JHEP05(2020)098 Springer May 21, 2020 April 29, 2020 : : February 17, 2020 : Published Accepted , recovering the condition Received 0 ≥ Z ∈ r , for this local anomaly to cancel. The Published for SISSA by 0 https://doi.org/10.1007/JHEP05(2020)098 2, for ≥ / Z ∈ + 3 gauge theories r r 2, for / U(2) +1 r [email protected] , . 3 2001.07731 The Authors. c Anomalies in Field and String Theories, Gauge Symmetry

We discuss anomaly cancellation in U(2) gauge theories in four dimensions. , [email protected] U(2)) implies that there can be no global anomalies, in contrast to the related B ( Department of Applied MathematicsUniversity and of Theoretical Physics, Cambridge, Wilberforce Road, Cambridge, U.K. E-mail: Spin 5 Open Access Article funded by SCOAP ArXiv ePrint: cancelled by awhich Wess-Zumino can term, itself leaving be a cancelled by low-energy coupling theory to topological withKeywords: degrees a of global freedom. anomaly, SU(2) global anomaly’ is alsoto equivalent, though a only perturbative at anomaly the ingauge level the anomaly of U(2) with the theory, a partition which gauge-gravity function, isis anomaly. this an This time even perturbative a number anomaly combination of vanishesfor of if fermions a cancelling there with mixed isospin the 4 new SU(2) anomaly. Alternatively, this perturbative anomaly can be is replaced by anumber local of anomaly fermions with when isospin SU(2) 2 case is of a embedded U(2) in theory U(2). definedstructure, without which There a choice is must of possible be spinand structure an when but odd even all rather using fermions (even) a (bosons) U(1) spin-U(2) have charge, half-integer is (integer) more isospin subtle. We find that the recently-discovered ‘new Abstract: For a U(2) gauge theoryΩ defined with a spincase structure, of the an vanishing of SU(2) the gauge bordism theory. group We show explicitly that the familiar SU(2) global anomaly Joe Davighi and Nakarin Lohitsiri Anomaly interplay in JHEP05(2020)098 2 represen- / which becomes + 1 1 r ], 3 , 2 ]), the exponentiated 6 , 5 to a five-manifold that bounds 13 / D i ]. -invariant [ 4 η 7 SU(2), in the background of a single 2 also, and given that U(2) is locally / ∈ 5 Z 1 3 − 10 11 U(2)) = 0 (1.1) – 1 – B 15 ( (U(2)) = Spin 5 4 13 π Ω . 2 2. Equivalently, the anomaly can be seen from a constant / Z 16 ]. Such a theory is anomalous because the (Euclidean) partition U(1) which has a global anomaly associated with the SU(2) factor. 1 [ -invariant of an extension of the Dirac operator η × (SU(2)) = 0 4 ≥ 1 π Z -invariant of a Dirac operator is a regularized sum of its positive eigenvalues minus its ∈ η r One might be forgiven for guessing that a U(2) chiral gauge theory suffers from a Here we refer to the 5.1 Case I:5.2 with a spin Case structure II: without a spin structure 4.1 Interpretation of4.2 the U(2) anomalies Disentangling the anomaly interplay 1 -invariant must be trivial on all closed spin five-manifolds equipped with a U(2) gauge (which can be straightforwardlyη adapted from calculations in [ spacetime. The negative eigenvalues, as introduced by Atiyah, Patodi, and Singer [ It turns out thatthat this global is anomalies not are the detecteda by case. bordism the A exponentiated invariant when quick perturbative way anomalies of vanish. reaching Because this the conclusion spin-bordism is group to recall gauge transformation by theinstanton, central as element we review in section similar global anomaly, givenequivalent to that SU(2) An SU(2) chiral gaugeanomaly theory when there in is fourtations, an dimensions odd for suffers number from offunction fermion a changes multiplets non-perturbative in sign global isospin undertrivial 2 an element SU(2) in gauge transformation that corresponds to the non- 1 Introduction 5 Cobordism and the absence of U(2) global anomalies A Spin-U(2) bordism 3 U(2) gauge theory with4 a spin structure U(2) gauge theory without a spin structure Contents 1 Introduction 2 Review of the SU(2) global anomalies JHEP05(2020)098 ]. 8 (1.2) U(1). 2, for . The / 3 × + 3 r U(1). Thus, SU(2) ]. For example, × 7 ∈ ) 2, which allows for a iπ / SU(2) , e Z -invariant using the Atiyah- ∈ 1 η ) − iπ 2, by reducing the anomaly / , e 1 + 1 SU(2)) = r , B ( U(1) corresponds to a representation of Spin 5 U(1) 2 ) that there can be no other new global × 2 multiplets, is of some phenomenological × / 2) arises simply because there is a mixed / Z / 1.1 + 1 + 1 – 2 – r SU(2) r ∼ = Note that this is only true when the global structure of U(2) 2 ]. 4 ), changes sign under the combined action of a diffeomorphism 2 P C U(1), even though the formula for the perturbative anomaly is the × U(1) in the background of a single instanton, and might thus be tempted × 2 quotient is generated by the central element ( we arrive at the same conclusion by directly computing the / 5 Z SU(2) . The partition function for such a theory, defined on certain manifolds that are not ∈ 0 structure, and perform a similar analysis relating to the ‘new SU(2) (global) anomaly’ ≥ 1) We then turn to the more subtle case of a U(2) gauge theory defined without a spin or Understanding the absence of global anomalies in a U(2) gauge theory, but nonetheless We show explicitly that the (perturbative) mixed triangle anomaly can vanish only In this paper, our first goal is to explain why there is no global anomaly in a U(2) c , Z In section 1 2 ∈ − is anomalous if therer is an oddspin number (in of particular, on fermion multiplets with isospin 4 Patodi-Singer (APS) index theorem [ spin that afflicts an SU(2)Recall gauge that theory fields that in is suchrequires a similarly that theory defined all are without fermions instead a (bosons) defined spin using have structure a half-integer [ spin-SU(2) (integer) structure, isospin. which The SU(2) theory anomalies in a U(2) theory (defined with a spinthe structure). necessity of the conditioninterest, because on U(2) isospin could be 2 theanomaly gauge group cancellation for the in electroweak theory suchnumbers [ a of field theory content provides in constraints the on context of the going electroweak beyond quantum the Standard Model. the gauge group is strictlygauge U(2). group The argument SU(2) does notsame, follow because for not the every (locally representationU(2). isomorphic) of Having SU(2) realised that theas apparently a global SU(2) local anomaly anomaly, is manifest we in may U(2) conclude rather from ( odd number of multipletstriangle with anomaly. isospin 2 if there is ancancellation even condition number modulo of 2. multiplets with isospin 2 to reach the sametransformation conclusion is that equivalently there describedthe can by anomalous transformation be the is element a inthe ( global fact variation anomaly. a of local However, the U(1)the this transformation, effective fermionic and gauge action. partition we can function The compute using