Aspects of Symmetries and RG Flow Constraints Ken Intriligator (UCSD) ICMP Montreal 2018 Thank the organizers for the opportunity to attend this conference and give this talk. Based on work with Clay Cordova (IAS / U. Chicago) and Thomas Dumitrescu (UCLA), esp. 1802.04790
RG flows “# d.o.f.” UV CFT (+ relevant deformations) (Gen’ly difficult. E.g. open Clay Prize RG course graining. problem for QCD. Can instead guess in special cases, do non-trivial checks.) IR CFT (+ irrelevant deformations) . (OK even if SCFT is non-Lagrangian) . Move on the moduli space of (susy) vacua. . Gauge a (e.g. UV or IR free) global symmetry.
RG flow constraints . ’t Hooft anomaly matching for global symmetries + gravity. . Reducing # of d.o.f. intuition. For d=2,4 (& d=6 susy) : a-theorem They must be constant on RG flows; match at endpoints. For unitary thys a UV ≥ a IR a ≥ 0 conformal a-theorem proof of X h T µ µ i ⇠ aE d + c i I i anomaly : Komargodski + Schwimmer via i conf’l anomaly matching. . Additional power from supersymmetry. Supermultiplets and (d=odd: via sphere partition function / entanglement entropy.) supermultiplets of anomalies .
q-form global currents • Conserved flavor current: . Source: bkgd. ∂ µ J a A a µ = 0 µ (a = g Lie alg. index) µ = ( D µ λ ) a δ A a = “q=0-form” global symmetry. • Conserved higher q-form global symmms: Gaiotto, Kapustin, Seiberg, Willett and refs therein. d ∗ j ( q +1) = 0 with ∂ µ 1 j ( q +1) I.e. j ( q +1) [ µ 1 ...µ q +1 ] = 0 . [ µ 1 ...µ q +1 ] Z ∗ j ( q +1) Q ( Σ d − q − 1 ) = q>0: only abelian, U(1) (q) Σ d − q − 1 or discrete subgp. ∆ exact ( j q +1 ) = d − q − 1 Is q>0 possible for (S)CFTs? Often, “no”. E.g. we show that no q>0 conserved current multiplets for 6d unitary SCFTs.
Couple all currents to background fields • Poincare’: Source = bkgd metric g µ ν = δ ab e a µ e b ν δ e (1) a = − θ (0) a e (1) b b • Conserved flavor current: . Source: bkgd. ∂ µ J a A a µ = 0 µ Invariance: µ = ( D µ λ ) a δ A a • Conserved q>0 current: Z Z B ( q +1) ∧ ? j ( q +1) B µ 1 ...µ q +1 j [ µ 1 ...µ q +1 ] dV = S ⊃ invariance since d ? j ( q +1) = 0 δ B ( q +1) = d Λ q Background gauge invariance encodes conservation laws.
Recall various anomalies Effective action as fn � [ d ψ ][ dA ] e − S [ B , ψ ,A ] / � ) W [ B ] = − log( of background fields: Z I ( d ) [ B , δ B ] W [ B + δ B ] − W [ B ] = A [ B ] = 2 π i d I ( d ) [ B , δ B ] = δ I ( d +1) [ B ] , d I ( d +1) [ B ] = I ( d +2) [ B ] (descent procedure) For d=2n, the matter content must be gauge anomaly free. Anomalies encoded in a topological d+2 form in gauge and global background field strength Chern classes, and Pontryagin classes for the background metric curvature. Compute via (n+1)-gon diagram, or inflow, etc. Calculable via various methods. We discuss mixed gauge+ global anomalies. They quantum- deform the global symmetry group into a``2-group.”
