text: epl draft Nonlinear scattering of atomic bright solitons in disorder A. Boissé, G. Berthet, L. Fouché, G. Salomon, A. Aspect, S. Lepoutre and T. Bourdel1 1 Laboratoire Charles Fabry, Institut d’Optique, CNRS, Univ Paris-Sud - 2, Avenue Augustin Fresnel, 91127 PALAISEAU CEDEX, France PACS 03.75.Lm – Tunneling, Josephson eﬀect, Bose-Einstein condensates in periodic potentials, solitons, vortices, and topological excitations PACS 05.45.Yv – Solitons PACS 67.85.-d – Ultracold gases, trapped gases Abstract –We observe nonlinear scattering of 39K atomic bright solitons launched in a one- dimensional (1D) speckle disorder. We directly compare it with the scattering of non-interacting particles in the same disorder. The atoms in the soliton tend to be collectively either reﬂected or transmitted, in contrast with the behavior of independent particles in the single scattering regime, thus demonstrating a clear nonlinear eﬀect in scattering. The observed strong ﬂuctuations in the reﬂected fraction, between zero and 100%, are interpreted as a consequence of the strong sensitivity of the system to the experimental conditions and in particular to the soliton velocity. This behavior is reproduced in a mean-ﬁeld framework by Gross-Pitaevskii simulations, and mesoscopic quantum superpositions of the soliton being fully reﬂected and fully transmitted are not expected for our parameters. We discuss the conditions for observing such superpositions, which would ﬁnd applications in atom interferometry beyond the standard quantum limit. The physics of transport of particles in disorder is asso- quantum particles: nonlinear scattering of atomic bright ciated with diﬀerent scenarios. In absence of interaction, solitons in an optical disorder. the simplest description is based on diﬀusion [1], but the A soliton is a stable non-spreading wave-packet, solu- coherence of the matter waves describing the particles can tion of a nonlinear wave equation, where a strong nonlin- play a role, as in the phenomena of coherent backscat- earity compensates dispersion. Solitons are ubiquitous in tering [2–4] and Anderson localization [5–14]. However, nonlinear wave physics [34,35]. Their propagation in a dis- in many physical systems, interactions cannot be ignored. ordered medium is intriguing since the eﬀect of the nonlin- In condensed matter physics, interactions between elec- earity cannot be treated as a small perturbation of the non trons can strongly aﬀect electric conductivity [15] and in interacting problem [36]. An atomic bright soliton is a 1D optics, high intensity light induces a nonlinear response of Bose-Einstein condensate of atoms with attractive inter- dielectrics, leading for instance to the optical Kerr eﬀect, actions [37,38]. At the mean ﬁeld level, it is described by and thus spatial and/or temporal ﬂuctuations of the index the Gross-Pitaevskii equation, which is identical to the so of refraction. Understanding the interplay between disor- called "nonlinear Schrödinger equation" used to describe der and interactions in the transport of quantum particles the 1D propagation of light in Kerr media. This approach is thus an important challenge. has been used to numerically study soliton non-linear scat- In a mean-ﬁeld approach, one can use nonlinear wave tering on a narrow barrier [39–44]. Qualitatively similar equations in disordered media [16,17] in order to describe results for a 1D disordered potential in the single scatter- experimental observations of the competition between a ing regime can be expected. Experimentally, atomic bright weak nonlinearity and localization, in optics [18, 19] or soliton scattering has only been studied in the regime of in ultra-cold quantum gases [20]. Beyond the mean-ﬁeld negligible interaction energy, where the behavior resem- approximation, many-body localization phenomena, lead- bles the one of non-interacting particles [45–47]. ing to non-ergodic behavior, are predicted [21,22]. In this In this paper, we report the observation of nonlinear context, several problems of transport of interacting quan- scattering of an atomic bright soliton in the regime where tum gases in disorder have been studied [23–33]. We re- the interaction energy is of the order of the center of mass port here on a new phenomenon of nonlinear transport of kinetic energy [48]. As the experiment is repeated, we ﬁnd p-1 ##LTLine## ##LTLine##

text: epl

text: draft

text: Nonlinear

text: scattering

text: of

text: atomic

text: bright

text: solitons

text: in

text: disorder

text: A.

text: Boissé,

text: G.

text: Berthet,

text: L.

text: Fouché,

text: G.

text: Salomon,

text: A.

text: Aspect,

text: S.

text: Lepoutre

text: and

text: T.

text: Laboratoire

text: Charles

text: Fabry,

text: Institut

text: d’Optique,

text: CNRS,

text: Univ

text: Paris-Sud

text: Avenue

text: Augustin

text: Fresnel,

text: PALAISEAU

text: CEDEX,

text: France

text: PACS

text: Tunneling,

text: Josephson

text: eﬀect,

text: Bose-Einstein

text: condensates

text: in

text: periodic

text: potentials,

text: solitons,

text: vortices,

text: and

text: topological

text: excitations

text: PACS

text: Solitons

text: PACS

text: Ultracold

text: gases,

text: trapped

text: gases

text: Abstract

text: –We

text: observe

text: nonlinear

text: scattering

text: of

text: atomic

text: bright

text: solitons

text: launched

text: in

text: a

text: one-

text: dimensional

text: speckle

text: disorder.

text: We

text: directly

text: compare

text: it

text: with

text: the

text: scattering

text: of

text: non-interacting

text: particles

text: in

text: the

text: same

text: disorder.

text: The

text: atoms

text: in

text: the

text: soliton

text: tend

text: to

text: be

text: collectively

text: either

text: reﬂected

text: or

text: transmitted,

text: in

text: contrast

text: with

text: the

text: behavior

text: of

text: independent

text: particles

text: in

text: the

text: single

text: scattering

text: regime,

text: thus

text: demonstrating

text: a

text: clear

text: nonlinear

text: eﬀect

text: in

text: scattering.

text: The

text: observed

text: strong

text: ﬂuctuations

text: in

text: the

text: reﬂected

text: fraction,

text: between

text: zero

text: and

text: are

text: interpreted

text: as

text: a

text: consequence

text: of

text: the

text: strong

text: sensitivity

text: of

text: the

text: system

text: to

text: the

text: experimental

text: conditions

text: and

text: in

text: particular

text: to

text: the

text: soliton

text: velocity.

text: This

text: behavior

text: is

text: reproduced

text: in

text: a

text: mean-ﬁeld

text: framework

text: by

text: Gross-Pitaevskii

text: simulations,

text: and

text: mesoscopic

text: quantum

text: superpositions

text: of

text: the

text: soliton

text: being

text: fully

text: reﬂected

text: and

text: fully

text: transmitted

text: are

text: not

text: expected

text: for

text: our

text: parameters.

text: We

text: discuss

text: the

text: conditions

text: for

text: observing

text: such

text: superpositions,

text: which

text: would

text: ﬁnd

text: applications

text: in

text: atom

text: interferometry

text: beyond

text: the

text: standard

text: quantum

text: limit.

text: The

text: physics

text: of

text: transport

text: of

text: particles

text: in

text: disorder

text: is

text: asso-

text: quantum

text: particles:

text: nonlinear

text: scattering

text: of

text: atomic

text: bright

text: ciated

text: with

text: diﬀerent

text: scenarios.

text: In

text: absence

text: of

text: interaction,

text: solitons

text: in

text: an

text: optical

text: disorder.

text: the

text: simplest

text: description

text: is

text: based

text: on

text: diﬀusion

text: but

text: the

text: A

text: soliton

text: is

text: a

text: stable

text: non-spreading

text: wave-packet,

text: solu-

text: coherence

text: of

text: the

text: matter

text: waves

text: describing

text: the

text: particles

text: can

text: tion

text: of

text: a

text: nonlinear

text: wave

text: equation,

text: where

text: a

text: strong

text: nonlin-

text: play

text: a

text: role,

text: as

text: in

text: the

text: phenomena

text: of

text: coherent

text: backscat-

text: earity

text: compensates

text: dispersion.

text: Solitons

text: are

text: ubiquitous

text: in

text: tering

text: and

text: Anderson

text: localization

text: However,

text: nonlinear

text: wave

text: physics

text: Their

text: propagation

text: in

text: a

text: dis-

text: in

text: many

text: physical

text: systems,

text: interactions

text: cannot

text: be

text: ignored.

text: ordered

text: medium

text: is

text: intriguing

text: since

text: the

text: eﬀect

text: of

text: the

text: nonlin-

text: In

text: condensed

text: matter

text: physics,

text: interactions

text: between

text: elec-

text: earity

text: cannot

text: be

text: treated

text: as

text: a

text: small

text: perturbation

text: of

text: the

text: non

text: trons

text: can

text: strongly

text: aﬀect

text: electric

text: conductivity

text: and

text: in

text: interacting

text: problem

text: An

text: atomic

text: bright

text: soliton

text: is

text: a

text: optics,

text: high

text: intensity

text: light

text: induces

text: a

text: nonlinear

text: response

text: of

text: Bose-Einstein

text: condensate

text: of

text: atoms

text: with

text: attractive

text: inter-

text: dielectrics,

text: leading

text: for

text: instance

text: to

text: the

text: optical

text: Kerr

text: eﬀect,

text: actions

text: At

text: the

text: mean

text: ﬁeld

text: level,

text: it

text: is

text: described

text: by

text: and

text: thus

text: spatial

text: and/or

text: temporal

text: ﬂuctuations

text: of

text: the

text: index

text: the

text: Gross-Pitaevskii

text: equation,

text: which

text: is

text: identical

text: to

text: the

text: so

text: of

text: refraction.

text: Understanding

text: the

text: interplay

text: between

text: disor-

text: called

text: "nonlinear

text: Schrödinger

text: equation"

text: used

text: to

text: describe

text: der

text: and

text: interactions

text: in

text: the

text: transport

text: of

text: quantum

text: particles

text: the

text: propagation

text: of

text: light

text: in

text: Kerr

text: media.

text: This

text: approach

text: is

text: thus

text: an

text: important

text: challenge.

text: has

text: been

text: used

text: to

text: numerically

text: study

text: soliton

text: non-linear

text: scat-

text: In

text: a

text: mean-ﬁeld

text: approach,

text: one

text: can

text: use

text: nonlinear

text: wave

text: tering

text: on

text: a

text: narrow

text: barrier

text: Qualitatively

text: similar

text: equations

text: in

text: disordered

text: media

text: in

text: order

text: to

text: describe

text: results

text: for

text: a

text: disordered

text: potential

text: in

text: the

text: single

text: scatter-

text: experimental

text: observations

text: of

text: the

text: competition

text: between

text: a

text: ing

text: regime

text: can

text: be

text: expected.

