202:
521:(i.e. without boundary) and oriented, then it is the boundary of some d+1 dimensional oriented manifold M. If we then arbitrarily extend the fields (including ε) as defined on M to M with the only condition being they match on the boundaries and the expression Ω, being the exterior product of p-forms, can be extended and defined in the interior, then
512:
as a functional of the nonintegrated fields and is linear in ε. Let us make the further assumption (which turns out to be valid in all the cases of interest) that this functional is local (i.e. Ω(x) only depends upon the values of the fields and their derivatives at x) and that it can be expressed as
1110:
1416:
912:
609:
54:
All gauge anomalies must cancel out. Anomalies in gauge symmetries lead to an inconsistency, since a gauge symmetry is required in order to cancel degrees of freedom with a negative norm which are unphysical (such as a
1199:
503:
927:
364:
1279:
1290:
425:
236:
391:
1211:
Once again, if we assume Ω can be expressed as an exterior product and that it can be extended into a d+1 -form in a d+2 dimensional oriented manifold, we can define
171:
191:
90:
Anomalies occur only in even spacetime dimensions. For example, the anomalies in the usual 4 spacetime dimensions arise from triangle
Feynman diagrams.
620:
527:
1121:
1105:{\displaystyle \delta _{\epsilon _{1}}d\Omega ^{(d)}(\epsilon _{2})-\delta _{\epsilon _{2}}d\Omega ^{(d)}(\epsilon _{1})=d\Omega ^{(d)}(\left).}
433:
1115:
Because of the
Frobenius consistency condition, this means that there exists a d+1-form Ω (not depending upon ε) defined over M satisfying
248:
239:
1411:{\displaystyle \delta _{\epsilon }\Omega ^{(d+2)}=d\delta _{\epsilon }\Omega ^{(d+1)}=d^{2}\Omega ^{(d)}(\epsilon )=0}
1217:
99:
1501:
394:
400:
1444:
214:
1439:
372:
211:. If there is a gauge anomaly, the resulting action will not be gauge invariant. If we denote by
1205:
136:
71:
44:
8:
1434:
119:
20:
176:
509:
32:
28:
514:
238:
the operator corresponding to an infinitesimal gauge transformation by ε, then the
36:
907:{\displaystyle \leftS=\int _{M^{d+1}}\left=\int _{M^{d+1}}d\Omega ^{(d)}(\left).}
518:
115:
70:
is usually used for vector gauge anomalies. Another type of gauge anomaly is the
208:
111:
103:
75:
60:
40:
604:{\displaystyle \delta _{\epsilon }S=\int _{M^{d+1}}d\Omega ^{(d)}(\epsilon ).}
1495:
1449:
1194:{\displaystyle \delta _{\epsilon }\Omega ^{(d+1)}=d\Omega ^{(d)}(\epsilon ).}
207:
Let us look at the (semi)effective action we get after integrating over the
48:
498:{\displaystyle \delta _{\epsilon }S=\int _{M^{d}}\Omega ^{(d)}(\epsilon )}
130:
107:
79:
194:
59:
polarized in the time direction). Indeed, cancellation occurs in the
201:
359:{\displaystyle \left{\mathcal {F}}=\delta _{\left}{\mathcal {F}}}
122:
56:
114:, and can be calculated exactly at one loop level, via a
393:, including the (semi)effective action S where is the
1293:
1220:
1124:
930:
921:
arbitrary extension of the fields into the interior,
623:
530:
436:
403:
375:
251:
217:
179:
139:
1410:
1273:
1193:
1104:
906:
603:
497:
419:
385:
358:
230:
185:
165:
74:, because coordinate reparametrization (called a
1493:
1274:{\displaystyle \Omega ^{(d+2)}=d\Omega ^{(d+1)}}
614:The Frobenius consistency condition now becomes
16:Breakdown of gauge symmetry at the quantum level
85:
1483:Gauge Theory of Elementary Particle Physics
1284:in d+2 dimensions. Ω is gauge invariant:
93:
1467:Treiman, Sam, and Roman Jackiw, (2014).
1480:
1494:
917:As the previous equation is valid for
420:{\displaystyle \delta _{\epsilon }S}
231:{\displaystyle \delta _{\epsilon }}
13:
1378:
1343:
1305:
1250:
1222:
1164:
1136:
1045:
1007:
952:
847:
781:
726:
574:
517:of p-forms. If the spacetime M is
471:
378:
351:
301:
14:
1513:
200:
240:Frobenius consistency condition
110:is a vector), the anomaly is a
1485:. Oxford Science Publications.
1481:Cheng, T.P.; Li, L.F. (1984).
