text: ∗ Characterization of a correlated topological Kondo insulator in one dimension I. Hagym´asi and O¨. Legeza † ‡ (Dated: May 28, 2020) We investigate the ground-state of a p-wave Kondo-Heisenberg model introduced by Alexandrov and Coleman with an Ising-type anisotropy in the Kondo interaction and correlated conduction electrons. Our aim is to understand how they aﬀect the stability of the Haldane state obtained in the SU(2) symmetric case without the Hubbard interaction. By applying the density-matrix renormalization group algorithm and calculating the entanglement entropy we show that in the anisotropic case a phase transition occurs and a N´eel state emerges above a critical value of the Coulomb interaction. These ﬁndings are also corroborated by the examination of the entanglement spectrum and the spin proﬁle of the system which clarify the structure of each phase. PACS numbers: 71.10.Fd, 71.10.Pm, 71.27.+a, 73.20.-r, 75.30.Mb I. INTRODUCTION In contrast, in the present case since they become nonlo- cal, these singlets are broken at the boundary, and edge states appear.24 This p-wave Kondo-Heisenberg model Kondo insulators are semiconductors at low tempera- has been investigated with several methods. The large-N tures with a few meV gap in contrast to the conventional expansion has revealed that magnetic end states appear semiconductors in which the gap is in the order of a few at the boundaries.24 Analyzing the ground state with eV.1 It is now generally accepted that strong correlations Abelian bosonization28 and density-matrix renormaliza- among the electrons lead to the stabilization of this state tion group (DMRG) algorithm,37 it has turned out that with such a small gap.2 the ground state is actually the Haldane phase.38,39 It In the last decade a new group of insulators were dis- is also interesting to mention that Haldane phase has covered, which are gapped in the bulk, but they have been previously observed in the ordinary Kondo lat- soft, topologically protected surface states.3–5 They are tice model with ferromagnetic Kondo coupling40 and the topological insulators whose basic properties can be in the extended periodic Anderson model with Hund’s understood within the framework of ordinary band the- coupling.41 The facts above clearly indicates that one ory. The question naturally arises how the correlation has to apply sophisticated many-body techniques to ac- among the electrons aﬀects the properties of topolog- curately describe the ground state. ical insulators. This aspect has attracted signiﬁcant In this paper our aim is to study the stability of the attention6–13 since the existence of topological phases Haldane phase against perturbations. The most obvi- was suggested in strongly correlated systems.14–18 These ous and physically relevant choices are either to intro- observations made it necessary to understand, inter alia, duce anisotropy in the Kondo interaction or the inclu- the theoretical properties of topological Mott6,19–22 and sion of a Coulomb interaction in the conduction band. Kondo insulators.23–29 The most famous example of The role of the Coulomb interaction was explored in the topological Kondo insulators is SmB6, which was discov- case of the conventional Kondo lattice model,42,43 but ered more than four decades ago.30 Recent experimental only the weak coupling limit of the present model has observations31–33 that SmB6 can host robust conducting been studied so far with bosonization.28 In contrast, the surface states renewed the interest in this compound in anisotropy was studied only in the conventional model particular and in Kondo insulators in general.34–36 The and its phase diagram was determined.44–46 We address most important aspect of topological Kondo insulators is these problems using the DMRG algorithm,47–51 which that hybridization occurs between d and f bands whose is the state-of-the-art tool to determine the ground state parities are opposite to each other.29 This leads to the and it makes possible to go beyond the weakly inter- appearance of a symmetry-protected topological ground acting limit. Since the DMRG calculation is closely re- state.28 lated to the quantum information theory, we can deter- To capture the eﬀects of strong correlations and topol- mine the entanglement entropy52–57 and entanglement ogy simultaneously, Alexandrov and Coleman suggested spectrum,58 which are very useful to detect quantum a Kondo lattice model with a special form of the Kondo phase transitions59–62 and symmetry-protected topolog- exchange.24 The model consists of a free one-dimensional ical order.63–65 Our analysis of the anisotropic model re- electron gas coupled to an S = 1/2 Heisenberg chain via veals that (i) a phase transition occurs as the interaction a Kondo exchange with p-wave character. In the conven- is increased within the conduction band and the Haldane tional Kondo lattice model, where the Kondo exchange state transforms into a N´eel state, and (ii) the entangle- is on-site, local singlets are formed in the strong coupling ment spectrum can be used as a ﬁngerprint to identify a limit and the boundaries do not play an important role. topological Kondo insulator in one dimension.

text: Characterization

text: of

text: a

text: correlated

text: topological

text: Kondo

text: insulator

text: in

text: one

text: dimension

text: I.

text: Hagym´asi

text: and

text: O¨.

text: Legeza

text: (Dated:

text: May

label: date

text: We

label: abstract

text: investigate

text: the

text: ground-state

text: of

text: a

text: p-wave

text: Kondo-Heisenberg

text: model

text: introduced

text: by

text: Alexandrov

text: and

text: Coleman

text: with

text: an

text: Ising-type

text: anisotropy

text: in

text: the

text: Kondo

text: interaction

text: and

text: correlated

text: conduction

text: electrons.

text: Our

text: aim

text: is

text: to

text: understand

text: how

text: they

text: aﬀect

text: the

text: stability

text: of

text: the

text: Haldane

text: state

text: obtained

text: in

text: the

text: symmetric

text: case

text: without

text: the

text: Hubbard

text: interaction.

text: By

text: applying

text: the

text: density-matrix

text: renormalization

text: group

text: algorithm

text: and

text: calculating

text: the

text: entanglement

text: entropy

text: we

text: show

text: that

text: in

text: the

text: anisotropic

text: case

text: a

text: phase

text: transition

text: occurs

text: and

text: a

text: N´eel

text: state

text: emerges

text: above

text: a

text: critical

text: value

text: of

text: the

text: Coulomb

text: interaction.

text: These

text: ﬁndings

text: are

text: also

text: corroborated

text: by

text: the

text: examination

text: of

text: the

text: entanglement

text: spectrum

text: and

text: the

text: spin

text: proﬁle

text: of

text: the

text: system

text: which

text: clarify

text: the

text: structure

text: of

text: each

text: phase.

text: PACS

text: numbers:

text: I.

text: INTRODUCTION

text: In

text: contrast,

text: in

text: the

text: present

text: case

text: since

text: they

text: become

text: nonlo-

text: cal,

text: these

text: singlets

text: are

text: broken

text: at

text: the

text: boundary,

text: and

text: edge

text: states

text: This

text: p-wave

text: Kondo-Heisenberg

text: model

text: Kondo

text: insulators

text: are

text: semiconductors

text: at

text: low

text: tempera-

text: has

text: been

text: investigated

text: with

text: several

text: methods.

text: The

text: large-N

text: tures

text: with

text: a

text: few

text: meV

text: gap

text: in

text: contrast

text: to

text: the

text: conventional

text: expansion

text: has

text: revealed

text: that

text: magnetic

text: end

text: states

text: appear

text: semiconductors

text: in

text: which

text: the

text: gap

text: is

text: in

text: the

text: order

text: of

text: a

text: few

text: at

text: the

text: Analyzing

text: the

text: ground

text: state

text: with

text: It

text: is

text: now

text: generally

text: accepted

text: that

text: strong

text: correlations

text: Abelian

text: and

text: density-matrix

text: renormaliza-

text: among

text: the

text: electrons

text: lead

text: to

text: the

text: stabilization

text: of

text: this

text: state

text: tion

text: group

text: (DMRG)

text: it

text: has

text: turned

text: out

text: that

text: with

text: such

text: a

text: small

text: the

text: ground

text: state

text: is

text: actually

text: the

text: Haldane

text: It

text: In

text: the

text: last

text: decade

text: a

text: new

text: group

text: of

text: insulators

text: were

text: dis-

text: is

text: also

text: interesting

text: to

text: mention

text: that

text: Haldane

text: phase

text: has

text: covered,

text: which

text: are

text: gapped

text: in

text: the

text: bulk,

text: but

text: they

text: have

text: been

text: previously

text: observed

text: in

text: the

text: ordinary

text: Kondo

text: lat-

text: soft,

text: topologically

text: protected

text: surface

text: They

text: are

text: tice

text: model

text: with

text: ferromagnetic

text: Kondo

text: and

text: the

text: topological

text: insulators

text: whose

text: basic

text: properties

text: can

text: be

text: in

text: the

text: extended

text: periodic

text: Anderson

text: model

text: with

text: Hund’s

text: understood

text: within

text: the

text: framework

text: of

text: ordinary

text: band

text: the-

text: The

text: facts

text: above

text: clearly

text: indicates

text: that

text: one

text: ory.

text: The

text: question

text: naturally

text: arises

text: how

text: the

text: correlation

text: has

text: to

text: apply

text: sophisticated

text: many-body

text: techniques

text: to

text: ac-

text: among

text: the

text: electrons

text: aﬀects

text: the

text: properties

text: of

text: topolog-

text: curately

text: describe

text: the

text: ground

text: state.

text: ical

text: insulators.

text: This

text: aspect

text: has

text: attracted

text: signiﬁcant

text: In

text: this

text: paper

text: our

text: aim

text: is

text: to

text: study

text: the

text: stability

text: of

text: the

text: since

text: the

text: existence

text: of

text: topological

text: phases

text: Haldane

text: phase

text: against

text: perturbations.