Anomalies (4d case) gauge Gauge anomalies must vanish for a healthy theory. Constrains chiral matter content. gauge gauge Global ABJ anomaly, only for global U(1)s. If non-zero, global U(1) is just not a symm (explicitly broken by instantons, perhaps to a discrete subgp). gauge gauge Global ’t Hooft anomalies. Useful if non-zero: must be constant along RG flow, match at ends. Global Global Does not violate either symmetry. Deforms gauge Our global symmetry to a 2-group symmetry. star: (2 π ) 2 F global ∧ F gauge = κ κ d ∗ j (1) 2 π F global ∧ ∗ J (2) global = Global Global B
4d QED example Consider a 4d (non-susy) QED, i.e. u(1) gauge theory, with N flavors of massless Dirac Fermion (IR free, needs a UV cutoff). SU ( N ) (0) L × SU ( N ) (0) R × U (1) (1) Global symmetry: B j µ ν U (1) (0) B ∝ ✏ µ νρσ f ρσ broken by ABJ anomaly. A global dyn. u(1) U (1) (0) → u (1) gauge current gauge field. V u (1) gauge ∼ ± 1 Non-zero mixed anomaly. As we will discuss, it deforms the global symmetry to a 2-group symm.: SU ( N ) L,R SU ( N ) L,R ⇣ ⌘ SU ( N ) (0) L − R × κ =1 U (1) (1) × SU ( N ) (0) L + R B
Chiral toy model examples Consider a 4d (non-susy) theory with two 0-form flavor symms U(1) A and U(1) C and matter chiral Fermions with charges (q A , q C ). ’t Hooft q A q C κ A 3 = Tr U (1) 3 C A = 1 1 3 ψ 1 mixed κ A 2 C = Tr U (1) 2 A U (1) C = 12 A A 1 4 ψ 2 ABJ=0 κ AC 2 = Tr U (1) A U (1) 2 C = 0 -1 5 ψ 3 0 -6 gauge=0 ψ 4 κ C 3 = Tr U (1) 3 C = 0 Take A=global and C=gauge symmetry. Non-zero ’t Hooft and mixed anomaly. I mixed = ( κ A 2 C c 2 ( F A ) + q C,tot p 1 ( T )) ∧ c 1 ( f c ) 6 gauge global
Likewise 6d anomalies gauge gauge Gauge anomalies must vanish. Can use a dyn GSWS mechanism to cancel reducible parts. gauge gauge Global Global ’t Hooft anomalies. Useful if non-zero. Must be constant along RG flow, match at ends. Global Global Does not violate any symmetry. Deforms global gauge gauge symmetry to a 2-group symmetry. Here the Global Global gauge group can be non-Abelian. (In 4d, there is only one gauge vertex, so it must be u(1).) Example: small SO(32) instanton theory (Witten ’95) I mixed � � = c 2 ( F sp ( N ) ) c 2 ( F SO (32) ) + (16 + N ) p 1 ( T ) 8
6d anomalies (aside) For 6d U(1) gauge theories, can also get a “4-group”: gauge Global I ⊃ k gGGG c 1 ( f gauge ) c 3 ( F Global ) Global Global Does not violate any symmetry. Deforms global symmetry to a 4-group symmetry. G Global × κ U (1) (3) δ A (1) G Global = D λ (0) B G U (1) (3) κ δ B (4) = d Λ (3) + B : ∗ j 4 = c 1 ( f gauge ) 3!(2 π ) 2 Tr ( λ G F G ∧ F G ) κ ∝ κ gGGG We will focus on the 2-group cases, i.e. involving 2-form bkgd gauge fields (can couple to strings).
