REVIEW TALK (2+1)d dualities with N = 2 supersymmetry Antonio Amariti INFN - Sezione di Milano Pisa, October 18, 2019 Antonio Amariti (INFN) (2+1)d dualities 1 / 30
Outline The contribution to the PRIN 1 Overview of the phases of (3+1)d SQCD 2 (2+1)d tools 3 (The) two main (2+1)d N = 2 dualities 4 A new duality from compactifications: KK monopoles. 5 I S 3 × S 1 → Z S 3 6 D-branes: reduction from T-duality. 7 Constructing the web 8 Antonio Amariti (INFN) (2+1)d dualities 2 / 30
The contribution to the PRIN Motivations Exchange of ideas and results between the high-energy and the condensed- matter communities, after the discovery of (2+1)d bosonization and it general- ization to a more general web . Project Shed some light on the web of (2+1)d dualities drawing some inspiration from the knowledge of supersymmetric infrared dualities. Techniques This research will rely on: Supersymmetry breaking, Localization, Branes, . . . Goals Tests and new 2+1 non-SUSY dualities; Generalization to other dimensions. Antonio Amariti (INFN) (2+1)d dualities 3 / 30
The contribution to the PRIN Today: review of (2+1)d dualities with four supercharges Q Why N = 2 ? A1 Holomorphy protects W from quantum corrections; (for N = 1 also time reversal needed). A2 (2+1)d N = 2 dualities can be derived from 4d N = 1 by a ”sensible” compactification. A3 There is a large web of dualities (here we are focusing on for (2+1)d U ( N c ) SQCD). This web will be the subject of the talk, we will show how to obtain it using (3+1)d results, dimensional reduction, brane engineering and localization (on the three sphere). Antonio Amariti (INFN) (2+1)d dualities 4 / 30
Overview of the phases of (3+1)d SQCD The phase structure of 4d SQCD G AUGINO CONDENSATION CONFINING, SUSY VACUA CLASSICAL MODULI SPACE IR FREE N =0 N = N c +1 f f N = N 3 f c N = N 3/2 c f N f N = N f c CONFORMAL UV FREE WINDOW CONFINING, ADS QUANTUM SUPERPOTENTIAL MODULI RUNAWAY SPACE Antonio Amariti (INFN) (2+1)d dualities 5 / 30
Overview of the phases of (3+1)d SQCD The phase structure of 4d SQCD SEIBERG DUALITY IR FREE N =0 N = N c +1 f f N = N 3 f c N = N 3/2 c f N f N = N f c Antonio Amariti (INFN) (2+1)d dualities 6 / 30
(2+1)d tools Tools in (2+1)d SUSY offers many tools to study (2+1)d models and check conjec- tured dualities Moduli space: HB and CB CB coordinates, monopoles and ”superpotentials” Localization: spheres, indices and topological twist Anomalies: gauge vs global, continuous vs discrete (parity) CS vs YM action Real masses and background symmetries Topological symmetry Antonio Amariti (INFN) (2+1)d dualities 7 / 30
