text: 14 subject to the boundary conditions (cid:48) (cid:48) u (0) = 0, au(1) + bu (1) = 0. (40) We consider the following two cases: Case one: a = b = 1. Case two: a = 0.1, b = 1. − − The Adomian polynomials of nonlinear term f(x, u) = e u(x) in this problem are computed as − − A = e U(0), 0 − A = U(1)e U(0), 1 − − 1 − A = U(2)e U(0) U2(1)e U(0), 2 2 − − − 1 − A = U(3)e U(0) U(1)U(2)e U(0) + U3(1)e U(0), 3 6 . . . Again no exact solution exists for this equation, hence it was handled numerically. Table 8 describes the numerical results of the ﬁrst case obtained by the proposed method at the order of approximation N = 12 and the other existing methods, including the FDM [6], the non-polynomial cubic spline method (NPCSM) [7], the CSM [8] and the SGM [9]. Meanwhile, a comparison for the approximate solutions of the second case obtained by the present method with the same approximation level as the ﬁrst case and the previous existing methods which include the CSM [8], the SGM [9], the BSDM [10] and the BSM [16] is presented in Table 9. One can seen from two Tables that our computations are in good line with the results obtained by the other approaches compared. In fact, at the approximation level for N = 12, the maximal absolute error is found to be order of − − magnitude O(10 7) for the ﬁrst case, and O(10 9) for the second case. Example 5. Consider the following SBVP with nonlinear term diﬀerent from the forms (4)-(7) which arises in the radial stress on a rotationally symmetric shallow membrane cap [12,13]: (cid:48)(cid:48) 3 (cid:48) 1 − 1 u (x) + u (x) = , (41) x 2 8u2(x) subject to the boundary conditions (cid:48) u (0) = 0, u(1) = 1. (42) ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: subject

text: to

text: the

text: boundary

text: conditions

label: equation

text: u

text: bu

text: We

text: consider

text: the

text: following

text: two

text: cases:

text: Case

text: one:

text: a

text: b

text: Case

text: two:

text: a

text: b

text: The

text: Adomian

text: polynomials

text: of

text: nonlinear

text: term

text: f(x,

text: u)

text: e

text: u(x)

text: in

text: this

text: problem

text: are

text: computed

text: as

text: A

text: e

text: A

text: A

text: A

text: Again

text: no

text: exact

text: solution

text: exists

text: for

text: this

text: equation,

text: hence

text: it

text: was

text: handled

text: numerically.

text: Table

text: describes

text: the

text: numerical

text: results

text: of

text: the

text: ﬁrst

text: case

text: obtained

text: by

text: the

text: proposed

text: method

text: at

text: the

text: order

text: of

text: approximation

text: N

text: and

text: the

text: other

text: existing

text: methods,

text: including

text: the

text: FDM

text: the

text: non-polynomial

text: cubic

text: spline

text: method

text: (NPCSM)

text: the

text: CSM

text: and

text: the

text: SGM

text: Meanwhile,

text: a

text: comparison

text: for

text: the

text: approximate

text: solutions

text: of

text: the

text: second

text: case

text: obtained

text: by

text: the

text: present

text: method

text: with

text: the

text: same

text: approximation

text: level

text: as

text: the

text: ﬁrst

text: case

text: and

text: the

text: previous

text: existing

text: methods

text: which

text: include

text: the

text: CSM

text: the

text: SGM

text: the

text: BSDM

text: and

text: the

text: BSM

text: is

text: presented

text: in

text: Table

text: One

text: can

text: seen

text: from

text: two

text: Tables

text: that

text: our

text: computations

text: are

text: in

text: good

text: line

text: with

text: the

text: results

text: obtained

text: by

text: the

text: other

text: approaches

text: compared.

text: In

text: fact,

text: at

text: the

text: approximation

text: level

text: for

text: N

text: the

text: maximal

text: absolute

text: error

text: is

text: found

text: to

text: be

text: order

text: of

text: magnitude

text: for

text: the

text: ﬁrst

text: case,

text: and

text: for

text: the

text: second

text: case.

text: Example

text: Consider

text: the

text: following

text: SBVP

text: with

text: nonlinear

text: term

text: diﬀerent

text: from

text: the

text: forms

text: which

text: arises

text: in

text: the

text: radial

text: stress

text: on

text: a

text: rotationally

text: symmetric

text: shallow

text: membrane

text: cap

text: u

text: (x)

text: u

text: (x)

text: x

text: subject

text: to

text: the

text: boundary

text: conditions

text: u

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 20 List of Figures | | Figure 1. The absolute residual error functions ER (x) for N = 2, 4, 6 (left) and N,2 8, 10, 12 (right) of Example 3. Figure 2. The logarithmic plots for the maximal error remainder parameters MER N,α for N = 2 through 12 by step 2 and α = 1 (up, left), α = 2 (up, right), α = 3 (down) of Example 3. | | Figure 3. The absolute residual error functions ER (x) for N = 4, 6, 8 (left) and N 10, 12, 14 (right) of Example 5. Figure 4. The logarithmic plot for the maximal error remainder parameters MER for N N = 2 through 14 by step 2 of Example 5.