non-invariance of the the appropriate path counterterms integral in measure (when there is an be written as where the As for the SU(2)( case, one could make a constant gauge transformation by the element when perturbative anomalies cancel.possible In contrast global Ω anomaly in the SU(2) theory. gauge theory, defined with aargument choice is of simple spin enough structure. to summarise This in is this the Introduction. subject Recall of firstly section that U(2) may bundle, which means that there can be no global anomalies in the 4d U(2) gauge theory JHEP05(2020)098 2 3 / (2.1) + 3 r ). The Atiyah- 5 γ 1. This gives rise to . , ) − 4 representation, coupled ) denote the number of cannot be replaced by a F − j ( in deriving the new SU(2) n 1 ( p ϕ W ) + j n ( and T we interpret our results in terms − ϕ 5 + 1) (2.2) . Let = j F F ∧ F + 1)(2 has the same effect on the fermionic partition j Tr ( – 3 – j W . This is the new SU(2) anomaly, which we shall M 2 3 Z 2. W / (which we take to be Euclidean) using a spin struc- 2 and 1 π ) = 8 j . Finally, in section ϕ M + 3 ( − r 4 T = − n − + We first review the global anomaly that occurs for an SU(2) gauge n . is the first Pontryagin number (or instanton number), and we review the pair of global anomalies in SU(2) gauge theory. In section 2 Z 2 ∈ ) F ( 1 ]. Consider a single fermion transforming in the isospin- p 1 In section It is important to stress that, in the U(2) theory, this condition on isospin 4 The new SU(2) anomaly enjoys a similar but subtly different fate to the old one. This The second goal of this paper is to understand what happens to the new SU(2) anomaly and an SU(2) gauge transformation Singer index theorem tells us that where The old anomaly. theory defined on ature four-manifold [ to a background SU(2)fermion gauge modes field with with curvature positive (negative) chirality (i.e. eigenvalue under without spin structure inof section cobordism invariants. We thenceU(2) explain theory why there defined are using no a other spin-U(2) global structure. anomalies in2 the Review of the SU(2) global anomalies multiplets must be satisfiednew simply SU(2) for anomaly, perturbative this anomaliesspin condition to manifolds. persists cancel; even thus, if unlike we the choose to restrictwe our discuss attention the to U(2) theory defined using a spin structure, before turning to the case gauge-gravity anomaly for theof U(1) perturbative current. anomalies reduced Byusing modulo considering a 4, this spin-U(2) we particular find structurefermion combination that multiplets can with the only isospin U(2) be 4 gauge anomaly-free theory defined when there is an even number of local U(2) gauge transformation,the as anomalous combined was action the of function case as for the a ‘old’ locala SU(2) local U(2) anomaly. anomaly, gauge that However, transformationa is with Feynman a determinant diagram combination with of two the external mixed triangle SU(2) anomaly currents (corresponding and to one U(1) current) with the using this time a spin-U(2)that a structure global to anomaly should parallel afflict transportand such a fields. again, theory, corresponding this Again, to turns one the out might new SU(2) not expect anomaly; to be thetime, case, because of as the we crucial showanomaly, role in we played section by find the that diffeomorphism the anomalous combination of recap in section in the analogous situation infield which content the is gauge such group that is(even) all enlarged U(1) fermions from charges, (bosons) SU(2) then have to the half-integer U(2). (integer) U(2) isospins If gauge and the theory odd can be defined without a spin structure, ϕ JHEP05(2020)098 ]. is 11 F (2.5) (2.3) (2.4) which 1) , where , where modulo − ϕ that the a F j is equipped 3 j 1) σ M − ≡ N = − A ) with an SU(2) n structure, in ref. [ to act by complex M c + ∗ i viz. + z n . A spin-SU(2) connec- denotes a basis for the The transition functions 2 7→ of SU(2) paired with the (2), } i 3 (going beyond the case where P a j z 1 t su C : G { − , in , using a spin ]. of ϕ 2 1 . a } P i SU(2). The new SU(2) anomaly C z { SU(2) ∈ . Here , and 2 2 × 1 ab / is congruent to . δ P 0 Z − ) − C ≥ j n ( Z ) (mod 2) connection T − structure – but one must assume that ∈ F 1 2 c ( Spin(4) c 2 can contribute to this anomaly, and only in + r 1 to be on the path integral measure, since ( / – 4 – n is equivalent to applying a gauge transformation p ≡ ) F F ) = M j + 1 b j (for certain non-spin manifolds ( t structure, in order to see the new SU(2) anomaly. 1) 1) (4) r 2, for a j T c t − − / M ≡ ]. = 2 SU(2) of j + 1 j 10 r , N ϕ ’ structures, for various Lie groups 9 (2). Because G Spin SU(2), this implies that SU(2) is anomalous in such a scenario. su ∈ . 1 Suppose now that there is no spin structure available, and that − W there is an even number of fermions transforming in repre- ) is odd. Thus, the anomaly vanishes if and only if the following holds 4 Spin(4). All fields must transform in representations of this group, which . 2 ∈ 2.2 P 2 quotient is generated by the central element F C / 1) Z representation of − j may be defined by embedding a spin Condition 1: sentations with isospin 2 is odd, then the partition function will change sign under the action of ( A To see this anomaly one may take In the simpler case that we discussed above, we saw how the usual SU(2) anomaly Only fermions with isospin j The idea of using such ‘spin- It was first observed that a fermionic theory can be defined on 3 4 is the fermion number. But since ( N = U(1)), was introduced in refs. [ tion G Indeed, every orientable four-manifold admitswith a a spin spin-SU(2) structure, and not a spin SU(2) anomaly is theis non-invariance of a the combined path diffeomorphism integralgauge under transformation a transformation ˆ conjugation on the homogeneous complex coordinates spin such as could be seenequivalent from to the an action SU(2)is gauge of more transformation ( subtle, by and cannot be seen from a pure gauge transformation. Rather, the new where the element ( requires that all fermionsSuch a have theory half-integer can isospin, be defined and on all all orientable bosons four-manifolds, have including those integer that isospin. are not The new anomaly. fermions are instead defined using afor weaker a spin-SU(2) structure. spin-SU(2) bundle are valued in the group This is the familiar SU(2) anomaly discovered by Witten [ F by the central element backgrounds with odd instantonDynkin number, index because ( it is only for these values of isospin- 2, the total number of fermion zero modes satisfies If is the Dynkin index defined via Tr( JHEP05(2020)098 . ) ϕ 1.1 with (2.8) (2.9) (2.6) (2.7) (2.10) ϕ defined ]. . 8 A