text: The

text: most

text: obvi-

text: was

text: suggested

text: in

text: strongly

text: correlated

text: These

text: ous

text: and

text: physically

text: relevant

text: choices

text: are

text: either

text: to

text: intro-

text: observations

text: made

text: it

text: necessary

text: to

text: understand,

text: inter

text: alia,

text: duce

text: anisotropy

text: in

text: the

text: Kondo

text: interaction

text: or

text: the

text: inclu-

text: the

text: theoretical

text: properties

text: of

text: topological

text: and

text: sion

text: of

text: a

text: Coulomb

text: interaction

text: in

text: the

text: conduction

text: band.

text: Kondo

text: The

text: most

text: famous

text: example

text: of

text: The

text: role

text: of

text: the

text: Coulomb

text: interaction

text: was

text: explored

text: in

text: the

text: topological

text: Kondo

text: insulators

text: is

text: which

text: was

text: discov-

text: case

text: of

text: the

text: conventional

text: Kondo

text: lattice

text: but

text: ered

text: more

text: than

text: four

text: decades

text: Recent

text: experimental

text: only

text: the

text: weak

text: coupling

text: limit

text: of

text: the

text: present

text: model

text: has

text: that

text: can

text: host

text: robust

text: conducting

text: been

text: studied

text: so

text: far

text: with

text: In

text: contrast,

text: the

text: surface

text: states

text: renewed

text: the

text: interest

text: in

text: this

text: compound

text: in

text: anisotropy

text: was

text: studied

text: only

text: in

text: the

text: conventional

text: model

text: particular

text: and

text: in

text: Kondo

text: insulators

text: in

text: The

text: and

text: its

text: phase

text: diagram

text: was

text: We

text: address

text: most

text: important

text: aspect

text: of

text: topological

text: Kondo

text: insulators

text: is

text: these

text: problems

text: using

text: the

text: DMRG

text: which

text: that

text: hybridization

text: occurs

text: between

text: d

text: and

text: f

text: bands

text: whose

text: is

text: the

text: state-of-the-art

text: tool

text: to

text: determine

text: the

text: ground

text: state

text: parities

text: are

text: opposite

text: to

text: each

text: This

text: leads

text: to

text: the

text: and

text: it

text: makes

text: possible

text: to

text: go

text: beyond

text: the

text: weakly

text: inter-

text: appearance

text: of

text: a

text: symmetry-protected

text: topological

text: ground

text: acting

text: limit.

text: Since

text: the

text: DMRG

text: calculation

text: is

text: closely

text: re-

text: lated

text: to

text: the

text: quantum

text: information

text: theory,

text: we

text: can

text: deter-

text: To

text: capture

text: the

text: eﬀects

text: of

text: strong

text: correlations

text: and

text: topol-

text: mine

text: the

text: entanglement

text: and

text: entanglement

text: ogy

text: simultaneously,

text: Alexandrov

text: and

text: Coleman

text: suggested

text: which

text: are

text: very

text: useful

text: to

text: detect

text: quantum

text: a

text: Kondo

text: lattice

text: model

text: with

text: a

text: special

text: form

text: of

text: the

text: Kondo

text: phase

text: and

text: symmetry-protected

text: topolog-

text: The

text: model

text: consists

text: of

text: a

text: free

text: one-dimensional

text: ical

text: Our

text: analysis

text: of

text: the

text: anisotropic

text: model

text: re-

text: electron

text: gas

text: coupled

text: to

text: an

text: S

text: Heisenberg

text: chain

text: via

text: veals

text: that

text: (i)

text: a

text: phase

text: transition

text: occurs

text: as

text: the

text: interaction

text: a

text: Kondo

text: exchange

text: with

text: p-wave

text: character.

text: In

text: the

text: conven-

text: is

text: increased

text: within

text: the

text: conduction

text: band

text: and

text: the

text: Haldane

text: tional

text: Kondo

text: lattice

text: model,

text: where

text: the

text: Kondo

text: exchange

text: state

text: transforms

text: into

text: a

text: N´eel

text: state,

text: and

text: (ii)

text: the

text: entangle-

text: is

text: on-site,

text: local

text: singlets

text: are

text: formed

text: in

text: the

text: strong

text: coupling

text: ment

text: spectrum

text: can

text: be

text: used

text: as

text: a

text: ﬁngerprint

text: to

text: identify

text: a

text: limit

text: and

text: the

text: boundaries

text: do

text: not

text: play

text: an

text: important

text: role.

text: topological

text: Kondo

text: insulator

text: in

text: one

text: dimension.


text: 1 Comparison-based Image Quality Assessment for Parameter Selection Haoyi Liang, Student Member, IEEE, and Daniel S. Weller, Member, IEEE Abstract— Image quality assessment (IQA) is traditionally IQA algorithms are classiﬁed based on whether the ref- classiﬁed into full-reference (FR) IQA and no-reference (NR) IQA erence image (the distortion-free image) is required: full- according to whether the original image is required. Although reference (FR), reduced-reference (RR) and no-reference NR-IQA is widely used in practical applications, room for (NR). FR-IQA [8]–[13] is a relatively well-studied area. improvement still remains because of the lack of the refer- ence image. Inspired by the fact that in many applications, Traditional methods like mean squared error (MSE) and such as parameter selection, a series of distorted images are signal-to-noise ratio (SNR) are used as the standard signal available, the authors propose a novel comparison-based image ﬁdelity indices [14]. A more sophisticated FR-IQA algorithm, quality assessment (C-IQA) method. The new comparison-based Structural Similarity Index Method (SSIM) [9], considers the framework parallels FR-IQA by requiring two input images, structure information in images and performs well in differ- and resembles NR-IQA by not using the original image. As a result, the new comparison-based approach has more application ent studies [14]–[17]. RR-IQA algorithms [18]–[20] require scenarios than FR-IQA does, and takes greater advantage of the some statistical features of the reference image, such as the accessible information than the traditional single-input NR-IQA power spectrum, and measure the similarity of these features does. Further, C-IQA is compared with other state-of-the-art NR- from the reference image and the distorted image. NR-IQA IQA methods on two widely used IQA databases. Experimental algorithms adopt two different approaches. The ﬁrst kind of results show that C-IQA outperforms the other NR-IQA methods for parameter selection, and the parameter trimming framework NR-IQA [21]–[27] has a similar approach to the RR-IQA. combined with C-IQA saves the computation of iterative image The difference is that rather than extracting the features from reconstruction up to 80%. the reference images, this kind of NR-IQA extracts statistical Index Terms—Image distortion, image quality assessment features from a training set. The second kind of NR-IQA (IQA), human visual system (HVS), comparison-based image algorithms [7], [17] adopts a local approach to quantifying quality assessment (C-IQA), parameter selection structure as a surrogate for quality. A common implementation of the second approach calculates local scores by analyzing the quality of gradient. The overall score is synthesized by taking I. INTRODUCTION the average of the local scores. Obtaining an image with high perceptual quality is the Among these three kinds of IQA algorithms, speed and ultimate goal of many image processing problems, such as accuracy generally decrease from FR-IQA, RR-IQA to NR- image reconstruction, denoising and inpaiting. However, mea- IQA progressively. Unfortunately, reference images do not suring the perceptual image quality by subjective experiment is exist in many cases. Noticing that in many applications, time-consuming and expensive, so designing an image quality including parameter selection and comparison of different assessment (IQA) algorithm that agrees with the human visual restoration algorithms, we compare a set of distorted images system (HVS) [1]–[5] is a foundational image processing with the same image content, a new comparison-based IQA objective. Moreover, most image restoration algorithms require (C-IQA) method is proposed in this paper. The prototype one or more parameters to regulate the restoration process, of comparison-based IQA is reﬂected in [23]. However, the and no-reference IQA methods can be used to guide selecting concept of comparison in [23] is just implicitly mentioned the parameters. For instance, the regularization parameter of by sorting the overall image qualities of a series of distorted image reconstruction [6] is selected by a no-reference image images. In our work, the comparison-based framework is built quality index [7]. However, most existing no-reference IQA al- from low level image structures, and the ﬁnal output is a graph gorithms output the estimated image quality based on a single that can illustrate the local relative quality. distorted image, ignoring that different degraded images can After proposing the comparison-based IQA method, we provide more information together to the quality estimation of demonstrate one of its applications, parameter selection. The each degraded image. This observation inspires us to develop framework of parameter trimming, ﬁrst proposed in [6], is a comparison-based IQA method to ﬁll the gap between the designed to boost the parameter selection by combining NR- increasing need of parameter selection for image processing IQA with parameter selection. In [6], parameters that do not algorithms and the lack of such a NR-IQA algorithm that show the potential to obtain the best result are cut during the makes full use of the available information. convergence process. The rest of the paper is organized as follows. Section Haoyi Liang and Daniel S. Weller are with the Charles L. Brown introduces and compares different NR-IQA methods. Section Department of Electrical and Computer Engineering, University of elaborates on the implementation details of C-IQA. The Virginia, Charlottesville, VA 22904 USA (email: hl2uc@virginia.edu, dweller@virginia.edu). framework of parameter trimming and the technique used

label: paragraph

text: Comparison-based

label: title

text: Image

text: Quality

text: Assessment

text: for

text: Parameter

text: Selection

text: Haoyi

label: author

text: Liang,

text: Student

text: Member,

text: IEEE,

text: and

text: Daniel

text: S.