label: paragraph

text: List

text: of

text: Figures

text: Figure

text: The

text: absolute

text: residual

text: error

text: functions

text: ER

text: (x)

text: for

text: N

text: (left)

text: and

text: (right)

text: of

text: Example

text: Figure

text: The

text: logarithmic

text: plots

text: for

text: the

text: maximal

text: error

text: remainder

text: parameters

text: MER

text: N,α

text: for

text: N

text: through

text: by

text: step

text: and

text: α

text: (up,

text: left),

text: α

text: (up,

text: right),

text: α

text: (down)

text: of

text: Example

text: Figure

text: The

text: absolute

text: residual

text: error

text: functions

text: ER

text: (x)

text: for

text: N

text: (left)

text: and

text: N

text: (right)

text: of

text: Example

text: Figure

text: The

text: logarithmic

text: plot

text: for

text: the

text: maximal

text: error

text: remainder

text: parameters

text: MER

text: for

text: N

text: N

text: through

text: by

text: step

text: of

text: Example


text: 5 which is eminently convenient for symbolic implementation by computer algebraic systems such as Maple and Mathematics. The method to generate the Adomian polynomials in [39] is described as follows: (cid:88) ≥ C1 = u , n 1, n n − 1 n k − ≤ ≤ (11) Ck = (j + 1)u Ck−1− , 2 k n, n n j+1 n 1 j j=0 such that (cid:88) A = f(u ), 0 0 n ≥ (12) A = Ckf(k)(u ), n 1. n n 0 k=1 It is worth mentioning that Duan’s algorithm involving (11) and (12) has been testiﬁed to be one of the fastest subroutines on record [39], including the fast generation method given by Adoamin and Rach [42]. 2.2. Diﬀerent(cid:20)ial tran(cid:21)sform The diﬀerential transform of the kth diﬀerentiable function u(x) at x = 0 is deﬁned by 1 dku(x) U(k) = , (13) k! dxk x=0 and the diﬀerential inverse transform of U(k) is described as (cid:88) ∞ u(x) = U(k)xk, (14) k=0 where u(x) is the original function and U(k) is the transformed function. For the practical applications, the function u(x) is expressed by a truncated series and Eq. (14) can be written as (cid:88) ≈ N u(x) u (x) = U(k)xk. (15) N k=0 It is not diﬃcult to deduce the transformed functions of the fundamental operations listed in Table 1. 3. Method of solution of SBVPs (1)-(3) (cid:88) We want to ﬁnd the approximate solution of the problem (1)-(3) with the type: N u (x) = U(k)xk, (16) N k=0 · · · where the coeﬃcients U(0), U(1), , U(N) are determined using the following steps: ##LTLine## ##LTLine## ##LTLine##

text: which

text: is

text: eminently

text: convenient

text: for

text: symbolic

text: implementation

text: by

text: computer

text: algebraic

text: systems

text: such

text: as

text: Maple

text: and

text: Mathematics.

text: The

text: method

text: to

text: generate

text: the

text: Adomian

text: polynomials

text: in

text: is

text: described

text: as

text: follows:

text: u

text: n

text: n

text: n

text: n

text: k

text: Ck

text: (j

text: k

text: n,

text: n

text: n

text: n

text: j

text: such

text: that

text: A

text: f(u

text: n

text: A

text: Ckf(k)(u

text: n

text: n

text: n

text: It

text: is

text: worth

text: mentioning

text: that

text: Duan’s

text: algorithm

text: involving

text: and

text: has

text: been

text: testiﬁed

text: to

text: be

text: one

text: of

text: the

text: fastest

text: subroutines

text: on

text: record

text: including

text: the

text: fast

text: generation

text: method

text: given

text: by

text: Adoamin

text: and

text: Rach

text: The

text: diﬀerential

text: transform

text: of

text: the

text: kth

text: diﬀerentiable

text: function

text: u(x)

text: at

text: x

text: is

text: deﬁned

text: by

text: dku(x)

text: U(k)

text: k!

text: dxk

text: and

text: the

text: diﬀerential

text: inverse

text: transform

text: of

text: U(k)

text: is

text: described

text: as

text: u(x)

text: U(k)xk,

text: where

text: u(x)

text: is

text: the

text: original

text: function

text: and

text: U(k)

text: is

text: the

text: transformed

text: function.

text: For

text: the

text: practical

text: applications,

text: the

text: function

text: u(x)

text: is

text: expressed

text: by

text: a

text: truncated

text: series

text: and

text: Eq.

text: can

text: be

text: written

text: as

text: N

text: u(x)

text: u

text: (x)

text: U(k)xk.

text: N

text: It

text: is

text: not

text: diﬃcult

text: to

text: deduce

text: the

text: transformed

text: functions

text: of

text: the

text: fundamental

text: operations

text: listed

text: in

text: Table

text: Method

text: of

text: solution

text: of

text: SBVPs

text: We

text: want

text: to

text: ﬁnd

text: the

text: approximate

text: solution

text: of

text: the

text: problem

text: with

text: the

text: type:

text: N

text: u

text: (x)

text: U(k)xk,

text: N

text: where

text: the

text: coeﬃcients

text: U(N)

text: are

text: determined

text: using

text: the

text: following

text: steps:

