Resurgence Structure to All Orders of Multi-bions in Deformed SUSY   Quantum Mechanics

Toshiaki Fujimori; Syo Kamata; Tatsuhiro Misumi; Muneto Nitta,; Norisuke Sakai

arXiv:1705.10483·hep-th·August 23, 2017

Resurgence Structure to All Orders of Multi-bions in Deformed SUSY Quantum Mechanics

Toshiaki Fujimori, Syo Kamata, Tatsuhiro Misumi, Muneto Nitta,, Norisuke Sakai

PDF

TL;DR

This paper explores the resurgence phenomena in deformed supersymmetric quantum mechanics models, demonstrating how semiclassical multi-bion solutions reproduce exact energy expansions and reveal intricate cancellation of ambiguities.

Contribution

It provides a detailed analysis of resurgence structures in nearly supersymmetric quantum mechanics, including exact coefficient calculations and the role of complex multi-bion solutions.

Findings

01

Semiclassical multi-bion solutions reproduce exact energy expansions.

02

Imaginary ambiguities cancel between perturbative and non-perturbative contributions.

03

Resurgence structure varies at different orders of deformation parameter.

Abstract

We investigate the resurgence structure in quantum mechanical models originating in 2d non-linear sigma models with emphasis on nearly supersymmetric and quasi-exactly solvable parameter regimes. By expanding the ground state energy in powers of a supersymmetry-breaking deformation parameter $δ ϵ$ , we derive exact results for the expansion coefficients. In the class of models described by real multiplets, the $O (δ ϵ)$ ground state energy has a non-Borel summable asymptotic series, which gives rise to imaginary ambiguities leading to rich resurgence structure. We discuss the sine-Gordon quantum mechanics (QM) as an example and show that the semiclassical contributions from complex multi-bion solutions correctly reproduce the corresponding part in the exact result including the imaginary ambiguities. As a typical model described by chiral multiplets, we…

Equations520

H Ψ = [- g^{2} Δ + V] Ψ, ΔΨ \equiv \frac{1}{det G} \partial_{i} (G^{ij} det G) \partial_{j} Ψ,

H Ψ = [- g^{2} Δ + V] Ψ, ΔΨ \equiv \frac{1}{det G} \partial_{i} (G^{ij} det G) \partial_{j} Ψ,

V = \frac{1}{g ^{2}} G^{ij} \partial_{i} W \partial_{j} W - ϵ Δ W,

V = \frac{1}{g ^{2}} G^{ij} \partial_{i} W \partial_{j} W - ϵ Δ W,

Ψ^{(0)} \equiv ⟨ φ ∣0 ⟩ = exp (- \frac{W}{g ^{2}}) .

Ψ^{(0)} \equiv ⟨ φ ∣0 ⟩ = exp (- \frac{W}{g ^{2}}) .

⟨ 0∣0 ⟩ = \int_{M} d v exp (- \frac{2 W}{g ^{2}}) < \infty,

⟨ 0∣0 ⟩ = \int_{M} d v exp (- \frac{2 W}{g ^{2}}) < \infty,

\displaystyle E^{(n)}=\frac{1}{n!}\frac{\partial^{n}}{\partial\epsilon^{n}}E\bigg{|}_{\epsilon=1},\hskip 28.45274pt\Psi^{(n)}=\frac{1}{n!}\frac{\partial^{n}}{\partial\epsilon^{n}}\Psi\bigg{|}_{\epsilon=1}.

\displaystyle E^{(n)}=\frac{1}{n!}\frac{\partial^{n}}{\partial\epsilon^{n}}E\bigg{|}_{\epsilon=1},\hskip 28.45274pt\Psi^{(n)}=\frac{1}{n!}\frac{\partial^{n}}{\partial\epsilon^{n}}\Psi\bigg{|}_{\epsilon=1}.

