A Monotonicity Theorem for Two-dimensional Boundaries and Defects - PowerPoint PPT Presentation

The g-theorem Put fixed point BCFT on a hemisphere g α ≡ Z ren. α “Boundary entropy” ln g α Counts DOF localized at boundary

The g-theorem Thermodynamic entropy Affleck and Ludwig PRL 67 (1991) 161 System size L Temperature T T � 1 /L S thermo = π 3 cLT + ln g α + . . .

The g-theorem Entanglement Entropy (EE) Calabrese + Cardy hep-th/0405152 Interval including the boundary S EE = c 6 ln 2 ` a + ln g α + . . . − 1 ln g α = S BCFT 2 S CFT EE EE

The g-theorem Affleck and Ludwig PRL 67 (1991) 161 Friedan and Konechny hep-th/0312197 Define a thermodynamic g-function S thermo = π 3 cLT + S ∂ ( T ) Euclidean symmetry, locality, reflection positivity ∂ S ∂ ⇒ ∂ T ≥ 0 g UV ≥ g IR Strong form Weak form

Generalizations? Higher-dimensional g-theorems? Proposals Yamaguchi hep-th/0207171 Takayanagi et al. 1105.5165, 1108.5152, 1205.1573 Estes, Jensen, O’B., Tsatis, Wrase 1403.6475 Gaiotto 1403.8052 Many tests in particular examples No proofs until now!

GOAL Prove a g-theorem for Local, reflection-positive BCFT in d = 3 Proposals Nozaki, Takayanagi, Ugajin 1205.1573 Estes, Jensen, O’B., Tsatis, Wrase 1403.6475 Trace Anomaly Matching Schwimmer and Theisen 1011.0696 Komargodski and Schwimmer 1107.3987 Komargodski 1112.4538

Examples Graphene with a boundary Critical Ising model in with a boundary d = 3 M-theory: M2-branes with a boundary String theory: various brane intersections Holographic BCFTs

Outline: • Review: Monotonicity Theorems • The Systems • The Trace Anomaly • The Proof • Summary and Outlook

The Systems BCFT in d = 3 With a planar boundary y z x = 0 x

The Systems SO ( d + 1 , 1) = SO (4 , 1) x µ → Λ µ Rotations ν x ν x µ → x µ + c µ Translations x µ → λ x µ Dilatations x µ + b µ x 2 Special Conformal x µ → 1 + 2 x ν b ν + b 2 x 2 Broken to subgroup that preserves x = 0

The Systems x µ → Λ µ Rotations ν x ν Broken to rotations in ( y, z ) y z x = 0 x

The Systems x µ → x µ + c µ Translations Broken to translations along ( y, z ) y z x = 0 x

The Systems x µ → λ x µ Dilatations Unbroken y z x = 0 x

The Systems x µ + b µ x 2 Special Conformal x µ → 1 + 2 x ν b ν + b 2 x 2 b x = 0 Broken to y z x = 0 x

The Systems SO ( d + 1 , 1) → SO ( d, 1) SO (4 , 1) → SO (3 , 1) NOT the full conformal group d = 2 y z x = 0 x

The Systems Boundary RG Flows Z S ( λ ) UV BCFT → S ( λ ) UV dx dy dz δ ( x ) λ O O ( y, z ) BCFT + scalar with O ∆ O < 2 T µ ν = [ T µ ν ] bulk + δ ( x ) [ T µ ν ] ∂ Invariance under diffeomorphisms along the boundary [ T µx ] ∂ = [ T xµ ] ∂ = [ T xx ] ∂ = 0 ∂ µ [ T µ ν ] ∂ ∝ [ T xx ] bulk

The Systems Boundary RG Flows Z S ( λ ) UV BCFT → S ( λ ) UV dx dy dz δ ( x ) λ O O ( y, z ) BCFT + scalar with O ∆ O < 2 T µ ν = [ T µ ν ] bulk + δ ( x ) [ T µ ν ] ∂ Bulk theory remains conformal ⇥ ⇤ ⇥ ⇤ T µ = δ ( x ) ∂ 6 = 0 T µ T µ bulk = 0 µ µ µ

UV BCFT UV Single free, massless, real scalar Neumann B.C. Boundary RG Flow Z S UV BCFT → S UV d 3 x � ( x ) m 2 Φ 2 ( ~ x ) BCFT + ∆ Φ 2 = 1 IR BCFT Single free, massless, real scalar IR Dirichlet B.C.