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 20 KATRIN WEHRHEIM AND CHRIS WOODWARD This duality implies non-degeneracy of the pairing(cid:90) ∗ × −∗ → R (cid:55)→ (cid:104) ∧ (cid:105) (11) H (X; d ) Hn (X, ∂X; d ) , (γ, β) γ β . α α X Replacing X with ∂X we have a non-degenerate pairing on H(∂X; d | ). α ∂X If dim(X) = 2k+1 is odd, the pairing in the last item restricts to a non-degenerate pairing on the middle dimensional homology Hk(∂X; d | ). The pairing is skew- α ∂X symmetric, hence symplectic, if k is odd. In that case, the following Lemma shows that the restriction of H(X; d ) to the boundary gives rise to a Lagrangian subspace. α ∈ A Lemma 3.2.4. (Lagrangian restriction of odd cohomology) Let α (P ) be a central curvature connection. The image ρ(H(X; d )) of H(X; d ) in H(∂X; d | ) α α α ∂X is maximally isotropic. In particular, if dim(X) = 2k + 1 for some integer k then ⊂ ρ(Hk(X; d )) Hk(∂X; d | ) is maximally isotropic, and if k is odd, α (cid:0) α ∂X (cid:1) (cid:0) (cid:1) (12) dim ρ(Hk(X; d )) = 1 dim Hk(∂X; d | ) . α 2 α ∂X ∈ Proof. Stokes’ theorem implies that the image ρ(H(X; d )) is isotropic: For β , β α 1 2 ker(dα) (cid:90) (cid:90) (cid:90) (cid:104) ∧ (cid:105) (cid:104) ∧ (cid:105) ρ(β ) ρ(β ) = d β β 1 2 1 2 ∂X X (cid:104) ∧ (cid:105) − | |(cid:104) ∧ (cid:105) = d β β + ( 1) β1 β d β = 0. α 1 2 1 α 2 X To see that the image is maximal isotropic, ﬁrst note that restriction to the boundary induces a long exact sequence → • → • → • → (13) . . . H (X, ∂X; d ) H (X; d ) H (∂X; d | ) . . . . α α α ∂X ##LTLine##

text: KATRIN

text: WEHRHEIM

text: AND

text: CHRIS

text: WOODWARD

text: This

label: list

text: duality

text: implies

text: non-degeneracy

text: of

text: the

text: R

text: H

text: (X;

text: d

text: Hn

text: (X,

text: ∂X;

text: d

text: (γ,

text: β)

text: γ

text: β

text: α

text: α

text: X

text: Replacing

text: X

text: with

text: ∂X

text: we

text: have

text: a

text: non-degenerate

text: pairing

text: on

text: H(∂X;

text: d

text: α

text: ∂X

text: If

text: dim(X)

text: is

text: odd,

text: the

text: pairing

text: in

text: the

text: last

text: item

text: restricts

text: to

text: a

text: non-degenerate

text: pairing

text: on

text: the

text: middle

text: dimensional

text: homology

text: Hk(∂X;

text: d

text: The

text: pairing

text: is

text: skew-

text: α

text: ∂X

text: symmetric,

text: hence

text: symplectic,

text: if

text: k

text: is

text: odd.

text: In

text: that

text: case,

text: the

text: following

text: Lemma

text: shows

text: that

text: the

text: restriction

text: of

text: H(X;

text: d

text: to

text: the

text: boundary

text: gives

text: rise

text: to

text: a

text: Lagrangian

text: subspace.

text: α

text: A

text: Lemma

text: (Lagrangian

text: restriction

text: of

text: odd

text: cohomology)

text: Let

text: α

text: (P

text: be

text: a

text: central

text: curvature

text: connection.

text: The

text: image

text: ρ(H(X;

text: d

text: of

text: H(X;

text: d

text: in

text: H(∂X;

text: d

text: α

text: α

text: α

text: ∂X

text: is

text: maximally

text: isotropic.

text: In

text: particular,

text: if

text: dim(X)

text: for

text: some

text: integer

text: k

text: then

text: ρ(Hk(X;

text: d

text: Hk(∂X;

text: d

text: is

text: maximally

text: isotropic,

text: and

text: if

text: k

text: is

text: odd,

text: α

text: α

text: ∂X

text: dim

text: ρ(Hk(X;

text: d

text: dim

text: Hk(∂X;

text: d

text: α

text: α

text: ∂X

text: Proof.

text: Stokes’

text: theorem

text: implies

text: that

text: the

text: image

text: ρ(H(X;

text: d

text: is

text: isotropic:

text: For

text: β

text: β

text: α

text: ker(dα)

text: ρ(β

text: ρ(β

text: d

text: β

text: β

text: ∂X

text: X

text: d

text: β

text: β

text: β1

text: β

text: d

text: β

text: α

text: α

text: X

text: To

text: see

text: that

text: the

text: image

text: is

text: maximal

text: isotropic,

text: ﬁrst

text: note

text: that

text: restriction

text: to

text: the

text: boundary

text: induces

text: a

text: long

text: exact

text: sequence

text: H

text: (X,

text: ∂X;

text: d

text: H

text: (X;

text: d

text: H

text: (∂X;