1 − is divisible by 0 1 0 j J = W . Hence, the fermionic ϕ ] , the index 8 , j . ) is the second Stiefel-Whitney 1 under ˆ by obeys the following quantisation For a single fermion multiplet in − + 3) . a ϕ ] TM 5 j ( A , and the theory is non-anomalous, if 2 [ equals the index of the Dirac operator ϕ w . Z 1 2 j 1)(2 ) (mod 1) , but is congruent to 2 mod 4 only when 0 2 j / ≥ N j − = J Z 2 TM π j 1) ( ∈ da 2 2 connection − (4 r , where 1 – 5 – ( w c 1 P S 24 has to arise from the path integral over the fermion M ˆ C ϕ Z −→ Z ϕ = ] ⊂ 1 2 2, for j connection reverses sign under the diffeomorphism which also flips its sign, such as A / [ S ≡ c N Z W π + 3 = da 2 r j S , however, is invariant under the combined action of J ], which demonstrates the subtle interplay between local and Z defines a properly-normalised U(1) gauge field. In particular, such that A 6 , a a 5 . For all other half-integer values of the Dirac operator only has zero modes of one chirality. 0 ] transforms under the action of ˆ 2 ≥ the number of zero modes . Such a spin P A Z 2 [ 2 there is an even number of fermions transforming in repre- C P Z ∈ P C r C connection ⊂ representation coupled to the background spin-SU(2) connection is even for all half-integer values of c 1 j j 2 for P J / C + 3 Condition 2: sentations with isospin 4 r An anomaly in the transformation ˆ This is because on is the diagonal Pauli matrix. The spin (they are not only congruent modulo 2 as before). 5 = 4 3 j theory defined with aimplies spin there structure, are no for global whichpreviously anomalies. noted the We in vanishing will refs. here of give [ global the a anomalies bordism physical in explanation group U(2). of ( this fact, This is the new SU(2) anomaly recently discovered by Wang,3 Wen, and Witten [ U(2) gauge theoryWe with now a turn spin structure to U(2) gauge theory. We begin with the simpler case of a U(2) gauge 4. Hence, the partitionand function only is if invariant the under following ˆ condition holds: partition function The index j above, the Atiyah-Singer index theorem implies the index is [ The zero modes come in pairs with eigenvalues +1 and any SU(2) gauge transformation zero modes. On J the isospin- for some The spin-SU(2) connection condition for any closed orientedclass, 2-manifold which is such thatchoose a 2 spin σ JHEP05(2020)098 )), U(1) (3.3) (3.1) (3.2) 3.1 × , where } j,α 2 is labelled q . Recall that / { 4 Z 2. The isospin- S / /  (satisfying ( which vanishes for , +1 q 0 R U(1)) ≥ , together with a U(1) ∧ j , which is a local gauge Z × U(2), this global SU(2) 2 R iπ must satisfy the following e ⊂ ∈ j j and defined on and charge and (SU(2) q j F ∼ = ) is violated. Thus, we find that . ] ) must hold for a U(2) gauge theory 2.4 A + gravitational piece , [ , we have that 2.4 θ Z F ) is satisfied by all the SM fermion fields, where ) ) = 1, this reduces to j ∧ ( F ) + gravitational piece] 3.1 ( . Namely, ] under a potentially anomalous U(1) gauge qT F 1 F j ( p A (mod 2) [ 1) 1 Tr p Z − – 6 – 4 and ( j ) S 2 j q Z ( → representations of U(2), with charges ≡ 2 ] ) is only odd for isospins j π SU(2). Embedding SU(2) q j A is also odd for these representations. Hence, there is iqθ 8 [ ( iqθ T ∈ q Z T − −  1 − ] exp ] exp [ A A [ [ 6 Z Z → = ) means that ] and the instanton number A 3.1 [ π Z = . We assume without loss of generality that all fermions have left-handed j θ ]). For a U(1) transformation by angle 12 , subject to a restriction relating 2; in other words, precisely when condition ( q ,...N / ) is even. Indeed, one can directly derive that condition ( Recall that the Dynkin index The variation of the partition function Consider a theory with a single fermion with isospin The representation theory of U(2) plays a crucial role in the arguments used in this j fermions transforming in isospin- We note in passing that this isospin-charge relation ( + 1 ( 6 . Setting = 1 j 4 r α chirality. The mixed triangle anomaly (that is, the triangle anomalyU(1) involving corresponds two to SU(2) hypercharge.or U(2). Hence the electroweak gauge symmetry could be either SU(2) the SU(2) global anomaly manifestsembedded itself in rather U(2). as There a are perturbative no anomaly global when SU(2) anomalies is inby considering the the U(2) equations theory. forN perturbative anomaly cancellation. Suppose that we have qT charge relation ( necessarily an anomaly if2 there is an odd number of fermions in multiplets with isospin where the gravitational piece isS proportional to the integral of Tr We see that the path integral is invariant under this transformation if and only if transformation is equivalent to atransformation. U(1) gauge transformation by transformation can be computed using(see the e.g. appropriate counterterms [ in the effective action in convenient units where both gauge couplings are setcoupled to to one. a background U(2)the gauge usual field SU(2) with anomaly curvature occursthe when gauge the transformation fermionic by partition function changes sign under paper. Recall that anan irreducible irreducible representation representation of of U(2) SU(2),charge itself labelled by an‘isospin-charge isospin relation’ JHEP05(2020)098 , . c π 5 (4.1) (4.2) (4.3) (3.4) (3.5) , as we W is the spin a Spin(4) with the 2 quotient within / ∈ Z , , F , where a 1) 3 to be the combination of . odd even σ − ϕ q q ϕ 1 when acting on fermions. = − A . , , = ←→ ←→ 2 . define ˆ = 0 ] U(2) ϕ with the U(2) gauge transformation 2 A 2 2 plus gauge transformation [ × ∗ j,α i / / q z Z ϕ Z 2 j / =1 0 j 7→ N + 1) X α J 2, together with the isospin-charge relation, ≥ 0 (mod 2) i ) / 0 Z z 1) j ≥ Spin(4) ≡ ( : − ∈ Z +1 – 7 – ( T 1 ϕ ≡ j 0 2 (2 ≥ / ˆ j ϕ −→ ∈ Z X (4) ] +1 2 0 j acts on the partition function as A ≡ ), which is invariant under ˆ ≥ [ ∈ X , and as in section Z ϕ U(2) 2 Z 2 j 2.7 -invariant explicitly. We give such an account in section ∈ mix P j η A ←→ ←→ U(2). Recalling also the effects of the C ), ( Spin ) must be satisfied to avoid a perturbative mixed anomaly. It ∈ (on its own) is such that 2.6 2.4 1 ϕ to be − boson M . Recall that ˆ can be thought of as a certain spatial rotation through an angle fermion 2 ϕ ) is odd only for j ( T that is a combined diffeomorphism . Moreover, we define a spin-U(2) connection