text: Weller,

text: Member,

text: IEEE

text: Abstract—

text: Image

text: quality

text: assessment

text: (IQA)

text: is

text: traditionally

text: IQA

text: algorithms

text: are

text: classiﬁed

text: based

text: on

text: whether

text: the

text: ref-

text: classiﬁed

text: into

text: full-reference

text: (FR)

text: IQA

text: and

text: no-reference

text: (NR)

text: IQA

text: erence

text: image

text: (the

text: distortion-free

text: image)

text: is

text: required:

text: full-

text: according

text: to

text: whether

text: the

text: original

text: image

text: is

text: required.

text: Although

text: reference

text: (FR),

text: reduced-reference

text: (RR)

text: and

text: no-reference

text: NR-IQA

text: is

text: widely

text: used

text: in

text: practical

text: applications,

text: room

text: for

text: (NR).

text: FR-IQA

text: is

text: a

text: relatively

text: well-studied

text: area.

text: improvement

text: still

text: remains

text: because

text: of

text: the

text: lack

text: of

text: the

text: refer-

text: ence

text: image.

text: Inspired

text: by

text: the

text: fact

text: that

text: in

text: many

text: applications,

text: Traditional

text: methods

text: like

text: mean

text: squared

text: error

text: (MSE)

text: and

text: such

text: as

text: parameter

text: selection,

text: a

text: series

text: of

text: distorted

text: images

text: are

text: signal-to-noise

text: ratio

text: (SNR)

text: are

text: used

text: as

text: the

text: standard

text: signal

text: available,

text: the

text: authors

text: propose

text: a

text: novel

text: comparison-based

text: image

text: ﬁdelity

text: indices

text: A

text: more

text: sophisticated

text: FR-IQA

text: algorithm,

text: quality

text: assessment

text: (C-IQA)

text: method.

text: The

text: new

text: comparison-based

text: Structural

text: Similarity

text: Index

text: Method

text: (SSIM)

text: considers

text: the

text: framework

text: parallels

text: FR-IQA

text: by

text: requiring

text: two

text: input

text: images,

text: structure

text: information

text: in

text: images

text: and

text: performs

text: well

text: in

text: differ-

text: and

text: resembles

text: NR-IQA

text: by

text: not

text: using

text: the

text: original

text: image.

text: As

text: a

text: result,

text: the

text: new

text: comparison-based

text: approach

text: has

text: more

text: application

text: ent

text: studies

text: RR-IQA

text: algorithms

text: require

text: scenarios

text: than

text: FR-IQA

text: does,

text: and

text: takes

text: greater

text: advantage

text: of

text: the

text: some

text: statistical

text: features

text: of

text: the

text: reference

text: image,

text: such

text: as

text: the

text: accessible

text: information

text: than

text: the

text: traditional

text: single-input

text: NR-IQA

text: power

text: spectrum,

text: and

text: measure

text: the

text: similarity

text: of

text: these

text: features

text: does.

text: Further,

text: C-IQA

text: is

text: compared

text: with

text: other

text: state-of-the-art

text: NR-

text: from

text: the

text: reference

text: image

text: and

text: the

text: distorted

text: image.

text: NR-IQA

text: IQA

text: methods

text: on

text: two

text: widely

text: used

text: IQA

text: databases.

text: Experimental

text: algorithms

text: adopt

text: two

text: different

text: approaches.

text: The

text: ﬁrst

text: kind

text: of

text: results

text: show

text: that

text: C-IQA

text: outperforms

text: the

text: other

text: NR-IQA

text: methods

text: for

text: parameter

text: selection,

text: and

text: the

text: parameter

text: trimming

text: framework

text: NR-IQA

text: has

text: a

text: similar

text: approach

text: to

text: the

text: RR-IQA.

text: combined

text: with

text: C-IQA

text: saves

text: the

text: computation

text: of

text: iterative

text: image

text: The

text: difference

text: is

text: that

text: rather

text: than

text: extracting

text: the

text: features

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text: quality.

text: distorted

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text: comparison-based

text: IQA

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text: applications,

text: parameter

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text: each

text: degraded

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text: framework

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text: parameter

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text: parameters

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text: algorithm

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text: best

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text: rest

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text: the

text: paper

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text: Section

text: Haoyi

text: Liang

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text: University

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text: the

text: implementation

text: details

text: of

text: C-IQA.

text: The

text: Virginia,

text: Charlottesville,

text: VA

text: USA

text: (email:

text: dweller@virginia.edu).

text: framework

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text: technique

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text: 9 Algorithm 5 Key Image DIIVINE Metric Q C-IQA(rank) BRISQUE Anisotropy CT-IQA(rank) Key Images Selection; LIVE CSIQ key img = [1] 0.4 0.4 key = 1 num for i = 1 : N do if MSE(I , I ) > K then 0.2 0.2 key imig p=re[kkeey(ykeiymnugm,)i] thresh key = key + 1 num num end if end for 0 0 0.85 0.9 0.95 0.85 0.9 0.95 Distortion level (average of SSIM indexes) Key Images Comparision; − for i = 2 : (key 1) do num Fig. 10: Weighted inversion numbers of different NR-IQA if CQ(I , I − ) > 0 and key img(i) key img(i 1) methods at 14 distortion levels CQ(I , I ) > 0 then key img(i) key img(i+1) best = i key break; eight distorted images in each set according to different IQA end if algorithms. The weighted inversion numbers [52] between the end for ranking results by NR-IQA algorithms and by SSIM are used to evaluate the performance of different NR-IQA algorithms. Best Image Selection; · · · − Assume (I , , I ) is the ranking sequence according to start = key img(best 1) 1 N num key a NR-IQA alg(cid:88)orithm (cid:88)from low quality to high quality, the end = key img(best + 1) num key weighted inversion number in our experiment is deﬁned as for i = start : end do num num − score (i) = CQ(I(i), I(start )) start num WInv = max(0, SSIM(I ) SSIM(I )). num i j score (i) = CQ(I(i), I(end )) end num i=1:N j=i+1:N end for Since C-IQA/CT-IQA are rank-based algorithms, distorted best = max(score + score ) img start end images in one set are sorted by the bubble sort algorithm. The bubble sort algorithm is the same as the traditional one, except that we use C-IQA/CT-IQA to compare each adjacent 0.25 pair of images in the ranking sequence. In our experiment, 0.25 0.2 the ﬁnal ranking results are not sensitive to the initial ranking 0.2 0.15 and we start the bubble sort from a random rank. Fig. 10 0.15 0.1 0.1 shows weighted inversion numbers at 14 distortion levels on 0.05 0.05 two databases. 0 0 In Table I, we provide the average weighted inversion num- DII BRI MQ Ani CQ CTQ DII BRI MQ Ani CQ CTQ Different IQAs Different IQAs bers over 14 distortion levels of different NR-IQA algorithms. (a) LIVE (b) CSIQ It is clear C-IQA and CT-IQA are two of the best NR-IQA algorithms. Fig. 11: The SSIM differences between the best images chosen by IQA method and by SSIM from denoised images A series of increasingly denoised images by bilateral ﬁl- · · · tering, I , I , , I , is created for each image the same as 1 2 30 the previous part. The SSIM index of the most oversmoothed From Fig. 11, we can see that MetricQ and CT-IQA are ± image I is between 0.85 0.01. Because the MSD between two of the best NR-IQA algorithms. Table II provides more 30 the adjacent images are below minimum resolution, Alg. 5 quantitative evaluations of different NR-IQA algorithms. is adopted to select the best result. Key images are a set of In order to have a better understanding of Comparison- images among which the MSD is greater than the minimum based IQA, we show two outliers of C-IQA/CT-IQA in Fig. resolution. Alg. 5 ﬁrst separates the 30 increasingly distorted 12. Without the knowledge of the contents in the scenes, images into a few parts by key images. Images in the two ﬁne textures are regarded as noise in these two images, and parts next to the best key image are evaluated based on the both C-IQA/CT-IQA choose oversmoothed images as the best two key images on the ends. By doing so, we avoid comparing denoised results. the adjacent images directly. We set K = 3.0 in this thresh experiment. The algorithm used for image reconstruction is introduced in The SSIM index difference between the best images chosen Section III-B0b. In the experiment, 70% Fourier transform data by a NR-IQA method and the one chosen by SSIM is used to are used to reconstruct the image and in order to be more re- evaluate NR-IQA methods. There are 59 original images in the alistic, Fourier transform data are distorted by Gaussian noise. two databases and 1770 distorted images in our experiment. The SNR is kept at 20 dB in all reconstruction experiments. 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text: original

text: images

text: in

text: the

text: alistic,

text: Fourier

text: transform

text: data

text: are

text: distorted

text: by

text: Gaussian

text: noise.

text: two

text: databases

text: and

text: distorted

text: images

text: in

text: our

text: experiment.

text: The

text: SNR

text: is

text: kept

text: at

text: dB

text: in

text: all

text: reconstruction

text: experiments.