text: d

text: α

text: α

text: α

text: ∂X

text: ##LTLine##


text: FLOER FIELD THEORY 21 Now consider the commutative diagram formed from the long exact sequence (13) and its dual: H•(X; d ) (cid:45) H•(∂X; d | ) c (cid:45) H•+1(X, ∂X; d ) α α ∂X α . . . . . . . (cid:63) (cid:63) (cid:63) −• ∨ (cid:45) − −• ∨ (cid:45) − −• ∨ Hn (X, ∂X; d ) Hn 1 (∂X; d | ) Hn 1 (X; d ) α α ∂X α Here the vertical arrows are given by (10) and the pairing (11). To check commu- tativity, use Stokes’ theorem and the fact that the connecting morphism c above is (cid:55)→ | given by β d β˜ for any extension β˜ = β. α ∈ • ∂X − −• Now suppose β H (∂X; d | ) lies in the annihilator of the image of Hn 1 (X; d ). ∈ • α ∂X − −• ∨ α By deﬁnition β H (∂X; d | ) maps to 0 in Hn 1 (X; d ) . Since the ver- α α ∂X tical maps are isomorphisms and the diagram is commutative, the image of β in • H +1(X, ∂X; d ) vanishes, so by exactness of the top sequence β lies in the image of • α H (X; d ). The last claim follows from the fact that maximally isotropic subspaces α of symplectic vector spaces are half-dimensional. For analogous results on Dirac (cid:3) operators on manifolds with boundary see [4, Theorem 22.24]. ⊂ Corollary 3.2.5. If X has dimension 3 then ρ(H1(X; d )) H1(∂X; d | ) is a α α ∂X → Lagrangian subspace. Furthermore, if H1(X, ∂X; d ) = 0 then ρ : H1(X; d ) α α H1(∂X; d | ) is a Lagrangian embedding. α ∂X Proof. The ﬁrst statement follows immediately from Lemma 3.2.4 with k = 1. The (cid:3) injectivity of ρ in the second statement follows from the long exact sequence (13). This subsection introduces the moduli space of connections with central curvature and ﬁxed determinant. As before, G is a compact connected group and X a compact oriented manifold with (possibly empty) boundary. Deﬁnition 3.2.6. (a) (Commutator subgroup) Let { | ∈ } G := [G, G] = [g , g ] g , g G 0 1 2 1 2 denote the commutator subgroup of G. In particular, [U(r), U(r)] = SU(r). More generally, G is the subgroup whose Lie algebra equals to the semisimple 0 part [g, g] of the Lie algebra g. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: FLOER

text: FIELD

text: THEORY

text: Now

text: consider

text: the

text: commutative

text: diagram

text: formed

text: from

text: the

text: long

text: exact

text: sequence

text: and

text: its

text: dual:

text: H•(X;

text: d

text: H•(∂X;

text: d

text: c

text: ∂X;

text: d

text: α

text: α

text: ∂X

text: α

text: Hn

text: (X,

text: ∂X;

text: d

text: Hn

text: (∂X;

text: d

text: Hn

text: (X;

text: d

text: α

text: α

text: ∂X

text: α

text: Here

text: the

text: vertical

text: arrows

text: are

text: given

text: by

text: and

text: the

text: pairing

text: To

text: check

text: commu-

text: tativity,

text: use

text: Stokes’

text: theorem

text: and

text: the

text: fact

text: that

text: the

text: connecting

text: morphism

text: c

text: above

text: is

text: given

text: by

text: β

text: d

text: β˜

text: for

text: any

text: extension

text: β˜

text: β.

text: α

text: ∂X

text: Now

text: suppose

text: β

text: H

text: (∂X;

text: d

text: lies

text: in

text: the

text: annihilator

text: of

text: the

text: image

text: of

text: Hn

text: (X;

text: d

text: α

text: ∂X

text: α

text: By

text: deﬁnition

text: β

text: H

text: (∂X;

text: d

text: maps

text: to

text: in

text: Hn

text: (X;

text: d

text: Since

text: the

text: ver-

text: α

text: α

text: ∂X

text: tical

text: maps

text: are

text: isomorphisms

text: and

text: the

text: diagram

text: is

text: commutative,

text: the

text: image

text: of

text: β

text: in

text: H

text: ∂X;

text: d

text: vanishes,

text: so

text: by

text: exactness

text: of

text: the

text: top

text: sequence

text: β

text: lies

text: in

text: the

text: image

text: of

text: α

text: H

text: (X;

text: d

text: The

text: last

text: claim

text: follows

text: from

text: the

text: fact

text: that

text: maximally

text: isotropic

text: subspaces

text: α

text: of

text: symplectic

text: vector

text: spaces

text: are

text: half-dimensional.

text: For

text: analogous

text: results

text: on

text: Dirac

text: operators

text: on

text: manifolds

text: with

text: boundary

text: see

text: Theorem

text: Corollary

text: If

text: X

text: has

text: dimension

text: then

text: d

text: d

text: is

text: a

text: α

text: α

text: ∂X

text: Lagrangian

text: subspace.

text: Furthermore,

text: if

text: ∂X;

text: d

text: then

text: ρ

text: d

text: α

text: α

text: d

text: is

text: a

text: Lagrangian

text: embedding.

text: α

text: ∂X

text: Proof.

text: The

text: ﬁrst

text: statement

text: follows

text: immediately

text: from

text: Lemma

text: with

text: k

text: The

text: injectivity

text: of

text: ρ

text: in

text: the

text: second

text: statement

text: follows

text: from

text: the

text: long

text: exact

text: sequence

text: This

text: subsection

text: introduces

text: the

text: moduli

text: space

text: of

text: connections

text: with

text: central

text: curvature

text: and

text: ﬁxed

text: determinant.