(2+1)d tools Multiplets Vector: V = ( A µ , λ α , σ, D ) where σ from dim. red. of A 3 Chiral : Φ = ( φ, ψ, F ) Coulomb branch (CB) Due to � σ � , combined with the dual photon ϕ = d ∗ F 3 + i ϕ i ; e Σ i CB coordinate (UV monopole) Chiral Σ i = σ i g 2 Monopole superpotentials: W ∝ e f (Σ i ) , lift some CB directions Abelian global symmetries Axial U (1) A (anomalous in (3+1)d); U (1) R R-symmetry; topological U (1) J shifting ϕ . Antonio Amariti (INFN) (2+1)d dualities 8 / 30
(2+1)d tools Chern-Simons (CS) action � 3 A 3 − λ ˜ Tr ( A ∧ dA − 2 k w/ k ∈ Z S CS = λ + 2 σ D ) 4 π Real masses R φ R | 2 , � σ bckg � real mass for φ Coupling | σ i bckg T i CS and fermions Integrating out fermions with large real masses generates an effective � = k ij + 1 CS: k eff I c i ( ψ I ) c j ( ψ I ) sgn ( m I ) ij 2 Antonio Amariti (INFN) (2+1)d dualities 9 / 30
(The) two main (2+1)d N = 2 dualities Aharony duality ’98 ELECTRIC : 3d N = 2 U ( N c ) SQCD, with N f ( > N c ) � Q and N f � ˜ Q ; W = 0 q , M = Q ˜ MAGNETIC : 3d N = 2 U ( N f − N c ) SQCD, with N f � q and � ˜ Q , V ± monopoles of U ( N c ) (singlets of the dual phase) and v ± monopoles of U ( N f − N c ) W = Mq ˜ q + v + V + + v − V − If N f = N c = 1 it reduces to mirror symmetry: U (1) with N f = 1 dual to 3 chirals interacting through W = XYZ Giveon-Kutasov duality ’08 ELECTRIC : 3d N = 2 U ( N c ) k SQCD, with N f + | k | > N c and N f � Q and and N f � ˜ Q ; W = 0 q and M = Q ˜ MAGNETIC : 3d N = 2 U ( N f − N c + | k | ) − k SQCD, with N f � q and � ˜ Q W = Mq ˜ q Antonio Amariti (INFN) (2+1)d dualities 10 / 30
A new duality from compactifications: KK monopoles. 4d/3d reduction of U ( N c ) SQCD [Aharony, Razamat, Willett, Seiberg ’13] On S 1 effective description with W BPS − monopole and W KK − monopole (reproduce SUSY vacua at N f = 0 ) KK and W mag If N f � = 0: W ele participate to a new duality: KK ARSW duality ’13 ELECTRIC : 3d N = 2 U ( N c ) SQCD, with N f ( > N c ) � Q and N f � ˜ Q W = W ele KK = η V + V − with η = e − 1 / ( rg 2 3 ) = e − 1 / g 2 4 = Λ b holo MAGNETIC : 3d N = 2 U ( N f − N c ) SQCD, with N f � q and N f � ˜ q W = Mq ˜ q + ˜ η v + v − U (1) A broken (as in 4d) by KK monopoles GK and A from this new duality by real mass flows and Higgsing Antonio Amariti (INFN) (2+1)d dualities 11 / 30
I S 3 × S 1 → Z S 3 This procedure can be reproduced by Localization Here we focus on the S 1 reduction of the 4d superconformal index: r = I mag S 3 = Z mag I ele Z ele → S 3 × S 1 S 3 × S 1 S 3 r r → 0 This procedure requires the cancellation of divergent pre-factors. It is not guaranteed to work (e.g. N = 4 SYM and SO ( N c ) dualities). Antonio Amariti (INFN) (2+1)d dualities 12 / 30