text: All

text: noisrevnI

text: dethgieW

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text: dethgieW

text: ecnereffiD

text: MISS

text: ecnereffiD

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text: 24 SVEN MEINHARDT → → \ → ⊂ (1) [V X] = [Z X] + [V Z X] for all closed Z V, Con Con Con C −i→d C (2) 1 = [Spec Spec ] , Con ∗ → × → → (3) u ([W Y ] ) = [X W X] for all u : X Y, Con Y Con −→ −−◦→ → v u v (4) u ([V X] ) = [V Y ] if u : X Y is of ﬁnite type, ! Con Con −→ (cid:2) −→ × −−×−→ × v w v w (5) [V X] [W Y ] = [V W X Y ] , Con Con Con → → (6) σn([V X] ) = [Symn(V ) Symn(X)] . Con Con −→ (cid:48) −→(cid:48) v v (7) [V X] = [V X] if there is an isomorphism Con Con (cid:48)(cid:48) → (cid:48) (cid:48) ◦ (cid:48)(cid:48) v : V V such that v = v v , Obviously, the same must hold in every motivic category as they share the same properties, and so the same applies to the initial object if it exists. Moreover, for → connected X the group Con(X) is generated by all classes [V X] satisfying Con relation (1). The same must be true for the initial motivic theory since otherwise → the subgroup spanned by the elements [V X] for connected X and extended init by Property 4.3(i) for non-connected X is a proper subtheory of the initial theory which will lead to a contradiction. However, there are more relations in Con(X) as G −z→d G G −i→d G for example [ ] = d[ ] which might not hold in other motivic m m m m theories as for example the initial one. Dropping the subscript “init” we will, there- fore, deﬁne our (hopefully) initial theory by associating to every connected scheme −→ v X the group K (Sch ) generated by symbols [V X] for every isomorphism class 0 X → (due to equation (7)) of morphisms v : V (cid:89)X of ﬁnite type, subject to the relation (1). For non-connected X we simply put K (Sch ) := K (Sch ). 0 X 0 Xi ∈ Xi π0(X) ∈ ∗ (cid:2) To obtain a motivic theory K (Sch), we must deﬁne 1 K (SchC), u , u , and 0 0 ! σn as in equations (2)–(6), at least over connected components. It has been shown in [10] that σn-operations satisfying these properties do indeed exist. Moreover, L L ∈ the authors prove that σn(a ) = σn(a) n holds for every a K (X) and every n ∈ N, were L = c c∗(1) = [A1 →−c Spec C] ∈ K (SchC) is considered0 as an element ! 0 → of K (Sch ) via the embedding 0 : K (SchC) (cid:44) K (Sch ). 0 Sym(X) ! 0 0 Sym(X) Exercise 4.8. Use the properties of a morphism between motivic theories to show → (cid:55)→ → → that [V X] [V X] deﬁnes a homomorphism η : K (Sch ) R(X) for R X→ 0 X connected X which extends to a morphism η : K (Sch) R of motivic theories. 0 Prove that this morphism is the only possible one. Hence, K (Sch) is the initial 0 object in the category of motivic theories. Moreover, show that η is a λ-morphism. The initial property of K (Sch) is just a generalization of the well-known property → ∈ 0 (cid:81) that X [X] K (SchC) is the universal Euler characteristics. ⊂ 0 → Let Rgm(X) R(X) be the subgroup generated by all elements [V X] if X is R connected and Rgm(X) := ∈ Rgm(X ) for general X. One should think of Xi π0(X) i Z elements in Rgm(X) as “geometric” since they are -linear combinations of elements obtained by geometric constructions, namely pull-backs and push-forwards of the ∈ C unit 1 R(Spec ). ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: SVEN

text: MEINHARDT

label: equation

text: [V

text: X]

text: [Z

text: X]

text: [V

text: Z

text: X]

text: for

text: all

text: closed

text: Z

text: V,

text: Con

text: Con

text: Con

text: C

text: −i→d

text: C

text: [Spec

text: Spec

text: Con

text: u

text: ([W

text: Y

text: [X

text: W

text: X]

text: for

text: all

text: u

text: X

text: Y,

text: Con

text: Y

text: Con

text: v

text: u

text: v

text: u

text: ([V

text: X]

text: [V

text: Y

text: if

text: u

text: X

text: Y

text: is

text: of

text: ﬁnite

text: type,

text: Con

text: Con

text: v

text: w

text: v

text: w

text: [V

text: X]

text: [W

text: Y

text: [V

text: W

text: X

text: Y

text: Con

text: Con

text: Con

text: σn([V

text: X]

text: [Symn(V

text: Symn(X)]

text: Con

text: Con

text: v

text: v

text: [V

text: X]

text: [V

text: X]

text: if

text: there

text: is

text: an

text: isomorphism

text: Con

text: Con

text: v

text: V

text: V

text: such

text: that

text: v

text: v

text: v

text: Obviously,

text: the

text: same

text: must

text: hold

text: in

text: every

text: motivic

text: category

text: as

text: they

text: share

text: the

text: same

text: properties,

text: and

text: so

text: the

text: same

text: applies

text: to

text: the

text: initial

text: object

text: if

text: it

text: exists.

text: Moreover,

text: for

text: connected

text: X

text: the

text: group

text: Con(X)

text: is

text: generated

text: by

text: all

text: classes

text: [V

text: X]

text: satisfying

text: Con

text: relation

text: The

text: same

text: must

text: be

text: true

text: for

text: the

text: initial

text: motivic

text: theory

text: since

text: otherwise

text: the

text: subgroup

text: spanned

text: by

text: the

text: elements

text: [V

text: X]

text: for

text: connected

text: X

text: and

text: extended

text: init

text: by

text: Property

text: for

text: non-connected

text: X

text: is

text: a

text: proper

text: subtheory

text: of

text: the

text: initial

text: theory

text: which

text: will

text: lead

text: to

text: a

text: contradiction.

text: However,

text: there

text: are

text: more

text: relations

text: in

text: Con(X)

text: as

text: G

text: −z→d

text: G

text: G

text: −i→d

text: G

text: for

text: example

text: d[

text: which

text: might

text: not

text: hold

text: in

text: other

text: motivic

text: m

text: m

text: m

text: m

text: theories

text: as

text: for

text: example

text: the

text: initial

text: one.

text: Dropping

text: the

text: subscript

text: “init”

text: we

text: will,

text: there-

text: fore,

text: deﬁne

text: our

text: (hopefully)

text: initial

text: theory

text: by

text: associating

text: to

text: every

text: connected

text: scheme

text: v

text: X

text: the

text: group

text: K

text: (Sch

text: generated

text: by

text: symbols

text: [V

text: X]

text: for

text: every

text: isomorphism

text: class

text: X

text: (due

text: to

text: equation

text: of

text: morphisms

text: v

text: V

text: of

text: ﬁnite

text: type,

text: subject

text: to

text: the

text: relation

text: For

text: non-connected

text: X

text: we

text: simply

text: put

text: K

text: (Sch

text: K

text: (Sch

text: X

text: Xi

text: Xi

text: To

text: obtain

text: a

text: motivic

text: theory

text: K

text: (Sch),

text: we

text: must

text: deﬁne

text: K

text: (SchC),

text: u

text: u

text: and

text: σn

text: as

text: in

text: equations

text: at

text: least

text: over

text: connected

text: components.

text: It

text: has

text: been

text: shown

text: in

text: that

text: σn-operations

text: satisfying

text: these

text: properties

text: do

text: indeed

text: exist.

text: Moreover,

text: L

text: L

text: the

text: authors

text: prove

text: that

text: σn(a

text: σn(a)

text: n

text: holds

text: for

text: every

text: a

text: K

text: (X)

text: and

text: every

text: n

text: N,

text: were

text: L

text: c

text: →−c

text: Spec

text: C]

text: K

text: (SchC)

text: is

text: as

text: an

text: element

text: of

text: K

text: (Sch

text: via

text: the

text: embedding

text: K

text: (SchC)

text: K

text: (Sch

text: Sym(X)

text: Sym(X)

text: Exercise

text: Use

text: the

text: properties

text: of

text: a

text: morphism

text: between

text: motivic

text: theories

text: to

text: show

text: that

text: [V

text: X]

text: [V

text: X]

text: deﬁnes

text: a

text: homomorphism

text: η

text: K

text: (Sch

text: R(X)

text: for

text: R

text: X→

text: X

text: connected

text: X

text: which

text: extends

text: to

text: a

text: morphism

text: η

text: K

text: (Sch)

text: R

text: of

text: motivic

text: theories.

text: Prove

text: that

text: this

text: morphism

text: is

text: the

text: only

text: possible

text: one.

text: Hence,

text: K

text: (Sch)

text: is

text: the

text: initial

text: object

text: in

text: the

text: category

text: of

text: motivic

text: theories.

text: Moreover,

text: show

text: that

text: η

text: is

text: a

text: λ-morphism.

text: The

text: initial

text: property

text: of

text: K

text: (Sch)

text: is

text: just

text: a

text: generalization

text: of

text: the

text: well-known

text: property

text: that

text: X

text: [X]

text: K

text: (SchC)

text: is

text: the

text: universal

text: Euler

text: characteristics.

text: Let

text: Rgm(X)

text: R(X)

text: be

text: the

text: subgroup

text: generated

text: by

text: all

text: elements

text: [V

text: X]

text: if

text: X

text: is

text: R

text: connected

text: and

text: Rgm(X)

text: Rgm(X

text: for

text: general

text: X.

