The Bootstrap Program for Defect CFT Pedro Liendo October 17 2018 String Seminars in Trieste, ICTP/SISSA. 1 / 26
Motivation 2 / 26
IR UV Renormalization group flow Figure: Renormalization group (RG) flow. 3 / 26
UV IR Renormalization group flow Figure: Renormalization group (RG) flow. Figure: Kenneth G. Wilson (1936-2013). 3 / 26
CFTs are everywhere 4 / 26
CFTs are everywhere Critical phenomena. Gas-liquid, order-disorder, superfluids. 4 / 26
CFTs are everywhere Critical phenomena. Gas-liquid, order-disorder, superfluids. Mathematics. 2 d CFTs, vertex operator algebras (VOA). 4 / 26
CFTs are everywhere Critical phenomena. Gas-liquid, order-disorder, superfluids. Mathematics. 2 d CFTs, vertex operator algebras (VOA). String theory. Worldsheet dynamics. 4 / 26
CFTs are everywhere Critical phenomena. Gas-liquid, order-disorder, superfluids. Mathematics. 2 d CFTs, vertex operator algebras (VOA). String theory. Worldsheet dynamics. Holography. 4 / 26
CFTs are everywhere Critical phenomena. Gas-liquid, order-disorder, superfluids. Mathematics. 2 d CFTs, vertex operator algebras (VOA). String theory. Worldsheet dynamics. Holography. Black holes, quantum gravity. 4 / 26
Bootstrap basics 5 / 26
The conformal algebra The conformal algebra is SO ( d + 1 , 1) 6 / 26
The conformal algebra The conformal algebra is SO ( d + 1 , 1) It includes translations, rotations, and scale transformations (+ more) x → x + a , x → R · x , x → h x . 6 / 26
The conformal algebra The conformal algebra is SO ( d + 1 , 1) It includes translations, rotations, and scale transformations (+ more) x → x + a , x → R · x , x → h x . Conformal generators αα , M β α , K ˙ α , ¯ M ˙ α {P α ˙ β , D} ˙ O → { ∆ , j , ¯ j } 6 / 26
The conformal algebra The conformal algebra is SO ( d + 1 , 1) It includes translations, rotations, and scale transformations (+ more) x → x + a , x → R · x , x → h x . Conformal generators αα , M β α , K ˙ α , ¯ M ˙ α {P α ˙ β , D} ˙ O → { ∆ , j , ¯ j } Operators are organized in conformal families K ˙ αα O (0) = 0 Primary : P k Descendants : α O (0) α ˙ 6 / 26
CFT correlators The conformal algebra puts tight restrictions on correlation functions � 1 if ∆ 1 = ∆ 2 | x 1 − x 2 | 2∆ φ � φ 1 ( x 1 ) φ 2 ( x 2 ) � = , 0 if ∆ 1 � = ∆ 2 C 123 � φ 1 ( x 1 ) φ 2 ( x 2 ) φ 3 ( x 3 ) � = | x 12 | ∆ 1 +∆ 2 − ∆ 3 | x 23 | ∆ 2 +∆ 3 − ∆ 1 | x 13 | ∆ 1 +∆ 3 − ∆ 2 . 7 / 26
CFT correlators The conformal algebra puts tight restrictions on correlation functions � 1 if ∆ 1 = ∆ 2 | x 1 − x 2 | 2∆ φ � φ 1 ( x 1 ) φ 2 ( x 2 ) � = , 0 if ∆ 1 � = ∆ 2 C 123 � φ 1 ( x 1 ) φ 2 ( x 2 ) φ 3 ( x 3 ) � = | x 12 | ∆ 1 +∆ 2 − ∆ 3 | x 23 | ∆ 2 +∆ 3 − ∆ 1 | x 13 | ∆ 1 +∆ 3 − ∆ 2 . The collection { C , ∆ } is the CFT data. 7 / 26