text: Experimentally,

text: atomic

text: bright

text: weak

text: nonlinearity

text: and

text: localization,

text: in

text: optics

text: or

text: soliton

text: scattering

text: has

text: only

text: been

text: studied

text: in

text: the

text: regime

text: of

text: in

text: ultra-cold

text: quantum

text: gases

text: Beyond

text: the

text: mean-ﬁeld

text: negligible

text: interaction

text: energy,

text: where

text: the

text: behavior

text: resem-

text: approximation,

text: many-body

text: localization

text: phenomena,

text: lead-

text: bles

text: the

text: one

text: of

text: non-interacting

text: particles

text: ing

text: to

text: non-ergodic

text: behavior,

text: are

text: predicted

text: In

text: this

text: In

text: this

text: paper,

text: we

text: report

text: the

text: observation

text: of

text: nonlinear

text: context,

text: several

text: problems

text: of

text: transport

text: of

text: interacting

text: quan-

text: scattering

text: of

text: an

text: atomic

text: bright

text: soliton

text: in

text: the

text: regime

text: where

text: tum

text: gases

text: in

text: disorder

text: have

text: been

text: studied

text: We

text: re-

text: the

text: interaction

text: energy

text: is

text: of

text: the

text: order

text: of

text: the

text: center

text: of

text: mass

text: port

text: here

text: on

text: a

text: new

text: phenomenon

text: of

text: nonlinear

text: transport

text: of

text: kinetic

text: energy

text: As

text: the

text: experiment

text: is

text: repeated,

text: we

text: ﬁnd

text: ##LTLine##

text: ##LTLine##


text: A. Boissé et al. that the atoms tend to be collectively either reﬂected or transmitted. More precisely, the histogram of the reﬂected fraction shows two distinct peaks at low (close to 0) and high (close to 1) reﬂected fractions, in contrast with the t = 0 observed bell shaped histogram for non-interacting parti- cles in the single scattering regime. This behavior is a sig- 0 < t <10 ms nature of the non-linear behavior of solitons in scattering. v 0 We ﬁnd that Gross Pitaevskii simulations are suﬃcient 10 ms < t < 40 ms to account for our observed double peaked histogram, be- cause of their strong sensitivity to small ﬂuctuations of 40 ms < t < 90 ms the experimental parameters and in particular to the soli- ton velocity. We argue however, that mesoscopic quantum t ~ 90 ms superpositions of all atoms being reﬂected and all atoms t ~ 240 ms being transmitted [49–51], could be observable in similar conditions provided that the number of atoms is signiﬁ- cantly reduced. t = 262 ms z Our experiment starts with a 39K condensate in the | − (cid:105) F = 1, m = 1 state, produced by evaporative cooling F Fig. 1: (Color online) Schematic of the experimental sequence. in an optical trap [52] close to the 561 G Feshbach reso- nance [53]. A soliton, containing N =5500(800) atoms, A soliton is launched into a 1D waveguide along z (continu- ous blue line) from a longitudinal trap (dotted red line). The is then created by ramping the magnetic ﬁeld close to soliton is ﬁrst accelerated to a controlled velocity v before a the scattering length zero crossing at 504.4 G [48, 53]. 0 1D speckle at 532 nm (green curve) is shone on the atoms for The atoms then have a negative mean-ﬁeld interaction 50 ms. The reﬂected and transmitted parts are ﬁnally sepa- energy, which binds them together. The elongated trap rated and observed after an additional 150 ms wait time, when is made of two horizontal far-detuned optical beams (at an image of the density distribution is taken. 1064 nm and 1550 nm), and it has identical radial frequen- cies of ω⊥/2π = 195 Hz and a longitudinal frequency of ω /2π = 44 Hz. the transmitted and reﬂected components are well sepa- z The soliton scattering in a 1D disordered potential is rated, and the radial trap is switched oﬀ. Each cloud ex- studied through the measurement of the reﬂected fraction pands for another 22 ms, and the separated components of the cloud sent with a low velocity in a far oﬀ reso- are observed (ﬁg. 1) by resonant ﬂuorescence imaging as nance speckle ﬁeld. The sequence is the following (see presented in [48]. The atom numbers in each component Fig. 1). The longitudinal (along z) conﬁnement is sud- are directly obtained (within a multiplying constant) by denly removed and the soliton starts to propagate along integration over two zones corresponding to positive and z in a 1D tube. We control the initial longitudinal ac- negative velocities (see Fig. 1), whereas the background is celeration through the addition of a small magnetic ﬁeld estimated from neighboring zones. We thus have a mea- gradient. The latter is subsequently ramped down be- surement that is independent of any assumption on the tween 10 ms and 40 ms after trap release such that the cloud shapes. The accuracy of atom number detection acceleration then vanishes (see footnote1). We choose the permits us to determine the reﬂected fraction with a 10% initial acceleration in order to reach a velocity of either accuracy for each individual run. − − v = 0.51(16) mm.s 1 or v = 0.90(20) mm.s 1, corre- The disorder is created from a laser speckle at 532 nm, 0 0 sponding to a center-of-mass kinetic energy per particle which yields a repulsive conservative potential for the E /h = mv2/2h =13(8) Hz or 40(17) Hz, where m is the atoms [54]. The laser beam, propagating perpendicularly kin 0 atomic mass and h the Planck constant. ∼The ﬂuctuations to z, passes through a diﬀusing plate and is focused on of the initial velocity exceed, by a factor 25, those asso- the atoms. Its cross-section intensity distribution on the ciated with the quantum ﬂuctuations of the soliton center diﬀusing plate is elliptical, with long axis along z and of mass in the ground state of the initial trap. They are short axis perpendicular to z. The speckle pattern shone due to uncontrolled and undamped residual dipole oscil- on the atoms has an intensity autocorrelation function lations in the initial trap. whose widths along these two directions are √respectively A 1D disorder potential is then turned on for 50 ms and σ = 0.38 µm, and 2.4 µm (half-width at 1/ e). Along the atoms are partially scattered or reﬂected, since we thze propagation axis of the laser beam, this autocorre- are in a 1D situation. After a waiting time of 150 ms, lation width is 10 µm. The two correlation lengths per- (cid:112) pendicular to z exceed the r.m.s. radial size of the cloud 1In practice, we have an additional residual anti-trapping curva- × given by the ground state extension of the harmonic oscil- ture of frequency i 1.9Hz [48], which only plays a role on long time scales and that we take into account in the analysis. lator h/4πmω⊥ = 0.8 µm. The disordered potential is p-2 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: A.

label: paragraph

text: Boissé

text: et

text: al.

text: that

text: the

text: atoms

text: tend

text: to

text: be

text: collectively

text: either

text: reﬂected

text: or

text: transmitted.

text: More

text: precisely,

text: the

text: histogram

text: of

text: the

text: reﬂected

text: fraction

text: shows

text: two

text: distinct

text: peaks

text: at

text: low

text: (close

text: to

text: and

text: high

text: (close

text: to

text: reﬂected

text: fractions,

text: in

text: contrast

text: with

text: the

text: t

font: default

text: observed

text: bell

text: shaped

text: histogram

text: for

text: non-interacting

text: parti-

text: cles

text: in

text: the

text: single

text: scattering

text: regime.

text: This

text: behavior

text: is

text: a

text: sig-

text: t

text: ms

text: nature

text: of

text: the

text: non-linear

text: behavior

text: of

text: solitons

text: in

text: scattering.

text: v

text: We

text: ﬁnd

text: that

text: Gross

text: Pitaevskii

text: simulations

text: are

text: suﬃcient

text: ms

text: t

text: ms

text: to

text: account

text: for

text: our

text: observed

text: double

text: peaked

text: histogram,

text: be-

text: cause

text: of

text: their

text: strong

text: sensitivity

text: to

text: small

text: ﬂuctuations

text: of

text: ms

text: t

text: ms

text: the

text: experimental

text: parameters

text: and

text: in

text: particular

text: to

text: the

text: soli-

text: ton

text: velocity.

text: We

text: argue

text: however,

text: that

text: mesoscopic

text: quantum

text: t

text: ms

text: superpositions

text: of

text: all

text: atoms

text: being

text: reﬂected

text: and

text: all

text: atoms

text: t

text: ms

text: being

text: transmitted

text: could

text: be

text: observable

text: in

text: similar

text: conditions

text: provided

text: that

text: the

text: number

text: of

text: atoms

text: is

text: signiﬁ-

text: cantly

text: reduced.

text: t

text: ms

text: z

text: Our

text: experiment

text: starts

text: with

text: a

text: condensate

text: in

text: the

text: F

text: m

text: state,

text: produced

text: by

text: evaporative

text: cooling

text: F

text: Fig.

text: (Color

text: online)

text: Schematic

text: of

text: the

text: experimental

text: sequence.

text: in

text: an

text: optical

text: trap

text: close

text: to

text: the

text: G

text: Feshbach

text: reso-

text: nance

text: A

text: soliton,

text: containing

text: N

text: atoms,

text: A

text: soliton

text: is

text: launched

text: into

text: a

text: waveguide

text: along

text: z

text: (continu-

text: ous

text: blue

text: line)

text: from

text: a

text: longitudinal

text: trap

text: (dotted

text: red

text: line).

text: The

text: is

text: then

text: created

text: by

text: ramping

text: the

text: magnetic

text: ﬁeld

text: close

text: to

text: soliton

text: is

text: ﬁrst

text: accelerated

text: to

text: a

text: controlled

text: velocity

text: v

text: before

text: a

text: the

text: scattering

text: length

text: zero

text: crossing

text: at

text: G

text: speckle

text: at

text: nm

text: (green

text: curve)

text: is

text: shone

text: on

text: the

text: atoms

text: for

text: The

text: atoms

text: then

text: have

text: a

text: negative

text: mean-ﬁeld

text: interaction

text: ms.

text: The

text: reﬂected

text: and

text: transmitted

text: parts

text: are

text: ﬁnally

text: sepa-

text: energy,

text: which

text: binds

text: them

text: together.

text: The

text: elongated

text: trap

text: rated

text: and

text: observed

text: after

text: an

text: additional

text: ms

text: wait

text: time,

text: when

text: is

text: made

text: of

text: two

text: horizontal

text: far-detuned

text: optical

text: beams

text: (at

text: an

text: image

text: of

text: the

text: density

text: distribution

text: is

text: taken.

text: nm

text: and

text: nm),

text: and

text: it

text: has

text: identical

text: radial

text: frequen-

text: cies

text: of

text: ω⊥/2π

text: Hz

text: and

text: a

text: longitudinal

text: frequency

text: of

text: ω

text: Hz.

text: the

text: transmitted

text: and

text: reﬂected

text: components

text: are

text: well

text: sepa-

text: z

text: The

text: soliton

text: scattering

text: in

text: a

text: disordered

text: potential

text: is

text: rated,

text: and

text: the

text: radial

text: trap

text: is

text: switched

text: oﬀ.

text: Each

text: cloud

text: ex-

text: studied

text: through

text: the

text: measurement

text: of

text: the

text: reﬂected

text: fraction

text: pands

text: for

text: another

text: ms,

text: and

text: the

text: separated

text: components

text: of

text: the

text: cloud

text: sent

text: with

text: a

text: low

text: velocity

text: in

text: a

text: far

text: oﬀ

text: reso-

text: are

text: observed

text: (ﬁg.

text: by

text: resonant

text: ﬂuorescence

text: imaging

text: as

text: nance

text: speckle

text: ﬁeld.

text: The

text: sequence

text: is

text: the

text: following

text: (see

text: presented

text: in

text: The

text: atom

text: numbers

text: in

text: each

text: component

text: Fig.

text: The

text: longitudinal

text: (along

text: z)

text: conﬁnement

text: is

text: sud-

text: are

text: directly

text: obtained

text: (within

text: a

text: multiplying

text: constant)

text: by

text: denly

text: removed

text: and

text: the

text: soliton

text: starts

text: to

text: propagate

text: along

text: integration

text: over

text: two

text: zones

text: corresponding

text: to

text: positive

text: and

text: z

text: in

text: a

text: tube.

text: We

text: control

text: the

text: initial

text: longitudinal

text: ac-

text: negative

text: velocities

text: (see

text: Fig.

text: whereas

text: the

text: background

text: is

text: celeration

text: through

text: the

text: addition

text: of

text: a

text: small

text: magnetic

text: ﬁeld

text: estimated

text: from

text: neighboring

text: zones.

text: We

text: thus

text: have

text: a

text: mea-

text: gradient.

text: The

text: latter

text: is

text: subsequently

text: ramped

text: down

text: be-

text: surement

text: that

text: is

text: independent

text: of

text: any

text: assumption

text: on

text: the

text: tween

text: ms

text: and

text: ms

text: after

text: trap

text: release

text: such

text: that

text: the

text: cloud

text: shapes.