text: One

text: should

text: think

text: of

text: Xi

text: i

text: Z

text: elements

text: in

text: Rgm(X)

text: as

text: “geometric”

text: since

text: they

text: are

text: -linear

text: combinations

text: of

text: elements

text: obtained

text: by

text: geometric

text: constructions,

text: namely

text: pull-backs

text: and

text: push-forwards

text: of

text: the

text: C

text: unit

text: R(Spec

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 32 SVEN MEINHARDT Proof. To show that Igm is a λ-ideal, it suﬃces to look at generators [V ×A1 −f−×−id−A→1 X X × A1] and [W −(−g−,h→) Y × A1] of Igm and RGgm(Y × A1) respectively with [V −→f ∈ X (cid:2) m X] Rgm(X). (cf. Exercise 5.11) For the -product [V × A1 → X × A1] (cid:2) [W → Y × A1] = [V × A1 × W −→u X × Y × A1] with u(v, z, w) = (f(v), g(w), z + h(w)) we use the isomorphism × A × (cid:51) (cid:55)−→ ∈ × × A V 1 W (v, z, w) (v, w, z + h(w)) V W 1 to show [V × A1 × W −→u X × Y × A1] = pr∗ × ([V × W −f−×−→g X × Y ]) ∈ Igm× . X Y X Y We also have (cid:0) σn[V × A11 −f−×−id−A→1 (cid:1)X ×(cid:0) A1] = [Symn(V × A1) −→p˜ Symn(cid:1)(X) × A1] ×A → ×A with p˜ being induced by the S -invariant morphism p : (V 1)n Symn(X) 1 n with p (v , z ), . . . , (v , z ) = (f(v ), . . . , f(v )), z + . . . + z . We deﬁne 1 1 n n 1 n 1 n (V n × An)0 := {(v , . . . , v , z , . . . ,(cid:0)z ) ∈ V n × An | z + . . . z = 0}(cid:1) 1 n 1 n 1 n ×A ×A −→ × for all n > 0. There is a S -equivariant isomorphism ψ : (V n n)0 1 (V A n 1)n sending ((v , . . . , v , z , . . . , z ), z) to (v , z +z/n), . . . , (v , z +z/n) . Then, ◦ × 1 ×An 1×A −n→ 1 ×1A n n p ψ = q idA : (V n n)0 1 Symn(X) 1 for the S -invariant morphism 1 n q : (V n × An)0 (cid:16) (V n × An)0//S −→q˜ Symn(X) with q(v , . . . , v , z , . . . , z ) = n 1 n 1 n (f(v ), . . . , f(v )). Modding out the S -action, we see that 1 n n [Symn(V × A1) −→p˜ Symn(X) × A1] = pr∗ ([(V n × A1n)0//S −→q˜ Symn(X)]) Symn n (cid:3) is indeed in Igm . Symn(X) (cid:80) (cid:2) Exercise 5.11. Convince yourself using the formula σn(a+b) = n π(l)(σl(a) − − × A × l=0 −! −→ σn l(b)) for the motivic theory R( 1) with π(l) : Syml(X) Symn 1(X) ∈ ∈ Symn(X) being the natural map, that a Igm implies σn(a) Igm is indeed X Symn(X) true if it already holds for generators a = pr∗ ([V −→f X]) of Igm. X X Due to Lemma 5.9, we can form the quotient Rgm (X) = RGgm(X×A1)/ pr∗ Rgm(X) and obtain a new motivic theory together withmaomn orphism Rmgm → Rgm oXf motivic theories (cf. Exercise 4.9) given by Rgm(X) −(−id−×−0−→)! RGgm(X × A1) (cid:16)moRngm (X). mon m Exercise 5.12. Fix a motivic theory R and consider the map RGgm(X × A1) −→ (cid:55)−→ × ∗ − × ∗ mC → A Rgm(X) given by a (id 0) (a) (id 1) (a), where 0, 1 : Spec 1 are the ∗ obvious maps. As pr Rgm(X) is in the kernel, we obtain a well-deﬁned group X → → homomorphism Rgm (X) Rgm(X). Note that the composition Rgm(X) → mon Rgm (X) Rgm(X) is the identity. Hence, Rgm(X) is a direct summand of mon → Rgm (X). However, show that the retraction Rgm (X) Rgm(X) is not a mor- mon mon phism of motivic theories and Rgm is not a direct summand of the motivic theory Rgm . mon Exercise 5.13. Using the notation of the previous exercise, show that the kernel −→ of Congm (X) Congm(X) = Con(X) is trivial, i.e. Con(X) = Congm (X). mon C mon On the other hand, show that the kernel is nonzero for X = Spec and R = − Q − K (Db( , )), R = K (MHM( ) ) and R = K (Sch). 0 0 mon 0 ##LTLine## ##LTLine## ##LTLine##

text: SVEN

text: MEINHARDT

text: Proof.

text: To

text: show

text: that

text: Igm

text: is

text: a

text: λ-ideal,

text: it

text: suﬃces

text: to

text: look

text: at

text: generators

text: [V

text: −f−×−id−A→1

text: X

text: X

text: and

text: [W

text: −(−g−,h→)

text: Y

text: of

text: Igm

text: and

text: RGgm(Y

text: respectively

text: with

text: [V

text: −→f

text: X

text: m

text: X]

text: Rgm(X).

text: (cf.

text: Exercise

text: For

text: the

text: -product

text: [V

text: X

text: [W

text: Y

text: [V

text: W

text: −→u

text: X

text: Y

text: with

text: u(v,

text: z,

text: w)

text: (f(v),

text: g(w),

text: z

text: h(w))

text: we

text: use

text: the

text: isomorphism

text: A

text: A

text: V

text: W

text: (v,

text: z,

text: w)

text: (v,

text: w,

text: z

text: h(w))

text: V

text: W

text: to

text: show

text: [V

text: W

text: −→u

text: X

text: Y

text: pr∗

text: ([V

text: W

text: −f−×−→g

text: X

text: Y

text: Igm×

text: X

text: Y

text: X

text: Y

text: We

text: also

text: have

text: σn[V

text: −f−×−id−A→1

text: [Symn(V

text: −→p˜

text: ×A

text: ×A

text: with

text: p˜

text: being

text: induced

text: by

text: the

text: S

text: -invariant

text: morphism

text: p

text: (V

text: Symn(X)

text: n

text: with

text: p

text: (v

text: z

text: (v

text: z

text: (f(v

text: f(v

text: z

text: z

text: We

text: deﬁne

text: n

text: n

text: n

text: n

text: (V

text: n

text: {(v

text: v

text: z

text: V

text: n

text: An

text: z

text: z

text: n

text: n

text: n

text: ×A

text: ×A

text: for

text: all

text: n

text: There

text: is

text: a

text: S

text: -equivariant

text: isomorphism

text: ψ

text: (V

text: n

text: (V

text: A

text: n

text: sending

text: ((v

text: v

text: z

text: z

text: z)

text: to

text: (v

text: z

text: +z/n),

text: (v

text: z

text: +z/n)

text: Then,

text: ×An

text: −n→

text: n

text: n

text: p

text: ψ

text: q

text: idA

text: (V

text: n

text: Symn(X)

text: for

text: the

text: S

text: -invariant

text: morphism

text: n

text: q

text: (V

text: n

text: (V

text: n

text: −→q˜

text: Symn(X)

text: with

text: q(v

text: v

text: z

text: z

text: n

text: n

text: n

text: (f(v

text: f(v

text: Modding

text: out

text: the

text: S

text: -action,

text: we

text: see

text: that

text: n

text: n

text: [Symn(V

text: −→p˜

text: Symn(X)

text: pr∗

text: ([(V

text: n

text: −→q˜

text: Symn(X)])

text: Symn

text: n

text: is

text: indeed

text: in

text: Igm

text: Symn(X)

text: Exercise

text: Convince

text: yourself

text: using

text: the

text: formula

text: σn(a+b)

text: n

text: π(l)(σl(a)

text: A

text: σn

text: l(b))

text: for

text: the

text: motivic

text: theory

text: R(

text: with

text: π(l)

text: Syml(X)

text: Symn

text: Symn(X)

text: being

text: the

text: natural

text: map,

text: that

text: a

text: Igm

text: implies

text: σn(a)

text: Igm

text: is

text: indeed

text: X

text: Symn(X)

text: true

text: if

text: it

text: already

text: holds

text: for

text: generators

text: a

text: pr∗

text: ([V

text: −→f

text: X])

text: of

text: Igm.

text: X

text: X

text: Due

text: to

text: Lemma

text: we

text: can

text: form

text: the

text: quotient

text: Rgm

text: (X)

text: pr∗

text: Rgm(X)

text: and

text: obtain

text: a

text: new

text: motivic

text: theory

text: together

text: withmaomn

text: orphism

text: Rmgm

text: Rgm

text: oXf

text: motivic

text: theories

text: (cf.

text: Exercise

text: given

text: by

text: Rgm(X)

text: −(−id−×−0−→)!

text: RGgm(X

text: (X).

text: mon

text: m

text: Exercise

text: Fix

text: a

text: motivic

text: theory

text: R

text: and

text: consider

text: the

text: map

text: RGgm(X

text: mC

text: A

text: Rgm(X)

text: given

text: by

text: a

text: (id

text: (a)

text: (id

text: (a),

text: where

text: Spec

text: are

text: the

text: obvious

text: maps.