ϕ ) label the U(2) representations as before. 1 − 2 quotient is generated by the product of the element ( q, j 0 1 0 / Z The diffeomorphism In the analogous SU(2) theory, the new SU(2) anomaly is associated with a trans- = the complex conjugation diffeomorphism W connection satisfying eqs. ( More specifically, reviewed in section Let us first analysethat the end, behaviour again of take the U(2) theory under this same transformation. To where ( formation ˆ The order-2 central element U(2), we have the following constraints on the allowed representations: a spin-U(2) structurerepresentations to of parallel the group transport fields, provided that all fields transform in is possible to give ain this unified theory discussion by of computing the the perturbative and non-perturbative anomalies 4 U(2) gauge theoryWe without now a turn spin to structure the case where a spin structure is not available. Instead, we can use and hence that condition ( The fact that means that reducing mod 2 immediately yields gauge bosons and one U(1) gauge boson) is proportional to JHEP05(2020)098 , ] is = A 3 [ + 1 (4.6) (4.7) (4.8) (4.4) (4.5) da Z j µνστ = , taking ˜ R f M and coupled to the στ q , where F µν f 3 µνστ a 2 are the generators of  δ ). Consequently, cancel- / 1 2 x, a 4 , σ 4.3 = 2 = Z , , is g d x, = π ) g and charge a µν 4 √ µν 2 fermion. Because the matrix a d ∈ F j ˜ / F τ grav , so that there is no corresponding µν S µνστ ˜ Z F ˜ π R − , µν a ∈ , θ / F ψ θ / µνστ ! gauge with metric Tr i R S 0 U(2) ] is identical to ( − implies that M − M M A Z ∈ i Z [ 0 a q 3 ) – 8 – 6= 1 for Z

! 2 σ Q iθ π 0 ˜ ϕ ] exp ( iθ 24 e W 7−→ = 16 A Tr( a [ iθ A 0 2 − ψ e Z π

= iθ → 16 ] − ) = A representation. Recall that [ θ gauge ( = are the components of the Riemann tensor. Z j S ˜ W . This time the gravitational contribution will be non-vanishing grav µνστ S A R is necessarily not equivalent to a pure U(2) gauge transformation. Since ) is not proportional to the identity, this diffeomorphism cannot therefore : a ϕ ψ 4.4 has non-zero signature. Taking into account the contributions from both the , where 2 ), for a single fermion multiplet with isospin- στ θ P is the generator of the U(1) factor in U(2), and the trace sums over all 2 ( αβ C ˜ W Q R We can relate both these integrals to characteristic classes of bundles over However, what we can do instead is construct a local U(2) gauge transformation whose µναβ is inequivalent to a local gauge transformation, in contrast to the situation in section  ˆ 1 2 components of the isospin- care with the variousthe normalisation SU(2) factors. factor Noting of that U(2), the choice indices explicit for clarity), and where where in which the trace is only over the SU(2) gauge indices (we here choose to keep Lorentz spin-U(2) connection because mixed gauge anomaly and thefunction, gauge-gravity for anomaly, now the on shift a in general the four-manifold Euclidean partition i.e. by a pure U(1) phase.gauge Note transformation that in det SU(2) byunder design. Let us now compute the transformation of action on the fermionic partitionlation function of perturbative anomalies shallvanishes. guarantee To that wit, the consider suspected a global gauge anomaly transformation in by fact be subsumed by the U(1)transformation phase degree ˆ of freedom inϕ U(2). Thus, as inwe the might SU(2) suspect case, that the this new SU(2) global anomaly will stick around in the U(2) theory. Weyl fermion where the index labels Lorentzappearing SU(2) in ( indices of the spin-1 corresponding (in certain coordinates) to the following transformation on a 2-component JHEP05(2020)098 i 2; / (4.9) , and (4.13) (4.10) (4.14) (4.15) (4.11) (4.12) q + 3 0 ≥ Z plus a U(2) 4 ϕ ∈ , ) j f. M ∧ ( ; the only difference σ f 2 2) has the same action and odd charge . M 2) with eigenvalues + j  Z ) to an integral over the π/ q ] = 3 ) π/ ( ( j  ( ˜ 4.7 M ϕ W [ T 1 plus SU(2) gauge transforma- 2 p ) that detected the new SU(2) + 1) π . ϕ 2) of these two transformations. ] iqθ = 4 j 2.8 A q π/ [ − (2 R ( 1 2 Z 2 = q. ∧ ] of the diffeomorphism ) ˜ 2 σ. ) W π − , + 1) x A j q/ · 1 8 R [ 2 ( j 4 ) j (2 ˆ J ϕ d j Z T P = ( (2 Tr 1) 2, we see that there is a perturbative U(2) C µν 4 ≡ iθ ) is violated. T 2 ˜ f 8 / iθ − f ) . The argument proceeds almost exactly as M  ( = − π ∧ 2) Z µν – 9 – 4 + 3 2.10 f iθ f iW (2 = 1 2 2) M π/ = + ( Z − M π/ . We can thus reduce ( M = = 4 ϕ (  connection determines its first Pontryagin class in Z a Z ˜ x W ∈ gauge c  −−−−−→ 1 2 4 of the diffeomorphism W S ) 2 gives ] j gravity · j ] exp ϕ gd S A 2 ( [ π/ is the same index from ( A √ 2) T viz. [ j Z = , Z J  π/ θ ( to deduce that 2 M µνστ → connection π ˜ q ˜ W R iqθ ] 4 ) for the Dynkin index, we find that the factor in square brackets of c A − ) of the spin [ σ 2.2 + 1) , where µνστ Z we have that, when = q 2.7 j R j 2 J M P iθ Z . gauge C 2 − 2 ) = (2 S π P 2 (mod 4) only when 1 Q C 16 ≡ = ) − = 1 for is to consider the composition ˆ j ( M σ connection, T W Another way to see that the U(2) gauge transformation by The partition function therefore shifts by c tion In other words, consider thegauge combined action transformation on by the argument for theis new SU(2) that anomaly, now as the summarised fermion in zero section modes transform in pairs under ˆ Recalling that all fermions inthat this theory have half-integral isospin anomaly when there isin an other odd words, number precisely of when fermion condition multiplets ( with isospin on the path integral as the action ˆ is nothing but anomaly. Therefore, setting Using the expression ( and that Tr( when For the gravitational contribution, we use the fact that Since spin The normalisation ( terms of the signature the curvature of the spin JHEP05(2020)098 In )], 7 (4.16) (4.17) (4.18) 4.16 ( 1 2 )- 2) is always 1 as before). 3.4 − [( π/ ( 1 4 ˜ W · viz. ϕ noted above, we immedi- j J . , . = 0 ) emerges somewhat coincidentally ), for the cancellation of the old and j,α = 0 q j j,α 2.10 2.10 =1 N q X α α 0 (mod 2) X j + 1) 2) must contribute the same mod 2 anomaly. ≡ ) and ( J j 1 – 10 – π/ 2 ( (2 2.4 / ˜ j W +3 X 0 X ≥ ) enjoys a different ‘status’ in the SU(2) theory versus X ≡ Z half integer 4 and j ∈ 1) whose product is now +1 (rather than j 2.10 ϕ grav − ) on the U(2) theory is required by U(2) gauge invariance, ) that, in the SU(2) case, is required to cancel the new SU(2) A for the old SU(2) anomaly, we can again deduce the necessity 4.18 2.10 3 2. / ) directly from the equations for perturbative anomaly cancellation. This + 1 0 ≥ 2.10 Z 2 ∈ (rather than +1 and j i − As a result, the condition ( For the new SU(2) anomaly, however, the mixed diffeomorphism plus gauge transfor- As we saw in section In fact, the new SU(2) anomaly is not an insurmountable barrier to consistency on non-spin manifolds 7 the consistency of ancontrast, SU(2) the gauge constraint theory ( when formulated only oneither; in spin this manifolds. case, onehas can couple the to same a anomaly topological theory quantum (specifically, field this theory anomaly (tQFT), in theory the has same 5-form 4d lagrangian bulk, given which by the product partition function. Infrom this perturbative sense, anomaly the cancellationas condition in ‘trivialising’ ( the the U(2)interpretation new theory, is SU(2) which rather global that should there be anomaly; is thought for no of global the anomalythe old at U(2) all SU(2) theory. in anomaly, U(2). It the is correct important to recall that the new SU(2) anomaly is no barrier to there is an associated perturbativeisospin anomaly if there are an odd number ofmation multiplets is with not equivalentbe to equivalent a to a local local transformation transformation in in U(2). U(2) at It the nonetheless level transpires of its to action on the fermionic We have now seen how bothnew conditions SU(2) ( anomalies, do notsubgroup correspond of to U(2). global anomalies The whendifferent. arguments SU(2) used In is for the embedded case the as of two a the the anomalies global old transformation were, in SU(2) however, SU(2) anomaly, qualitatively corresponds for to a a theory local defined transformation using in a U(2), spin for structure, which recovering the condition ( anomaly. 