text: The

text: accuracy

text: of

text: atom

text: number

text: detection

text: acceleration

text: then

text: vanishes

text: (see

text: We

text: choose

text: the

text: permits

text: us

text: to

text: determine

text: the

text: reﬂected

text: fraction

text: with

text: a

text: initial

text: acceleration

text: in

text: order

text: to

text: reach

text: a

text: velocity

text: of

text: either

text: accuracy

text: for

text: each

text: individual

text: run.

text: v

text: mm.s

text: or

text: v

text: mm.s

text: corre-

text: The

text: disorder

text: is

text: created

text: from

text: a

text: laser

text: speckle

text: at

text: nm,

text: sponding

text: to

text: a

text: center-of-mass

text: kinetic

text: energy

text: per

text: particle

text: which

text: yields

text: a

text: repulsive

text: conservative

text: potential

text: for

text: the

text: E

text: /h

text: Hz

text: or

text: Hz,

text: where

text: m

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text: the

text: atoms

text: The

text: laser

text: beam,

text: propagating

text: perpendicularly

text: kin

text: atomic

text: mass

text: and

text: h

text: the

text: Planck

text: constant.

text: ∼The

text: ﬂuctuations

text: to

text: z,

text: passes

text: through

text: a

text: diﬀusing

text: plate

text: and

text: is

text: focused

text: on

text: of

text: the

text: initial

text: velocity

text: exceed,

text: by

text: a

text: factor

text: those

text: asso-

text: the

text: atoms.

text: Its

text: cross-section

text: intensity

text: distribution

text: on

text: the

text: ciated

text: with

text: the

text: quantum

text: ﬂuctuations

text: of

text: the

text: soliton

text: center

text: diﬀusing

text: plate

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text: elliptical,

text: with

text: long

text: axis

text: along

text: z

text: and

text: of

text: mass

text: in

text: the

text: ground

text: state

text: of

text: the

text: initial

text: trap.

text: They

text: are

text: short

text: axis

text: perpendicular

text: to

text: z.

text: The

text: speckle

text: pattern

text: shone

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text: uncontrolled

text: and

text: undamped

text: residual

text: dipole

text: oscil-

text: on

text: the

text: atoms

text: has

text: an

text: intensity

text: autocorrelation

text: function

text: lations

text: in

text: the

text: initial

text: trap.

text: whose

text: widths

text: along

text: these

text: two

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text: are

text: √respectively

text: A

text: disorder

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text: is

text: then

text: turned

text: on

text: for

text: ms

text: and

text: σ

text: µm,

text: and

text: µm

text: (half-width

text: at

text: e).

text: Along

text: the

text: atoms

text: are

text: partially

text: scattered

text: or

text: reﬂected,

text: since

text: we

text: thze

text: propagation

text: axis

text: of

text: the

text: laser

text: beam,

text: this

text: autocorre-

text: are

text: in

text: a

text: situation.

text: After

text: a

text: waiting

text: time

text: of

text: ms,

text: lation

text: width

text: is

text: µm.

text: The

text: two

text: correlation

text: lengths

text: per-

text: pendicular

text: to

text: z

text: exceed

text: the

text: r.m.s.

text: radial

text: size

text: of

text: the

text: cloud

text: practice,

text: we

text: have

text: an

text: additional

text: residual

text: anti-trapping

text: curva-

text: given

text: by

text: the

text: ground

text: state

text: extension

text: of

text: the

text: harmonic

text: oscil-

text: ture

text: of

text: frequency

text: i

text: which

text: only

text: plays

text: a

text: role

text: on

text: long

text: time

text: scales

text: and

text: that

text: we

text: take

text: into

text: account

text: in

text: the

text: analysis.

text: lator

text: µm.

text: The

text: disordered

text: potential

text: is

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: A. Boissé et al. [51] A. I. Streltsov, O. E. Alon, and L. S. Cederbaum, J. Phys. B 42, 091004 (2009). [52] G. Salomon, L. Fouché, S. Lepoutre, A. Aspect, and T. Bourdel, Phys. Rev. A 90, 033405 (2014). [53] C. D’Errico, M. Zaccanti, M. Fattori, G. Roati, M. Ingus- cio, G. Modugno and A. Simoni, New Journal of physics 9, 223 (2007). [54] D. Clément, A.F. Varón, J.A. Retter, L. Sanchez-Palencia, A. Aspect and P. Bouyer, New J. Phys. 8, 165 (2006). [55] L. D. Carr and Y. Castin, Phys. Rev. A 66, 063602 (2002). [56] V.A. Hopkins, J. Keat, G. D. Meegan, T. Zhang, and J. D. Maynard, Phys. Rev. Lett. 76, 1102 (1996). [57] R.G. McDonald and A.S. Bradley, Phys. Rev. A 93, 063604 (2016). [58] L.M. Aycock, H.M. Hurst, D. Genkina, H.-I Lu, V. Galit- ski, I. B. Spielman, arXiv: 1608.03916 [59] P. Medley, M.A. Minar, N.C. Cizek, D. Berryrieser, and M.A. Kasevich, Phys. Rev. Lett. 112, 060401 (2014). [60] M. Kasevich, Atom systems and Bose-Einstein condensates for metrology and navigation, ﬁrst NASA Quantum Future Technologies conference (2012):http://quantum.nasa.gov/materials/2012-01- 18-B1-Kasevich.pdf [61] C. Lee, J.H. Huang, H.M. Deng, H.Dai, J. Xu, Frontiers of physics 7, 109 (2012). [62] N. Veretenov, Y. Rozhdestvensky, N. Rosanov, V. Smirnov, S. Fedorov, The European Physical Journal D 42, 455 (2007). [63] G.-B. Jo, Y. Shin, S. Will, T.A. Pasquini, M. Saba, W. Ketterle, D.E. Pritchard, M. Vengalattore, and M. Pren- tiss, Phys. Rev. Lett. 98, 030407 (2007). [64] B. Gertjerenken, Phys. Rev. A 88, 053623 (2013). [65] B. Gertjerenken, T.P. Wiles, C. Weiss, arXiv:1508.00656 [66] K. Sacha, C.A. Müller, D. Delande, and J. Zakrzewski, Phys. Rev. Lett. 103, 210402 (2009). p-6 ##LTLine##

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text: Medley,

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text: Kasevich,

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text: Lee,

text: J.H.

text: Huang,

text: H.M.

text: Deng,

text: H.Dai,

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text: Smirnov,

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text: Fedorov,

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text: Jo,

text: Y.

text: Shin,

text: S.

text: Will,

text: T.A.

text: Pasquini,

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text: Saba,

text: W.

text: Ketterle,

text: D.E.

text: Pritchard,

text: M.

text: Vengalattore,

text: and

text: M.

text: Pren-

text: tiss,

text: Phys.

text: Rev.

text: Lett.

text: B.

text: Gertjerenken,

text: Phys.

text: Rev.

text: A

text: B.

text: Gertjerenken,

text: T.P.

text: Wiles,

text: C.

text: Weiss,

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text: − ≤ ≤ holds. Including the kernel weight β with C = β A + (1 β )B and 0 β 1, we obtain the 1 1 1 1 following inequality (cid:88) (cid:88) (cid:88) p p p ≤ − λ(C) β λ(A) + (1 β ) λ(B) . (5) i 1 i 1 i i=1 i=1 i=1 One can see, the the right hand side is maximized if the kernel matrix with the highest sum of the p largest eigenvalues has a weight of 1. In that setting, the right hand side is equal to the left hand side. The extension to more than two kernel matrices can be made recursively. Therefore, optimizing (cid:54) Problem 3 leads to weight vectors β with β = 1 and β = 0 for all j = i, where i is the index of the i j matrix with the p largest eigenvalues. Although this behavior maximizes the variance, it might not be the best choice for biological data, where we assume, that different data types can give complementing information and should therefore be considered jointly. Hence, in the following, we will introduce a scoring function, that combines the idea of kPCA with the assumption of different data supplementing each other. 2.4 Scoring function Besides preserving the global variance, the main goal of this approach is to be able to integrate data from different sources that can complement each other. So, for integrating two different kernel matrices A and B to an ensemble kerne(cid:18)l matrix K, we propose the f(cid:19)ollowing gain function: λ(K) − g = exp i 1 (6) i max(λ(A) , λ(B) , 1) (cid:80) i i for each dimension i. Then the overall score for a projection into a p-dimensional space is calculated { } as G = 1 p g . The main idea is that we deﬁne a baseline, i.e., max λ(A) , λ(B) , 1 , that p i=1 i i i represents the variance we can have by using only one matrix. Gains of variance in comparison to this baseline have a strong positive impact on the score while losses of variance are penalized only slightly. Thereby, we can account for the fact that small losses of variance in one direction often do not change the global structure of the data, but allow for more variance in a different direction. Additionally, we ensure that the baseline is not smaller than 1, which is the variance each direction would have in case of equal distribution. This scoring function is maximized to ﬁnd the best kernel weights β and can easily be extended to integrate more than two matrices. 2.5 Materials We applied the approach to ﬁve different cancer sets from TCGA [6], which were preprocessed by Wang et al. [7]. The data sets are breast invasive carcinoma (BIC; 105 samples), colon adenocarci- noma (COAD; 92 samples), glioblastoma multiforme (GBM; 213 samples), kidney renal clear cell carcinoma (KRCCC; 122 samples), and lung squamous cell carcinoma (LSCC; 106 samples). For all cancer types, gene expression, DNA methylation, and miRNA expression measurements, as well as survival data are available. The ﬁrst three data types are used in the dimensionality reduction and clustering process, the latter is used to perform survival analysis in the evaluation step. Each data type is represented by a kernel matrix generated using the Gaussian kernel function. The kernel width parameter γ was chosen according to the rule of thumb γ = 1/2d2, with d being the number of features in the matrix [8]. 3 Results For each cancer type, we generated results using a two-step procedure: First, we ran the dimensionality reduction approach in order to integrate the three data types and reduce the noise in the ﬁnal projection. In other approaches, the number of projection dimensions is usually determined either using the elbow method or based on a chosen threshold for the remaining variance. Here, we beneﬁt from the scoring function, which indicates if we gain variance in comparison to using only one matrix. Since this function does not have a global maximum, we use its ﬁrst local maximum to determine 3 ##LTLine## ##LTLine##

text: holds.

text: Including

text: the

text: kernel

text: weight

text: β

text: with

text: C

text: β

text: A

text: β

text: )B

text: and

text: β

text: we

text: obtain

text: the

text: following

text: inequality

label: equation

text: p

text: p

text: p

text: λ(C)

text: β

text: λ(A)

text: β

text: λ(B)

text: i

text: i

text: i

text: One

text: can

text: see,

text: the

text: the

text: right

text: hand

text: side

text: is

text: maximized

text: if

text: the

text: kernel

text: matrix

text: with

text: the

text: highest

text: sum

text: of

text: the

text: p

text: largest

text: eigenvalues

text: has

text: a

text: weight

text: of

text: In

text: that

text: setting,

text: the

text: right

text: hand

text: side

text: is

text: equal

text: to

text: the

text: left

text: hand

text: side.

text: The

text: extension

text: to

text: more

text: than

text: two

text: kernel

text: matrices

text: can

text: be

text: made

text: recursively.

text: Therefore,

text: optimizing

text: Problem

text: leads

text: to

text: weight

text: vectors

text: β

text: with

text: β

text: and

text: β

text: for

text: all

text: j

text: i,

text: where

text: i

text: is

text: the

text: index

text: of

text: the

text: i

text: j

text: matrix

text: with

text: the

text: p

text: largest

text: eigenvalues.