text: As

text: pr

text: Rgm(X)

text: is

text: in

text: the

text: kernel,

text: we

text: obtain

text: a

text: well-deﬁned

text: group

text: X

text: homomorphism

text: Rgm

text: (X)

text: Rgm(X).

text: Note

text: that

text: the

text: composition

text: Rgm(X)

text: mon

text: Rgm

text: (X)

text: Rgm(X)

text: is

text: the

text: identity.

text: Hence,

text: Rgm(X)

text: is

text: a

text: direct

text: summand

text: of

text: mon

text: Rgm

text: (X).

text: However,

text: show

text: that

text: the

text: retraction

text: Rgm

text: (X)

text: Rgm(X)

text: is

text: not

text: a

text: mor-

text: mon

text: mon

text: phism

text: of

text: motivic

text: theories

text: and

text: Rgm

text: is

text: not

text: a

text: direct

text: summand

text: of

text: the

text: motivic

text: theory

text: Rgm

text: mon

text: Exercise

text: Using

text: the

text: notation

text: of

text: the

text: previous

text: exercise,

text: show

text: that

text: the

text: kernel

text: of

text: Congm

text: (X)

text: Congm(X)

text: Con(X)

text: is

text: trivial,

text: i.e.

text: Con(X)

text: Congm

text: (X).

text: mon

text: C

text: mon

text: On

text: the

text: other

text: hand,

text: show

text: that

text: the

text: kernel

text: is

text: nonzero

text: for

text: X

text: Spec

text: and

text: R

text: Q

text: K

text: (Db(

text: R

text: K

text: (MHM(

text: and

text: R

text: K

text: (Sch).

text: mon

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: AN INTRODUCTION INTO (MOTIVIC) DONALDSON–THOMAS THEORY 47 by the identity on the factors M not being part of an extension. × × (cid:111)(cid:111) × (cid:47)(cid:47) × Md Md(cid:79)(cid:79) (cid:48) Md(cid:48)(cid:48) Md M(cid:79)(cid:79) d(cid:48),d(cid:48)(cid:48) Md M(cid:79)(cid:79) d(cid:48)+d(cid:48)(cid:48) × (cid:111)(cid:111) (cid:47)(cid:47) M (cid:48) M (cid:48)(cid:48) M (cid:48) (cid:48)(cid:48) M (cid:48) (cid:48)(cid:48) d,d d d,d ,d d,d +d (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) × (cid:111)(cid:111) (cid:47)(cid:47) M (cid:48) M (cid:48)(cid:48) M (cid:48) (cid:48)(cid:48) M (cid:48) (cid:48)(cid:48) d+d d d+d ,d d+d +d Show that the diagram commutes and that every square is cartesian. (2) Use this diagram and the base change property of a motivic theory to prove associativity of the Ringel–Hall product. C −→0 (3) The zero representation induces a map Spec M. Show that 1 := ∈ 0 0 (1) R(M) is a unit for the Ringel–Hall product. ! Let us form the following cartesian product Mζ−s(cid:48)s(cid:31) (cid:127) (cid:47)(cid:47) Exact π2 (cid:47)(cid:47) M µ,µ × (cid:15)(cid:15) (cid:15)(cid:15) π1 π3 Mζ−ss × Mζ−(cid:48) ss(cid:31) (cid:127) (cid:47)(cid:47) M × M. µ µ (cid:48) − → If µ > µ , the composition Mζ s(cid:48)s M is an isomorphism onto the image which is µ,µ the substack of M consisting of all representations whose Harder–Narasimhan ﬁl- − − tration has only one subquotient in Mζ ss and another one in Mζ(cid:48) ss. Indeed, the µ µ functoriality of the Harder–Narasimhan ﬁltration ensures that taking the Harder– − → Narasimhan ﬁltration provides an inverse morphism to Mζ s(cid:48)s M. Fixing an- (cid:48)(cid:48) (cid:48) (cid:48)(cid:48) µ,µ other slope µ with µ > µ we continue this way and take the ﬁber product Mζ−s(cid:48)s (cid:48)(cid:48)(cid:31) (cid:127) (cid:47)(cid:47) Exact π2 (cid:47)(cid:47) M µ,µ ,µ × (cid:15)(cid:15) (cid:15)(cid:15) π1 π3 Mζ−s(cid:48)s × Mζ−(cid:48)(cid:48)ss(cid:31) (cid:127) (cid:47)(cid:47) M × M. µ,µ µ which can be identiﬁed with the substack of representations having a Harder– − − − Narasimhan ﬁltration with subquotients in Mζ ss, Mζ(cid:48) ss, Mζ(cid:48)(cid:48)ss. As every quiver µ µ µ representation has a unique Harder–Narasimhan ﬁltration, we obtain a locally ﬁ- \{ } − \{ } → \{ } − nite stratiﬁcation of M 0 with strata Mζ ss 0 (cid:44) M 0 , where Mζ ss − µ1,...,µr µ1,...,µr is deﬁned as above by means of r 1 ﬁber products for every strictly decreasing −∞ ∞ sequence µ > . . . > µ in ( , + ] of length r. Using the notation 1 r − → ∈ δµζ1,...,µr := (Mζµ1,s..s.,µr(cid:88)(cid:44) M)!(1Mζµ−1,s.s..,µr ) R(M), this stratiﬁcation can be written as − 1M = 10 + ∈N (δµζ1,...,µr 10). 0<r µ1>...>µr ∗ ∗ Exercise 6.24. Prove the formula δζ = δζ . . . δζ for all sequences µ > µ1,...,µr µ1 µr 1 . . . > µ of real numbers. r Applying the formula proven in the exercise we can rewrite the inﬁnite sum as an inﬁnite product ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: AN

text: INTRODUCTION

text: INTO

text: (MOTIVIC)

text: DONALDSON–THOMAS

text: THEORY

text: by

label: list

text: the

text: identity

text: on

text: the

text: factors

text: M

text: not

text: being

text: part

text: of

text: an

text: extension.

text: Md

text: Md

text: Md

text: M

text: M

text: M

text: M

text: d,d

text: d

text: d,d

text: ,d

text: d,d

text: +d

text: M

text: M

text: M

text: M

text: d+d

text: d

text: d+d

text: ,d

text: d+d

text: +d

text: Show

text: that

text: the

text: diagram

text: commutes

text: and

text: that

text: every

text: square

text: is

text: cartesian.

text: Use

text: this

text: diagram

text: and

text: the

text: base

text: change

text: property

text: of

text: a

text: motivic

text: theory

text: to

text: prove

text: associativity

text: of

text: the

text: Ringel–Hall

text: product.

text: C

text: −→0

text: The

text: zero

text: representation

text: induces

text: a

text: map

text: Spec

text: M.

text: Show

text: that

text: R(M)

text: is

text: a

text: unit

text: for

text: the

text: Ringel–Hall

text: product.

text: Let

text: us

text: form

text: the

text: following

text: cartesian

text: product

text: Exact

text: π2

text: M

text: µ,µ

text: π1

text: π3

text: Mζ−ss

text: M

text: M.

text: µ

text: µ

text: If

text: µ

text: µ

text: the

text: composition

text: Mζ

text: M

text: is

text: an

text: isomorphism

text: onto

text: the

text: image

text: which

text: is

text: µ,µ

text: the

text: substack

text: of

text: M

text: consisting

text: of

text: all

text: representations

text: whose

text: Harder–Narasimhan

text: ﬁl-

text: tration

text: has

text: only

text: one

text: subquotient

text: in

text: Mζ

text: ss

text: and

text: another

text: one

text: in

text: ss.

text: Indeed,

text: the

text: µ

text: µ

text: functoriality

text: of

text: the

text: Harder–Narasimhan

text: ﬁltration

text: ensures

text: that

text: taking

text: the

text: Harder–

text: Narasimhan

text: ﬁltration

text: provides

text: an

text: inverse

text: morphism

text: to

text: Mζ

text: M.

text: Fixing

text: an-

text: µ,µ

text: other

text: slope

text: µ

text: with

text: µ

text: µ

text: we

text: continue

text: this

text: way

text: and

text: take

text: the

text: ﬁber

text: product

text: Exact

text: π2

text: M

text: µ,µ

text: ,µ

text: π1

text: π3

text: M

text: M.

text: µ,µ

text: µ

text: which

text: can

text: be

text: identiﬁed

text: with

text: the

text: substack

text: of

text: representations

text: having

text: a

text: Harder–

text: Narasimhan

text: ﬁltration

text: with

text: subquotients

text: in

text: Mζ

text: ss,

text: ss,

text: As

text: every

text: quiver

text: µ

text: µ

text: µ

text: representation

text: has

text: a

text: unique

text: Harder–Narasimhan

text: ﬁltration,

text: we

text: obtain

text: a

text: locally

text: ﬁ-

text: nite

text: stratiﬁcation

text: of

text: M

text: with

text: strata

text: Mζ

text: ss

text: M

text: where

text: Mζ

text: ss

text: is

text: deﬁned

text: as

text: above

text: by

text: means

text: of

text: r

text: ﬁber

text: products

text: for

text: every

text: strictly

text: decreasing

text: sequence

text: µ

text: µ

text: in

text: of

text: length

text: r.

