Anomalies and Supersymmetry Ioannis Papadimitriou “Integrability, Holography, Higher-Spin Gravity and Strings” QUARKS online workshop dedicated to A.D. Sakharov’s centennial May 31, 2021 Anomalous global symmetry (flavor, R-symmetry, higher form, higher spin) Supersymmetry Noether current multiplet (couples to vector/gravity multiplet) “Small multiplet” “Large multiplet” ● N=1 vector multiplet in WZ gauge ● N=1 vector multiplet in superspace ● Conformal current multiplet ● Ferrara-Zumino multiplet ● R-multiplet ● S-multiplet (?) ● Extended supersymmetry (N>1) ● ... ● ... Deformation of the supercharge algebra 1 / 40 References Ruben Minasian, IP and Piljin Yi [arXiv:2104.13391] IP [arXiv:1703.04299, arXiv:1902.06717, arXiv:1904.00347] Ok Song An [arXiv:1703.09607] Ok Song An, Jin U Kang, Jong Chol Kim, Yong Hae Ko and Sok-Hyon Won [arXiv:1812.10209, arXiv:1902.04525] George Katsianis, IP, Kostas Skenderis and Marika Taylor [arXiv:1902.06715, arXiv:2011.09506] Cyril Cosset, Lorenzo Di Pietro, Heeyeon Kim [arXiv:1905.05722] Sergei Kuzenko, Adam Schwimmer, Stefan Theisen [arXiv:1909.07084] Adam Bzowski,Guido Festuccia,Vladimir Prochazka [arXiv:2011.09978] Koichiro Nakagawa, Yu Nakayama [arXiv:2103.10048] 2 / 40 Outline 1 Global symmetries and anomalies 2 Diffeomorphism and supersymmetry Ward identities 3 Generalized anomaly descent 4 Supersymmetric Chern-Simons and anomaly inflow 5 Summary 3 / 40 Outline 1 Global symmetries and anomalies 2 Diffeomorphism and supersymmetry Ward identities 3 Generalized anomaly descent 4 Supersymmetric Chern-Simons and anomaly inflow 5 Summary 4 / 40 Noether’s theorem Continuous global symmetries∗ lead to a conserved current µ @µJ = 0 and an associated conserved charge Q = dd−1x J t ˆCt The conserved charges satisfy an algebra K QI ; QJ = fIJ QK ; I; J; K = 1; 2; 3; ··· ∗ I will focus exclusively on symmetries connected to the identity 5 / 40 (Perturbative) Anomalies Quantum anomalies arise due to the breaking of classical symmetries by the UV regulator or the path integral measure They are manifest in Ward identities and the charge algebra µ DµhJ i = −A µ ν ρ @µhJ (x1)J (x2)J (x3)i = contact terms K QI ; QJ = fIJ QK + kIJ The structure of anomalies is determined by the Wess-Zumino (WZ) consistency conditions up to overall numerical coefficients The algebra deformation kIJ is determined by the anomaly A 6 / 40 Background fields Global symmetries can be studied by turning on background gauge µ fields Aµ that couple to the Noether currents J , i.e. (to linear order) Z[A] = eiW [A] = Dφ exp iS[φ] + i ddx A ·J ˆ ˆ The full, non-linear, functional W [A] is obtained by gauging the global symmetry and integrating over the matter fields only – not Aµ W [A] is the generating function of connected current correlators δnW [A] µ1 µ2 µn hJ (x1)J (x2) ···J (xn)i = δAµ1 (x1)δAµ2 (x2) ··· δAµn (xn) 7 / 40 ’t Hooft anomalies If Z[A] transforms with a phase under a gauge transformation Aµ ! A + D# and the phase cannot be removed by a local counterterm the theory has a ’t Hooft anomaly Z[A + D#] = Z[A] exp i ddx # ·A(A) ˆ For infinitesimal gauge parameter #(x) this is equivalent to d δ#W [A] = G(#; A) = d x # ·A(A) | {z } ˆ | {z } integrated anomaly local anomaly The gauge dependence of the effective action is equivalent to the non conservation of the current d δW a d µ δ#W [A] = d x a (Dµ#) = d x #·A(A) )DµhJ i = −A(A) ˆ δAµ ˆ |{zµ} hJa i 8 / 40 Wess-Zumino conditions The anomalies must satisfy the Wess-Zumino (WZ) consistency conditions, whose infinitesimal form is (δ#1 δ#2 − δ#2 δ#1 )W [A] = δ[#1;#2]W [A] or δ#1 G(#2;A) − δ#2 G(#1;A) = G([#1;#2];A) Any local functional Wloc[A] is a trivial solution of the WZ conditions Nontrivial solutions of the WZ conditions are local functionals G(#; A) = ddx # ·A(A) ˆ that are not the gauge variation of a local functional Wloc[A] 9 / 40 BRST transformations Solving the WZ conditions is a cohomology problem This is formalized by replacing the infinitesimal gauge parameter #a with a Grassmann-valued Faddeev-Popov ghost va so that sA = − dv − Av − vA = −Dv δ#A = dA + [A; #] = D# ! sF = F v − vF δ F = [F; #] # 1 2 sv = − 2 [v; v] = −v where F = dA + A2 is the field strength of A The BRST operator s is nilpotent and so the WZ condition becomes s2W [A] = 0 , sG(v; A) = 0 Nontrivial consistent anomalies are elements of the cohomology of s 10 / 40 BRST algebra The cohomology problem that determines the consistent anomaly can be solved using the anomaly descent procedure that follows from the BRST algebra [Stora ’77, ’84; Zumino ’84; Manes,˜ Stora, Zumino ’85] d2 = 0; s2 = 0; (d + s)2 = 0 We start by defining the quantities Ab ≡ A + v; Fb ≡ (d + s)Ab + Ab2 The nilpotency of d and d + s imply respectively the Bianchi identities dF + AF − FA = 0; (d + s)Fb + AbFb − FbAb = 0 These imply that the corresponding anomaly polynomials are closed dPd+2(F ) = 0; (d + s)Pd+2(Fb) = 0 11 / 40 Russian formula and anomaly descent The next key ingredient is the “Russian formula” Fb = F Together with the above results, this implies that 0 = Pd+2(Fb) − Pd+2(F ) = (d + s)Ωd+1(A;b Fb) − dΩd+1(A; F ) (0) (1) 2 = sΩd+1(A; F ) + dΩd (v; A; F ) + O(v ) (k) where Ωd+1−k determine the expansion of the Chern-Simons form X (k) Ωd+1(A;b Fb) = Ωd+1(A;b F ) = Ωd+1−k(v; A; F ) k≥0 Grouping terms of equal ghost number gives the descent equations (k) (k+1) sΩd+1−k(v; A; F ) + dΩd−k (v; A; F ) = 0 12 / 40 Consistent anomaly The solution to the BRST cohomology problem is obtained from the k = 1 descent relation, which integrated over Md, with @Md = ;, gives s Ω(1)(v; A; F ) = − dΩ(2) (v; A; F ) = 0 ˆ d ˆ d−1 Md Md We identify the consistent anomaly G(v; A) = N Ω(1)(v; A; F ) ˆ d Md where N is a normalization factor determined by a 1-loop calculation d i 2 N = d (nL − nR); d even; nL;R = # of chiral fermions 2 d+2 (2π) 2 ! 13 / 40 Anomaly inflow The descent relation for k = 0 leads to a toy version of anomaly inflow. Integrating over Md+1 with @Md+1 = Md we obtain s NΩ(0) (A; F ) = −N Ω(1)(v; A; F ) = −G(v; A) ˆ d+1 ˆ d Md+1 Md (0) Hence, the Chern-Simons form Ωd+1(A; F ) “cancels” the anomaly s W [A] + N Ω(0) (A; F ) = 0 ˆ d+1 Md+1 This relation provides a constructive way to determine the consistent anomaly, given the general expression for the Chern-Simons form 1 d (0) d+2 2 Ωd+1(A; F ) = Ωd+1(A; F ) = 2 dt Pd+2 A; Ft ˆ0 2 2 where Ft ≡ tF + (t − t)A 14 / 40 BZ polynomial and