text: As

text: before,

text: G

text: is

text: a

text: compact

text: connected

text: group

text: and

text: X

text: a

text: compact

text: oriented

text: manifold

text: with

text: (possibly

text: empty)

text: boundary.

text: Deﬁnition

text: (a)

text: (Commutator

text: subgroup)

text: Let

text: G

text: [G,

text: G]

text: [g

text: g

text: g

text: g

text: G

text: denote

text: the

text: commutator

text: subgroup

text: of

text: G.

text: In

text: particular,

text: [U(r),

text: U(r)]

text: SU(r).

text: More

text: generally,

text: G

text: is

text: the

text: subgroup

text: whose

text: Lie

text: algebra

text: equals

text: to

text: the

text: semisimple

text: part

text: [g,

text: g]

text: of

text: the

text: Lie

text: algebra

text: g.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 32 KATRIN WEHRHEIM AND CHRIS WOODWARD A ⊂ A ≥ we restrict ω to (P ) (P ). The latter is a Banach submanifold for s 2 by δ Proposition 3.2.8. Since the de Rham operator commutes with pull-back, | ∈ A | ω A Ω2( (P )), dω A = 0 (P) δ (P) δ δ | G is again a closed two-form. Finally, note that ω A is (P )-invariant and vanishes ∈ δ(P) 0 on the vertical vectors d ξ, ξ Ω0(X, P ([g, g])) by (20). Hence this two-form is the α pull-back of a two-form on the quotient M(P ). The two-form form ω on the M(P) quotient is close(cid:90)d because the de Rham operator commutes with pull-back and the A → map (P ) M(P ) is a submersion. Non-degeneracy follows from the identity δ ∼ (cid:104) ∧ ∗ (cid:105) (cid:107) (cid:107) ∀ ∈ a a = a 2 , a H1(X; d ) = T M(P ). L2 α [α] X The assertion on monotonicity is a variation on the Drezet-Narasimhan theorem [12, Theorem F], see also [26] and [24]. A symplectic proof is given in [33]. There it is shown, using the description in Remark 3.3.7 below, that the ﬁrst Chern class (cid:3) c (M(P )) is 2c times the symplectic class [ω ]. 1 M(P) Now restricting to G = U(r) we have the following main result of this section. Theorem 3.3.3. (Properties of moduli spaces of connections on a surface) Let P be a principal U(r)-bundle of degree d coprime to r over a compact, connected, oriented surface X without boundary. ≥ (a) If X has genus g(X) 1, then the moduli space M(P ) is a nonempty compact symplectic manifold of dimension − − dim(M(P )) = (2g(X) 2)(r2 1) with even Chern numbers (cid:104) (cid:105) ⊂ Z c (M(P )), H (M(P )) 2 1 2 and satsifyingt the monotonicity relation c (M(P )) = 2r[ω ]. 1 M(P) ∅ (b) If X has genus g(X) = 0, then M(P ) = . (c) If X has genus g(X) = 1, then M(P ) = pt is a point. Moreover, M(P ) is always connected and simply-connected. Remark 3.3.4. (Non-coprime case) Suppose we are in the situation of Theorem 3.3.3, ≥ except that r, d do have a common divisor. Then for g(X) 1 there exist connections ∈ A (cid:54) α (P ) with non-discrete automorphism groups, i.e. H0(X; d[g,g]) = 0. For δ ∅ ∈ Z α g(X) = 0 we have M(P ) = iﬀ d/r , and otherwise M(P ) is a point.

text: KATRIN

text: WEHRHEIM

text: AND

text: CHRIS

text: WOODWARD

text: A

text: A

text: we

text: restrict

text: ω

text: to

text: (P

text: (P

text: The

text: latter

text: is

text: a

text: Banach

text: submanifold

text: for

text: s

text: by

text: δ

text: Proposition

text: Since

text: the

text: de

text: Rham

text: operator

text: commutes

text: with

text: pull-back,

text: A

text: ω

text: A

text: (P

text: dω

text: A

text: (P)

text: δ

text: (P)

text: δ

text: δ

text: G

text: is

text: again

text: a

text: closed

text: two-form.

text: Finally,

text: note

text: that

text: ω

text: A

text: is

text: (P

text: )-invariant

text: and

text: vanishes

text: δ(P)

text: on

text: the

text: vertical

text: vectors

text: d

text: ξ,

text: ξ

text: P

text: ([g,

text: g]))

text: by

text: Hence

text: this

text: two-form

text: is

text: the

text: α

text: pull-back

text: of

text: a

text: two-form

text: on

text: the

text: quotient

text: M(P

text: The

text: two-form

text: form

text: ω

text: on

text: the

text: M(P)

text: quotient

text: is

text: because

text: the

text: de

text: Rham

text: operator

text: commutes

text: with

text: pull-back

text: and

text: the

text: A

text: map

text: (P

text: M(P

text: is

text: a

text: submersion.

