Exact results with defects based on: 1805.04111, 1910.06332, 1911.05082, 2004.07849 with M. Lemos, M. Meineri; M. Bill´ o, F. Galvagno, A. Lerda; G. Bliard, V. Forini, L. Griguolo, D. Seminara Lorenzo Bianchi May 28 th , 2020. Cortona Young Lorenzo Bianchi (INFN) Exact results with defects 28/05/2020 1 / 13
Why (super)conformal defects? There are several physically relevant examples Wilson and ’t Hooft lines 1 Boundaries and interfaces 2 R´ enyi entropy 3 Surface defects 4 Lorenzo Bianchi (INFN) Exact results with defects 28/05/2020 2 / 13
Why (super)conformal defects? There are several physically relevant examples Wilson and ’t Hooft lines 1 Boundaries and interfaces 2 R´ enyi entropy 3 Surface defects 4 Interplay of various techniques AdS/CFT correspondence 1 Supersymmetric localization 2 Integrability 3 Conformal bootstrap 4 Lorenzo Bianchi (INFN) Exact results with defects 28/05/2020 2 / 13
Why (super)conformal defects? There are several physically relevant examples Wilson and ’t Hooft lines 1 Boundaries and interfaces 2 R´ enyi entropy 3 Surface defects 4 Interplay of various techniques AdS/CFT correspondence 1 Supersymmetric localization 2 Integrability 3 Conformal bootstrap 4 They probe aspects of the theory that are not accessible to correlation functions of local operators, e.g. global structure of the gauge group. Lorenzo Bianchi (INFN) Exact results with defects 28/05/2020 2 / 13
Why (super)conformal defects? There are several physically relevant examples Wilson and ’t Hooft lines 1 Boundaries and interfaces 2 R´ enyi entropy 3 Surface defects 4 Interplay of various techniques AdS/CFT correspondence 1 Supersymmetric localization 2 Integrability 3 Conformal bootstrap 4 They probe aspects of the theory that are not accessible to correlation functions of local operators, e.g. global structure of the gauge group. They preserve part of the original (super)symmetry, leading to constraints on physical observables. Lorenzo Bianchi (INFN) Exact results with defects 28/05/2020 2 / 13
CFT Set of conformal primary operators O ∆ ,ℓ ( x ) plus descendants ∂ µ 1 . . . ∂ µ n O ∆ ,ℓ . Operator product expansion (OPE) � c ijk | x | ∆ k − ∆ i − ∆ j O k (0) + x µ ∂ µ O k (0) + . . . O i ( x ) O j (0) = � �� � k ∈ prim. all fixed The set of data { ∆ i , c ijk } fully specifies the CFT, up to extended probes. Lorenzo Bianchi (INFN) Exact results with defects 28/05/2020 3 / 13
CFT Set of conformal primary operators O ∆ ,ℓ ( x ) plus descendants ∂ µ 1 . . . ∂ µ n O ∆ ,ℓ . Operator product expansion (OPE) � c ijk | x | ∆ k − ∆ i − ∆ j O k (0) + x µ ∂ µ O k (0) + . . . O i ( x ) O j (0) = � �� � k ∈ prim. all fixed The set of data { ∆ i , c ijk } fully specifies the CFT, up to extended probes. Two- and three-point functions �O ∆ ( x ) O ∆ (0) � → ∆ �O i ( x 1 ) O j ( x 2 ) O k ( x 3 ) � → c ijk Lorenzo Bianchi (INFN) Exact results with defects 28/05/2020 3 / 13