E^{(1)} = \frac{⟨ 0∣ δ H ∣0 ⟩}{⟨ 0∣0 ⟩}, E^{(2)} = - \frac{⟨ Ψ ^{(1)} ∣ H _{ϵ = 1} ∣ Ψ ^{(1)} ⟩}{⟨ 0∣0 ⟩}, \dots,

E^{(1)} = \frac{⟨ 0∣ δ H ∣0 ⟩}{⟨ 0∣0 ⟩}, E^{(2)} = - \frac{⟨ Ψ ^{(1)} ∣ H _{ϵ = 1} ∣ Ψ ^{(1)} ⟩}{⟨ 0∣0 ⟩}, \dots,

⟨ 0∣0 ⟩ = \int_{M} d v exp (- \frac{2 W}{g ^{2}}) = s \in S \sum F_{s} (g^{2}) exp (- \frac{2 W _{s}}{g ^{2}}),

⟨ 0∣0 ⟩ = \int_{M} d v exp (- \frac{2 W}{g ^{2}}) = s \in S \sum F_{s} (g^{2}) exp (- \frac{2 W _{s}}{g ^{2}}),

S = \int d t (\frac{1}{4 g ^{2}} G_{ij} \overset{φ}{˙}^{i} \overset{φ}{˙}^{j} - \frac{1}{g ^{2}} G^{ij} \partial_{i} W \partial_{j} W + Δ W) .

S = \int d t (\frac{1}{4 g ^{2}} G_{ij} \overset{φ}{˙}^{i} \overset{φ}{˙}^{j} - \frac{1}{g ^{2}} G^{ij} \partial_{i} W \partial_{j} W + Δ W) .

⟨ 0∣0 ⟩ = \int_{M} d v exp (- \frac{2 W}{g ^{2}}) .

⟨ 0∣0 ⟩ = \int_{M} d v exp (- \frac{2 W}{g ^{2}}) .

G_{i \overset{ˉ}{j}} \frac{d φ ^{i}}{d t} = \overline{\frac{1}{g ^{2}} \frac{\partial W}{\partial φ ^{j}}}, t \to - \infty lim φ^{i} = φ_{s}^{i},

G_{i \overset{ˉ}{j}} \frac{d φ ^{i}}{d t} = \overline{\frac{1}{g ^{2}} \frac{\partial W}{\partial φ ^{j}}}, t \to - \infty lim φ^{i} = φ_{s}^{i},

G_{i \overset{ˉ}{j}} \frac{d φ ^{i}}{d t} = \overline{\frac{1}{g ^{2}} \frac{\partial W}{\partial φ ^{j}}}, t \to \infty lim φ^{i} = φ_{s}^{i} .

G_{i \overset{ˉ}{j}} \frac{d φ ^{i}}{d t} = \overline{\frac{1}{g ^{2}} \frac{\partial W}{\partial φ ^{j}}}, t \to \infty lim φ^{i} = φ_{s}^{i} .

Re W ∣_{J_{s}} \geq Re W_{s} \geq Re W ∣_{K_{s}}, Im W ∣_{J_{s}} = Im W_{s} = Im W ∣_{K_{s}} .

Re W ∣_{J_{s}} \geq Re W_{s} \geq Re W ∣_{K_{s}}, Im W ∣_{J_{s}} = Im W_{s} = Im W ∣_{K_{s}} .

⟨ J_{s}, K_{s^{'}} ⟩ = δ_{s s^{'}} .

⟨ J_{s}, K_{s^{'}} ⟩ = δ_{s s^{'}} .

M = s \in S \sum n_{s} J_{s}, n_{s} = ⟨ M, K_{s} ⟩,

M = s \in S \sum n_{s} J_{s}, n_{s} = ⟨ M, K_{s} ⟩,

⟨ 0∣0 ⟩ = s \in S \sum F_{s} (g^{2}) exp (- \frac{2 W _{s}}{g ^{2}}),

⟨ 0∣0 ⟩ = s \in S \sum F_{s} (g^{2}) exp (- \frac{2 W _{s}}{g ^{2}}),

F_{s} (g^{2}) = n_{s} \int_{J_{s}} d v exp [- \frac{2 ( W - W _{s} )}{g ^{2}}] .

