Gravitational Decoupling II: Picard-Lefschetz Theory Jonathan Brown University of Wisconsin - Madison Great Lakes Strings, University of Chicago April 14, 2018 Based on arXiv:1710.04737 with Alex Cole, William Cottrell, and Gary Shiu Jonathan Brown (UW Madison) Gravitational Decoupling and PL GLS 4/14/18 1 / 13
Overview A Lorentzian prescription for semiclassical physics: Review of Picard-Lefschetz theory Tunneling with Picard-Lefschetz theory between non-metastable states ‘Euclidean’ solutions with Picard-Lefschetz theory Gravitational decoupling in the Picard-Lefschetz prescription vs. the Euclidean prescription Conclusions Jonathan Brown (UW Madison) Gravitational Decoupling and PL GLS 4/14/18 2 / 13
A Lorentzian Prescription: Picard-Lefschetz The Euclidean prescription generically leads to a failure of decoupling Instead we should use a Lorentzian prescription: Picard-Lefschetz Witten ‘11 Euclideanization: complexify time � � D φ e iS → D φ e − S E Z = Picard-Lefschetz: complexify fields � � D φ e iS → D φ e iS Z = C R J σ Jonathan Brown (UW Madison) Gravitational Decoupling and PL GLS 4/14/18 3 / 13
A Lorentzian Prescription: Picard-Lefschetz Start with 1d integral � dz e iS ( z ) /λ Z ( λ ) = C R Define the exponent I ≡ iS ( z ) /λ = h + iH Find a contour with lim z →∞ h → −∞ and H = const These are the steepest descent contours J σ around saddle points z σ Jonathan Brown (UW Madison) Gravitational Decoupling and PL GLS 4/14/18 4 / 13
A Lorentzian Prescription: Picard-Lefschetz Enumerate all saddle points and define the Lefschetz decomposition � � dz e iS ( z ) /λ = � dz e iS ( z ) /λ n σ C R J σ σ The n σ are topologically determined intersection numbers depending on steepest ascent contours K σ n σ = �C R , K σ � mod 2 Claim: Lefschetz decomposition defines a semiclassical expansion for the path integral � � D φ e iS [ φ ] /λ = � D φ e iS [ φ ] /λ Z ( λ ) = n σ C R J σ σ � n σ e iS [ φ σ ] /λ � a σ,j λ j = σ j Jonathan Brown (UW Madison) Gravitational Decoupling and PL GLS 4/14/18 5 / 13
A Lorentzian Prescription: Picard-Lefschetz n 1 = n 2 = 1 n 3 = 0 Jonathan Brown (UW Madison) Gravitational Decoupling and PL GLS 4/14/18 6 / 13
Tunneling with Picard-Lefschetz To illustrate PL consider the model Garay, Halliwell, Mena Marugan ‘91 � �� �� M 2 � d 4 x √− g � 2 R − 1 3 2 φ 2 ( ∂φ ) 2 − p S g = 2 α cosh + S GHY 3 M p Take a minisuperspace ansatz ds 2 = − N 2 a ( t ) 2 dt 2 + a ( t ) 2 d Ω 2 3 Define t ∈ [0 , 1] and physical time dτ = N a dt Choosing the gauge ˙ N = 0, the path integral takes the form � � D a D φ e iS g Z = dN C R Jonathan Brown (UW Madison) Gravitational Decoupling and PL GLS 4/14/18 7 / 13
Tunneling with Picard-Lefschetz The decoupled theory is free �� � � � 3 2 φ 3 V d ( φ ) = lim 2 α cosh = 2 α 3 M p M p →∞ Given arbitrary boundary conditions a (0) = a 1 φ (0) = φ 1 a (1) = a 2 φ (1) = φ 2 There are four saddle points labeled N ±± First example: let a 1 = a 2 Jonathan Brown (UW Madison) Gravitational Decoupling and PL GLS 4/14/18 8 / 13
Tunneling with Picard-Lefschetz Only the saddle point N ++ (in blue) contributes with action = ( φ 2 − φ 1 ) 2 S g [ N ++ ] M − 1 � � + O p V 2∆ τ In the decoupling limit this is the action for a free theory: decoupling is successful For α < 0 this can be viewed as the field tunneling through the tip of an inverted cosh Jonathan Brown (UW Madison) Gravitational Decoupling and PL GLS 4/14/18 9 / 13
‘Euclidean’ Solutions with Picard-Lefschetz M p →∞ − − − − − → Second example: reproduce Euclidean solutions with PL Assume boundary conditions a 1 = 0 and a 2 > 0 Decoupled theory is Euclidean with metric ds 2 = dt 2 + t 2 d Ω 2 3 The potential is constant and so there is only a constant solution φ ( t ) = const The action per unit spacetime volume is then � S d,E 3 = 2 α V Jonathan Brown (UW Madison) Gravitational Decoupling and PL GLS 4/14/18 10 / 13
‘Euclidean’ Solutions with Picard-Lefschetz = 12 iM 2 S g [ N − + ] p M 0 � � + O p a 2 V 2 � S g [ N −− ] 3 M − 1 � � = i 2 α + O p V With gravity two saddle points contribute: N − + (green) and N −− (red) In the decoupling limit N − + ceases to contribute and N −− agrees exactly with the decoupled theory Jonathan Brown (UW Madison) Gravitational Decoupling and PL GLS 4/14/18 11 / 13
‘Euclidean’ Solutions with Picard-Lefschetz Solutions found with PL prescription have conical singularities at the poles This means ∂φ ∂τ � = 0 and so these are excluded by the Euclidean prescription Conical singularity does not spoil decoupling Smoothing out one pole necessarily creates an HT singularity at the opposite pole Euclidean prescription requires solutions that are pathological and forbids solutions that are not! Jonathan Brown (UW Madison) Gravitational Decoupling and PL GLS 4/14/18 12 / 13
Conclusions The Euclidean prescription is not an indiscriminate tool for tunneling computations Generically, the Euclidean prescription leads to a failure of decoupling due to HT-type singularities The Euclidean prescription severely restricts the allowed boundary conditions A potential alternative Lorentzian prescription is Picard-Lefschetz PL typically allows for successful decoupling The solutions in PL for which decoupling succeeds are generically the solutions forbidden by the Euclidean prescription Jonathan Brown (UW Madison) Gravitational Decoupling and PL GLS 4/14/18 13 / 13
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