CFT correlators The conformal algebra puts tight restrictions on correlation functions � 1 if ∆ 1 = ∆ 2 | x 1 − x 2 | 2∆ φ � φ 1 ( x 1 ) φ 2 ( x 2 ) � = , 0 if ∆ 1 � = ∆ 2 C 123 � φ 1 ( x 1 ) φ 2 ( x 2 ) φ 3 ( x 3 ) � = | x 12 | ∆ 1 +∆ 2 − ∆ 3 | x 23 | ∆ 2 +∆ 3 − ∆ 1 | x 13 | ∆ 1 +∆ 3 − ∆ 2 . The collection { C , ∆ } is the CFT data. The four-point function is not completely fixed, for identical fields. g ( u , v ) � φ ( x 1 ) φ ( x 2 ) φ ( x 3 ) φ ( x 4 ) � = | x 12 | 2∆ φ | x 34 | 2∆ φ , 7 / 26
Operator Product Expansion The product of two primary fields can be replaced by a sum: � φ ( x ) φ (0) ∼ C O d ( x , ∂ ) O (0) . O 8 / 26
Operator Product Expansion The product of two primary fields can be replaced by a sum: � φ ( x ) φ (0) ∼ C O d ( x , ∂ ) O (0) . O Four-point functions can then be expanded � 1 C 2 � φ ( x 1 ) φ ( x 2 ) φ ( x 3 ) φ ( x 4 ) � = | x 12 | 2∆ φ | x 34 | 2∆ φ (1 + O g O ( u , v )) O where the “conformal block” g O ( u , v ) is known ( Dolan-Osborn ). 8 / 26
Crossing Symmetry Four-point functions satisfy crossing symmetry : � � v ∆ φ (1 + C 2 O g O ( u , v )) = u ∆ φ (1 + C 2 O g O ( v , u )) O O 9 / 26
Crossing Symmetry Four-point functions satisfy crossing symmetry : � � v ∆ φ (1 + C 2 O g O ( u , v )) = u ∆ φ (1 + C 2 O g O ( v , u )) O O It can be represented pictorially, � � C 2 C 2 = ∆ O O O ∆ O O O 9 / 26
Crossing Symmetry Four-point functions satisfy crossing symmetry : � � v ∆ φ (1 + C 2 O g O ( u , v )) = u ∆ φ (1 + C 2 O g O ( v , u )) O O It can be represented pictorially, � � C 2 C 2 = ∆ O O O ∆ O O O Very constraining system for the CFT data. 9 / 26
Bootstrap techniques 10 / 26
Bootstrap techniques The numerical bootstrap. Powerful numerical techniques that constrain the low-lying spectrum. (Poland, Rattazi, Rychkov, Simmons-Duffin, Tonni, Vichi) 10 / 26
Bootstrap techniques The numerical bootstrap. Powerful numerical techniques that constrain the low-lying spectrum. (Poland, Rattazi, Rychkov, Simmons-Duffin, Tonni, Vichi) The lightcone bootstrap. Analytic constraints for operators with high spin. (Poland, Kaplan, Komargodski, Fitzpatrick, Simmons-Duffin, Caron-Huot) 10 / 26
Bootstrap techniques The numerical bootstrap. Powerful numerical techniques that constrain the low-lying spectrum. (Poland, Rattazi, Rychkov, Simmons-Duffin, Tonni, Vichi) The lightcone bootstrap. Analytic constraints for operators with high spin. (Poland, Kaplan, Komargodski, Fitzpatrick, Simmons-Duffin, Caron-Huot) Solvable truncation. In supersymmetric theories there is a solvable truncation of the crossing equations. (C. Beem, M. Lemos, PL, W. Peelaers, L. Rastelli, B. van Rees.) 10 / 26
Defect CFT 11 / 26
Defect CFT Extended objects are important observables in CFT: Wilson and ’t Hoft lines, surface operators, boundaries, interfaces, . . . 12 / 26
