Non-Perturbative Corrections from Complex Saddle Points in CP N - - PowerPoint PPT Presentation

Syo Kamata (Keio U.) Tatsuhiro Misumi (Akita U.) Muneto Nitta (Kieo U.) Norisuke Sakai (Keio U.)

Strings and Fields @ YITP, Aug. 8th 2016 Toshiaki Fujimori (Keio University)

Non-Perturbative Corrections from Complex Saddle Points in CPN Models

based on arXiv:1607.04205

Saddle Point Method

saddle point = solution of e.o.m. · path integral

· trivial saddle (vacuum) perturbative series

Saddle Point Method

saddle point = solution of e.o.m. · path integral

· trivial saddle (vacuum) perturbative series

Saddle Point Method

· non-trivial saddle (e.g. instanton) non-perturbative saddle point = solution of e.o.m. · path integral

• 4
• 2

2 4

• 4
• 2

2 4

· e.g. Airy function saddle points : not only on original integration path

Complex Saddle Point

· complex saddle points in physical system

[Behtash, Dunne, Schäfer, Sulejmanpasic, Ünsal, 2015]

Complex Saddle Point

instanton-anti instanton pair with ``complex separation”

complex bion

· cancelation of imaginary ambiguity from perturbative asymptotic series

resurgence

· complex saddles ··· imaginary contribution

pert. non-pert.

full partition function : real and no ambiguity

· instanton in 2d CPN Sigma model

CPN Sigma model

• n R×S1 with twisted b.c.

· N fractional instantons · instanton in 2d CPN Sigma model

CPN Sigma model

non-perturbative in 2d fractional inst-inst

bion

• n R×S1 with twisted b.c.

· N fractional instantons · instanton in 2d CPN Sigma model

CPN Sigma model

reduction to CPN QM · kinks in CPN QM non-perturbative in 2d non-perturbative in 1d fractional inst-inst fractional kink-kink

bion bion

• n R×S1 with twisted b.c.

complex φ-plane

π 4 π 2 3π 4

π θ V

· Quantum mechanics of particle on sphere

twisted b.c. fermion

CP1 Quantum Mechas

N S

potential

φ-plane

· SUSY case

: ground state energy is exactly zero

SUSY ground state energy

N =(2,0) model in 2d

· near SUSY case

near SUSY ground state energy

· SUSY case

: ground state energy is exactly zero

N =(2,0) model in 2d

· Euclidean action

Euclidean e.o.m

Saddle point equation

· symmetry : time and phase shift

conservation law

Solution of E.O.M.

solution

··· moduli parameters

: position

: phase

· kink-antikink pair

kink

antikink

Solution of E.O.M.

solution

relative distance (stabilized) ··· moduli parameters

: position

: phase

real bion solution

: height

“real” bion : saddle point on original integration contour

· this does not vanish in the supersymmetric case There should be other saddle points which cancel the real bion contribution

contribution of real bion

non-perturbative

· Analytic continuation of integrand

Complexification

holomorphic

complexification of CP1

· real and imaginary parts of complex

: “complex relative distance”

Complex bion solution

solution

kink

antikink · kink-antikink pair

τ Im(Σ) Re(Σ)

real bion

Kink profile of bion

height Σ

complexification

complex bion

τ Im(Σ) Re(Σ)

Kink profile of bion

complexification

height Σ

regularized complex bion

τ Im(Σ) Re(Σ)

Kink profile of bion

complexification

height Σ

π 2 π 2

Contribution of complex bion

contribution of complex bion has imaginary ambiguity depending on arg g

· correction to ground state energy

Leading non-perturbative correction

· Gaussian integration

• ne loop determinant

= i(1 − e±2πϵi)16ω4 g2λ ω + m ω − m 2ϵ exp

• −2ω

g2

• · correction to ground state energy

Leading non-perturbative correction

· Gaussian integration

• ne loop determinant

· asymptotic form in the limit

g → 0

· cancelation of ambiguities?

· supersymmetric case

= i(1 − e±2πϵi)16ω4 g2λ ω + m ω − m 2ϵ exp

• −2ω

g2

• SUSY case

· cancelation of real and complex bion contributions · consistent with the exact result

· near supersymmetric case

= i(1 − e±2πϵi)16ω4 g2λ ω + m ω − m 2ϵ exp

• −2ω

g2

• near SUSY case

incompatible with exact result

··· nearly flat directions appear

Gaussian approximation is not valid

· near supersymmetric case

= i(1 − e±2πϵi)16ω4 g2λ ω + m ω − m 2ϵ exp

• −2ω

g2

• near SUSY case

· with fixed

incompatible with exact result

· nearly flat directions : quasi-moduli parameters

effective action on complexified quasi-moduli space

relative kink position and phase

complexified complexified quasi-moduli integral

Quasi-Moduli Integral

(for well-separated kinks)

Lefschetz Thimble Method

· decomposition of integration contour : set of saddle points intersection number : upward flow : downward flow thimble dual thimble intersection pairing flow equation

: real bion saddle points

Quasi-Moduli Integral

· application of Lefschetz thimble method : complex bion

: real bion saddle points

Quasi-Moduli Integral

· application of Lefschetz thimble method solution of flow eq. : complex bion

Thimble and Dual Thimble

thimble Jσ · Thimbles are surfaces in 4d space dual thimble Kσ 3d projection

Quasi Moduli Integral

Stokes phenomenon · integral along Jσ · intersection number of original contour and Kσ ambiguity

consistent with exact result in (near) SUSY case

Non-Perturbative Correction

correction to ground state energy

· complex saddle points · non-perturbative correction from bion solutions in CP1 quantum mechanics · Lefschetz thimble method ··· consistent result · cancelation of ambiguity? resurgence? · field theory

Summary

Future work

Eliminating fermion

· fermionic part of Lagrangian

induced potential

· partition function of f=0 sector · fermion number : conserved charge

Lefschetz thimble

• 1.0
• 0.5

0.0 0.5 1.0

• 1.0
• 0.5

0.0 0.5 1.0

· trivial example Gaussian integral · saddle point

Example

• 4
• 2

2 4

• 4
• 2

2 4

· Airy function

Stokes phenomena

• 4
• 2

2 4

• 4
• 2

2 4

• 4
• 2

2 4

• 4
• 2

2 4

jumps but does not