text: Although

text: this

text: behavior

text: maximizes

text: the

text: variance,

text: it

text: might

text: not

text: be

text: the

text: best

text: choice

text: for

text: biological

text: data,

text: where

text: we

text: assume,

text: that

text: different

text: data

text: types

text: can

text: give

text: complementing

text: information

text: and

text: should

text: therefore

text: be

text: considered

text: jointly.

text: Hence,

text: in

text: the

text: following,

text: we

text: will

text: introduce

text: a

text: scoring

text: function,

text: that

text: combines

text: the

text: idea

text: of

text: kPCA

text: with

text: the

text: assumption

text: of

text: different

text: data

text: supplementing

text: each

text: other.

text: Scoring

text: function

text: Besides

text: preserving

text: the

text: global

text: variance,

text: the

text: main

text: goal

text: of

text: this

text: approach

text: is

text: to

text: be

text: able

text: to

text: integrate

text: data

text: from

text: different

text: sources

text: that

text: can

text: complement

text: each

text: other.

text: So,

text: for

text: integrating

text: two

text: different

text: kernel

text: matrices

text: A

text: and

text: B

text: to

text: an

text: ensemble

text: matrix

text: K,

text: we

text: propose

text: the

text: gain

text: function:

text: λ(K)

text: g

text: exp

text: i

text: i

text: max(λ(A)

text: λ(B)

text: i

text: i

text: for

text: each

text: dimension

text: i.

text: Then

text: the

text: overall

text: score

text: for

text: a

text: projection

text: into

text: a

text: p-dimensional

text: space

text: is

text: calculated

text: as

text: G

text: p

text: g

text: The

text: main

text: idea

text: is

text: that

text: we

text: deﬁne

text: a

text: baseline,

text: i.e.,

text: max

text: λ(A)

text: λ(B)

text: that

text: p

text: i

text: i

text: i

text: represents

text: the

text: variance

text: we

text: can

text: have

text: by

text: using

text: only

text: one

text: matrix.

text: Gains

text: of

text: variance

text: in

text: comparison

text: to

text: this

text: baseline

text: have

text: a

text: strong

text: positive

text: impact

text: on

text: the

text: score

text: while

text: losses

text: of

text: variance

text: are

text: penalized

text: only

text: slightly.

text: Thereby,

text: we

text: can

text: account

text: for

text: the

text: fact

text: that

text: small

text: losses

text: of

text: variance

text: in

text: one

text: direction

text: often

text: do

text: not

text: change

text: the

text: global

text: structure

text: of

text: the

text: data,

text: but

text: allow

text: for

text: more

text: variance

text: in

text: a

text: different

text: direction.

text: Additionally,

text: we

text: ensure

text: that

text: the

text: baseline

text: is

text: not

text: smaller

text: than

text: which

text: is

text: the

text: variance

text: each

text: direction

text: would

text: have

text: in

text: case

text: of

text: equal

text: distribution.

text: This

text: scoring

text: function

text: is

text: maximized

text: to

text: ﬁnd

text: the

text: best

text: kernel

text: weights

text: β

text: and

text: can

text: easily

text: be

text: extended

text: to

text: integrate

text: more

text: than

text: two

text: matrices.

text: Materials

text: We

text: applied

text: the

text: approach

text: to

text: ﬁve

text: different

text: cancer

text: sets

text: from

text: TCGA

text: which

text: were

text: preprocessed

text: by

text: Wang

text: et

text: al.

text: The

text: data

text: sets

text: are

text: breast

text: invasive

text: carcinoma

text: (BIC;

text: samples),

text: colon

text: adenocarci-

text: noma

text: (COAD;

text: samples),

text: glioblastoma

text: multiforme

text: (GBM;

text: samples),

text: kidney

text: renal

text: clear

text: cell

text: carcinoma

text: (KRCCC;

text: samples),

text: and

text: lung

text: squamous

text: cell

text: carcinoma

text: (LSCC;

text: samples).

text: For

text: all

text: cancer

text: types,

text: gene

text: expression,

text: DNA

text: methylation,

text: and

text: miRNA

text: expression

text: measurements,

text: as

text: well

text: as

text: survival

text: data

text: are

text: available.

text: The

text: ﬁrst

text: three

text: data

text: types

text: are

text: used

text: in

text: the

text: dimensionality

text: reduction

text: and

text: clustering

text: process,

text: the

text: latter

text: is

text: used

text: to

text: perform

text: survival

text: analysis

text: in

text: the

text: evaluation

text: step.

text: Each

text: data

text: type

text: is

text: represented

text: by

text: a

text: kernel

text: matrix

text: generated

text: using

text: the

text: Gaussian

text: kernel

text: function.

text: The

text: kernel

text: width

text: parameter

text: γ

text: was

text: chosen

text: according

text: to

text: the

text: rule

text: of

text: thumb

text: γ

text: with

text: d

text: being

text: the

text: number

text: of

text: features

text: in

text: the

text: matrix

text: Results

text: For

text: each

text: cancer

text: type,

text: we

text: generated

text: results

text: using

text: a

text: two-step

text: procedure:

text: First,

text: we

text: ran

text: the

text: dimensionality

text: reduction

text: approach

text: in

text: order

text: to

text: integrate

text: the

text: three

text: data

text: types

text: and

text: reduce

text: the

text: noise

text: in

text: the

text: ﬁnal

text: projection.

text: In

text: other

text: approaches,

text: the

text: number

text: of

text: projection

text: dimensions

text: is

text: usually

text: determined

text: either

text: using

text: the

text: elbow

text: method

text: or

text: based

text: on

text: a

text: chosen

text: threshold

text: for

text: the

text: remaining

text: variance.

text: Here,

text: we

text: beneﬁt

text: from

text: the

text: scoring

text: function,

text: which

text: indicates

text: if

text: we

text: gain

text: variance

text: in

text: comparison

text: to

text: using

text: only

text: one

text: matrix.

text: Since

text: this

text: function

text: does

text: not

text: have

text: a

text: global

text: maximum,

text: we

text: use

text: its

text: ﬁrst

text: local

text: maximum

text: to

text: determine

text: ##LTLine##

text: ##LTLine##


text: 5.2 Construction (1) In this and the next subsection, we construct an exact symplectic manifold M and F → a collection of two-tailed Lagrangian branes L(cid:122) = (L(cid:122), L(cid:122), . . . L(cid:122)) so that (L(cid:122)) is A 0 1 n isomorphic to . In this subsection, we construct them for the case that S = T = ∅ S,T as a prototype of all the construction. Thanks to the previous two lemmas, what we have to construct is reduced to a two-dimensional manifold M with non-empty boundary, a perfect collection of two- ± tailed Lagrangian submanifolds L , and brane structures on the underlying spaces of the two-tailed Lagrangian submanifolds. ≤ ≤ We set Cj (cid:66) S 1 = ×R/−Z for 0 j (cid:32) n and deﬁne (cid:33)M0 by the plumbing (and smoothing) −of Dj (cid:66) Cj [ ε3, ε3]. Namely, let ϕj and xj be the natural coordinat∈e of Cj and [ ε3, ε3]. Our plumbing is deﬁned to identify two points (ϕj+1, xj+1) { | − } ⊂ 1 − ∈ ≤ (ϕj+1, xj+1) ε3 < ϕj+1 < ε3 Dj+1 and 2 + xj+1, ϕj+1 Dj for every 0 j < n. thatFτiCxj|sCmj oisotthhe(raignhtitpohdanaldemda)pD. eWhne tdweiﬁsntseτsCujbam∈loanngifoC⊂ldj ssubpypoLr(cid:48)jte(cid:66)d iτnC−D11τjC−.12W· ·e· τaC−s1sj−u1Cmje. Then, all these submanifolds pass through (0, 0) D0 M0 and there is no other inter- section point. We deform them to avoid (0, 0) and pass through the left side of the point with respect to their orientation as in Figure 3 and obtain the resulting submanifolds L0, L1, . . . , Ln. (cid:48) Figure 3: L ’s and L’s Next, we give tails to L ’s such that they form a compatible collection of two-tailed j Lagrangian submanifolds. Let us consider the following sequence of collections of sub- − − − · · · − ≤ m≤anifolds Γj (cid:66) (C1,C2, . . . ,Cj, τC−j1Cj+1, τC1j τC1j+1Cj+2, . . . , τC1j τC1n−(cid:48)1Cn(cid:48)) for 1(cid:48) j n. Here, the two extreme cases are Γn =−(C1,C2, . . . ,Cn)−and Γ1 = (L1, L2, . . . , Ln). The collection Γj is obtained by applying τC1j o∈n the latter (n j) submanifolds of Γj+1. Now, the submanifolds in Γj+1 around (0, 0) Dj+1 is disposed as in the left half of Figure 4. 20 ##LTFigure## ##LTLine##

label: section

text: Construction

text: In

text: this

text: and

text: the

text: next

text: subsection,

text: we

text: construct

text: an

text: exact

text: symplectic

text: manifold

text: M

text: and

text: F

font: GEVSKC+txsy

text: a

text: collection

text: of

text: two-tailed

text: Lagrangian

text: branes

font: BMLHDW+rtxr

font: BPVBKG+rtxmi

text: so

text: that

text: is

text: A

text: n

text: isomorphic

text: to

text: In

text: this

text: subsection,

text: we

text: construct

text: them

text: for

text: the

text: case

text: that

text: S

text: T

font: DKWIDA+txsyb

text: S,T

text: as

text: a

text: prototype

text: of

text: all

text: the

text: construction.

text: Thanks

text: to

text: the

text: previous

text: two

text: lemmas,

text: what

text: we

text: have

text: to

text: construct

text: is

text: reduced

text: to

text: a

text: two-dimensional

text: manifold

text: M

text: with

text: non-empty

text: boundary,

text: a

text: perfect

text: collection

text: of

text: two-

text: tailed

text: Lagrangian

text: submanifolds

text: L

text: and

text: brane

text: structures

text: on

text: the

text: underlying

text: spaces

text: of

text: the

text: two-tailed

text: Lagrangian

text: submanifolds.

text: We

text: set

text: Cj

text: S

text: ×R/−Z

text: for

text: j

text: n

text: and

text: deﬁne

text: by

text: the

text: plumbing

text: (and

text: smoothing)

text: −of

text: Dj

text: Cj

text: Namely,

text: let

text: ϕj

text: and

text: xj

text: be

text: the

text: natural

text: coordinat∈e

text: of

text: Cj

text: and

text: Our

text: plumbing

text: is

text: deﬁned

text: to

text: identify

text: two

text: points

text: ε3

text: ε3

text: and

text: Dj

text: for

text: every

text: j

text: n.

text: thatFτiCxj|sCmj

text: oisotthhe(raignhtitpohdanaldemda)pD.

text: eWhne

text: tdweiﬁsntseτsCujbam∈loanngifoC⊂ldj

text: ·e·

text: Then,

text: all

text: these

text: submanifolds

text: pass

text: through

text: and

text: there

text: is

text: no

text: other

text: inter-

text: section

text: point.

text: We

text: deform

text: them

text: to

text: avoid

text: and

text: pass

text: through

text: the

text: left

text: side

text: of

text: the

text: point

text: with

text: respect

text: to

text: their

text: orientation

text: as

text: in

text: Figure

text: and

text: obtain

text: the

text: resulting

text: submanifolds

text: Ln.

text: Figure

text: L

text: ’s

text: and

text: L’s

text: Next,

text: we

text: give

text: tails

text: to

text: L

text: ’s

text: such

text: that

text: they

text: form

text: a

text: compatible

text: collection

text: of

text: two-tailed

text: j

text: Lagrangian

text: submanifolds.