the covariant current The anomaly G(#; A) implies that the consistent current J µ does not transform covariantly under gauge transformations µ δW [A] δ δG(#; A) δ#hJa i = δ# a = δ# a W [A]+ a δAµ δAµ δAµ µ δG(#; A) = [hJ i;#]a+ a δAµ Bardeen and Zumino (BZ) [Bardeen, Zumino ’84] showed that there exists a local polynomial X µ(A) that transforms as µ µ δG(#; A) δ#Xa (A) = [X (A);#]a− a δAµ It follows that the sum of J µ and X µ(A) does transform covariantly µ µ µ µ µ hJcovi ≡ hJ i + X (A); δ#hJcovi = [hJcovi;#] 15 / 40 Covariant anomaly The BZ polynomial is also determined by the anomaly polynomial 1 d−2 ddx δA ·X (A) = −N d(d+2) dt t P δA; A; F 2 ˆ 4 ˆ ˆ d+2 t 0 Md 2 2 where again Ft ≡ tF + (t − t)A and δA is a general variation of A The BZ polynomial, and hence the covariant current, cannot be expressed as the variation of an effective action in Md The divergence of the covariant current is µ DµhJcovi = −Acov(A) where the covariant anomaly d G (v; A) = ddx v ·A (A) = N d+2 P v; F 2 cov ˆ cov 2 ˆ d+2 Md Md depends only on the field strength F 16 / 40 An identity There is a general local relation between the BZ polynomial and the consistent anomaly that plays an important role in the following a ν a Fµν Xa (A) − AµAa(A) = 0 or equivalently, for any vector field κµ, 1 d−2 (1) d(d + 2) 2 Ωd (ikA; A; F ) + dt tPd+2 ikF; A; Ft = 0 4 ˆ0 e.g. for the Abelian case in d = 4 ν µνρσ Fτν X (A) − Aτ A(A) / Fµν FρσAτ + 4Aν FρσFτµ = 0 since antisymmetrizing five indices in four dimensions gives zero F[µν FρσAτ] = 0 17 / 40 Outline 1 Global symmetries and anomalies 2 Diffeomorphism and supersymmetry Ward identities 3 Generalized anomaly descent 4 Supersymmetric Chern-Simons and anomaly inflow 5 Summary 18 / 40 Ward identities The anomalous conservation of the consistent current in the presence of an arbitrary gauge field Aµ is an example of a Ward identity µ DµhJ i = −A(A)(∗) As we saw, this is obtained by writing δ#W [A] in two different ways d d δW a δ#W [A] = d x # ·A(A) = d x a (Dµ#) ˆ ˆ δAµ | {z } δ Aa |{z} # µ hJ µi where the first equality is a consequence of the WZ conditions, while the second follows from the chain rule and the definition of hJ µi Successive derivatives of (∗) with respect to Aµ lead to the Ward identities for any correlator of consistent currents µ1 µ2 µn @µ1 hJ (x1)J (x2) ···J (xn)i = contact terms 19 / 40 Spacetime symmetries Spacetime symmetries, such as diffeomorphisms and local Lorentz transformations, act on multiple background fields As a result, the Ward identities for these symmetries contain several operators, besides the energy-momentum tensor As a warmup for supersymmetry, let us consider the diffeomorphism Ward identity in the presence of an anomalous flavor symmetry The current operators related to these symmetries are µ −1 δW µ −1 δW a hTa i = e a ; hJ i = e ; e ≡ det(eµ) δeµ δAµ a b (Recall that gµν = eµ eν ηab) 20 / 40 Gravitational anomalies The transformations of W [e; A] under diffeomorphisms and Lorentz transformations define the diffeomorphism and Lorentz anomalies δξW [e; A] = GD(ξ; Γ; R; F ); δλW [e; A] = GL(λ, !; R; F ) where Γ and ! are respectively the Christoffel and spin connections GL can be determined