Defect CFT Extended objects are important observables in CFT: Wilson and ’t Hoft lines, surface operators, boundaries, interfaces, . . . O 1 O 2 Figure: Local operatos in the presence of a defect. 12 / 26
Defect CFT Extended objects are important observables in CFT: Wilson and ’t Hoft lines, surface operators, boundaries, interfaces, . . . O 1 O 2 Figure: Local operatos in the presence of a defect. We have SO (1 , d + 1) → SO (1 , p + 1) × SO ( q ) where q + p = d . 12 / 26
Defect CFT correlators The SO (1 , p + 1) × SO ( q ) symmetry preserved by the defect implies that one-point functions are non-zero: a O �O ( x ) � = ( x i ) ∆ . 13 / 26
Defect CFT correlators The SO (1 , p + 1) × SO ( q ) symmetry preserved by the defect implies that one-point functions are non-zero: a O �O ( x ) � = ( x i ) ∆ . Two-point functions depend on two conformal invariants 1 � φ ( x 1 ) φ ( x 2 ) � = z ) ∆ φ/ 2 g ( z , ¯ z ) , ( z ¯ z = z ∗ in Euclidean signature where ¯ 13 / 26
Defect CFT correlators The SO (1 , p + 1) × SO ( q ) symmetry preserved by the defect implies that one-point functions are non-zero: a O �O ( x ) � = ( x i ) ∆ . Two-point functions depend on two conformal invariants 1 � φ ( x 1 ) φ ( x 2 ) � = z ) ∆ φ/ 2 g ( z , ¯ z ) , ( z ¯ z = z ∗ in Euclidean signature where ¯ Remark. Compare with the four-point function in the bulk CFT. 13 / 26
b b Two-point function configuration z 0 = = 1 ¯ z defect ⊗ O (1 , 1) z, ¯ z = 0 O ( z, ¯ z ) z 1 = = 0 z ¯ Figure: Configuration of the system in the plane orthogonal to the defect. 14 / 26
Bulk OPE Bulk channel: We had � φ ( x ) φ (0) ∼ C φφ O d ( x , ∂ ) O (0) . O recall that in the presence of a defect a scalar can have a non-zero one-point function 15 / 26
Bulk OPE Bulk channel: We had � φ ( x ) φ (0) ∼ C φφ O d ( x , ∂ ) O (0) . O recall that in the presence of a defect a scalar can have a non-zero one-point function The expansion for the two-point function is � (1 − z )(1 − ¯ � − ∆ φ � z ) � φ ( x 1 ) φ ( x 2 ) � = C φφ O a O f ∆ , J ( z , ¯ z ) z ) 1 / 2 ( z ¯ ∆ , J where the sum goes over the bulk spectrum. 15 / 26
Defect OPE Defect channel: We can also write a bulk operator as a sum of defect operators � x ) ˆ O D ( x i , ∂ � φ ( x ) = b φ � O ( � x ) ˆ O where the “hat” denotes a boundary quantity. 16 / 26
Defect OPE Defect channel: We can also write a bulk operator as a sum of defect operators � x ) ˆ O D ( x i , ∂ � φ ( x ) = b φ � O ( � x ) ˆ O where the “hat” denotes a boundary quantity. Plugging this expansion into the two-point function, � O ) 2 � � φ ( x 1 ) φ ( x 2 ) � = ( b φ � f ˆ ∆ , s ( z , ¯ z ) . ˆ ∆ , s where the sum goes over the boundary spectrum. 16 / 26
Crossing symmetry Equality of both expansions implies � (1 − z )(1 − ¯ � − ∆ φ � � z ) O � b 2 C φφ O a O f ∆ , J ( z , ¯ z ) = f � ∆ , s ( z , ¯ z ) φ � z ) 1 / 2 ( z ¯ ∆ , J � ∆ , s 17 / 26
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