text: Let

text: us

text: consider

text: the

text: following

text: sequence

text: of

text: collections

text: of

text: sub-

text: m≤anifolds

text: Γj

text: ,Cj,

text: for

text: j

text: n.

text: Here,

text: the

text: two

text: extreme

text: cases

text: are

text: Γn

text: ,Cn)−and

text: Γ1

text: Ln).

text: The

text: collection

text: Γj

text: is

text: obtained

text: by

text: applying

text: o∈n

text: the

text: latter

text: (n

text: j)

text: submanifolds

text: of

text: Now,

text: the

text: submanifolds

text: in

text: around

text: is

text: disposed

text: as

text: in

text: the

text: left

text: half

text: of

text: Figure

text: ##LTFigure##

text: ##LTLine##


text: Figure 8: Bypass diagram Figure 9: Bypass diagram 2 25 ##LTFigure## ##LTFigure##

text: Figure

text: Bypass

label: caption

text: diagram

text: Figure

text: Bypass

text: diagram

text: ##LTFigure##

label: figure

text: ##LTFigure##


text: (cid:48) − ∩ (cid:48) − ≥ \ intersections L (p 1) L (p 1) for distinct i, j p remain in M D and the intersections − ∩ − i j − L(p 1) L(p 1) for distinct i, j < p 1 remain in K. i j (cid:48) − ∩ (cid:48) − Now, we study the intersection of L (p− 1) L (p 1). By the condition (ii) of property (cid:48) ∩ (cid:48) ∈ ⊂ p 1 i − Pp, we ha∩ve L ∩(pp−)1 L (ip) K (cid:48) D− fo∩r an(cid:48)y−i (cid:44)∈ p \ 1. By the assumption that τLp−1(Lp−1 D) D(cid:48) =− ∅, we have L (pp−11) L (ip 1) M D. Thus we have proved the property P − of L (p 1). (cid:50) p 1 (cid:48) (cid:48) Now we modify L (p) by isotopies. First, we set L(n) = L (n). Before we construct isotopy, we prove the following lemma: (cid:48) − Lemma 7.3 L (n 1) can be a collection of underlying spaces of a perfect collection of tailed Lagrangian submanifolds. − ± ∩ Proof For j (cid:44) n 1, we can deﬁ−ne tails γj so that the±ir image are in Hj since Li Hj (cid:44) ∅ if and only if i = j. For j = n 1, we can deﬁne γ − as in Figure 12. (cid:50) j 1 (cid:48) − Figure 12: L (n 1) (cid:48) − ± Now we can deform L (nn 1) freely away fr(cid:48)om− γn as in the sense of the ﬁrst dis- cussion of this section. The ﬁrst isotopy for L (n 1) is constructed as follows. By the (cid:48) − ∩ ∩ assumptions, the connected component ξ of L (nn 1) D = τLn−1Ln D which lies in the left side of ln−1 in K has the∈shape ∩that ξ comes from ln(0), go along ln, turn right just before l reaches the p − L − L , go along the left side (with respect to the n n 1,n n 1 n orientation of l − ) of l − , and ﬁnally go out from D at the left side of l − (0) as in n 1 n 1 n 1 Figure 1−3. We call such a path ξ whose endpoints are near ln−1(0) and ln(0) a path of type (n 1, n). (We always write the smaller one at the left and bigger one at the right independent of the orientation of the path.) (cid:101) (cid:48) − We make some observations about the intersections of L (n 1). We can see that − ξ intersects Lj for d(n) < j < n 1 twice. We deform ξ into ξ so that the pairs 29 ##LTFigure##

text: intersections

text: L

text: (p

text: L

text: (p

text: for

text: distinct

text: i,

text: j

text: p

text: remain

text: in

text: M

text: D

text: and

text: the

text: intersections

text: i

text: j

text: L(p

text: L(p

text: for

text: distinct

text: i,

text: j

text: p

text: remain

text: in

text: K.

text: i

text: j

text: Now,

text: we

text: study

text: the

text: intersection

text: of

text: L

text: (p−

text: L

text: (p

text: By

text: the

text: condition

text: (ii)

text: of

text: property

text: p

text: i

text: Pp,

text: we

text: ha∩ve

text: L

text: L

text: (ip)

text: K

text: D−

text: fo∩r

text: p

text: By

text: the

text: assumption

text: that

text: D)

text: we

text: have

text: L

text: L

text: (ip

text: M

text: D.

text: Thus

text: we

text: have

text: proved

text: the

text: property

text: P

text: of

text: L

text: (p

text: p

text: Now

text: we

text: modify

text: L

text: (p)

text: by

text: isotopies.

text: First,

text: we

text: set

text: L(n)

text: L

text: (n).

text: Before

text: we

text: construct

text: isotopy,

text: we

text: prove

text: the

text: following

text: lemma:

text: Lemma

text: L

text: (n

text: can

text: be

text: a

text: collection

text: of

text: underlying

text: spaces

text: of

text: a

text: perfect

text: collection

text: of

text: tailed

text: Lagrangian

text: submanifolds.

text: Proof

text: For

text: j

text: n

text: we

text: can

text: deﬁ−ne

text: tails

text: γj

text: so

text: that

text: the±ir

text: image

text: are

text: in

text: Hj

text: since

text: Li

text: Hj

text: if

text: and

text: only

text: if

text: i

text: j.

text: For

text: j

text: n

text: we

text: can

text: deﬁne

text: γ

text: as

text: in

text: Figure

text: j

text: Figure

text: L

text: (n

text: Now

text: we

text: can

text: deform

text: L

text: (nn

text: freely

text: away

text: γn

text: as

text: in

text: the

text: sense

text: of

text: the

text: ﬁrst

text: dis-

text: cussion

text: of

text: this

text: section.

text: The

text: ﬁrst

text: isotopy

text: for

text: L

text: (n

text: is

text: constructed

text: as

text: follows.

text: By

text: the

text: assumptions,

text: the

text: connected

text: component

text: ξ

text: of

text: L

text: (nn

text: D

text: D

text: which

text: lies

text: in

text: the

text: left

text: side

text: of

text: in

text: K

text: has

text: the∈shape

text: ∩that

text: ξ

text: comes

text: from

text: go

text: along

text: ln,

text: turn

text: right

text: just

text: before

text: l

text: reaches

text: the

text: p

text: L

text: L

text: go

text: along

text: the

text: left

text: side

text: (with

text: respect

text: to

text: the

text: n

text: n

text: n

text: n

text: orientation

text: of

text: l

text: of

text: l

text: and

text: ﬁnally

text: go

text: out

text: from

text: D

text: at

text: the

text: left

text: side

text: of

text: l

text: as

text: in

text: n

text: n

text: n

text: Figure

text: We

text: call

text: such

text: a

text: path

text: ξ

text: whose

text: endpoints

text: are

text: near

text: and

text: a

text: path

text: of

text: type

text: (n

text: n).

text: (We

text: always

text: write

text: the

text: smaller

text: one

text: at

text: the

text: left

text: and

text: bigger

text: one

text: at

text: the

text: right

text: independent

text: of

text: the

text: orientation

text: of

text: the

text: path.)

text: We

text: make

text: some

text: observations

text: about

text: the

text: intersections

text: of

text: L

text: (n

text: We

text: can

text: see

text: that

text: ξ

text: intersects

text: Lj

text: for

text: d(n)

text: j

text: n

text: twice.

text: We

text: deform

text: ξ

text: into

text: ξ

text: so

text: that

text: the

text: pairs

text: ##LTFigure##


text: notions and deﬁne the A∞-Koszul dual. In section 3, we prepare the geometric notions, e.g. exact symplectic manifolds and their Fukaya categories. At the last part of the section, we present the key theorem proved by Seidel which states the equivalence of algebraic twists and Dehn twists. In section 4, we state the main theorem. In section 5 and 6, we construct exact Riemann surfaces whose Fukaya categories are the targets of the “embedding” from directed A∞-categories. In section 7, we do the computation of A∞-Koszul duals, i.e. the computation of Dehn twists. The computation and the formula of A∞-Koszul duals are the main ingredients of this paper. Acknowledgement I would like to thank my supervisor Toshitake Kohno for giving me great advice and navigating me and this study to an appropriate direction. I also want to thank A. Ikeda for teaching me about Koszul duality theory and to F. Sanda, M. Kawasaki, T. Kuwagaki, J. Yoshida, and R. Sato for fruitful discussion. Finally, I am deeply greatful to my friend I. Hoshimiya, A. Kiriya, R. Shibuki, and Y. Todo for supporting me when I was in diﬃcult situations. This work was supported by the Program for Leading Graduate Schools, MEXT, Japan. 2 Algebraic preliminaries In this section, we review the deﬁnitions of algebraic objects we use in this paper, and deﬁne the A∞-Koszul dual, the key concept in this paper. For the notation of signs, we follow Seidel’s notation in [Se08]. The deﬁnition of the Koszul dual for A∞- algebras with some properties already exists [EL16], [LPWZ04]. Our construction is a generalisation to directed A∞-categories. 2.1 Basic deﬁnitions and properties of A∞-categories A Deﬁnition 2.1 (A∞-category) An A∞-category , consists of the following data: (cid:77) A 1. a set Ob( ), ∈ A 2. a Z-graded vector space homA(X, Y) = homiA(X, Y) for each X, Y Ob( ), ∈ i Z 3. maps called higher composition maps ⊗ ⊗ · · · ⊗ µd : homA(Xd−1, Xd) homA(Xd−2, Xd−1) homA(X0, X1) → − homA(X0, Xd)[2 d], ≥ ∈ A for d 1 and X0, X1, . . . , Xd Ob( ). (cid:88) We impose that the µ’s satisfy the A∞-associativity relation: − ( 1)(cid:70)(cid:88)iµl(ad, . . . , ai+j+1, µj(ai+j, . . . , ai+1), ai, . . . a1) = 0 i,j,l ≥ | | − | | for d 1, where (cid:70) = ( a 1) , ( a = deg(a )). i l i i ≤ ≤ 1 l i 4

text: notions

text: and

text: deﬁne

text: the

text: A∞-Koszul

text: dual.

text: In

text: section

text: we

text: prepare

text: the

text: geometric

text: notions,

text: e.g.

text: exact

text: symplectic

text: manifolds

text: and

text: their

text: Fukaya

text: categories.

text: At

text: the

text: last

text: part

text: of

text: the

text: section,

text: we

text: present

text: the

text: key

text: theorem

text: proved

text: by

text: Seidel

text: which

text: states

text: the

text: equivalence

text: of

text: algebraic

text: twists

text: and

text: Dehn

text: twists.

text: In

text: section

text: we

text: state

text: the

text: main

text: theorem.

text: In

text: section

text: and

text: we

text: construct

text: exact

text: Riemann

text: surfaces

text: whose

text: Fukaya

text: categories

text: are

text: the

text: targets

text: of

text: the

text: “embedding”

text: from

text: directed

text: A∞-categories.

text: In

text: section

text: we

text: do

text: the

text: computation

text: of

text: A∞-Koszul

text: duals,

text: i.e.

text: the

text: computation

text: of

text: Dehn

text: twists.

text: The

text: computation

text: and

text: the

text: formula

text: of

text: A∞-Koszul

text: duals

text: are

text: the

text: main

text: ingredients

text: of

text: this

text: paper.

text: Acknowledgement

text: I

text: would

text: like

text: to

text: thank

text: my

text: supervisor

text: Toshitake

text: Kohno

text: for

text: giving

text: me

text: great

text: advice

text: and

text: navigating

text: me

text: and

text: this

text: study

text: to

text: an

text: appropriate

text: direction.

text: I

text: also

text: want

text: to

text: thank

text: A.

text: Ikeda

text: for

text: teaching

text: me

text: about

text: Koszul

text: duality

text: theory

text: and

text: to

text: F.