F_{s} (g^{2}) = n_{s} \int_{J_{s}} d v exp [- \frac{2 ( W - W _{s} )}{g ^{2}}] .

G_{ij} \frac{\partial φ ^{j}}{\partial τ} = \frac{\partial W}{\partial φ ^{i}},

G_{ij} \frac{\partial φ ^{j}}{\partial τ} = \frac{\partial W}{\partial φ ^{i}},

J_{s}^{+} \neq = J_{s}^{-}, K_{s}^{+} \neq = K_{s}^{-},

J_{s}^{+} \neq = J_{s}^{-}, K_{s}^{+} \neq = K_{s}^{-},

⟨ 0∣0 ⟩ = s \in S \sum F_{s}^{\pm} (g^{2}) exp (- \frac{2 W _{s}}{g ^{2}}),

⟨ 0∣0 ⟩ = s \in S \sum F_{s}^{\pm} (g^{2}) exp (- \frac{2 W _{s}}{g ^{2}}),

F_{s}^{\pm} (g^{2}) = n_{s}^{\pm} \int_{J_{s}^{\pm}} d v exp [- \frac{2 ( W - W _{s} )}{g ^{2}}], n_{s}^{\pm} \equiv ⟨ M, K_{s}^{\pm} ⟩ .

F_{s}^{\pm} (g^{2}) = n_{s}^{\pm} \int_{J_{s}^{\pm}} d v exp [- \frac{2 ( W - W _{s} )}{g ^{2}}], n_{s}^{\pm} \equiv ⟨ M, K_{s}^{\pm} ⟩ .

⟨ 0∣0 ⟩ = e^{- \frac{2 W _{0}}{g ^{2}}} s \in S \sum F_{s}^{\pm} (g^{2}) exp (- 2 S_{kink, s}),

⟨ 0∣0 ⟩ = e^{- \frac{2 W _{0}}{g ^{2}}} s \in S \sum F_{s}^{\pm} (g^{2}) exp (- 2 S_{kink, s}),

G_{i \overset{ˉ}{j}} = \partial_{i} \overset{ˉ}{\partial}_{\overset{ˉ}{j}} K,

G_{i \overset{ˉ}{j}} = \partial_{i} \overset{ˉ}{\partial}_{\overset{ˉ}{j}} K,

\partial_{i} ξ_{\overset{ˉ}{j}} + \overset{ˉ}{\partial}_{\overset{ˉ}{j}} \overset{ˉ}{ξ}_{i} = \overset{ˉ}{\partial}_{\overset{ˉ}{j}} ξ^{i} = \partial_{i} \overset{ˉ}{ξ}^{\overset{ˉ}{j}} = 0.

\partial_{i} ξ_{\overset{ˉ}{j}} + \overset{ˉ}{\partial}_{\overset{ˉ}{j}} \overset{ˉ}{ξ}_{i} = \overset{ˉ}{\partial}_{\overset{ˉ}{j}} ξ^{i} = \partial_{i} \overset{ˉ}{ξ}^{\overset{ˉ}{j}} = 0.

\partial_{i} μ = - i G_{i \overset{ˉ}{j}} \overset{ˉ}{ξ}^{\overset{ˉ}{j}}, \overset{ˉ}{\partial}_{\overset{ˉ}{i}} μ = i G_{j \overset{ˉ}{i}} ξ^{j} .

\partial_{i} μ = - i G_{i \overset{ˉ}{j}} \overset{ˉ}{ξ}^{\overset{ˉ}{j}}, \overset{ˉ}{\partial}_{\overset{ˉ}{i}} μ = i G_{j \overset{ˉ}{i}} ξ^{j} .

\displaystyle S_{2d}=\frac{1}{g_{2d}^{2}}\int d^{2}x\,G_{i\bar{j}}\Big{[}-\partial_{\mu}\varphi^{i}\partial^{\mu}\bar{\varphi}^{\bar{j}}+i\bar{\psi}^{\bar{j}}(\mathcal{D}_{t}-\mathcal{D}_{x})\psi^{i}\Big{]}.