text: Non-degeneracy

text: follows

text: from

text: the

text: identity

text: δ

text: a

text: a

text: a

text: a

text: d

text: T

text: M(P

text: α

text: [α]

text: X

text: The

text: assertion

text: on

text: monotonicity

text: is

text: a

text: variation

text: on

text: the

text: Drezet-Narasimhan

text: theorem

text: Theorem

text: F],

text: see

text: also

text: and

text: A

text: symplectic

text: proof

text: is

text: given

text: in

text: There

text: it

text: is

text: shown,

text: using

text: the

text: description

text: in

text: Remark

text: below,

text: that

text: the

text: ﬁrst

text: Chern

text: class

text: c

text: (M(P

text: is

text: times

text: the

text: symplectic

text: class

text: [ω

text: M(P)

text: Now

text: restricting

text: to

text: G

text: U(r)

text: we

text: have

text: the

text: following

text: main

text: result

text: of

text: this

text: section.

text: Theorem

text: (Properties

text: of

text: moduli

text: spaces

text: of

text: connections

text: on

text: a

text: surface)

text: Let

text: P

text: be

text: a

text: principal

text: U(r)-bundle

text: of

text: degree

text: d

text: coprime

text: to

text: r

text: over

text: a

text: compact,

text: connected,

text: oriented

text: surface

text: X

text: without

text: boundary.

text: (a)

text: If

text: X

text: has

text: genus

text: g(X)

text: then

text: the

text: moduli

text: space

text: M(P

text: is

text: a

text: nonempty

text: compact

text: symplectic

text: manifold

text: of

text: dimension

text: dim(M(P

text: with

text: even

text: Chern

text: numbers

text: Z

text: c

text: (M(P

text: H

text: (M(P

text: and

text: satsifyingt

text: the

text: monotonicity

text: relation

text: c

text: (M(P

text: M(P)

text: (b)

text: If

text: X

text: has

text: genus

text: g(X)

text: then

text: M(P

text: (c)

text: If

text: X

text: has

text: genus

text: g(X)

text: then

text: M(P

text: pt

text: is

text: a

text: point.

text: Moreover,

text: M(P

text: is

text: always

text: connected

text: and

text: simply-connected.

text: Remark

text: (Non-coprime

text: case)

text: Suppose

text: we

text: are

text: in

text: the

text: situation

text: of

text: Theorem

text: except

text: that

text: r,

text: d

text: do

text: have

text: a

text: common

text: divisor.

text: Then

text: for

text: g(X)

text: there

text: exist

text: connections

text: A

text: α

text: (P

text: with

text: non-discrete

text: automorphism

text: groups,

text: i.e.

text: d[g,g])

text: For

text: δ

text: Z

text: α

text: g(X)

text: we

text: have

text: M(P

text: iﬀ

text: d/r

text: and

text: otherwise

text: M(P

text: is

text: a

text: point.


text: 34 KATRIN WEHRHEIM AND CHRIS WOODWARD Remark 3.3.5. (Holomorphic description of the moduli spaces) The moduli spaces M(P ) have a holomorphic description, for G = U(r) due to a famous theorem of Narasimhan-Seshadri [36], generalized to arbitrary groups in Ramanathan’s thesis [39], [40]. However, we never use the holomorphic description. Lemma 3.3.6. (Symplectomorphisms of moduli spaces induced by bundle isomor- phisms) Let r, d be coprime integers. → → (a) For any two U(r)-bundles P X , P X of degree d, any bundle iso- → 0 0 1 1 morphism ψ : P P covering an orientation-preserving diﬀeomorphism 0 1 ∗ → from X to X induces a symplectomorphism ψ : M(P ) M(P ) given by 0 1 1 0 pull-back of representatives. (b) Any two such bundle isomorphisms from P to P covering the same diﬀeo- 0 1 morphism from X to X induce the same symplectomorphism. → → 0 1 (c) If P X, P X are U(r)-bundles of degree d over the same surface, then 0 1 the moduli spaces M(P ), M(P ) are canonically symplectomorphic. 0 1 Proof. For manifolds of dimension at most three, bundles are classiﬁed up to iso- morphism by their ﬁrst Chern class. For connected surfaces, the Chern class is → determined by the degree. Hence if π : P X , j = 0, 1 are bundles of the same j j →j ◦ degree then there exists an isomorphism φ : P P with π φ = π . This induces A A 0 1 1 0 a map from (P ) to (P ), given on the level of tangent spaces by the pull-back of ∗ 1 0 → basic forms φ : Ω1(P ,(cid:90)g) Ω1(P , g(cid:90)) . This pull-back acts symplectically: ∈ 1 basic 0 basic for a, b Ω1(P , g) we have 1 basic (cid:104) ∧ (cid:105) (cid:104) ∗ ∧ ∗ (cid:105) π ∗ a b = π ∗ φ a φ b 1, 0, X1 X0 since φ covers an orientation-preserving diﬀeomorphism from X to X . Moreover A A 0 1 pull-back by φ maps (P ) to ∗ (P ) and is equivariant with respect to the G δ 1 G φ δ 0 ∗G ∗ gauge actions of (P ) and (P ) = φ (P ). Hence φ descends to a sym- 0 1 → 0 0 0 1 plectomorphism M(P ) M(P ). Any two such isomorphisms covering the same 1 0 diﬀeomorphism diﬀer by a gauge transformation of P . Since any gauge transforma- 1 tion of P induces the identity on M(P ) and M(P ) is independent of the choice 1 1 1 → of determinant connection (see Remark 3.2.7), the identiﬁcation M(P ) M(P ) 1 0(cid:3) depends only on the choice of diﬀeomorphism from X to X . 0 1 Remark 3.3.7. (Moduli spaces as representations of fundamental group of the punc- tured surface) Atiyah-Bott [2] provide a description of M(P ) in terms of representa- tions, which makes Lemma 3.2.10 more precise in the case of bundles over surfaces. In that case, M(P ) can be identiﬁed with the moduli space of G -representations − { } 0 of the fundamental group of the punctured surface X x , whose value on a small ∈ loop around x is equal to a certain central element z G . Here z is determined 0