4.1 Interpretation of the U(2) anomalies Reducing this equation modulo 4,ately and obtain using the properties of If we take a particularwe linear obtain combination of local anomaly equations, Thus, since there isnon-anomalous, and an so even each number of of ˆ zero modes,of the condition ( action oftime, ˆ however, we also need to use the cancellation of the gauge-gravity anomaly, and JHEP05(2020)098 φ (4.20) (4.19) as an 4 ]. Note that the , 13 and is a singlet 9 µνστ , ˜ iθ R e = µνστ , then let us modify the g ) emerges only coinciden- } and for any fermion content. gR ] j,α √ M for q 2.10 16 φ { , θ 2 is circle-valued; under the ‘large gauge π 15 grav φ , ), requires that the pseudoscalar ) + A x x 384 i 12 ( ( , θ φ 2 + | 2-valued global anomaly. This kind of anomaly b / → Z aµν − ) ˜ F x If we consider again a general spectrum with ( dφ a µν | φ 10 8 1 2 – 11 – and with charges φF j 2 viz. π ), the effective lagrangian is now invariant; the shifts mix L ⊃ A 32 i 4.5 = , the phase of the exponentiated action shifts by an integer multiple of WZ π ], a terminology that stems from a famous application to cancelling mixed ), for a smooth function L x 14 ( θ ) + 2 x ( , φ ) + x WZ → ( ) L φ x ( + φ → ) x ) is a dimensionless (circle-valued) pseudoscalar field which enjoys a shift sym- ( x φ ( L → L φ of Stiefel-Whitney classes), and thereby cancel the One can check explicitly that under any U(2) gauge transformation, including generic For instance, if one interprets the U(2) gauge theory described in section fermions transforming with isospin- 3 We remark that these WZ terms are well-defined even though We might imagine that heavy masses could arise from Yukawa-like interactions with a Higgs field. The mechanism we describe here for cancelling anomalies at low-energies might also be referred to as a and so the path integral is unchanged, for any orientable 4-manifold 9 8 w 10 j 2 π anomalies in string theory. transformation’ 2 However, the precise constructionventure of the a details suitable of Yukawa a sector UV is completion not here. immediately obvious, and we do not w cancellation mechanism was introduced astQFT a ‘topological to Green-Schwartz which mechanism’ we inthe [ couple theory. has no propagating degrees of freedom‘Green-Schwartz mechanism’ that [ would alter the phenomenology of longer linearly-realised. To seemations this, note that invariance undershould local have a U(1) kinetic gauge term transfor- of the Stueckelberg form, that is U(1) transformations of the formof ( the WZ terms preciselyof cancel the the path shift integral in measureHowever, the for gauge effective the action invariance chiral due comes fermions, to at as the a is non-invariance the price, purpose which of is the that construction. the full U(2) symmetry is no where metry under the U(1)under factor the in U(2), SU(2)out part. a “mirroring” These set WZrepresentations of but terms heavy with conveniently opposite chiral encode chirality. fermions, which the transform effects in of the integrating same set of U(2) cancel anomalies in theN low-energy theory. effective lagrangian by adding the pair of WZ terms [ leave behind a theoryanomaly with interplay described the above. ‘new’ type of globaleffective anomaly, field thereby theory disentangling the cutoff of scale the Λ, light then excitations Wess-Zumino that (WZ) terms is may valid be only included up in to the some lagrangian which momentum U(2) theory inconsistent (even on spin manifolds). 4.2 Disentangling theIt anomaly is interplay possible to maketally rigorous in the the claim U(2) thatleast theory the at condition without ( the spin level structure. of In effective fact, field in theory, this the section perturbative we anomaly show may that, be at cancelled to and so its violation, like the violation of the original Witten anomaly, would render the JHEP05(2020)098 .
→ 1 − b 0 1 0 , being R . So both ∧ 2 = π R ϕ 8 W Tr decoupling, is 2-valued global b ), one may now / Z 4.19 and hence invariant ϕ is the combination of a ϕ due to the chiral fermion , that led to the new SU(2) ϕ ). Hence, we conclude that ϕ is invariant under ˆ 2.9 φ ) without violating perturbative . ϕ . The Pontryagin class 2.10 ? Recall that ˆ F ϕ , at least in the absence of couplings to 2 P C – 12 – with a U(2) gauge transformation by ]. 8 ϕ defined earlier in this section, which should now be a becomes massive, meaning that at low-energies only a 3 b σ = reveal that there can be no further global anomalies). Unlike A is locally equivalent to a spatial rotation (in four dimensions), . We remark that a similar trick cannot be performed to restore ], is invariant under the diffeomorphism 2, then the effective field theory, which is free of perturbative ϕ 17 / 5.2 5.2 . . The global anomaly that remains would then have precisely the is an SU(2)-singlet, the pseudoscalar 2 φ + 3 φ P U(2) is linearly-realised. ) are invariant under the action of ˆ 0 C , and hence so is the field strength ) is violated, in other words if there is an odd number of fermions with ≥ ⊂ ϕ = Z 4.19 4 . Up to the effects of the WZ terms, we have arrived at precisely the SU(2) 2.10 M behaves like a U(1) gauge field. ϕ ∈ b Thus, the component is the U(1) component of the spin-U(2) connection, which transforms as j . Finally, given 11 b ϕ . It is worth spelling out the fact that, as is the case for the new SU(2) anomaly, this In this way, one can in fact disentangle the effects of perturbative anomalies in the U(2) We already know how the partition function varies under ˆ How do the pair of WZ terms transform under ˆ Interestingly, adding WZ terms to the effective lagrangian is not guaranteed to cancel dθ Locally, 11 + the old SU(2) global anomaly in the U(2) theory. residual global anomaly can always beof cancelled cobordism by coupling in to section a tQFT (and considerations same physical interpretation asthe the theory new SU(2) on anomaly;topological non-spin it degrees manifolds of presents such freedom. a as is barrier This in fact to a that defining sense the stillas new there we SU(2) in explain anomaly, U(2), in unlike may section the also old be one, understood from the perspective of cobordism, gauge theory with spin-U(2) structure,new and SU(2) isolate an anomaly effective at theorybe low that achieved