\displaystyle S_{2d}=\frac{1}{g_{2d}^{2}}\int d^{2}x\,G_{i\bar{j}}\Big{[}-\partial_{\mu}\varphi^{i}\partial^{\mu}\bar{\varphi}^{\bar{j}}+i\bar{\psi}^{\bar{j}}(\mathcal{D}_{t}-\mathcal{D}_{x})\psi^{i}\Big{]}.

S = \frac{1}{g ^{2}} \int d t G_{i \overset{ˉ}{j}} [\overset{φ}{˙}^{i} \dot{\overset{φ}{ˉ}}^{\overset{ˉ}{j}} - ξ^{i} \overset{ˉ}{ξ}^{\overset{ˉ}{j}} + i \overset{ˉ}{ψ}^{\overset{ˉ}{j}} D_{t} ψ^{i} - i \nabla_{k} ξ^{i} \overset{ˉ}{ψ}^{\overset{ˉ}{j}} ψ^{k}],

S = \frac{1}{g ^{2}} \int d t G_{i \overset{ˉ}{j}} [\overset{φ}{˙}^{i} \dot{\overset{φ}{ˉ}}^{\overset{ˉ}{j}} - ξ^{i} \overset{ˉ}{ξ}^{\overset{ˉ}{j}} + i \overset{ˉ}{ψ}^{\overset{ˉ}{j}} D_{t} ψ^{i} - i \nabla_{k} ξ^{i} \overset{ˉ}{ψ}^{\overset{ˉ}{j}} ψ^{k}],

\frac{1}{g ^{2}} = \frac{1}{g _{2 d}^{2}} \times 2 π {\mbox co m p a c t i f i c a t i o n r a d i u s} .

\frac{1}{g ^{2}} = \frac{1}{g _{2 d}^{2}} \times 2 π {\mbox co m p a c t i f i c a t i o n r a d i u s} .

S_{E, inst} = \frac{1}{g _{2 d}^{2}} \int i G_{i \overset{ˉ}{j}} d φ^{i} \land d \overset{φ}{ˉ}^{\overset{ˉ}{j}} \to S_{E, kink} = \frac{1}{g ^{2}} \int (\partial_{i} μ d φ^{i} + \overset{ˉ}{\partial}_{\overset{ˉ}{i}} μ d \overset{φ}{ˉ}^{\overset{ˉ}{i}}) .

S_{E, inst} = \frac{1}{g _{2 d}^{2}} \int i G_{i \overset{ˉ}{j}} d φ^{i} \land d \overset{φ}{ˉ}^{\overset{ˉ}{j}} \to S_{E, kink} = \frac{1}{g ^{2}} \int (\partial_{i} μ d φ^{i} + \overset{ˉ}{\partial}_{\overset{ˉ}{i}} μ d \overset{φ}{ˉ}^{\overset{ˉ}{i}}) .

W = μ .

W = μ .

H Ψ = G^{\overset{ˉ}{j} i} [- g^{2} \partial_{i} \overset{ˉ}{\partial}_{\overset{ˉ}{j}} + \frac{1}{g ^{2}} \partial_{i} μ \overset{ˉ}{\partial}_{\overset{ˉ}{j}} μ - ϵ \partial_{i} \overset{ˉ}{\partial}_{\overset{ˉ}{j}} μ] Ψ.

H Ψ = G^{\overset{ˉ}{j} i} [- g^{2} \partial_{i} \overset{ˉ}{\partial}_{\overset{ˉ}{j}} + \frac{1}{g ^{2}} \partial_{i} μ \overset{ˉ}{\partial}_{\overset{ˉ}{j}} μ - ϵ \partial_{i} \overset{ˉ}{\partial}_{\overset{ˉ}{j}} μ] Ψ.

Peer Reviews

No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.

Videos

No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.

Full text

Resurgence Structure to All Orders of Multi-bions

in Deformed SUSY Quantum Mechanics

Toshiaki Fujimori

toshiaki.fujimori018(at)gmail.com