The Systems Boundary RG Flows Z S ( λ ) UV BCFT → S ( λ ) UV dx dy dz δ ( x ) λ O O ( y, z ) BCFT + Trace Anomaly Matching Schwimmer and Theisen 1011.0696 Komargodski and Schwimmer 1107.3987 Komargodski 1112.4538 Reflection Positivity Weak form

Outline: • Review: Monotonicity Theorems • The Systems • The Trace Anomaly • The Proof • Summary and Outlook

Trace Anomaly CFT in any d g µ ν = δ µ ν Conformal invariance T µ = 0 µ

Trace Anomaly CFT in any d Non-trivial g µ ν Quantum Effects Break Conformal Invariance T µ µ 6 = 0

Trace Anomaly What is the general form of ? T µ µ Step #1 Write down all curvature invariants built from g µ ν with the correct dimension d = 4 = c 1 R µ νρσ R µ νρσ + c 2 R µ ν R µ ν + c 3 R 2 + . . . T µ µ

Trace Anomaly What is the general form of ? T µ µ Step #2 Wess-Zumino consistency g µ ν → e 2 Ω 1 e 2 Ω 2 g µ ν g µ ν → e 2 Ω 2 e 2 Ω 1 g µ ν = Fixes some coefficients = c 1 R µ νρσ R µ νρσ + c 2 R µ ν R µ ν + c 3 R 2 + . . . T µ µ

Trace Anomaly What is the general form of ? T µ µ Step #3 Add local counterterms to S ( g µ ν , λ ) Determine how they enter T µ µ Fixes more coefficients = c 1 R µ νρσ R µ νρσ + c 2 R µ ν R µ ν + c 3 R 2 + . . . T µ µ

Trace Anomaly CFT in any d T µ = 0 d odd µ T µ 6 = 0 d even µ c T µ = 24 π R d = 2 µ

Trace Anomaly d = 4 T µ = a E − c W µ νρσ W µ νρσ µ Euler density E = R µ νρσ R µ νρσ − 4 R µ ν R µ ν + R 2 Weyl tensor W µ νρσ central charges and c a

Trace Anomaly g µ ν ( x ) → e 2 Ω ( x ) g µ ν ( x ) Z µ is invariant d d x √ g T µ T µ = a E − c W µ νρσ W µ νρσ µ Type A Type B √ g W 2 √ g E Changes by a total derivative Invariant

Trace Anomaly BCFT in d = 3 ⇥ ⇤ ⇥ ⇤ µ = bulk + δ ( x ) T µ T µ T µ µ µ ∂ ⇥ ⇤ T µ bulk = 0 µ ⇥ ⇤ What is the general form of ? T µ µ ∂

Geometry of Submanifolds σ 1 , σ 2 “worldsheet” “target space” x µ Embedding x µ ( σ a ) Induced metric ∂ x µ ∂ x ν g ab = g µ ν ˆ ∂σ a ∂σ b ˆ ˆ ˆ R abcd R ab R

Geometry of Submanifolds Extrinsic Curvature “Second Fundamental Form” Gaussian Normal Coordinates g µ ν dx µ dx ν = dx 2 + ˆ g ab ( x, σ a ) d σ a d σ b K ab = 1 2 ∂ x ˆ g ab ( x, σ ) Mean curvature g ab K ab K ≡ ˆ

Trace Anomaly ⇥ ⇤ What is the general form of ? T µ µ ∂ Schwimmer + Theisen 0802.1017 See also: Berenstein, Corrado, Fischler, Maldacena hep-th/9809188 Graham + Witten hep-th/9901021 Henningson + Skenderis hep-th/9905163 Gustavsson hep-th/0310037, 0404150 Asnin 0801.1469

Trace Anomaly R + c 2 ( K ab K ab − 1 ∂ = c 1 ˆ T µ 2 K 2 ) ⇥ ⇤ µ Boundary “central charges” and c 1 c 2

Trace Anomaly g µ ν → e 2 Ω g µ ν R + c 2 ( K ab K ab − 1 ∂ = c 1 ˆ T µ 2 K 2 ) ⇥ ⇤ µ Type A Type B g ( K ab K ab − 1 h i p p g ˆ ˆ R � 2 r 2 Ω p 2 K 2 ) R ! g ˆ ˆ ˆ Changes by a total derivative Invariant

Trace Anomaly R + c 2 ( K ab K ab − 1 ∂ = c 1 ˆ T µ 2 K 2 ) ⇥ ⇤ µ g ( K ab K ab − 1 Z p 2 K 2 ) d 2 σ ˆ “Rigid String” Action Physics Polyakov, NPB 268 (1986) 406 Mathematics Willmore functional

Trace Anomaly GOAL R + c 2 ( K ab K ab − 1 ∂ = c 1 ˆ T µ 2 K 2 ) ⇥ ⇤ µ Boundary RG Flow c UV ≥ c IR 1 1 Nozaki, Takayanagi, Ugajin 1205.1573 Estes, Jensen, O’B., Tsatis, Wrase 1403.6475

Trace Anomaly GOAL R + c 2 ( K ab K ab − 1 ∂ = c 1 ˆ T µ 2 K 2 ) ⇥ ⇤ µ ∂ µ [ T µ ν ] ∂ ∝ [ T xx ] bulk We can’t just copy Zamolodchikov’s proof!