1474:
1461:
1399:
1393:
1388:
1382:
1359:
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1254:
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1226:
1185:
1179:
1174:
1168:
1152:
1140:
1096:
1060:
1055:
1049:
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851:
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785:
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741:
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589:
584:
578:
492:
486:
481:
475:
386:{\displaystyle {\mathcal {F}}}
1:
1471:. Princeton University Press.
1469:Current algebra and anomalies
1455:
427:is linear in ε, we can write
7:
1428:
133:attached to the loop where
78:) is the gauge symmetry of
10:
1518:
1440:Anomaly matching condition
86:Calculation of the anomaly
125:running in the loop with
1445:Green–Schwarz mechanism
166:{\displaystyle n=1+D/2}
1412:
1275:
1195:
1106:
908:
605:
499:
421:
387:
360:
232:
187:
167:
94:Vector gauge anomalies
39:—that invalidates the
1413:
1276:
1196:
1107:
909:
606:
500:
422:
388:
361:
233:
188:
168:
72:gravitational anomaly
31:: it is a feature of
1291:
1218:
1204:Ω is often called a
1122:
928:
621:
528:
434:
401:
373:
249:
215:
177:
137:
102:gauge anomalies (in
45:quantum field theory
27:is an example of an
1502:Anomalies (physics)
1435:Chiral gauge theory
369:for any functional
21:theoretical physics
1408:
1271:
1191:
1102:
904:
601:
495:
417:
383:
356:
228:
183:
163:
1206:Chern–Simons form
186:{\displaystyle D}
33:quantum mechanics
1509:
1487:
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1409:
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1111:
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610:
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566:
540:
539:
515:exterior product
504:
502:
501:
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104:gauge symmetries
37:one-loop diagram
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1119:
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1081:
1072:
1068:
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629:
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619:
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577:
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531:
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474:
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246:
222:
218:
216:
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209:chiral fermions
178:
175:
174:
155:
138:
135:
134:
116:Feynman diagram
96:
88:
17:
12:
11:
5:
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1234:
1231:
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1224:
1202:
1201:
1190:
1187:
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1170:
1166:
1162:
1159:
1154:
1151:
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1142:
1138:
1132:
1128:
1113:
1112:
1101:
1098:
1094:
1088:
1084:
1080:
1075:
1071:
1066:
1062:
1057:
1054:
1051:
1047:
1043:
1040:
1037:
1032:
1028:
1024:
1019:
1016:
1013:
1009:
1005:
998:
994:
989:
985:
982:
977:
973:
969:
964:
961:
958:
954:
950:
943:
939:
934:
915:
914:
903:
900:
896:
890:
886:
882:
877:
873:
868:
864:
859:
856:
853:
849:
845:
838:
835:
832:
828:
823:
819:
815:
811:
806:
802:
798:
793:
790:
787:
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779:
772:
768:
763:
759:
756:
751:
747:
743:
738:
735:
732:
728:
724:
717:
713:
708:
703:
695:
692:
689:
685:
680:
676:
673:
669:
661:
657:
652:
648:
641:
637:
632:
627:
612:
611:
600:
597:
594:
591:
586:
583:
580:
576:
572:
565:
562:
559:
555:
550:
546:
543:
538:
534:
506:
505:
494:
491:
488:
483:
480:
477:
473:
465:
461:
456:
452:
449:
444:
440:
416:
411:
407:
380:
367:
366:
353:
345:
339:
335:
331:
326:
322:
317:
312:
308:
303:
297:
289:
285:
280:
276:
269:
265:
260:
255:
242:requires that
225:
221:
182:
162:
158:
154:
151:
148:
145:
142:
112:chiral anomaly
95:
92:
87:
84:
76:diffeomorphism
61:Standard Model
41:gauge symmetry
15:
9:
6:
4:
3:
2:
1514:
1503:
1500:
1499:
1497:
1484:
1477:
1470:
1464:
1460:
1451:
1450:Mixed anomaly
1448:
1446:
1443:
1441:
1438:
1436:
1433:
1432:
1426:
1405:
1402:
1396:
1385:
1372:
1368:
1364:
1356:
1353:
1350:
1337:
1333:
1329:
1326:
1318:
1315:
1312:
1299:
1295:
1287:
1286:
1285:
1263:
1260:
1257:
1246:
1243:
1235:
1232:
1229:
1214:
1213:
1212:
1209:
1207:
1188:
1182:
1171:
1160:
1157:
1149:
1146:
1143:
1130:
1126:
1118:
1117:
1116:
1099:
1092:
1086:
1082:
1078:
1073:
1069:
1064:
1052:
1041:
1038:
1030:
1026:
1014:
1003:
996:
992:
987:
983:
975:
971:
959:
948:
941:
937:
932:
924:
923:
922:
920:
901:
894:
888:
884:
880:
875:
871:
866:
854:
843:
836:
833:
830:
826:
821:
817:
813:
804:
800:
788:
777:
770:
766:
761:
757:
749:
745:
733:
722:
715:
711:
706:
701:
693:
690:
687:
683:
678:
674:
671:
667:
659:
655:
650:
646:
639:
635:
630:
625:
617:
616:
615:
598:
592:
581:
570:
563:
560:
557:
553:
548:
544:
541:
536:
532:
524:
523:
522:
520:
516:
511:
489:
478:
463:
459:
454:
450:
447:
442:
438:
430:
429:
428:
414:
409:
405:
396:
343:
337:
333:
329:
324:
320:
315:
310:
306:
295:
287:
283:
278:
274:
267:
263:
258:
253:
245:
244:
243:
241:
223:
219:
210:
205:
203:
198:
196:
180:
160:
156:
152:
149:
146:
143:
140:
132:
128:
124:
121:
117:
113:
109:
105:
101:
91:
83:
81:
77:
73:
69:
68:gauge anomaly
64:
62:
58:
52:
50:
46:
42:
38:
34:
30:
26:
25:gauge anomaly
22:
1482:
1476:
1468:
1463:
1420:
1283:
1210:
1203:
1114:
918:
916:
613:
507:
368:
206:
199:
131:gauge bosons
126:
97:
89:
67:
65:
53:
49:gauge theory
47:; i.e. of a
24:
18:
508:where Ω is
395:Lie bracket
197:dimension.