CFT Set of conformal primary operators O ∆ ,ℓ ( x ) plus descendants ∂ µ 1 . . . ∂ µ n O ∆ ,ℓ . Operator product expansion (OPE) � c ijk | x | ∆ k − ∆ i − ∆ j O k (0) + x µ ∂ µ O k (0) + . . . O i ( x ) O j (0) = � �� � k ∈ prim. all fixed The set of data { ∆ i , c ijk } fully specifies the CFT, up to extended probes. Two- and three-point functions �O ∆ ( x ) O ∆ (0) � → ∆ �O i ( x 1 ) O j ( x 2 ) O k ( x 3 ) � → c ijk O 1 ( x 1 ) O 4 ( x 4 ) Crossing O 1 ( x 1 ) O 4 ( x 4 ) � O ∆ ,ℓ � = O ∆ ,ℓ ∆ ,ℓ ∆ ,ℓ O 2 ( x 2 ) O 3 ( x 3 ) O 2 ( x 2 ) O 3 ( x 3 ) Lorenzo Bianchi (INFN) Exact results with defects 28/05/2020 3 / 13
DCFT [Bill` o, Goncalves, Lauria, Meineri, 2016] Defect operators ˆ ℓ, s with parallel (ˆ O ˆ ℓ ) and orthogonal spin ( s ) and descendants ∆ , ˆ ∂ a 1 . . . ∂ a n ˆ O ˆ ℓ, s . ∆ , ˆ Lorenzo Bianchi (INFN) Exact results with defects 28/05/2020 4 / 13
DCFT [Bill` o, Goncalves, Lauria, Meineri, 2016] Defect operators ˆ ℓ, s with parallel (ˆ O ˆ ℓ ) and orthogonal spin ( s ) and descendants ∆ , ˆ ∂ a 1 . . . ∂ a n ˆ O ˆ ℓ, s . ∆ , ˆ One-point functions �O ( x ) � W ≡ �O ( x ) W � = a O r ∆ � W � Lorenzo Bianchi (INFN) Exact results with defects 28/05/2020 4 / 13
DCFT [Bill` o, Goncalves, Lauria, Meineri, 2016] Defect operators ˆ ℓ, s with parallel (ˆ O ˆ ℓ ) and orthogonal spin ( s ) and descendants ∆ , ˆ ∂ a 1 . . . ∂ a n ˆ O ˆ ℓ, s . ∆ , ˆ One-point functions �O ( x ) � W ≡ �O ( x ) W � = a O r ∆ � W � Bulk to defect coupling b O ˆ �O ( x ) ˆ O O ( y ) � W = r ∆ − ˆ ∆ ( r 2 + y 2 ) ˆ ∆ Lorenzo Bianchi (INFN) Exact results with defects 28/05/2020 4 / 13
DCFT [Bill` o, Goncalves, Lauria, Meineri, 2016] Defect operators ˆ ℓ, s with parallel (ˆ O ˆ ℓ ) and orthogonal spin ( s ) and descendants ∆ , ˆ ∂ a 1 . . . ∂ a n ˆ O ˆ ℓ, s . ∆ , ˆ One-point functions �O ( x ) � W ≡ �O ( x ) W � = a O r ∆ � W � Bulk to defect coupling b O ˆ �O ( x ) ˆ O O ( y ) � W = r ∆ − ˆ ∆ ( r 2 + y 2 ) ˆ ∆ Defect OPE � ˆ ∆ − ∆ ˆ O ( x ) = b O ˆ O | x ⊥ | O (0) +def. desc. � �� � def. prim. all fixed Lorenzo Bianchi (INFN) Exact results with defects 28/05/2020 4 / 13
DCFT [Bill` o, Goncalves, Lauria, Meineri, 2016] Defect operators ˆ ℓ, s with parallel (ˆ O ˆ ℓ ) and orthogonal spin ( s ) and descendants ∆ , ˆ ∂ a 1 . . . ∂ a n ˆ O ˆ ℓ, s . ∆ , ˆ One-point functions �O ( x ) � W ≡ �O ( x ) W � = a O r ∆ � W � Bulk to defect coupling b O ˆ �O ( x ) ˆ O O ( y ) � W = r ∆ − ˆ ∆ ( r 2 + y 2 ) ˆ ∆ Defect OPE � ˆ ∆ − ∆ ˆ O ( x ) = b O ˆ O | x ⊥ | O (0) +def. desc. � �� � def. prim. all fixed O , ˆ The naive set of defect CFT data is { a O , b O ˆ ∆ ˆ O , ˆ c ˆ O 3 } . O 1 ˆ O 2 ˆ Lorenzo Bianchi (INFN) Exact results with defects 28/05/2020 4 / 13
DCFT Defect crossing O 1 ( x 1 ) O 1 ( x 1 ) � � O ∆ ,ℓ ˆ = O ˆ ∆ , ˆ ℓ, s ∆ , ˆ ˆ ∆ ,ℓ ℓ, s O 2 ( x 2 ) O 2 ( x 2 ) Lorenzo Bianchi (INFN) Exact results with defects 28/05/2020 5 / 13