Trace Anomaly GOAL R + c 2 ( K ab K ab − 1 ∂ = c 1 ˆ T µ 2 K 2 ) ⇥ ⇤ µ Trace Anomaly Matching Schwimmer and Theisen 1011.0696 Komargodski and Schwimmer 1107.3987 Komargodski 1112.4538 Reflection Positivity Weak form

Outline: • Review: Monotonicity Theorems • The Systems • The Trace Anomaly • The Proof • Summary and Outlook

The Proof UV Trace Anomaly Matching Schwimmer and Theisen 1011.0696 Komargodski and Schwimmer 1107.3987 Komargodski 1112.4538 local, reflection-positive QFT in any d RG flow between fixed point CFTs IR

Dilaton Non-dynamical background metric g µ ν ( x ) Non-dynamical background scalar τ ( x ) g µ ν → e 2 Ω g µ ν τ → τ + Ω

Dilaton Non-dynamical background metric g µ ν ( x ) Non-dynamical background scalar τ ( x ) λ O O ( x ) → e ( ∆ O − d ) τ ( x ) λ O O ( x )

Dilaton Non-dynamical background metric g µ ν ( x ) Non-dynamical background scalar τ ( x ) Z S ( λ ) τ = d d x √ g L ( � , ~ x ) τ Z h i d d x √ g T µ τ =0 + O ( ⌧ 2 ) ⇥ ⇤ L ( � , ~ x ) τ =0 + ⌧ = µ

Dilaton Non-dynamical background metric g µ ν ( x ) Non-dynamical background scalar τ ( x ) Trace Anomaly Matching d even

Dilaton Integrate out massive DOF Obtain effective action S e ff ≡ − ln Z Regular and local in τ Expand in τ S e ff = S τ =0 + S τ e ff e ff

Dilaton g µ ν → e 2 Ω g µ ν τ → τ + Ω δ S e ff δ Ω = − δ δ Ω ln Z Expand in τ δ S e ff τ =0 + δ S τ δ Ω = −√ g e ff ⇥ ⇤ T µ µ δ Ω

δ S e ff UV ⇤ UV ⇥ T µ δ Ω = − τ =0 = 0 µ g µ ν = δ µ ν τ = 0 δ S e ff ⇥ ⇤ δ Ω = � T µ τ =0 6 = 0 µ δ S e ff ⇤ IR ⇥ IR T µ δ Ω = − τ =0 = 0 µ

δ S e ff UV ⇤ UV ⇥ T µ δ Ω = − τ =0 = 0 µ g µ ν = δ µ ν τ 6 = 0 δ S e ff τ =0 + δ S τ e ff ⇥ ⇤ T µ δ Ω = − δ Ω = 0 µ δ S e ff ⇤ IR ⇥ IR T µ δ Ω = − τ =0 = 0 µ

δ Ω = �p g δ S e ff UV ⇤ UV ⇥ T µ τ =0 6 = 0 µ g µ ν 6 = δ µ ν τ = 0 ⇤ UV ⇤ IR ⇥ ⇥ T µ T µ τ =0 6 = µ µ τ =0 δ Ω = �p g δ S e ff ⇤ IR ⇥ IR T µ τ =0 6 = 0 µ

δ Ω = �p g δ S e ff UV ⇤ UV ⇥ T µ τ =0 6 = 0 µ g µ ν 6 = δ µ ν τ 6 = 0 δ Ω = �p g δ Ω = �p g δ S e ff τ =0 + δ S τ ⇤ UV e ff ⇥ ⇤ ⇥ T µ T µ τ =0 6 = 0 µ µ δ Ω = �p g δ Ω = �p g δ S e ff τ =0 + δ S τ IR ⇤ IR ⇤ UV e ff ⇥ ⇥ T µ T µ τ =0 6 = 0 µ µ

Dilaton Trace Anomaly Matching Schwimmer and Theisen 1011.0696 At the IR fixed point, must produce S τ e ff δ S τ h⇥ i ⇤ UV ⇤ IR δ Ω = −√ g e ff ⇥ T µ T µ τ =0 − µ µ τ =0

Dilaton Boundary RG Flow Z S UV BCFT → S UV dx dy dz δ ( x ) λ O O ( y, z ) BCFT + λ O O ( y, z ) → e ( ∆ O − 2) τ ( y,z ) λ O O ( y, z ) τ is localized at the boundary, and depends only on boundary coordinates!

( K ab K ab − 1 UV ⇤ UV ˆ T µ = c UV R + c UV 2 K 2 ) ⇥ µ 1 2 ∂ ⇥ ⇤ ⇥ ⇤ µ = bulk + δ ( x ) T µ T µ T µ µ µ ∂ Boundary RG Flow Z S UV BCFT → S UV dx dy dz δ ( x ) λ O O ( y, z ) BCFT + ⇥ ⇤ T µ bulk = 0 µ 2 ( K ab K ab − 1 IR ⇤ IR ˆ T µ ∂ = c IR R + c IR 2 K 2 ) ⇥ µ 1