Non-Perturbative Corrections from Complex Saddle Points in CP N Models Toshiaki Fujimori (Keio University) based on arXiv:1607.04205 Syo Kamata (Keio U.) Tatsuhiro Misumi (Akita U.) Muneto Nitta (Kieo U.) Norisuke Sakai (Keio U.) Strings and Fields @ YITP, Aug. 8th 2016
Saddle Point Method · path integral saddle point = solution of e.o.m.
Saddle Point Method · path integral saddle point = solution of e.o.m. · trivial saddle (vacuum) perturbative series
Saddle Point Method · path integral saddle point = solution of e.o.m. · trivial saddle (vacuum) perturbative series · non-trivial saddle (e.g. instanton) non-perturbative
Complex Saddle Point saddle points : not only on original integration path · e.g. Airy function 4 2 0 - 2 - 4 - 4 - 2 0 2 4
Complex Saddle Point · complex saddle points in physical system [Behtash, Dunne, Schäfer, Sulejmanpasic, Ünsal, 2015] complex bion instanton-anti instanton pair with ``complex separation” · complex saddles ··· imaginary contribution · cancelation of imaginary ambiguity resurgence from perturbative asymptotic series full partition function : real and no ambiguity pert. non-pert.
CP N Sigma model · instanton in 2d CP N Sigma model on R × S 1 with twisted b.c.
CP N Sigma model · instanton in 2d CP N Sigma model on R × S 1 with twisted b.c. bion fractional inst-inst · N fractional instantons non-perturbative in 2d
CP N Sigma model · instanton in 2d CP N Sigma model on R × S 1 with twisted b.c. bion fractional inst-inst · N fractional instantons non-perturbative in 2d reduction to CP N QM bion fractional kink-kink · kinks in CP N QM non-perturbative in 1d
CP 1 Quantum Mechas · Quantum mechanics of particle on sphere φ -plane complex φ -plane V potential S θ fermion 3 π π π twisted b.c. π 4 2 4 N
SUSY ground state energy · SUSY case N =(2,0) model in 2d : ground state energy is exactly zero
near SUSY ground state energy · SUSY case N =(2,0) model in 2d : ground state energy is exactly zero · near SUSY case
Saddle point equation · Euclidean action Euclidean e.o.m conservation law · symmetry : time and phase shift
Solution of E.O.M. solution : phase ··· moduli parameters : position
Solution of E.O.M. solution : phase ··· moduli parameters : position · kink-antikink pair antikink kink relative distance (stabilized)
real bion solution : height “real” bion : saddle point on original integration contour
contribution of real bion non-perturbative · this does not vanish in the supersymmetric case There should be other saddle points which cancel the real bion contribution
Complexification complex · real and imaginary parts of complexification of CP 1 · Analytic continuation of integrand holomorphic
Complex bion solution solution · kink-antikink pair antikink kink : “complex relative distance”
Kink profile of bion height Σ complexification Im ( Σ ) Re ( Σ ) τ real bion
Kink profile of bion height Σ complexification Im ( Σ ) Re ( Σ ) τ complex bion
Kink profile of bion height Σ complexification Im ( Σ ) Re ( Σ ) τ regularized complex bion
Contribution of complex bion π π 2 2 0 0 contribution of complex bion has imaginary ambiguity depending on arg g
Leading non-perturbative correction one loop determinant · Gaussian integration · correction to ground state energy
Leading non-perturbative correction one loop determinant · Gaussian integration · correction to ground state energy � 2 ϵ = i (1 − e ± 2 πϵ i )16 ω 4 � ω + m � � − 2 ω exp g 2 λ g 2 ω − m · asymptotic form in the limit g → 0 · cancelation of ambiguities?
SUSY case � 2 ϵ = i (1 − e ± 2 πϵ i )16 ω 4 � ω + m � − 2 ω � exp g 2 λ g 2 ω − m · supersymmetric case · cancelation of real and complex bion contributions · consistent with the exact result
near SUSY case � 2 ϵ = i (1 − e ± 2 πϵ i )16 ω 4 � ω + m � − 2 ω � exp g 2 λ g 2 ω − m · near supersymmetric case incompatible with exact result
near SUSY case � 2 ϵ = i (1 − e ± 2 πϵ i )16 ω 4 � ω + m � − 2 ω � exp g 2 λ g 2 ω − m · near supersymmetric case incompatible with exact result · with fixed ··· nearly flat directions appear Gaussian approximation is not valid
Quasi-Moduli Integral · nearly flat directions : quasi-moduli parameters complexified relative kink position and phase complexified quasi-moduli integral effective action on complexified quasi-moduli space (for well-separated kinks)
Lefschetz Thimble Method · decomposition of integration contour : set of saddle points thimble : upward flow : downward flow dual thimble flow equation intersection number intersection pairing
Quasi-Moduli Integral · application of Lefschetz thimble method saddle points : real bion : complex bion
Quasi-Moduli Integral · application of Lefschetz thimble method saddle points : real bion : complex bion solution of flow eq.
Thimble and Dual Thimble · Thimbles are surfaces in 4d space thimble J σ dual thimble K σ 3d projection
Quasi Moduli Integral · integral along J σ · intersection number of original contour and K σ Stokes phenomenon ambiguity
Non-Perturbative Correction correction to ground state energy consistent with exact result in (near) SUSY case
Summary · non-perturbative correction from bion solutions in CP 1 quantum mechanics · complex saddle points · Lefschetz thimble method ··· consistent result Future work · field theory · cancelation of ambiguity? resurgence?
Eliminating fermion · fermionic part of Lagrangian · fermion number : conserved charge · partition function of f=0 sector induced potential
Lefschetz thimble 1.0 · trivial example Gaussian integral 0.5 0.0 · saddle point - 0.5 - 1.0 - 1.0 - 0.5 0.0 0.5 1.0
Example · Airy function 4 2 0 - 2 - 4 - 4 - 2 0 2 4
Stokes phenomena 4 4 2 2 0 0 - 2 - 2 - 4 - 4 - 4 - 2 0 2 4 - 4 - 2 0 2 4 jumps but does not
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