text: KATRIN

text: WEHRHEIM

text: AND

text: CHRIS

text: WOODWARD

text: Remark

text: (Holomorphic

text: description

text: of

text: the

text: moduli

text: spaces)

text: The

text: moduli

text: spaces

text: M(P

text: have

text: a

text: holomorphic

text: description,

text: for

text: G

text: U(r)

text: due

text: to

text: a

text: famous

text: theorem

text: of

text: Narasimhan-Seshadri

text: generalized

text: to

text: arbitrary

text: groups

text: in

text: Ramanathan’s

text: thesis

text: However,

text: we

text: never

text: use

text: the

text: holomorphic

text: description.

text: Lemma

text: (Symplectomorphisms

text: of

text: moduli

text: spaces

text: induced

text: by

text: bundle

text: isomor-

text: phisms)

text: Let

text: r,

text: d

text: be

text: coprime

text: integers.

text: (a)

text: For

text: any

text: two

text: U(r)-bundles

text: P

text: X

text: P

text: X

text: of

text: degree

text: d,

text: any

text: bundle

text: iso-

text: morphism

text: ψ

text: P

text: P

text: covering

text: an

text: orientation-preserving

text: diﬀeomorphism

text: from

text: X

text: to

text: X

text: induces

text: a

text: symplectomorphism

text: ψ

text: M(P

text: M(P

text: given

text: by

text: pull-back

text: of

text: representatives.

text: (b)

text: Any

text: two

text: such

text: bundle

text: isomorphisms

text: from

text: P

text: to

text: P

text: covering

text: the

text: same

text: diﬀeo-

text: morphism

text: from

text: X

text: to

text: X

text: induce

text: the

text: same

text: symplectomorphism.

text: (c)

text: If

text: P

text: X,

text: P

text: X

text: are

text: U(r)-bundles

text: of

text: degree

text: d

text: over

text: the

text: same

text: surface,

text: then

text: the

text: moduli

text: spaces

text: M(P

text: M(P

text: are

text: canonically

text: symplectomorphic.

text: Proof.

text: For

text: manifolds

text: of

text: dimension

text: at

text: most

text: three,

text: bundles

text: are

text: classiﬁed

text: up

text: to

text: iso-

text: morphism

text: by

text: their

text: ﬁrst

text: Chern

text: class.

text: For

text: connected

text: surfaces,

text: the

text: Chern

text: class

text: is

text: determined

text: by

text: the

text: degree.

text: Hence

text: if

text: π

text: P

text: X

text: j

text: are

text: bundles

text: of

text: the

text: same

text: j

text: j

text: →j

text: degree

text: then

text: there

text: exists

text: an

text: isomorphism

text: φ

text: P

text: P

text: with

text: π

text: φ

text: π

text: This

text: induces

text: A

text: A

text: a

text: map

text: from

text: (P

text: to

text: (P

text: given

text: on

text: the

text: level

text: of

text: tangent

text: spaces

text: by

text: the

text: pull-back

text: of

text: basic

text: forms

text: φ

text: This

text: pull-back

text: acts

text: symplectically:

text: basic

text: basic

text: for

text: a,

text: b

text: g)

text: we

text: have

text: basic

text: π

text: a

text: b

text: π

text: φ

text: a

text: φ

text: b

text: since

text: φ

text: covers

text: an

text: orientation-preserving

text: diﬀeomorphism

text: from

text: X

text: to

text: X

text: Moreover

text: A

text: A

text: pull-back

text: by

text: φ

text: maps

text: (P

text: to

text: (P

text: and

text: is

text: equivariant

text: with

text: respect

text: to

text: the

text: G

text: δ

text: G

text: φ

text: δ

text: ∗G

text: gauge

text: actions

text: of

text: (P

text: and

text: (P

text: φ

text: (P

text: Hence

text: φ

text: descends

text: to

text: a

text: sym-

text: plectomorphism

text: M(P

text: M(P

text: Any

text: two

text: such

text: isomorphisms

text: covering

text: the

text: same

text: diﬀeomorphism

text: diﬀer

text: by

text: a

text: gauge

text: transformation

text: of

text: P

text: Since

text: any

text: gauge

text: transforma-

text: tion

text: of

text: P

text: induces

text: the

text: identity

text: on

text: M(P

text: and

text: M(P

text: is

text: independent

text: of

text: the

text: choice

text: of

text: determinant

text: connection

text: (see

text: Remark

text: the

text: identiﬁcation

text: M(P

text: M(P

text: depends

text: only

text: on

text: the

text: choice

text: of

text: diﬀeomorphism

text: from

text: X

text: to

text: X

text: Remark

text: (Moduli

text: spaces

text: as

text: representations

text: of

text: fundamental

text: group

text: of

text: the

text: punc-

text: tured

text: surface)

text: Atiyah-Bott

text: provide

text: a

text: description

text: of

text: M(P

text: in

text: terms

text: of

text: representa-

text: tions,

text: which

text: makes

text: Lemma

text: more

text: precise

text: in

text: the

text: case

text: of

text: bundles

text: over

text: surfaces.