108:gauge boson
80:gravitation
35:—usually a
1456:References
1421:as d and δ
1425:commute.
1397:ϵ
1379:Ω
1344:Ω
1338:ϵ
1334:δ
1306:Ω
1300:ϵ
1296:δ
1251:Ω
1223:Ω
1183:ϵ
1165:Ω
1137:Ω
1131:ϵ
1127:δ
1083:ϵ
1070:ϵ
1046:Ω
1027:ϵ
1008:Ω
993:ϵ
988:δ
984:−
972:ϵ
953:Ω
938:ϵ
933:δ
885:ϵ
872:ϵ
848:Ω
822:∫
801:ϵ
782:Ω
767:ϵ
762:δ
758:−
746:ϵ
727:Ω
712:ϵ
707:δ
679:∫
656:ϵ
651:δ
636:ϵ
631:δ
593:ϵ
575:Ω
549:∫
537:ϵ
533:δ
490:ϵ
472:Ω
455:∫
443:ϵ
439:δ
410:ϵ
406:δ
334:ϵ
321:ϵ
311:δ
284:ϵ
279:δ
264:ϵ
259:δ
224:ϵ
220:δ
195:spacetime
129:external
66:The term
1496:Category
1429:See also
193:is the
123:fermion
118:with a
29:anomaly
519:closed
510:d-form
173:where
120:chiral
106:whose
100:vector
57:photon
397:. As
43:of a
513:the
23:, a
919:any
98:In
19:In
1498::
1208:.
82:.
63:.
51:.
1423:ε
1406:0
1403:=
1400:)
1394:(
1389:)
1386:d
1383:(
1373:2
1369:d
1365:=
1360:)
1357:1
1354:+
1351:d
1348:(
1330:d
1327:=
1322:)
1319:2
1316:+
1313:d
1310:(
1267:)
1264:1
1261:+
1258:d
1255:(
1247:d
1244:=
1239:)
1236:2
1233:+
1230:d
1227:(
1189:.
1186:)
1180:(
1175:)
1172:d
1169:(
1161:d
1158:=
1153:)
1150:1
1147:+
1144:d
1141:(
1100:.
1097:)
1093:]
1087:2
1079:,
1074:1
1065:[
1061:(
1056:)
1053:d
1050:(
1042:d
1039:=
1036:)
1031:1
1023:(
1018:)
1015:d
1012:(
1004:d
997:2
981:)
976:2
968:(
963:)
960:d
957:(
949:d
942:1
902:.
899:)
895:]
889:2
881:,
876:1
867:[
863:(
858:)
855:d
852:(
844:d
837:1
834:+
831:d
827:M
818:=
814:]
810:)
805:1
797:(
792:)
789:d
786:(
778:d
771:2
755:)
750:2
742:(
737:)
734:d
731:(
723:d
716:1
702:[
694:1
691:+
688:d
684:M
675:=
672:S
668:]
660:2
647:,
640:1
626:[
599:.
596:)
590:(
585:)
582:d
579:(
571:d
564:1
561:+
558:d
554:M
545:=
542:S
493:)
487:(
482:)
479:d
476:(
464:d
460:M
451:=
448:S
415:S
379:F
352:F
344:]
338:2
330:,
325:1
316:[
307:=
302:F
296:]
288:2
275:,
268:1
254:[
181:D
161:2
157:/
153:D
150:+
147:1
144:=
141:n
127:n
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