energies. suffers by from But including the itof WZ is the terms important theory (or to – somethinga emphasize for similar), pseudoscalar that instance, which this field in enriches can the the only gauge dynamics we have chosen one must include the effects of anomalies by virtue of theanomaly effective in WZ ˆ term, doestheory indeed defined suffer with from spin-SU(2) a structureto that illustrate was the introduced new by SU(2) Wang, anomaly Wen, [ and Witten WZ terms in ( contribution, which is preciselyif the condition variation ( given inisospins eq. ( interpreted as a spin-SU(2) connectioninvariant due under to the ˆ massive U(1)a component topological invariant [ under ˆ and given also that anomaly cancellation. Forcombined such diffeomorphism plus a gauge theory, transformation, denotedanomaly we on ˆ should reconsider its behaviour under the complex conjugation diffeomorphism The spin-U(2) connection b subgroup SU(2) the more subtle globalconsider anomalies. fermion In content the which presence violates of the condition WZ ( terms ( where JHEP05(2020)098 2 / 1 I U(2) (5.1) (5.2) B ] 6 , 5 ]. In the current 20 – 18 ]. 2 is necessarily in a non- -invariant must be trivial. / 2. For anomalous fermion 8 , η / -invariant becomes a cobor- 1 -invariant, which reduces in η [ η 1 S equipped with spin structure, the × . 2 M . / M Z equipped with a U(2)-bundle structure is a U(2)) = 0 X – 13 – B SU(2)) = ( , which realised the potentially anomalous global B ( 2 Spin 5 ), and thus in a complex representation. Hence, the Ω Spin 5 3.1 Ω with the U(2) and spin structures extended appropriately. Y is non-vanishing on the mapping torus 2 / 1 structure, if one includesanomalies, a and pair couples of to WZ a terms tQFT to to cancel cancel the the residual perturbative global anomaly. It is possible to3/2 write fermion, down that a can consistent be U(2) defined on theory non-spin of manifolds a using a single spin-U(2) isospin- I These statements can be seen from a slightly different perspective. The exponentiated When SU(2) is embedded in U(2), a fermion with isospin-1 One might distil the various ideas at play in this section into the following statement: -invariant captures both the global and perturbatives anomalies [ -invariant no longer reduces to a mod 2 index in this case. But this does not matter in η case, this can bemeans seen quite that explicitly. any The closedboundary vanishing spin of of five-manifold the a fifth six-manifold bordism group of Hence, in the case that perturbativedism anomalies invariant, vanish there and are the noWe cobordism therefore invariants and deduce thus that the with there our are explicit no calculation globalSU(2) in anomalies gauge section in transformation this to theory. be This equivalent to is a consistent local U(2) gauge transformation. trivial representation of U(1)η by ( the end, because one may calculate the bordism group directly to find that [ There is a correspondingthis cobordism case to invariant, namely adenote 5d the this mod 5d 2 mod indexcontent, 2 because index the for a fermions single are fermion in with real isospin-1 representations. Let For an SU(2) gauge theory definedoriginal on SU(2) a anomaly four-manifold is detected by the bordism group Finally, we discuss thedimensions. connection Such between considerations our willanomalies also results in enable the and us U(2) cobordism to gaugespin invariants theories conclude structure. in we that have five there considered, are defined no either further with or5.1 without a Case I: with a spin structure 5 Cobordism and the absence of U(2) global anomalies theory, but would rather embue thepostpone theory such with considerations topological for order future in work. the deep infrared. We the WZ term, such topological degrees of freedom would not alter the dynamics of the JHEP05(2020)098 2 ) ) is D 2 and 5.5 5.3 (5.5) (5.6) (5.7) (5.3) (5.4) U(1), D 2 S . Note × 1 ) denote a ] S , cannot be x 4 1 S ) is a multiple -genus (some- ˆ A R as ( 1 a j . We may choose , p t is gauge equivalent to 6 a # I , 3 M A ) F R j  ( . This is the boundary = , π 1 under the U(1) part and F 6 2

S qT F is the SU(2) connection written q parametrises the  1 2  φ × φ π . This can be expanded out A Tr = F ) becomes trivial on all closed 2 F M 1 3! X  F = . + ∧ on the equator, πiη , where X , where π 2 X F U 2 η F 2 dφ A F. − D ) Tr has unit magnetic flux through − ∼ π × ) Tr exp is a U(1) gauge field supported only on the ) coming from the first term in eq. ( + 2 6 2 f ) Tr a I 1 X π a R M ( 8 φ/ +1 R Y ˆ j ( = A Z 2 πiη 1 is spin, due to a signature theorem of Rochlin, M + ( (SU(2)), recall that a 5d gauge field on the mapping 2 1 2 p with instanton number one and letting U( Y 4 Z S = ), and A – 14 – π − 1 M ) × fq
24 π M f π 2 / M ∂Y " 2 D = such that -invariant on such a five-manifold and the anomaly i Z 2 2 and the SU(2) gauge field = φ/ η S S 1 2 F F Z × f − X q = ind M extended to 1 2 to which the U(2) bundle may be extended, where Z 6 φ I = (1 2 1 2 = A 2 φ D 6 + D = to be the mapping torus A I a , whence we obtain × × 6 2 to be the connection for a Dirac monopole with twice the smallest unit of I . M = X D M 2 Z a × M A D ∂Y M × = = Z M X Y -invariant explicitly, from the anomaly polynomial Z of the form η 1 is then fixed by the Atiyah-Patodi-Singer (APS) index theorem [ S 6 denote an SU(2) gauge field on × I ) is the first Pontryagin class of the tangent bundle. Now, We have that A M R under the SU(2) part of the gauge group U(2), we can write the U(2) gauge field ( 12 1 j = p X On the other hand, when the perturbative anomaly doesn’t vanish, we can use ( Letting on the boundary in terms of the U(1) gauge field is a one-instanton on 12 φ torus extended to any boundingmay six-manifold. consider a If connection such anabove SU(2) (supported configuration only is onfactor. embedded the in boundary In U(2), however, particular,charge we take placed at theA centre of the hemisphere. Because gauge transformation in the non-trivial class of F To see the anomaly, weF can choose where of 48 when the (orientable) four-manifold so we can ignore theand contribution focus to only exp( onisospin- the second term. For a fermion with charge where we have expressed the anomalytimes polynomial called explicitly in the terms ‘Diracto of the give genus’) and the U(2) gauge field that, importantly, this cannotbundles. be done in general for SU(2), or indeed for SU(2) to compute the the closed five-manifold of a six-manifold a hemisphere (topologically a disc) whose equator coincides with the original polynomial Whenever the perturbative anomalyspin vanishes, five-manifolds exp( and so there can be no additional anomaly. The direct relationship between the JHEP05(2020)098 we (5.8) (5.9) (5.11) (5.10) A , we cannot use . 1 S -invariant for this 5.1 η . In appendix 4 ). 3 , ) . 