text: Using

text: the

text: notation

text: r

text: R(M),

text: this

text: stratiﬁcation

text: can

text: be

text: written

text: as

text: ∈N

text: Exercise

text: Prove

text: the

text: formula

text: δζ

text: δζ

text: δζ

text: for

text: all

text: sequences

text: µ

text: µ1

text: µr

text: µ

text: of

text: real

text: numbers.

text: r

text: Applying

text: the

text: formula

text: proven

text: in

text: the

text: exercise

text: we

text: can

text: rewrite

text: the

text: inﬁnite

text: sum

text: as

text: an

text: inﬁnite

text: product

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 8 SVEN MEINHARDT C ⊂ ⊂ ⊂ ⊂ (5) Every object E in has a unique ﬁltration 0 E E . . . E = 1 2 n E, the Harder–Narasimhan ﬁltration, with semistable quotients E /E − of i i 1 strictly decreasing slopes. Example 2.13 (The r-Kronecker quiver). Let us illustrate this idea with a simple → example. Consider the abelian category of r-tuples x¯ of linear maps x : V V for ≤ ≤ i 1 2 1 i r between ﬁnite dimensional vector spaces V , V . Choosing two complex ∈ H 1 2 numbers ζ , ζ , we get a central charge by putting Z(x¯) = ζ dim V + 1 2 + 1 1 ζ dim V . Assume ﬁrst that arg(ζ ) = arg(ζ ). Then, all objects are semistable of 2 2 − 1 2 the same slope µ = cot(arg ζ ), and we have to face the old problems. Choose for 1 instance dim V = dim V = 1. The isomorphism type of such objects is determined 1 2 ∈ × by the choice of r complex numbers x , . . . , x up to rescaling by (g , g ) GL(V ) C∗ × C∗ − 1 r { 1 | 2 ∈ C∗} 1 GL(V ) = via g x g 1. As the diagonal group (g, g) g acts 2 1 i 2 A C∗ trivially, we have the take the GIT quotient of r by which is just a point C C∗ C as Spec [x , . . . , x ] = Spec . This corresponds to the fact that all objects 1 r → → have the same Jordan–H¨older factors x = 0 : V 0 and x = 0 : 0 V . i ⊕ 1 −→ i 2 0 Thus, all objects are S-equivalent to “V V ”= V V . If arg ζ > arg ζ , 1 2 1 2 2 1 non of our objects with dim V = dim V = 1 are semistable as the central charge → 1 2 ζ of the subobject 0 : 0 V has a bigger argument than the central charge 2 2 ζ + ζ of our given object. If, however, arg ζ < arg ζ , all objects except for the 1 2 ⊕ 2 1≤ ≤ semisimple V V corresponding to x = 0 for all 1 i r are semistable of −(cid:60)1 2 (cid:61) i M slope µ = e(ζ + ζ )/ m(ζ + ζ ), and even stable. The moduli space ζ1,ζ2 = A \{ } C∗ P 1− 2 1 2 (1C,1) r 0 / = r 1 parameterizing isomorphism classes of simple objects in of µ dimension vector (dim V , dim V ) = (1, 1) is even a ﬁne moduli space if we identify 1 2 → two families of r-tuples of line bundle morphisms x : V V on a parameter i 1 2 space S as soon as they become isomorphic after twisting V and V with some line 1 2 bundle L. Note that coarse moduli spaces parameterizing S-equivalence classes of objects in C might not exist for all central charges, but one can show the existence for generic µ central charges and reasonable abelian categories. We should also keep in mind that we paid a price for getting a reﬁned version of S-equivalence, namely S-equivalence in subcategories. Indeed, coarse moduli spaces of (S-equivalence classes of) semistable objects can only “see” semistable objects but no objects with a non-trivial Harder–Narasimhan ﬁltration. Hence, the construction of (coarse) moduli spaces remains unsatisfying. There is, however, an alternative way to overcome all the problems seen in the pre- vious examples. Following this approach, one can construct a ﬁne moduli “space” with a universal family parameterizing all objects - not only simple or stable ones - up to isomorphism. According to the conservative law of mathematical diﬃculties, we also have to pay a price for getting such a beautiful solution of our moduli prob- lem. It is hidden in the word “space”. In fact, we have to leave our comfort zone of varieties or schemes and have to dive into the universe of more general spaces known as “Artin stacks”. Recall that a scheme X is uniquely characterized by its set-valued functor h : (cid:55)→ X S Mor(S, X) of points. We have seen many set-valued functors before while studying moduli problems. The general pattern was the following. We considered (cid:55)−→ { C} set-valued contravariant functorsMF : S fam∼ilies of objects in /∼ and a ﬁne moduli space would be a scheme such that F = hM, while a coarse moduli space

text: SVEN

text: MEINHARDT

text: C

text: Every

text: object

text: E

text: in

text: has

text: a

text: unique

text: ﬁltration

text: E

text: E

text: E

text: n

text: E,

text: the

text: Harder–Narasimhan

text: ﬁltration,

text: with

text: semistable

text: quotients

text: E

text: /E

text: of

text: i

text: i

text: strictly

text: decreasing

text: slopes.

text: Example

text: (The

text: r-Kronecker

text: quiver).

text: Let

text: us

text: illustrate

text: this

text: idea

text: with

text: a

text: simple

text: example.

text: Consider

text: the

text: abelian

text: category

text: of

text: r-tuples

text: x¯

text: of

text: linear

text: maps

text: x

text: V

text: V

text: for

text: i

text: i

text: r

text: between

text: ﬁnite

text: dimensional

text: vector

text: spaces

text: V

text: V

text: Choosing

text: two

text: complex

text: H

text: numbers

text: ζ

text: ζ

text: we

text: get

text: a

text: central

text: charge

text: by

text: putting

text: Z(x¯)

text: ζ

text: dim

text: V

text: ζ

text: dim

text: V

text: Assume

text: ﬁrst

text: that

text: arg(ζ

text: arg(ζ

text: Then,

text: all

text: objects

text: are

text: semistable

text: of

text: the

text: same

text: slope

text: µ

text: cot(arg

text: ζ

text: and

text: we

text: have

text: to

text: face

text: the

text: old

text: problems.

text: Choose

text: for

text: instance

text: dim

text: V

text: dim

text: V

text: The

text: isomorphism

text: type

text: of

text: such

text: objects

text: is

text: determined

text: by

text: the

text: choice

text: of

text: r

text: complex

text: numbers

text: x

text: x

text: up

text: to

text: rescaling

text: by

text: (g

text: g

text: GL(V

text: C∗

text: C∗

text: r

text: C∗}

text: GL(V

text: via

text: g

text: x

text: g

text: As

text: the

text: diagonal

text: group

text: (g,

text: g)

text: g

text: acts

text: i

text: A

text: C∗

text: trivially,

text: we

text: have

text: the

text: take

text: the

text: GIT

text: quotient

text: of

text: r

text: by

text: which

text: is

text: just

text: a

text: point

text: C

text: C∗

text: C

text: as

text: Spec

text: [x

text: x

text: Spec

text: This

text: corresponds

text: to

text: the

text: fact

text: that

text: all

text: objects

text: r

text: have

text: the

text: same

text: Jordan–H¨older

text: factors

text: x

text: V

text: and

text: x

text: V

text: i

text: i

text: Thus,

text: all

text: objects

text: are

text: S-equivalent

text: to

text: “V

text: V

text: V

text: V

text: If

text: arg

text: ζ

text: arg

text: ζ

text: non

text: of

text: our

text: objects

text: with

text: dim

text: V

text: dim

text: V

text: are

text: semistable

text: as

text: the

text: central

text: charge

text: ζ

text: of

text: the

text: subobject

text: V

text: has

text: a

text: bigger

text: argument

text: than

text: the

text: central

text: charge

text: ζ

text: ζ

text: of

text: our

text: given

text: object.