text: Sanda,

text: M.

text: Kawasaki,

text: T.

text: Kuwagaki,

text: J.

text: Yoshida,

text: and

text: R.

text: Sato

text: for

text: fruitful

text: discussion.

text: Finally,

text: I

text: am

text: deeply

text: greatful

text: to

text: my

text: friend

text: I.

text: Hoshimiya,

text: A.

text: Kiriya,

text: R.

text: Shibuki,

text: and

text: Y.

text: Todo

text: for

text: supporting

text: me

text: when

text: I

text: was

text: in

text: diﬃcult

text: situations.

text: This

text: work

text: was

text: supported

text: by

text: the

text: Program

text: for

text: Leading

text: Graduate

text: Schools,

text: MEXT,

text: Japan.

text: Algebraic

text: preliminaries

text: In

text: this

text: section,

text: we

text: review

text: the

text: deﬁnitions

text: of

text: algebraic

text: objects

text: we

text: use

text: in

text: this

text: paper,

text: and

text: deﬁne

text: the

text: A∞-Koszul

text: dual,

text: the

text: key

text: concept

text: in

text: this

text: paper.

text: For

text: the

text: notation

text: of

text: signs,

text: we

text: follow

text: Seidel’s

text: notation

text: in

text: The

text: deﬁnition

text: of

text: the

text: Koszul

text: dual

text: for

text: A∞-

text: algebras

text: with

text: some

text: properties

text: already

text: exists

text: Our

text: construction

text: is

text: a

text: generalisation

text: to

text: directed

text: A∞-categories.

text: Basic

text: deﬁnitions

text: and

text: properties

text: of

text: A∞-categories

text: A

text: Deﬁnition

text: (A∞-category)

text: An

text: A∞-category

text: consists

text: of

text: the

text: following

text: data:

text: A

text: a

text: set

text: Ob(

text: A

text: a

text: Z-graded

text: vector

text: space

text: homA(X,

text: Y)

text: homiA(X,

text: Y)

text: for

text: each

text: X,

text: Y

text: Ob(

text: i

text: Z

text: maps

text: called

text: higher

text: composition

text: maps

text: µd

text: Xd)

text: d],

text: A

text: for

text: d

text: and

text: Xd

text: Ob(

text: We

text: impose

text: that

text: the

text: µ’s

text: satisfy

text: the

text: A∞-associativity

text: relation:

text: µj(ai+j,

text: ai,

text: i,j,l

text: for

text: d

text: where

text: a

text: a

text: deg(a

text: i

text: l

text: i

text: i

text: l

text: i


text: (b) the− two endpoints of ξ are on ∂K near li1(0) and li2(0) respec−tively for some p 1 < i1 < i2, i.e. ξ is a path of type (i1, i2; i3) for some i3 < p 1. We can assume that all paths of type (i1, i2; i3) do not intersect with the support of τL − . ≥ (cid:48)(cid:48) − p ∩1 After applying of τLp−1 to L(jp) for j p−, any≤connected component ξ of L (jp 1) Klje−f1t is of type (i1, i2; i3) with some i3 < p 1 i1 < i2. Now,≤such a path ξ of type (i1, i2; i3) intersects with Lj more than once only when d(cid:101)(i2) < j i3. At that time, the number of intersections i(cid:101)s two and the intersection points can be removed by isotopy as in the(cid:101)case of p = n. After the isotopy, we obtain a path ξ(cid:101)of typ−e (i1, i2; d(i2)). Observe that if d(i1) = d(i2), then ξ does not intersects with Lj for j < p 1, and if d(i1) < d(i2), then ξ intersects with Ld(i ). We change all such ξ’s into ξ’s by the isotopies. 2 If we apply the isotopies for suitable order, all the isotopies just reduces intersec- tions and not create new intersection points. (Such an order can be constructed as follows. Any path ξ divides K into two regions and one is contained in Kle−ft . We name (cid:48) ⇔ p 1⊂ the contained region K . We deﬁne partial order of paths by ξ < ξ K K (cid:48). This ξ ξ ξ is well-deﬁned by the condition of property P − . We add more relation and make it a p 1 total order. This is what we want.) Finally, we apply the isotopies for corresponding (cid:48)(cid:48) − − L (p 1) and obtain L(p 1). j j − − By the construction, we have shown that the collection L(p 1) of such L(p 1)’s has − j the property Q − except for the property that L(p 1) can be a collection of underlying p 1 spaces of a perfect collection of tailed Lagrangian submanifolds. Therefore, it’s time (cid:48)(cid:48) − − to check the condition (ii) for L (p 1) and L(p 1). Since each isotopy takes place in Kleft − ≤ (cid:48)(cid:48) − ∩ − ∩ ∩ j for a suita−ble p 1 j < n, we have L (jp 1) Kpri−gh1t = L(jp 1) Kpr(cid:48)i−g(cid:48)h1t−= τLj−1Lj − Kpri−gh1t for j > p 1. Here, Kpri−gh1t is the right part of lp−1 in K±. Hence, L (p 1) and L(p 1) can be drown as in Figure 14. Therefore, we can deﬁne γ as in Figure 14. Thus, we have − j constructed a collection L(p 1) with property Q − . p 1 − Figure 14: L(p 1) and their tails ± Finally, we have a perfect collection of tailed Lagrangian submanifolds L(0) = 31 ##LTFigure##

text: (b)

label: list

text: the−

text: two

text: endpoints

text: of

text: ξ

text: are

text: on

text: ∂K

text: near

text: and

text: respec−tively

text: for

text: some

text: p

text: i.e.

text: ξ

text: is

text: a

text: path

text: of

text: type

text: for

text: some

text: p

text: We

text: can

text: assume

text: that

text: all

text: paths

text: of

text: type

text: do

text: not

text: intersect

text: with

text: the

text: support

text: of

text: τL

text: p

text: ∩1

text: After

text: applying

text: of

text: to

text: L(jp)

text: for

text: j

text: p−,

text: any≤connected

text: component

text: ξ

text: of

text: L

text: (jp

text: is

text: of

text: type

text: with

text: some

text: p

text: Now,≤such

text: a

text: path

text: ξ

text: of

text: type

text: intersects

text: with

text: Lj

text: more

text: than

text: once

text: only

text: when

text: j

text: At

text: that

text: time,

text: the

text: number

text: of

text: intersections

text: two

text: and

text: the

text: intersection

text: points

text: can

text: be

text: removed

text: by

text: isotopy

text: as

text: in

text: of

text: p

text: n.

text: After

text: the

text: isotopy,

text: we

text: obtain

text: a

text: path

text: typ−e

text: Observe

text: that

text: if

text: then

text: ξ

text: does

text: not

text: intersects

text: with

text: Lj

text: for

text: j

text: p

text: and

text: if

text: then

text: ξ

text: intersects

text: with

text: Ld(i

text: We

text: change

text: all

text: such

text: ξ’s

text: into

text: ξ’s

text: by

text: the

text: isotopies.

text: If

text: we

text: apply

text: the

text: isotopies

text: for

text: suitable

text: order,

text: all

text: the

text: isotopies

text: just

text: reduces

text: intersec-

text: tions

text: and

text: not

text: create

text: new

text: intersection

text: points.

text: (Such

text: an

text: order

text: can

text: be

text: constructed

text: as

text: follows.

text: Any

text: path

text: ξ

text: divides

text: K

text: into

text: two

text: regions

text: and

text: one

text: is

text: contained

text: in

text: Kle−ft

text: We

text: name

text: p

text: 1⊂

text: the

text: contained

text: region

text: K

text: We

text: deﬁne

text: partial

text: order

text: of

text: paths

text: by

text: ξ

text: ξ

text: K

text: K

text: This

text: ξ

text: ξ

text: ξ

text: is

text: well-deﬁned

text: by

text: the

text: condition

text: of

text: property

text: P

text: We

text: add

text: more

text: relation

text: and

text: make

text: it

text: a

text: p

text: total

text: order.

text: This

text: is

text: what

text: we

text: want.)

text: Finally,

text: we

text: apply

text: the

text: isotopies

text: for

text: corresponding

text: L

text: (p

text: and

text: obtain

text: L(p

text: j

text: j

text: By

text: the

text: construction,

text: we

text: have

text: shown

text: that

text: the

text: collection

text: L(p

text: of

text: such

text: L(p

text: has

text: j

text: the

text: property

text: Q

text: except

text: for

text: the

text: property

text: that

text: L(p

text: can

text: be

text: a

text: collection

text: of

text: underlying

text: p

text: spaces

text: of

text: a

text: perfect

text: collection

text: of

text: tailed

text: Lagrangian

text: submanifolds.

text: Therefore,

text: it’s

text: time

text: to

text: check

text: the

text: condition

text: (ii)

text: for

text: L

text: (p

text: and

text: L(p

text: Since

text: each

text: isotopy

text: takes

text: place

text: in

text: Kleft

text: j

text: for

text: a

text: suita−ble

text: p

text: j

text: n,

text: we

text: have

text: L

text: (jp

text: L(jp

text: for

text: j

text: p

text: Here,

text: is

text: the

text: right

text: part

text: of

text: in

text: K±.

text: Hence,

text: L

text: (p

text: and

text: L(p

text: can

text: be

text: drown

text: as

text: in

text: Figure

text: Therefore,

text: we

text: can

text: deﬁne

text: γ

text: as

text: in

text: Figure

text: Thus,

text: we

text: have

text: j

text: constructed

text: a

text: collection

text: L(p

text: with

text: property

text: Q

text: p

text: Figure

text: L(p

text: and

text: their

text: tails

text: Finally,

text: we

text: have

text: a

text: perfect

text: collection

text: of

text: tailed

text: Lagrangian

text: submanifolds

text: ##LTFigure##


text: ⇔ − Suppose th−at lp = 1 ( d(p) = d(p 1)). Then the order of the points is qp, a(1p). Moreover, L(p 1) is away from the support of remaining Dehn twists and isotopie(cid:101)s. p Hence, we have the proof for the case l = 1. (However, in fact, this case is trivial.) ≥ p (cid:101) Now, we consider the case lp 2. The Dehn twist τLd(p) = τLa(2p) acts on the path ξ1 and obtain ξ2. We can see that the path ξ2 comes from a point near ld(p)(0) = la(p)(0), 2 go to a point near l − (0) = l (0), and intersects with L(d(p)) = L(a(2p)) by the following p 1 a(p) j j order of subscripts, d(d(p)) +11, d(d(p)) + 2, . . . , d(p − 1). Hence, the points in L(pa(2p)) appear by −the following orde−r: qp, d(p) (= a(2p)), d(d(p)) + 1 (= d(a(2p)) + 1), d(d(p)) + 2, . . . , d(p 1) (= d(a(p))), p 1 (= a(p)). Figure 15 illustrates the situation. 1 ⇔ 1 Suppose that l = 2 ( d(a(p)) = d(a(p))). Then, the order of the points is p 1 2 qp, a(2p), a(1p). By almost the same discussion as in the case of lp = 1, we have proved the case of l = 2. p ≥ Now, consider the case for l 3. By the same discussion, the intersection points in p L(pa(3p)) appear by the following order: qp, a(2p), d(a(2p)), d(a(2p))−1, . . . , d(a(3p))+1, a(3p), a(1p). Again Figure 15 illustrates the situation. If lp = 3, then we can ﬁnish the proof by the same argument, and if lp > 3, then we can ﬁnish the proof by the iteration of the above discussion. Next, we study the order of q † ’s. We prove the statement about the order of a(jp) ,p them by induction on p. For the case of p = n, the statement is trivial. Now we assume that the statement is true for p > s and prove the case of p = s. † † Let us see the case with small l . In the case of l = 1, the statement is trivial. In † s † s † † the case of ls = 2, there exists a relation [s, d (s)]Z and we have a(0s) = s, a(1s) = s + 1, † † and a(s) = d (s). In this case, L(s+1) is as in the left part of Figure 16. The remaining 2 Dehn twists and isotopies do not change the order of the intersection points q † ’s, so a(js) ,s the statement for these points holds. † † † † In the case o†f ls =† 3, then there exist two relations [s, d (s)]Z and [c1, d (s + 1)]Z satisfying s < c1 < d (s). The submanifold La(s(3s+)†1) is isotopic to τc†1τa(3s)†−1La(3s)†, where τj = τLj. Hence, the intersection of L(ss+1) and La(s(s+)†1) is just left (with respect to the 3 orientation of Lc†1) to the intersection of L(ss+1) and Lc(s†1) as in the right part of Figure 16. By the construction of L(s), the remaining Dehn twists and isotopies do not change the order of the intersection points q † ’s, so the statement for this case holds. † a(js) ,s † † † † In† the case of ls = 4,†there† exist thr†ee re≤lati†ons [†s, d (s)]Z, [c1, d (s + 1)]Z, and [b2, d 2(s)] such that s < c1 < d (s) and d (s) b2 < d (s+1). The submanifold La(s(s+)†1) † 3 is sam†e−as in the case of ls = 3 and La(s(4s+)†1) is †is≤otopic t†o−τd(a(4s)†−1†)τb†2τa(4s)†−1La(4s)†. Now, d(a(s) 1) satisﬁes the following inequality c d(a(s) 1) < d (s). Here, the second 4 1 4 † inequality follows from Lemma 4.11. The ﬁrst inequality follows from the fact that c † − † − 1 is the smallest element in S which is greater than s and s < d(d 2(s) 1) = d(a(s) 1). 4 35