I S 3 × S 1 → Z S 3 Localization 2 � SCI = Tr ( − 1) F e − β E p J 1 + r 2 q J 2 + r i ∈ F µ q i i � N f � N c � R R 2 ν a z − 1 = ( p ; p ) N c ( q ; q ) N c 2 µ a z i )Γ e (( pq ) a =1 Γ e (( pq ) ) dz i I ( N f , N f ) i � U ( N c ) i < j Γ e (( z i / z j ) ± 1 ) N c ! 2 π iz i i =1 with ( p ; p ) = � ∞ a =1 (1 − p a +1 ) and Γ e ( x ) ≡ � ∞ 1 − p k +1 q k +1 / z k , m =0 1 − p k q m z Reduction : define p = e 2 π ir ω 1 , q = e 2 π ir ω 2 , µ a = e 2 π irm a , µ a = e 2 π irn a , z i = e 2 π ir σ i The reduction corresponds to the limit r → 0 with ω i , m a , n a fixed. 2 µ a z i ∝ Γ h ( ω 1 + ω 2 r r → 0 Γ e (( pq ) lim R + m a + σ i ) 2 Subtraction of a divergent term ∝ Tr R and Tr F Antonio Amariti (INFN) (2+1)d dualities 13 / 30
I S 3 × S 1 → Z S 3 Localization The 4d compact integral is now a non-compact integral over σ � � G � ik πσ 2 1 d σ i I Γ h ( ω ∆ I + ρ I ( σ ) + � ρ I ( µ )) ω 1 ω 2 + 2 π i λσ i i Z G ; k ( λ ; � µ ) = √− ω 1 ω 2 � e ω 1 ω 2 | W | α ∈ G + Γ h ( ± α ( σ )) i =1 Balancing condition: relations between the fugacities (3+1)d (due to anomalies) or (2+1)d real masses (due to W mon ); FI: real mass for 3d U (1) J (here λ ); Dualities as integral identities (subtleties due to ∞ ); Real mass flow and CS: lim x →∞ Γ h ( x ) = e i π sgn ( x )( x − ω ) 2 ; Flows to generate new dualities Holomorphic mass Γ h ( x )Γ h (2 ω − x ) = 1 Antonio Amariti (INFN) (2+1)d dualities 14 / 30
I S 3 × S 1 → Z S 3 Aharony F � Z U ( N );0 ( λ ; µ ; ν ) = Γ h ( µ a + ν a ) Z U ( F − N );0 ( − λ ; ω − ν ; ω − µ ) a , b =1 � � F � 1 × ( µ a + ν a ) ± λ ) + ω ( F − N + 1) Γ h ( 2 a =1 Giveon-Kutasov � F e φ Z U ( N ); k ( λ ; µ ; ν ) = Γ h ( µ a + ν a ) Z U ( F − N + | k | ); − k ( − λ ; ω − ν ; ω − µ ) a , b =1 where φ = φ ( ω, µ, ν, λ, F , N , k ) collects the (CS) contact terms. ARSW F � Z U ( N );0 ( λ ; µ ; ν ) = e φ Γ h ( µ a + ν a ) Z U ( F − N );0 ( − λ ; ω − ν ; ω − µ ) a , b =1 with � F a =1 ( µ a + ν a ) = 2 ω ( F − N ) (balancing condition, enforced by W KK ). Antonio Amariti (INFN) (2+1)d dualities 15 / 30
D-branes: reduction from T-duality. D-branes Duality due to the brane creation effect during the transition through infinite coupling (i.e. crossing of NS branes) 4d Seiberg duality and branes NS NS’ [89] ND4 F [6] F D6 [45] SU(N) SQCD Separate the w/ F flavors D6 along [6] N NS F D6 NS’ N F-N D4 F D4 F-N F-N Separate D4/D6 Exchange Seiberg dual SQCD along [45] NS and NS’ after recollecting Antonio Amariti (INFN) (2+1)d dualities 16 / 30
D-branes: reduction from T-duality. D-branes: reduction from T-duality 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 NS X X X X X X T-duality NS X X X X X X NS ′ NS ′ → X X X X X X X X X X X X N c D 4 [ X ] along x 3 N c D 3 [ X ] X X X X X X X N f D 6 N f D 5 X X X X X X X X X X X X X 3 (i+1)-th D3 6 For N f = 0 (SYM): N = 0 f W = W mon = W ( BPS ) + W ( KK ) mon mon NS E1 NS’ E1: Euclidean D1, 3 For N f � = 0 ( > N c ) W = W ( KK ) mon N > N c f i-th D3 σ i − σ i +1 σ N − σ 1 + i ( ϕ i − ϕ i +1) + i ( ϕ N − ϕ 1) g 2 g 2 W ∝ e + η e 3 3 with Nambu-Goto (for σ ) and Boundary action (for ϕ ) Antonio Amariti (INFN) (2+1)d dualities 17 / 30
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