2 , / TY ( . Z . Recall that any fermion with isospin 3 2 ) = 1 j / × w ( ) Z 2  qT / 2 is a suitable generator for the bordism 1 ) are Stiefel-Whitney classes. The crucial = ]. The former corresponds to the old SU(2) / Z , and as the antipodal map on TY S 1) 8 2 ( Z 2 2 − = / P . We thus arrive at the same physical outcome × 2 representations must have odd and thus non- TY . Moreover, unlike in section / ) -invariant for an arbitrary closed five-manifold U(2) q ( 2 w C / , the 5d mod 2 indices associated with a single 2 2 – 15 – 1 / Z ) implies that not all such manifolds are bordant 3 η ] in identifying a mod 2 cobordism invariant dual × / 2 , Z Y 3 / 8 S P 2 3 Z I 2 ) = ( SU(2) 5.9 C w Spin / I 5 × × Z X  2 or 3 or Ω 2 / ) = 2 J Spin 5 P πiη / and Y 1 2 Ω C ( I 2 ( − J / 2 respectively [ 1 / 14 I ) vanishes trivially on spin manifolds, it does not appear in Y ( ] ) to be 2 or 3 J ) is automatically a cobordism invariant of 5-manifolds with spin- / 23 Y 5.9 – ( -invariant (as we saw already in section J η ). 21 ) is a mod 2 cobordism invariant of 5-manifolds with no further structure 5.2 Y ( 2 necessarily has odd charge 2 acts as complex conjugation on J / / Z Hence, ). Because ) or ( is a closed 5-manifold, and + 1 0 13 5.9 5.1 Y ≥ Z Fermions in either the isospin-1 Now consider the case of a U(2) gauge theory formulated without a spin structure, 2 Indeed, the fact that the newHere the SU(2) anomaly can be cancelled by the topological Green-Schwartz 13 14 ∈ group ( either ( mechanism, as noted in footnote 7 above, follows essentially from this fact. defined. U(2) structure, albeit one that can only be detected on non-spin 5 manifolds. For example, and thus the Dold manifold where point is that theory to a mod 2 indexthe such APS as index theoremwith to spin-U(2) structure, compute because the eq.to ( zero. Fortunately, we may followto ref. the [ generator of ( What is the interpretationpossible of new this global 5d anomaly that mod we 2 have cobordism so far invariant?vanishing missed? charge And under does U(1). it signify Thus, a it is not clear how to relate the anomaly, and the latter corresponds to the new one. but rather using a spin-U(2)calculate structure, using the as Adams was the spectral subject sequence that of section Recall that for thebordism SU(2) group gauge is theory [ defined without spin structure the corresponding A possible basis isfermion given with by isospin-1 j as in the usualcontributes SU(2) to global the anomaly, only that it5.2 is now the Case perturbative anomaly II: that without a spin structure and thereby conclude that exp( JHEP05(2020)098 , by . In (A.1) (A.2) (A.3) 4.2 BO generated → A BG by (pt) can be evaluated s BO − G t a stable bundle of virtual V , 8, this simplifies the Adams , (pt), using the Adams spec- (pt) U(2) 2 (pt) s s < / s × − Z ), with , G t − G t − V G Ω Spin 5 t Ω denotes the subalgebra of − X ⇒ 1 ∧ ⇒ ). Since there are no other independent A 2) on the corresponding U(2) theory is equiv- BG, 2) / 5.9 / ϕ Z is the Madsen-Tillmann spectrum defined in Z , , and thus with the new SU(2) anomaly, since ) , ) was identified, for any five-manifold with spin- 2 ) G – 16 – / G . To make the presentation clearer, we will write 3 Y G 2 = MSpin ( 2 in ( I X / J ( = Thom( G Z • (MT • G H ( MT H 1 ( and Sq ) in the rest of this appendix. s,t A 1 s,t A n Ext Ext . Since the action of ˆ 2 P C ) and SO( n ]. on 24 can be written as ϕ G for U( ], the cobordism invariant 8 n a Thom spectrum to be determined. For is the Steenrod algebra and MT , the potential global anomaly corresponding to this cobordism invariant necessar- G 4 A X When there is no odd-torsion involved, the bordism group Ω In ref. [ and SO n by the Anderson-Brown-Peterson theorem. Here by the Steenrod operationsU Sq our case, MT with spectral sequence above to where terms of the Thom spectrum bydimension MT 0 pulled back from the tautological stable bundle over In this appendix wetral sequence. calculate the For a bordismrecommend guide ref. group to [ Ω using the Adams sequence tovia compute the bordism Adams groups, spectral we sequence suggestions. JDsupported is by supported the DPST by Scholarship the from the STFC Thai consolidated Government. A grant ST/P000681/1. Spin-U(2) NL bordism is Acknowledgments We thank Ben Gripaiosreading and the David manuscript. Tong for We discussions, also and thank Pietro the Benetti anonymous Genolini referee for for their very interesting ily vanishes by perturbative anomalyincluding cancellation. WZ terms That to said, cancel as theit perturbative we is anomalies saw in possible in the to section lowanomaly’, energy reveal effective which a theory, corresponds low-energy to theorycobordism which the invariants, we does conclude indeed thatU(2) there suffer gauge are from theory no this defined other ‘new using possible U(2) a global spin-U(2) anomalies in structure. the SU(2) structure, with thethe mod 2 Dold index manifold correspondstransformation precisely ˆ to thealent, action at of the level the ofsection diffeomorphism the plus partition gauge function, to a local U(2) transformation as described in JHEP05(2020)098 + + × O. 3 3 B V V (A.5) (A.6) (A.7) (A.4) (A.9) + + −→ , . V 1 W = (Spin , such that SO − 1 G B , establishing MU U ∧ ), we can think f W B −→ . Therefore, the 3 , 00 × when A.5 f 2) 3 ) arises as a Puppe , BG G ◦ 2) 2 : MSO , 00 2 X SO / 2 ) = 0 (A.8) A.4 ∧ ). Therefore, the stable w Z V / B 2 2) ( , Z V + ( × 2 ( K A.5 0 2 / SO to be K f w Z 00 2 B ( SO , ◦ MSpin w ) are the second Stiefel-Whitney 5 5 B 0 2 + K 1 2 to ) + 0 2 1 − 00 2 w − w 3 U w ∈ w U → V 2 + 2) + B ) ( 2 B V , 0 2 ( 2 BG 2 w 2 2 gives rise to the following fibration of w −−−−−−−→ ]. g w × + / ,V BG 5) = Σ SO H 1 1 3 3 3 Z Spin : 21 ( U U B V ∈ − B 00 ) + from SO B K 2 00 f 2 + × V,V V h V ( B ◦ w 3 × . We follow the calculation of related examples 2 h V 00 1 2 ) + 3 ). w – 17 – 2 SO, giving rise to the identification of the Thom 00 SO w 3 w 7→ ,f B V 0 SO A.6 ) 1 MU SO is homotopy equivalent to the map + × is to be determined. This can be seen by finding a f fits into the homotopy pullback ( 2 ), and ( 0 B + U ∧ B 3 5) = h f ) 3 ,V B × 00 into f SO , respectively. The fibration ( ◦ 3 − W → SO 1 SO ) with ,f BG 2 0 2 0 2 × BG − U B B V V U SO ; 3 w ( V,V MSO f,f 1 −→ B 2 − ( ( B B BG 5 + + ; U is homotopy equivalent to : SO − H 3 G ) × BG f B 00 B V f 3 V ∈ ◦ BG ,f ◦ × ). We take the map 0 5 can be lifted to a stable spin bundle, denoted by × + 0 − = Σ 2 2 , and 2 We will now show that the Thom spectrum 3 −→ 2 SO 3 w − fits into the homotopy pullback V G f,f V , w w ( ( B 2 2 ( −−−−−→ SO . 