DCFT Defect crossing O 1 ( x 1 ) O 1 ( x 1 ) � � O ∆ ,ℓ ˆ = O ˆ ∆ , ˆ ℓ, s ∆ , ˆ ˆ ∆ ,ℓ ℓ, s O 2 ( x 2 ) O 2 ( x 2 ) Subset of defect CFT data: Physically (or geometrically) relevant 1 Universal (present in any defect CFT) 2 Lorenzo Bianchi (INFN) Exact results with defects 28/05/2020 5 / 13
DCFT Defect crossing O 1 ( x 1 ) O 1 ( x 1 ) � � O ∆ ,ℓ ˆ = O ˆ ∆ , ˆ ℓ, s ∆ , ˆ ˆ ∆ ,ℓ ℓ, s O 2 ( x 2 ) O 2 ( x 2 ) Subset of defect CFT data: Physically (or geometrically) relevant 1 Universal (present in any defect CFT) 2 Stress-tensor one-point function � T ab � W = − h ( q − 1) δ ab | x ⊥ | d � T ij � W = h ( p + 1) δ ij − d n i n j | x ⊥ | d Lorenzo Bianchi (INFN) Exact results with defects 28/05/2020 5 / 13
Displacement operator A defect breaks translation invariance ∂ µ T µ i ( x ⊥ , x � ) = δ q ( x ⊥ ) D i ( x � ) Lorenzo Bianchi (INFN) Exact results with defects 28/05/2020 6 / 13
Displacement operator A defect breaks translation invariance ∂ µ T µ i ( x ⊥ , x � ) = δ q ( x ⊥ ) D i ( x � ) D i ( x � ) is the displacement operator It implements small modifications of the defect � d p x � δ x i ( x � ) � D i ( x � ) X � W δ � X � W = − Its two-point function is fixed by conformal symmetry δ ij � D i ( x � ) D j (0) � W = C D | x � | 2( p +1) . Normalization fixed by Ward identity, C D is physical. Like � T µν T ρσ � ∼ c . Lorenzo Bianchi (INFN) Exact results with defects 28/05/2020 6 / 13
Relation between C D and h For superconformal defects [LB, Lemos, 2019] q Γ( p +1 C D = 2 p +1 ( q + p − 1)( p + 2) 2 ) π 2 2 ) h . p +1 Γ( q q − 1 π 2 Lorenzo Bianchi (INFN) Exact results with defects 28/05/2020 7 / 13
Relation between C D and h For superconformal defects [LB, Lemos, 2019] q Γ( p +1 C D = 2 p +1 ( q + p − 1)( p + 2) 2 ) π 2 2 ) h . p +1 Γ( q q − 1 π 2 The relation is theory independent, but C D and h are non-trivial functions of the parameters (e.g. λ , N ). Lorenzo Bianchi (INFN) Exact results with defects 28/05/2020 7 / 13
Relation between C D and h For superconformal defects [LB, Lemos, 2019] q Γ( p +1 C D = 2 p +1 ( q + p − 1)( p + 2) 2 ) π 2 2 ) h . p +1 Γ( q q − 1 π 2 The relation is theory independent, but C D and h are non-trivial functions of the parameters (e.g. λ , N ). Conjectured for Wilson lines in N = 4 SYM and ABJM theory [Lewkowycz, Maldacena, 2014] . Proven for d = 4 and any q > 1 (any SUSY) [LB, Lemos, Meineri, 2018; LB, Lemos, 2019] . Proof is general, no conceptual difficulty in its generalization. Examples p = 1 , q = 3 C D = 36 h p = 1 , q = 2 C D = 24 h p = 2 , q = 2 C D = 48 h Lorenzo Bianchi (INFN) Exact results with defects 28/05/2020 7 / 13
Wilson lines A µ dx µ � W = Tr P e i In any conformal gauge theory, the Wilson line is a conformal defect. Lorenzo Bianchi (INFN) Exact results with defects 28/05/2020 8 / 13
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