Mixed gauge/global anomalies and 2-groups ? j (2) = c 1 ( f gauge ) = f gauge Z ? j (2) ∈ Z U (1) (1) 4d: q J = , 2 ⇡ B Σ 2 1 Z ? j (2) = c 2 ( f gauge ) = U (1) (1) 6d: ? j (2) ∈ Z 8 ⇡ 2 Tr f gauge ∧ f gauge , q J = B Σ 2 Conserved since , charged objects = e.g. d ? j (2) = 0 ANO vortex strings (4d), instanton strings (6d). Couple the 1-form global symmetries to Z 2-form background gauge fields B. S 4 d, 6 d ⊃ B ∧ ? j The mixed “anomaly” means that B shifts under a bkgd flavor or metric gauge transformation A 0 = A + d λ A , B 0 = B + d Λ + κ 2 π λ A F A
“2-group” global symmetry If a non-trivial structure function interplay between a conserved q=2-form current and the other currents. Analogous to the Green-Schwarz mechanism for the global background fields coupled to the currents. (See e.g. Kapustin and Thorngren papers, and refs therein.) Global symmetry: G (0) × ˆ κ U (1) (1) bkgd gauge transfs κ δ B (2) = d Λ (1) + ˆ 2 π λ (0) dA (1) µ = ( D µ λ ) a δ A a + ˆ κ P + analog for Poincare’ SO(4) frame rotation of spin connection: 16 π tr( θ (0) d ω (1) ) H (3) = dB (2) − ˆ 2 π CS ( A ) − ˆ κ A κ P dH sourced by background 16 π CS ( ω ) , gauge & gravity instanton.
2-group structure constants Global 0-form and 1-form symmetries: G (0) G (1) β ∈ H 3 ( G (0) , G (1) ) we call them ˆ ˆ κ G (0) , κ P . Kapustin & Thorngren: Postnikov class. We also call them 2-group structure constants. Coefficients of CS terms in invariant field strength H (3) . For quantized charges, compact global groups, these coefficients must be integers: They are scheme indep κ P ∈ Z ˆ ˆ κ G (0) , physical properties of the QFT. Can only arise at tree-level level or one-loop. Mixed anomaly terms give this symmetry. U (1) (0) A u (1) (0) U (1) (0) κ P U (1) (1) E.g.: A × ˆ κ A , ˆ C B “Mixed anomaly” κ A = − 1 2 κ A 2 C ∈ Z ˆ GLOBAL gauge coeffs., so 2-group κ P = − 1 with no anomaly. ˆ 6 κ P 2 C ∈ Z
2-group affects reducible ’t Hooft anomaly matching E.g. only has ’t Hooft U (1) (0) κ A U (1) (1) Tr U (1) 3 A × ˆ A B anomaly matching mod , because of a possible 6ˆ κ A counterterm: , S SG = in � B (2) ∧ F (2) U (1) (1) B : n ∈ Z A 2 π E.g. can gap if Tr U (1) 3 κ A 3 → κ A 3 + 6 n ˆ A = 0 mod 6ˆ κ A κ A TQFTs can give similar, but physical (non-counterterm) terms with fractional n. They can be used to match ’t Hooft anomalies via a gapped TQFT. E.g. u(1) C gauge thy broken to TQFT by Higgs Z q C mechanism of field with charge . Allows to Tr U (1) 3 A � = 0 q C > 1 be matched by gapped TQFT if Tr U (1) 3 A = 0 mod 6 n ˆ κ A , q C n ∈ Z
2-group vs CFT Phrased in terms of Ward identities, contact term e.g.: ρσ ( x ) � = ˆ ∂ κ A 2 π ∂ λ δ (4) ( x � y ) � J B � j A µ ( x ) j A ν ( y ) J B νλ ( y ) J B ρσ ( z ) � ∂ x µ Implies a non-zero 3-point function also at separated points. Incompatible with additional constraints of CFT. Tension between 2-group vs CFT. 2-group can be an emergent symmetry, subject to constraints. E.g. the 4d u(1) gauge theory and 6d small SO(32) instanton examples are IR free, non-CFTs. Can UV complete if U(1) B is broken, accidental symmetry in IR. E.g. if u(1) is part of a non-Abelian UV completion, then U(1) B is broken (monopoles). Likewise for little string UV completion of small SO(32) instanton theory.
2-group RG flows, e.g.
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