text: Suppose

text: th−at

text: lp

text: d(p)

text: d(p

text: Then

text: the

text: order

text: of

text: the

text: points

text: is

text: qp,

text: Moreover,

text: L(p

text: is

text: away

text: from

text: the

text: support

text: of

text: remaining

text: Dehn

text: twists

text: and

text: p

text: Hence,

text: we

text: have

text: the

text: proof

text: for

text: the

text: case

text: l

text: (However,

text: in

text: fact,

text: this

text: case

text: is

text: trivial.)

text: p

text: Now,

text: we

text: consider

text: the

text: case

text: lp

text: The

text: Dehn

text: twist

text: τLd(p)

text: acts

text: on

text: the

text: path

text: ξ1

text: and

text: obtain

text: We

text: can

text: see

text: that

text: the

text: path

text: ξ2

text: comes

text: from

text: a

text: point

text: near

text: go

text: to

text: a

text: point

text: near

text: l

text: l

text: and

text: intersects

text: with

text: L(d(p))

text: by

text: the

text: following

text: p

text: a(p)

text: j

text: j

text: order

text: of

text: subscripts,

text: d(d(p))

text: d(d(p))

text: d(p

text: Hence,

text: the

text: points

text: in

text: appear

text: by

text: −the

text: following

text: orde−r:

text: qp,

text: d(p)

text: d(d(p))

text: d(d(p))

text: d(p

text: d(a(p))),

text: p

text: a(p)).

text: Figure

text: illustrates

text: the

text: situation.

text: Suppose

text: that

text: l

text: d(a(p))

text: d(a(p))).

text: Then,

text: the

text: order

text: of

text: the

text: points

text: is

text: p

text: qp,

text: By

text: almost

text: the

text: same

text: discussion

text: as

text: in

text: the

text: case

text: of

text: lp

text: we

text: have

text: proved

text: the

text: case

text: of

text: l

text: p

text: Now,

text: consider

text: the

text: case

text: for

text: l

text: By

text: the

text: same

text: discussion,

text: the

text: intersection

text: points

text: in

text: p

text: appear

text: by

text: the

text: following

text: order:

text: qp,

text: Again

text: Figure

text: illustrates

text: the

text: situation.

text: If

text: lp

text: then

text: we

text: can

text: ﬁnish

text: the

text: proof

text: by

text: the

text: same

text: argument,

text: and

text: if

text: lp

text: then

text: we

text: can

text: ﬁnish

text: the

text: proof

text: by

text: the

text: iteration

text: of

text: the

text: above

text: discussion.

text: Next,

text: we

text: study

text: the

text: order

text: of

text: q

text: ’s.

text: We

text: prove

text: the

text: statement

text: about

text: the

text: order

text: of

text: a(jp)

text: ,p

text: them

text: by

text: induction

text: on

text: p.

text: For

text: the

text: case

text: of

text: p

text: n,

text: the

text: statement

text: is

text: trivial.

text: Now

text: we

text: assume

text: that

text: the

text: statement

text: is

text: true

text: for

text: p

text: s

text: and

text: prove

text: the

text: case

text: of

text: p

text: s.

text: Let

text: us

text: see

text: the

text: case

text: with

text: small

text: l

text: In

text: the

text: case

text: of

text: l

text: the

text: statement

text: is

text: trivial.

text: In

text: s

text: s

text: the

text: case

text: of

text: ls

text: there

text: exists

text: a

text: relation

text: [s,

text: d

text: (s)]Z

text: and

text: we

text: have

text: s,

text: s

text: and

text: a(s)

text: d

text: (s).

text: In

text: this

text: case,

text: is

text: as

text: in

text: the

text: left

text: part

text: of

text: Figure

text: The

text: remaining

text: Dehn

text: twists

text: and

text: isotopies

text: do

text: not

text: change

text: the

text: order

text: of

text: the

text: intersection

text: points

text: q

text: ’s,

text: so

text: a(js)

text: ,s

text: the

text: statement

text: for

text: these

text: points

text: holds.

text: In

text: the

text: case

text: o†f

text: ls

text: then

text: there

text: exist

text: two

text: relations

text: [s,

text: d

text: (s)]Z

text: and

text: d

text: (s

text: satisfying

text: s

text: d

text: (s).

text: The

text: submanifold

text: is

text: isotopic

text: to

text: where

text: τj

text: τLj.

text: Hence,

text: the

text: intersection

text: of

text: and

text: is

text: just

text: left

text: (with

text: respect

text: to

text: the

text: orientation

text: of

text: to

text: the

text: intersection

text: of

text: and

text: as

text: in

text: the

text: right

text: part

text: of

text: Figure

text: By

text: the

text: construction

text: of

text: L(s),

text: the

text: remaining

text: Dehn

text: twists

text: and

text: isotopies

text: do

text: not

text: change

text: the

text: order

text: of

text: the

text: intersection

text: points

text: q

text: ’s,

text: so

text: the

text: statement

text: for

text: this

text: case

text: holds.

text: a(js)

text: ,s

text: In†

text: the

text: case

text: of

text: ls

text: exist

text: thr†ee

text: re≤lati†ons

text: [†s,

text: d

text: (s)]Z,

text: d

text: (s

text: and

text: d

text: such

text: that

text: s

text: d

text: (s)

text: and

text: d

text: (s)

text: d

text: The

text: submanifold

text: is

text: sam†e−as

text: in

text: the

text: case

text: of

text: ls

text: and

text: is

text: †is≤otopic

text: Now,

text: d(a(s)

text: satisﬁes

text: the

text: following

text: inequality

text: c

text: d(a(s)

text: d

text: (s).

text: Here,

text: the

text: second

text: inequality

text: follows

text: from

text: Lemma

text: The

text: ﬁrst

text: inequality

text: follows

text: from

text: the

text: fact

text: that

text: c

text: is

text: the

text: smallest

text: element

text: in

text: S

text: which

text: is

text: greater

text: than

text: s

text: and

text: s

text: d(d

text: d(a(s)


text: (cid:91) − ∩ By Prop∪osition 7.7,−we have cS∪j = [−j 1, j + 1]∩j. Now, we have ∂Yi Xp−1 = [si + 1, ti]s [j 1, j + 1]j ([p 1, p + 1]p Xp−1). Therefore it suﬃces to i −si<j<ti=p ∩ \ { } show that ([p 1, p + 1]p Xp−−1) qp,p−1∩is connec{ted. } the Iffolllpow=in2g, tdhiesncuwsesiohnav. eT[hpe in1te,rpse+ct1io]pn cSXp−∩1(=∪ qpc,pS−1), qisp,ath(2p)es.eTthwios fpoollionwtsshfernocme − p j<p j the interval [p 1, a(2p)]p is either contained in Xp−1 or just in∪tersects with Xp−1 only at the two end points. Since q − and q are away from Y by Proposition 7.7 p−,p 1 p+∩1,p { ti<}p i andNL−oewm,mwae7c.8o,nwsideehratvhee[cpase1l,pp≥+31.]Ipn thXisp−c1as=e, qthpe,pr−e1,eqxpis,at(2pr)e.lations [d(p), p]Z and [d(p⊂ 1), c1]Z w(cid:48)ith d(p) < c1 < p. Let us write c1 = ti(cid:48). Since ti(cid:48) = c1 < p−, we have YpCToiho(cid:48)iennnstiXγdwepsi−rttha1a.rtptLhsaaeftthrtosγγmetwbqiehspi,aqcp−hp1aisﬁthar.snftBreioenxmctteaeruqnssspee,ipco−ton1wfottiofhthγeq(cid:48)∪pfs,oajo(2<lpl)ptohacwalSotinnγjggagttSohqrpbpe,eawey(3pi)oft,ahnacdnittsdqs:pti,qhma(2epa)ﬁga−nelai[itlstplienawtae1laro,synaegc(2fprtS)io]omppn.. PYm{ ri(cid:48)uo,spttohsset}iatiyinotinenr7sY.e7ic(cid:48),tuiwonenlecsoasfnpiS,tac(2rpope)naaccnlhdueds∂eYthtih(cid:48)eaiptsoγtirngatnoqsivpne,atro(2ps)athsl,oeawinnedtehr{iaqovpre,ao(2(pf)[,pYqi−p(cid:48) ,aa1(3tp,)pq},ppp=,+a1(3p)1S.]pIpf∩∩lp∂X=Yp−i(cid:48)31,)bγy\ qp,p−1 = [a(2p), a(3p)]p and th≥is is connected. ∪ Finall, the c≥ase of lp 4, we consider the intersection points qp,a(rp) of γ and j<p cS j with r 4. Since a(rp) < a(3p) = si(cid:48), these points are contained in−Y˚i(cid:48). Therefor∩e again γ\ {must s}tay in Yi(cid:48) unless it reaches the point qp,a(3p) so we have ([p 1, p + 1]p Xp−1) qp,p−1 = [a(2p), a(3p)] and this is connected. (cid:50) Remark 7.10 By the above lemmas and propositions, the core cS of S has the follow- ing properties. • − There exists a (tj sj + 1)-gon Yj corresponds to a relation [sj, tj]Z. • The root and intersection points in S is distributed by the order displayed in j Proposition 7.7. • − Especially, the core satisﬁes cS j = [j 1, j + 1]j for 0 < j < n. • † ≥ † { } If l0 2, then cS0 = [1, d (cid:91)(0)]0, else c(cid:91)S0 = q1,0 . • ≥ − { } If ln 2, then cSn = [d(n), n 1]n, else cSn = qn.n−1 • ≤ ≤ ∪ For any 0 p n, X = cS Y is contractible. p j i ≤ ≤ ≤ ≤ 0 j p 1 t p i In many concrete examples, such a “diagram” is unique up to “isotopy”, and the author couldn’t ﬁnd any counter examples. Therefore, the author believes that the uniqueness holds under some justiﬁcation. However, we don’t go into this direction. 39

text: By

text: Prop∪osition

text: have

text: cS∪j

text: [−j

text: j

text: Now,

text: we

text: have

text: ∂Yi

text: [si

text: ti]s

text: [j

text: j

text: ([p

text: p

text: Therefore

text: it

text: suﬃces

text: to

text: i

text: −si<j<ti=p

text: show

text: that

text: ([p

text: p

text: connec{ted.