2 X SO 2 / V SO × G B w Z w B B BG + SO) X 3 BG = Thom( × B V ) + , as follows. Since ( 3 Spin 2 G h is null-homotopic and + SO, ], whose method was based on ref. [ V B ( H g B V Spin 2 23 ◦ ∈ , w B 2 6 2 2 is given by w w / ) = 5 is given by 5 from Z Equivalently, Therefore, the map The fibration / V − − ) ( Thom( 2 2 2 2 is a homotopy pullback square, which we also use to define the map the existence of a homotopy pullback ( V spectrum MT Whitney product formula, the secondV Stiefel-Whitney class of the virtual bundle where we obtain theSO-bundle last equality using the pullback square ( of its element asw a triplet of vector bundles ( which sends three bundles into a stable SO-bundle of virtual dimension 0. Using the where suitable map is null-homotopic. Moreover, sincesaying these classes that are the valued map modulofollowing diagram 2, this is equivalent to where classes for sequence, so the composite map U in refs. [ classifying spaces Calculation of JHEP05(2020)098 5) , -tower (A.11) (A.12) (A.10) , it can 0 0 h but with ) = (0 h s s − We will now − , t } s, t is the generator V ) 0 { 1 , h ] . ] 00 2 } 00 2 ) respectively. The 1 w w (MU 2[ 2[ UV • / / { , up to degree ten. commutes with ] , and an infinite (MU } Z H Z 00 2 , where 2 r m s of the same d ∼ = ⊗ ∼ = H α , w 2) UV ) ) 0 ) up to degree 5 can be ex- 0 ) UV 3 { / 3 ] 1 0 3 1 h +1 from the entry ( G Z 00 2 w s , w U 0 2 r X 0 = α 2 w ( d , w B being the first Chern class modulo . Recall that 0 • 3 w ( (MU ) and G • • 00 3 +1 2 (MSO 2[ H s • , w X / w and H H 0 2 α H Z w s 5 5 2[ α − − UV (MSO / , with the corresponding Adams chart for 2 3 Σ Σ 0 2 Z and w 1 6. H ∼ = ∼ = ] and ) – 18 – } 0 3 1 s < U . In the Adams chart, each dot corresponds to a is given by 6), the entries are too sparse and all the differentials { , w ] − 0 2 2 0 MU 3 ) and Adams spectral sequence. are in t +1 2 w G UV s dots gives a factor of ( UV s < − 2 , w 2[ 00 2 α V 00 2 0 2 X Σ / − w w m ( t w Z • ∧ 2[ UV -module structure of 3 ∼ = / and 1 H 4). However, using the fact that page for ) Z A 3 r, U 2 -module structure for ∼ = MSO 1 E SO ) 3 A 3 − ) = ( B ( s • (Σ 2) shown in figure • − H / 2). . The H (MSO -module structure of the spectrum Z / • 1 , s, t Z ) -tower containing are the Stiefel-Whitney classes, with A H , 0 G 2 0 3 h / X ( Z , w • ( 0 2 Figure 1 1 1 , H w 1 A ( 1 Finally, the rule for extracting the bordism groups can be roughly summarised as In the range of our interest ( A s,t = 1 means that the generator -module structure of 2 generator. A line joining two generators 1 s / to the entries ( be shown that thesecollapses differentials already are at trivial, the too. Therefore, the Adams spectralfollows: sequence an Ext Z ∆ of Ext are trivial, apart from a possible non-trivial differential Using the relations betweenclasses, Thom classes, we the find Steenrodpressed squares, that as and the the the Stiefel-Whitney cell diagram shown in figure where the ThomK¨unneththeorem classes for the cohomology ring of a Thom space implies that where 2, which coincides withhave the the second identifications Stiefel-Whitney class. By the Thom isomorphism, we A work out the JHEP05(2020)098 ]. (A.13) (A.14) SPIRE IN [ (1994) 5155 35 ]. , SPIRE (1985) 197 3 IN Z [ = 100 G 4 5 Ω J. Math. Phys. . , ]. , ), from which one can read off the 4 = 0 A.3 G 3 SPIRE . Ω IN (1982) 324 3 . arXiv:1910.14668 ][ s 2 , , / − Z t Z Spectral asymmetry and Riemannian geometry. 2 B 117 = Commun. Math. Phys. = (1976) 71 , Global anomalies in the standard model(s) and G 2 – 19 – G 5 Ω 79 Ω 1 ]. , hep-th/9405012 to be Phys. Lett. = 0 SPIRE 0 , ), which permits any use, distribution and reproduction in 2 G 1 IN invariants and determinant lines [ Ω η 0 5 4 3 2 1 ,

Beyond standard models and grand unifications: anomalies, topological Z anomaly s (pt). = (2001) 2343] [ CC-BY 4.0 U(2) 2 G 0 / × Z . With this rule, the bordism groups of degree lower than six can be Ω page of the Adams spectral sequence ( SU(2) 5 42 This article is distributed under the terms of the Creative Commons Z ≤ 2 Spin d E An Global gravitational anomalies arXiv:1910.11277 , . The Math. Proc. Cambridge Phil. Soc. , Erratum ibid. III beyond terms and dynamical constraints via cobordisms [ J. Davighi, B. Gripaios and N. Lohitsiri, Z. Wan and J. Wang, E. Witten, E. Witten, X.-z. Dai and D.S. Freed, M.F. Atiyah, V.K. Patodi and I.M. Singer, [5] [6] [1] [2] [3] [4] Attribution License ( any medium, provided the original author(s) and source are credited. References and, crucially for us, Open Access. gives a factor of read off from the chart in figure Figure 2 bordism groups Ω JHEP05(2020)098 B ] Izv. (1971) , , , D , 37B gauge theory Top. Quant. (1984) 269 Phys. Lett. (2019) 052301 , 3 + 1 (2016) 035001 , (2016) 022 60 (1991) 323 = 10 88 09 D B 234 (2019) 107 210 arXiv:1705.01853 Phys. Lett. 4 [ , arXiv:1909.08775 JHEP , , ]. ]. J. Math. Phys. Nucl. Phys. ]. , , (2017) 104 Rev. Mod. Phys. SPIRE Annals Phys. Yang-Mills and cobordisms: , IN SPIRE , -Invariant ) 07 η ][ IN SPIRE N [ IN anomaly ][ SU( JHEP ]. , Ann. Math. Sci. Appl. , SU(2) (1984) 117 – 20 – SPIRE New gravitational instantons and universal spin . IN [ A new 149B Gravitational anomalies arXiv:1710.04218 Time reversal, [ (1966) 207. A guide for computing stable homotopy groups Generalized spin structures in quantum gravity Anomaly cancellation in supersymmetric gauge anomalies and the topological Green-Schwarz Reflection positivity and invertible topological phases arXiv:1711.11587 ]. ]. ]. ]. 30 Anomaly inflow and the [ (1978) 181 ]. d (1980) 103 Generalized spin structures on four-dimensional space-times Consequences of anomalous Ward identities SPIRE SPIRE SPIRE SPIRE Phys. Lett. 72 B 77 (2017) 177 SPIRE , Higher anomalies, higher symmetries and cobordisms I: classification IN IN IN IN (2018) 89. IN [ ][ ][ ][ ][ 11 (2018) 244 718 On manifolds with free abelian fundamental group and their application Dai-Freed theorem and topological phases of matter 394 . JHEP ]. Gauge anomalies in an effective field theory Fermion path integrals and topological phases Phys. Lett. , , Line operators in the standard model ]. ]. ]. ]. SPIRE IN (1978) 42 ´ Alvarez-Gaum´eand E. Witten, SPIRE SPIRE SPIRE SPIRE [ arXiv:1812.11967 IN arXiv:1508.04715 arXiv:1607.01873 IN IN IN arXiv:1810.00844 95 [ mechanism and superstring theory structures 73 [ [ Commun. Math. Phys. Theor. Interact. interacting topological superconductors/insulators and quantumAnnals spin Phys. liquids in of higher-symmetry-protected topological states andanomalies their via boundary a fermionic/bosonic generalized[ cobordism theory [ arXiv:1604.06527 Ross. Akad. Nauk. Ser. Mat. [ [ [ I. Garc´ıa-Etxebarriaet al., 8 M.B. Green and J.H. Schwarz, A. Back, P.G.O. Freund and M. Forger, S.W. Hawking and C.N. Pope, J. Preskill, J. Wang, X.-G. Wen and E. Witten, S.J. Avis and C.J. Isham, D. Tong, A. Beaudry and J.A. Campbell, Z. Wan and J. Wang, E. Witten and K. Yonekura, D.S. Freed and M.J. Hopkins, M. Guo, P. Putrov and J. Wang, E. Witten, K. Yonekura, J. Wess and B. Zumino, L. S.P. Novikov, [8] [9] [7] [13] [14] [10] [11] [12] [24] [23] [20] [21] [22] [18] [19] [15] [16] [17]