448](https://doi.org/10.1007/BF01212448), [euclid:1103943444](http://projecteuclid.org/euclid.cmp/1103943444)) * [[Michael Atiyah]], [[Isadore Singer]], _Dirac operators coupled to vector potentials_, Proc. Nat. Acad. Sci. USA __81__, 2597-2600 (1984) ([doi:10.1073/pnas.81.8.2597]( https://doi.org/10.1073/pnas.81.8.2597), [jstor:23378](https://www.jstor.org/stable/23378)) * [[Jean-Michel Bismut]], [[Daniel Freed]], _The analysis of elliptic families. I. Metrics and connections on determinant bundles_ , Comm. Math. Phys. 106 (1986), no. 1, 159&#8211;176 ([doi:10.1007/BF01210930](https://doi.org/10.1007/BF01210930), [euclid:1104115586](https://projecteuclid.org/euclid.cmp/1104115586)) * [[Jean-Michel Bismut]], [[Daniel Freed]], _The analysis of elliptic families. II. Dirac operators, eta invariants, and the holonomy theorem_ , Comm. Math. Phys. 107 (1986), no. 1, 103&#8211;163 ([doi:10.1007/BF01206955](https://doi.org/10.1007/BF01206955), [euclid:1104115934](https://projecteuclid.org/euclid.cmp/1104115934)) * [[Daniel Freed]], _Determinants, torsion, and strings_, Comm. Math. Phys. Volume 107, Number 3 (1986), 483-513. ([doi:10.1007/BF01221001](https://doi.org/10.1007/BF01221001), [euclid:1104116145](http://projecteuclid.org/euclid.cmp/1104116145)) <a href="https://ncatlab.org/nlab/revision/diff/quantum+anomaly/55">diff</a>, <a href="https://ncatlab.org/nlab/revision/quantum+anomaly/55">v55</a>, <a href="https://ncatlab.org/nlab/show/quantum+anomaly">current</a>

   added pointers to electronic repositories (doi:, euclid:, jstor:) for all of the following items:

   • Luis Alvarez-Gaumé, Edward Witten, Gravitational Anomalies Nucl. Phys. B234 (1984) 269 (doi:10.1016/0550-3213(84)90066-X, spire:192309)

   • Luis Alvarez-Gaumé, Paul Ginsparg, The structure of gauge and gravitational anomalies, Ann. Phys. 161 (1985) 423. (doi:10.1016/0003-4916(85)90087-9, spire:202565)

   • Edward Witten, Global gravitational anomalies, Commun. Math. Phys. 100 (1985) 197. (doi:10.1007/BF01212448, euclid:1103943444)

   • Michael Atiyah, Isadore Singer, Dirac operators coupled to vector potentials, Proc. Nat. Acad. Sci. USA 81, 2597-2600 (1984) (doi:10.1073/pnas.81.8.2597, jstor:23378)

   • Jean-Michel Bismut, Daniel Freed, The analysis of elliptic families. I. Metrics and connections on determinant bundles , Comm. Math. Phys. 106 (1986), no. 1, 159–176 (doi:10.1007/BF01210930, euclid:1104115586)

   • Jean-Michel Bismut, Daniel Freed, The analysis of elliptic families. II. Dirac operators, eta invariants, and the holonomy theorem , Comm. Math. Phys. 107 (1986), no. 1, 103–163 (doi:10.1007/BF01206955, euclid:1104115934)

   • Daniel Freed, Determinants, torsion, and strings, Comm. Math. Phys. Volume 107, Number 3 (1986), 483-513. (doi:10.1007/BF01221001, euclid:1104116145)

   diff, v55, current

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeJan 21st 2020
   • (edited Jan 21st 2020)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to this review * [[Jeffrey Harvey]], _TASI 2003 Lectures on Anomalies_ ([arXiv:hep-th/0509097v1](https://arxiv.org/abs/hep-th/0509097), [spire:692082](http://inspirehep.net/record/692082)) and references on anomaly inflow: * [[Curtis Callan]], [[Jeffrey Harvey]], _Anomalies and Fermion Zero Modes on Strings and Domain Walls_, Nucl. Phys. B250 (1985) 427-436 (<a href="https://doi.org/10.1016/0550-3213(85)90489-4">doi:10.1016/0550-3213(85)90489-4</a>, [spire:15691](http://inspirehep.net/record/15691)) * [[Edward Witten]], Kazuya Yonekura, _Anomaly Inflow and the $\etas$-Invariant_ ([arXiv:1909.08775](https://arxiv.org/abs/1909.08775), [spire:1755070](http://inspirehep.net/record/1755070)) <a href="https://ncatlab.org/nlab/revision/diff/quantum+anomaly/55">diff</a>, <a href="https://ncatlab.org/nlab/revision/quantum+anomaly/55">v55</a>, <a href="https://ncatlab.org/nlab/show/quantum+anomaly">current</a>

   added pointer to this review

   • Jeffrey Harvey, TASI 2003 Lectures on Anomalies (arXiv:hep-th/0509097v1, spire:692082)

   and references on anomaly inflow:

   • Curtis Callan, Jeffrey Harvey, Anomalies and Fermion Zero Modes on Strings and Domain Walls, Nucl. Phys. B250 (1985) 427-436 (doi:10.1016/0550-3213(85)90489-4, spire:15691)

   • Edward Witten, Kazuya Yonekura, Anomaly Inflow and the $\etas$-Invariant (arXiv:1909.08775, spire:1755070)

   diff, v55, current

6. • CommentRowNumber6.
   • CommentAuthornLab edit announcer
   • CommentTimeFeb 23rd 2021
   • PermaLink
   Author: nLab edit announcer
   Format: MarkdownItexThe Berenzin integral is an integral over fermion fields. Andrea Russo <a href="https://ncatlab.org/nlab/revision/diff/quantum+anomaly/58">diff</a>, <a href="https://ncatlab.org/nlab/revision/quantum+anomaly/58">v58</a>, <a href="https://ncatlab.org/nlab/show/quantum+anomaly">current</a>

   The Berenzin integral is an integral over fermion fields.

   Andrea Russo

   diff, v58, current

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeFeb 24th 2021
   • (edited Feb 24th 2021)
   • PermaLink
   Author: Urs
   Format: MarkdownItexYes, that's why after having performed the Berezin integral, what remains is a function of the bosonic fields (since the fermions have been integrated out!), or rather is, globally, a section of a line bundle over the bosonic field values, whose non-triviality is the anomaly. So as stated it was correct and to the point. Hence I have rolled back your edit. (I can, of course, see that there is plenty of room to expand on what the entry says at this point, be more pedagogical, add examples and details on actual formulas. Maybe later.) <a href="https://ncatlab.org/nlab/revision/diff/quantum+anomaly/59">diff</a>, <a href="https://ncatlab.org/nlab/revision/quantum+anomaly/59">v59</a>, <a href="https://ncatlab.org/nlab/show/quantum+anomaly">current</a>

   Yes, that’s why after having performed the Berezin integral, what remains is a function of the bosonic fields (since the fermions have been integrated out!), or rather is, globally, a section of a line bundle over the bosonic field values, whose non-triviality is the anomaly.

   So as stated it was correct and to the point.

   Hence I have rolled back your edit.

   (I can, of course, see that there is plenty of room to expand on what the entry says at this point, be more pedagogical, add examples and details on actual formulas. Maybe later.)

   diff, v59, current

8. • CommentRowNumber8.
   • CommentAuthorDmitri Pavlov
   • CommentTimeFeb 15th 2023
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexAdded: Anomalies originate in the 1949 article by Steinberger: * J. Steinberger, _On the Use of Subtraction Fields and the Lifetimes of Some Types of Meson Decay_, Phys. Rev. 76, 1180 – Published 15 October 1949. [doi](https://doi.org/10.1103/PhysRev.76.1180). Another early reference is * Stephen L. Adler, _Axial-Vector Vertex in Spinor Electrodynamics_, Phys. Rev. 177, 2426 – Published 25 January 1969. [doi](https://doi.org/10.1103/PhysRev.177.2426). <a href="https://ncatlab.org/nlab/revision/diff/quantum+anomaly/62">diff</a>, <a href="https://ncatlab.org/nlab/revision/quantum+anomaly/62">v62</a>, <a href="https://ncatlab.org/nlab/show/quantum+anomaly">current</a>

   Added:

   Anomalies originate in the 1949 article by Steinberger:

   • J. Steinberger, On the Use of Subtraction Fields and the Lifetimes of Some Types of Meson Decay, Phys. Rev. 76, 1180 – Published 15 October 1949. doi.

   Another early reference is

   • Stephen L. Adler, Axial-Vector Vertex in Spinor Electrodynamics, Phys. Rev. 177, 2426 – Published 25 January 1969. doi.

   diff, v62, current