text: the

text: tdhiesncuwsesiohnav.

text: eT[hpe

text: fpoollionwtsshfernocme

text: p

text: j<p

text: j

text: the

text: interval

text: [p

text: is

text: either

text: contained

text: in

text: or

text: just

text: in∪tersects

text: with

text: only

text: at

text: the

text: two

text: end

text: points.

text: Since

text: q

text: and

text: q

text: are

text: away

text: from

text: Y

text: by

text: Proposition

text: p−,p

text: ti<}p

text: i

text: [d(p),

text: p]Z

text: and

text: [d(p⊂

text: d(p)

text: p.

text: Let

text: us

text: write

text: Since

text: p−,

text: we

text: have

text: PYm{

text: and

text: th≥is

text: is

text: connected.

text: Finall,

text: the

text: c≥ase

text: of

text: lp

text: we

text: consider

text: the

text: intersection

text: points

text: qp,a(rp)

text: of

text: γ

text: and

text: j<p

text: cS

text: j

text: with

text: r

text: Since

text: a(rp)

text: these

text: points

text: are

text: contained

text: Therefor∩e

text: again

text: γ\

text: {must

text: s}tay

text: in

text: unless

text: it

text: reaches

text: the

text: point

text: so

text: we

text: have

text: ([p

text: p

text: and

text: this

text: is

text: connected.

text: Remark

text: By

text: the

text: above

text: lemmas

text: and

text: propositions,

text: the

text: core

text: cS

text: of

text: S

text: has

text: the

text: follow-

text: ing

text: properties.

text: There

text: exists

text: a

text: (tj

text: sj

text: Yj

text: corresponds

text: to

text: a

text: relation

text: [sj,

text: tj]Z.

text: The

text: root

text: and

text: intersection

text: points

text: in

text: S

text: is

text: distributed

text: by

text: the

text: order

text: displayed

text: in

text: j

text: Proposition

text: Especially,

text: the

text: core

text: satisﬁes

text: cS

text: j

text: [j

text: j

text: for

text: j

text: n.

text: If

text: then

text: d

text: else

text: If

text: ln

text: then

text: cSn

text: [d(n),

text: n

text: else

text: cSn

text: For

text: any

text: p

text: n,

text: X

text: cS

text: Y

text: is

text: contractible.

text: p

text: j

text: i

text: j

text: p

text: t

text: p

text: i

text: In

text: many

text: concrete

text: examples,

text: such

text: a

text: “diagram”

text: is

text: unique

text: up

text: to

text: “isotopy”,

text: and

text: the

text: author

text: couldn’t

text: ﬁnd

text: any

text: counter

text: examples.

text: Therefore,

text: the

text: author

text: believes

text: that

text: the

text: uniqueness

text: holds

text: under

text: some

text: justiﬁcation.

text: However,

text: we

text: don’t

text: go

text: into

text: this

text: direction.


text:  setting Ob(A) = {Y1, Y2, . . . , Yn},  homA(Yi, Yj) (i < j) · homA→(Y)(Yi, Yj) = k ei (i = j) 0 (i > j), A→ A and µ’s of (Y) are canonically induced from those of . A Now, we begin the deﬁnition of A∞-Koszul duals. For an A∞-category , we M A A call an A∞-functor from op to C(k) (cid:66) Cb (k) a (right) A∞-module, where op A dg is the opposite A∞-category of and Cb (k) is the dg category of bounded cochain dg complexes of ﬁnite dimensional k vector spaces considered as an A∞-category. It is Q A known that such A∞-modules form a triangulated dg category (cid:66) mod( ). (Note A ∞ that all the hom spaces of this category are ﬁnite dimensional iﬀ #Ob( ) < .) Let A { · · · } be a directed A∞-category with its object set 0 < 1 < < n . We deﬁne an A∞-module S(j) for j ∈ Ob(A) determined by the data  S k if i = j, (j)(i) = 0 if i (cid:44) j A and call it a simple -module corresponds to j, where we consider k as a one-dimensional cochain complex concentrated in the degree zero part. Then, it is known that the A Q {S S − · · · S } full sub A∞-category !∞ of with object set (n) < (n 1) < < (0) A Q→ S forms a directed A∞-category. Hence, !∞ is naturally isomorphic to ( ), where S S S − S Q = ( (n), (n 1), . . . , (0)) is a collection of objects in . The details can be found in (5j) and (5o) in [Se08]. A { · · · } Deﬁnition 2.4 Let be a directed A∞-category with object set 0 < 1 < < n . A B A A directed A∞-category quasi-isomorphic to !∞ is called an A∞-Koszul dual of . Remark 2.5 The above deﬁnition is an analogy or a category version of the deﬁnition in [BGS96]. In that paper, they deal with Koszul ring A and give a diﬀerent deﬁnition • of its Koszul dual A!. However, Theorem 2.10.1 in that paper states that Ext (k, k) (cid:27) A (A!)opp canonically. Even though there exist many diﬀerent notations, we can translate from one to the other. Also, our deﬁnition is an analogy of the deﬁnition in [LPWZ04]. In that paper, they deﬁne Koszul dual E(A) (in their notation) for Adams connected A∞-algebra A by E(A) (cid:66) RHomA◦(k, k). The right hand side of the deﬁnition is a straightforward generalisation of the deﬁnition in [BGS96], hence the deﬁnitions in that paper and in our paper shares the common origin. In this paper, we treat with A∞-categories, not A∞-algebras and we focus on the very special case, directed A∞-categories. →− Example 2→.−6 Let R = k(∆, ρ) be a path algebra with relations over a ﬁnite direc- ted quiver ∆. Here, a ﬁnite quiver is a quiver with a ﬁnite set of vertices (which we write ∆ ) and a ﬁnite set of arrows (which we write ∆ ); a directed quiver is a 0 1 6

font: WDVHDL+txex

text: setting

text: Ob(A)

text: Yn},

text: homA(Yi,

text: Yj)

text: (i

text: j)

text: homA→(Y)(Yi,

text: Yj)

text: k

text: ei

text: (i

text: j)

text: (i

text: j),

text: A→

text: A

text: and

text: µ’s

text: of

text: (Y)

text: are

text: canonically

text: induced

text: from

text: those

text: of

text: A

text: Now,

text: we

text: begin

text: the

text: deﬁnition

text: of

text: A∞-Koszul

text: duals.

text: For

text: an

text: A∞-category

text: we

text: M

text: A

text: A

text: call

text: an

text: A∞-functor

text: from

text: op

text: to

text: C(k)

font: AHERPT+txsyc

text: Cb

text: (k)

text: a

text: (right)

text: A∞-module,

text: where

text: op

text: A

text: dg

text: is

text: the

text: opposite

text: A∞-category

text: of

text: and

text: Cb

text: (k)

text: is

text: the

text: dg

text: category

text: of

text: bounded

text: cochain

text: dg

text: complexes

text: of

text: ﬁnite

text: dimensional

text: k

text: vector

text: spaces

text: considered

text: as

text: an

text: A∞-category.

text: It

text: is

text: Q

text: A

text: known

text: that

text: such

text: A∞-modules

text: form

text: a

text: triangulated

text: dg

text: category

text: mod(

text: (Note

text: A

text: that

text: all

text: the

text: hom

text: spaces

text: of

text: this

text: category

text: are

text: ﬁnite

text: dimensional

text: iﬀ

text: #Ob(

text: Let

text: A

text: be

text: a

text: directed

text: A∞-category

text: with

text: its

text: object

text: set

text: n

text: We

text: deﬁne

text: an

text: A∞-module

text: S(j)

text: for

text: j

text: Ob(A)

text: determined

text: by

text: the

text: data

text: S

text: k

text: if

text: i

text: j,

text: (j)(i)

text: if

text: i

text: j

text: A

text: and

text: call

text: it

text: a

text: simple

text: -module

text: corresponds

text: to

text: j,

text: where

text: we

text: consider

text: k

text: as

text: a

text: one-dimensional

text: cochain

text: complex

text: concentrated

text: in

text: the

text: degree

text: zero

text: part.

text: Then,

text: it

text: is

text: known

text: that

text: the

text: A

text: Q

text: {S

text: S

text: S

text: full

text: sub

text: A∞-category

text: of

text: with

text: object

text: set

text: (n)

text: (n

text: A

text: Q→

text: S

text: forms

text: a

text: directed

text: A∞-category.

text: Hence,

text: is

text: naturally

text: isomorphic

text: to

text: where

text: S

text: S

text: S

text: S

text: Q

text: (n),

text: (n

text: is

text: a

text: collection

text: of

text: objects

text: in

text: The

text: details

text: can

text: be

text: found

text: in

text: and

text: in

text: A

text: Deﬁnition

text: Let

text: be

text: a

text: directed

text: A∞-category

text: with

text: object

text: set

text: n

text: A

text: B

text: A

text: A

text: directed

text: A∞-category

text: quasi-isomorphic

text: to

text: is

text: called

text: an

text: A∞-Koszul

text: dual

text: of

text: Remark

text: The

text: above

text: deﬁnition

text: is

text: an

text: analogy

text: or

text: a

text: category

text: version

text: of

text: the

text: deﬁnition

text: in

text: In

text: that

text: paper,

text: they

text: deal

text: with

text: Koszul

text: ring

text: A

text: and

text: give

text: a

text: diﬀerent

text: deﬁnition

text: of

text: its

text: Koszul

text: dual

text: A!.

text: However,

text: Theorem

text: in

text: that

text: paper

text: states

text: that

text: Ext

text: (k,

text: k)

text: A

text: (A!)opp

text: canonically.

text: Even

text: though

text: there

text: exist

text: many

text: diﬀerent

text: notations,

text: we

text: can

text: translate

text: from

text: one

text: to

text: the

text: other.

text: Also,

text: our

text: deﬁnition

text: is

text: an

text: analogy

text: of

text: the

text: deﬁnition

text: in

text: In

text: that

text: paper,

text: they

text: deﬁne

text: Koszul

text: dual

text: E(A)

text: (in

text: their

text: notation)

text: for

text: Adams

text: connected

text: A∞-algebra

text: A

text: by

text: E(A)

text: RHomA◦(k,

text: k).

text: The

text: right

text: hand

text: side

text: of

text: the

text: deﬁnition

text: is

text: a

text: straightforward

text: generalisation

text: of

text: the

text: deﬁnition

text: in

text: hence

text: the

text: deﬁnitions

text: in

text: that

text: paper

text: and

text: in

text: our

text: paper

text: shares

text: the

text: common

text: origin.

text: In

text: this

text: paper,

text: we

text: treat

text: with

text: A∞-categories,

text: not

text: A∞-algebras

text: and

text: we

text: focus

text: on

text: the

text: very

text: special

text: case,

text: directed

text: A∞-categories.

text: Example

text: Let

text: R

text: k(∆,

text: ρ)

text: be

text: a

text: path

text: algebra

text: with

text: relations

text: over

text: a

text: ﬁnite

text: direc-

text: ted

text: quiver

text: Here,

text: a

text: ﬁnite

text: quiver

text: is

text: a

text: quiver

text: with

text: a

text: ﬁnite

text: set

text: of

text: vertices

text: (which

text: we

text: write

text: and

text: a

text: ﬁnite

text: set

text: of

text: arrows

text: (which

text: we

text: write

text: a

text: directed

text: quiver

text: is

text: a