text: If,

text: however,

text: arg

text: ζ

text: arg

text: ζ

text: all

text: objects

text: except

text: for

text: the

text: 1≤

text: semisimple

text: V

text: V

text: corresponding

text: to

text: x

text: for

text: all

text: i

text: r

text: are

text: semistable

text: of

text: i

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text: → − → − f (980) production in D+ π+ π+ π and D+ π+ K+ K decays 0 s s ∗ † ‡ J.M. Dias, F. S. Navarra, and M. Nielsen Instituto de F´ısica, Universidade de Sa˜o Paulo, C.P. 66318, 05389-970 Sa˜o Paulo, SP, Brazil § E. Oset Departamento de F´ısica Teo´rica and IFIC, Centro Mixto Universidad de Valencia-CSIC, Institutos de Investigaci´on de Paterna, Aptdo. 22085, 46071 Valencia, Spain (Dated: May 28, 2020) → − → − We study the D+ π+ π+ π and D+ π+ K+ K decays adopting a mechanism in which the s s D+ meson decays weakly into a π+ and a qq¯ component, which hadronizes into two pseudoscalar s mesons. The ﬁnal state interaction between these two pseudoscalar mesons is taken into account by using the Chiral Unitary approach in coupled channels, which gives rise to the f (980) resonance. − − 0 Hence, we obtain the invariant mass distributions of the pairs π+π and K+K after the decay of that resonance and compare our theoretical amplitudes with those available from the experimental data. Our results are in a fair agreement with the shape of these data, within large experimental − − uncertainty, and a f (980) signal is seen in both the π+π and K+K distributions. Predictions 0 − − for the relative size of π+π and K+K distributions are made. PACS numbers: Keywords: I. INTRODUCTION The analysis of heavy meson weak decays measured in B-factories and at the LHC has been very important for the study of new hadronic states and ultimately for the understanding of hadron dynamics. In these reactions, the weak decay leads to hadronic states, in general composed by two or three hadrons, which undergo “ﬁnal state interactions” (FSI), through which they form the ﬁnal particles. The FSI is very complex and can inﬂuence all the conclusions → − concerning new states and even provide the strength of CP violation [1]. In this work we consider the D+ π+ π+ π → − s and D+ π+ K+ K decays and we study the eﬀect of FSI on the measured invariant mass spectra. These decays s have been studied by several experimental groups [2–9] and they have been considered excellent tools to study FSI. Their distinctive feature is the fact that they are Cabibbo favored. From the experimental data we know the branching fractions [10]: → − Γ(D+ π+ π+ π ) ± × − s = (1.09 0.05) 10 2 ; (1) Γ total → − Γ(D+ π+ K+ K ) ± × − s = (5.39 0.21) 10 2 . (2) Γ total → − → − (cid:39) ± The corresponding ratio Γ(D+ π+ π+ π ))/(Γ(D+ π+ K+ K ) 0.2 is in agreement with the value 0.265 ± s s 0.041 0.031 found in a previous estimate [6]. While the diﬀerences between these numbers require some quantitive analysis, the qualitative relation between these decay rates can be easily understood when we look at the Cabibbo favoured decay diagram in Fig. 1, which is also helicity and color favoured, where the d¯u makes up a π+ and one has an extra ss¯ pair. The ﬁnal ss¯ pair hadronizes by − creating extra q¯q pairs, which lead to KK¯ or ηη but not ππ. The ﬁnal state π+ K+ K can be produced directly and → − − through rescattering (K0 K¯ 0 or ηη K+ K ). In contrast, the ﬁnal state π+ π+ π can only be produced through − → − rescattering (K+ K or K0 K¯ 0 or ηη π+ π ). ∗ Electronic address: \markcomm{footercolor}{jdias@if.usp.br} † Electronic address: \markcomm{footercolor}{navarra@if.usp.br} ‡ Electronic address: \markcomm{footercolor}{mnielsen@if.usp.br} § Electronic address: \markcomm{footercolor}{oset@ific.uv.es} ##LTLine## ##LTLine## ##LTLine##

text: f

text: production

text: in

text: D+

text: π+

text: π+

text: π

text: and

text: D+

text: π+

text: K+

text: K

text: decays

text: s

text: s

text: J.M.

text: Dias,

text: F.

text: S.

text: Navarra,

text: and

text: M.

text: Nielsen

text: Instituto

text: de

text: F´ısica,

text: Universidade

text: de

text: Sa˜o

text: Paulo,

text: C.P.

text: Sa˜o

text: Paulo,

text: SP,

text: Brazil

text: E.

text: Oset

text: Departamento

text: de

text: F´ısica

text: Teo´rica

text: and

text: IFIC,

text: Centro

text: Mixto

text: Universidad

text: de

text: Valencia-CSIC,

text: Institutos

text: de

text: Investigaci´on

text: de

text: Paterna,

text: Aptdo.

text: Valencia,

text: Spain

text: (Dated:

text: May

text: We

text: study

text: the

text: D+

text: π+

text: π+

text: π

text: and

text: D+

text: π+

text: K+

text: K

text: decays

text: adopting

text: a

text: mechanism

text: in

text: which

text: the

text: s

text: s

text: D+

text: meson

text: decays

text: weakly

text: into

text: a

text: π+

text: and

text: a

text: qq¯

text: component,

text: which

text: hadronizes

text: into

text: two

text: pseudoscalar

text: s

text: mesons.

text: The

text: ﬁnal

text: state

text: interaction

text: between

text: these

text: two

text: pseudoscalar

text: mesons

text: is

text: taken

text: into

text: account

text: by

text: using

text: the

text: Chiral

text: Unitary

text: approach

text: in

text: coupled

text: channels,

text: which

text: gives

text: rise

text: to

text: the

text: f

text: resonance.

text: Hence,

text: we

text: obtain

text: the

text: invariant

text: mass

text: distributions

text: of

text: the

text: pairs

text: π+π

text: and

text: K+K

text: after

text: the

text: decay

text: of

text: that

text: resonance

text: and

text: compare

text: our

text: theoretical

text: amplitudes

text: with

text: those

text: available

text: from

text: the

text: experimental

text: data.

text: Our

text: results

text: are

text: in

text: a

text: fair

text: agreement

text: with

text: the

text: shape

text: of

text: these

text: data,

text: within

text: large

text: experimental

text: uncertainty,

text: and

text: a

text: f

text: signal

text: is

text: seen

text: in

text: both

text: the

text: π+π

text: and

text: K+K

text: distributions.

text: Predictions

text: for

text: the

text: relative

text: size

text: of

text: π+π

text: and

text: K+K

text: distributions

text: are

text: made.

text: PACS

text: numbers:

text: Keywords:

text: I.

text: INTRODUCTION

text: The

text: analysis

text: of

text: heavy

text: meson

text: weak

text: decays

text: measured

text: in

text: B-factories

text: and

text: at

text: the

text: LHC

text: has

text: been

text: very

text: important

text: for

text: the

text: study

text: of

text: new

text: hadronic

text: states

text: and

text: ultimately

text: for

text: the

text: understanding

text: of

text: hadron

text: dynamics.

text: In

text: these

text: reactions,

text: the

text: weak

text: decay

text: leads

text: to

text: hadronic

text: states,

text: in

text: general

text: composed

text: by

text: two

text: or

text: three

text: hadrons,

text: which

text: undergo

text: “ﬁnal

text: state

text: interactions”

text: (FSI),

text: through

text: which

text: they

text: form

text: the

text: ﬁnal

text: particles.

text: The

text: FSI

text: is

text: very

text: complex

text: and

text: can

text: inﬂuence

text: all

text: the

text: conclusions

text: concerning

text: new

text: states

text: and

text: even

text: provide

text: the

text: strength

text: of

text: CP

text: violation

text: In

text: this

text: work

text: we

text: consider

text: the

text: D+

text: π+

text: π+

text: π

text: s

text: and

text: D+

text: π+

text: K+

text: K

text: decays

text: and

text: we

text: study

text: the

text: eﬀect

text: of

text: FSI

text: on

text: the

text: measured

text: invariant

text: mass

text: spectra.

text: These

text: decays

text: s

text: have

text: been

text: studied

text: by

text: several

text: experimental

text: groups

text: and

text: they

text: have

text: been

text: considered

text: excellent

text: tools

text: to

text: study

text: FSI.

text: Their

text: distinctive

text: feature

text: is

text: the

text: fact

text: that

text: they

text: are

text: Cabibbo

text: favored.

text: From

text: the

text: experimental

text: data

text: we

text: know

text: the

text: branching

text: fractions

text: Γ(D+

text: π+

text: π+

text: π

text: s

text: Γ

text: total

text: Γ(D+

text: π+

text: K+

text: K

text: s

text: Γ

text: total

text: The

text: corresponding

text: ratio

text: Γ(D+

text: π+

text: π+

text: π

text: ))/(Γ(D+

text: π+

text: K+

text: K

text: is

text: in

text: agreement

text: with

text: the

text: value

text: s

text: s

text: found

text: in

text: a

text: previous

text: estimate

text: While

text: the

text: diﬀerences

text: between

text: these

text: numbers

text: require

text: some

text: quantitive

text: analysis,

text: the

text: qualitative

text: relation

text: between

text: these

text: decay

text: rates

text: can

text: be

text: easily

text: understood

text: when

text: we

text: look

text: at

text: the

text: Cabibbo

text: favoured

text: decay

text: diagram

text: in

text: Fig.

text: which

text: is

text: also

text: helicity

text: and

text: color

text: favoured,

text: where

text: the

text: d¯u

text: makes

text: up

text: a

text: π+

text: and

text: one

text: has

text: an

text: extra

text: ss¯

text: pair.

text: The

text: ﬁnal

text: ss¯

text: pair

text: hadronizes

text: by

text: creating

text: extra

text: q¯q

text: pairs,

text: which

text: lead

text: to

text: KK¯

text: or

text: ηη

text: but

text: not

text: ππ.

text: The

text: ﬁnal

text: state

text: π+

text: K+

text: K

text: can

text: be

text: produced

text: directly

text: and

text: through

text: rescattering

text: K¯

text: or

text: ηη

text: K+

text: K

text: In

text: contrast,

text: the

text: ﬁnal

text: state

text: π+

text: π+

text: π

text: can

text: only

text: be

text: produced

text: through

text: rescattering

text: (K+

text: K

text: or

text: K¯

text: or

text: ηη

text: π+

text: π

text: Electronic

text: address:

text: \markcomm{footercolor}{jdias@if.usp.br}

text: Electronic

text: address:

text: Electronic

text: address:

text: Electronic

text: address:

text: \markcomm{footercolor}{oset@ific.uv.es}

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##