text: In

text: that

text: case,

text: M(P

text: can

text: be

text: identiﬁed

text: with

text: the

text: moduli

text: space

text: of

text: G

text: -representations

text: of

text: the

text: fundamental

text: group

text: of

text: the

text: punctured

text: surface

text: X

text: x

text: whose

text: value

text: on

text: a

text: small

text: loop

text: around

text: x

text: is

text: equal

text: to

text: a

text: certain

text: central

text: element

text: z

text: G

text: Here

text: z

text: is

text: determined


text: FLOER FIELD THEORY 37 The assertions on zeroth cohomology follow from its interpretation as inﬁnitesimal automorphisms of a ﬂat connection, which are the Lie algebras of the centralizers of its holonomy groups, and part (a). The remaining assertions follow from the Mayer-Vietoris principle for the cohomology of d . To apply it, we consider Y as ∪ α the union U V where U is a neighborhood of X± and V is a neighborhood of the ∩ union of stable manifolds of the critical points, so that U V is a neighborhood → ∩ of the attaching cycles. Then H0(V ; d[g,g]) H0(X V ; d[g,g]) is surjective, since α + α any inﬁnitesimal automorphism of a ﬂat connection over the boundary of a disk ex- → tends over the interior. By the long exact sequence H1(Y ; d[g,g]) H1(X ; d[g|,g] ) ∩ α ∩ + α P+ is injective. On the other hand H1(X− V ; d[g,g]) vanishes, since X− V is ho- α motopic to a ﬁnite set of points and the Poincar´e lemma holds for cohomology with twisted coeﬃcients. Hence H1(Y ; d[g,g]; d[g,g]) surjects onto H1(X−; d[g|,g] ). α α α P− ∩ ∩ Since H1(X− V ; d[g,g]) and H2(X− V ; d[g,g]) vanish, H2(Y ; d[g,g]) is isomorphic ∼ α α α to H2(X−; d[g|,g] ) = H0(X−; d[g|,g] ). Finally surjectivity onto H2(X ; d[g|,g] ) follows α P− α P− + α P+ (cid:3) from the exact sequence. In order to understand the topology of L(Q) we use an explicit description in terms of representations of the extended fundamental group as in Remark 3.3.7. Lemma 3.4.5. Suppose that Y is an elementary bundle bordism from a surface X− of genus g to a surface X of genus g + 1. Let Q be a G-bundle over Y . Then in + ⊂ − × the descrip(cid:0)tion of Remark 3.3.7, the Lagrangian(cid:1)L(Q) M(P−) M(P ) is the + set of pairs of equivalence classes (cid:81) ∈ × [a , b , . . . , a , b ], [a , b , . . . , a , Id] G2g/G G2g+2/G 1 1 g g 1 1 g+1 0 0 0 0 → satisfying the relation g [a , b ] = z. In particular, the projection π : L(Q) j=1 j j M(P−) gives L(Q) the structure of a G -bundle over M(P−), whosevertical tangent 0 space is the associated bundle (cid:89)    ∼ g ∈ × | ker(Dπ) = [a , b , . . . , a , b , ξ] G2g g [a , b ] = z /G . 1 1 g g 0 j j 0 j=1 Proof. By Lemma 3.2.10, the moduli space M(Q) is a Hom(π (Y ), Zss) ﬁber bundle 1 (cid:89) over a component of Hom(π (Y ), Ad(G))/ Ad(G). Choose generators 1 g+1 ∈ α , β , . . . , α , β π (X ), [α , β ] = Id 1 1 g+1 g+1 1 + j j π1(X+) j=1 such that the maps of fundamental groups in Lemma 3.4.4 (a) realize (a) π (Y ) as the quotient of π (X ) by the subgroup generated by β , and 1 1 + g+1 (b) π (X−) as the subgroup of Y generated by α , β , . . . , α , β . 1 1 1 g g

text: FLOER

text: FIELD

text: THEORY

text: The

text: assertions

text: on

text: zeroth

text: cohomology

text: follow

text: from

text: its

text: interpretation

text: as

text: inﬁnitesimal

text: automorphisms

text: of

text: a

text: ﬂat

text: connection,

text: which

text: are

text: the

text: Lie

text: algebras

text: of

text: the

text: centralizers

text: of

text: its

text: holonomy

text: groups,

text: and

text: part

text: (a).

text: The

text: remaining

text: assertions

text: follow

text: from

text: the

text: Mayer-Vietoris

text: principle

text: for

text: the

text: cohomology

text: of

text: d

text: To

text: apply

text: it,

text: we

text: consider

text: Y

text: as

text: α

text: the

text: union

text: U

text: V

text: where

text: U

text: is

text: a

text: neighborhood

text: of

text: X±

text: and

text: V

text: is

text: a

text: neighborhood

text: of

text: the

text: union

text: of

text: stable

text: manifolds

text: of

text: the

text: critical

text: points,

text: so

text: that

text: U

text: V

text: is

text: a

text: neighborhood

text: of

text: the

text: attaching

text: cycles.

text: Then

text: d[g,g])

text: V

text: d[g,g])

text: is

text: surjective,

text: since

text: α

text: α

text: any

text: inﬁnitesimal

text: automorphism

text: of

text: a

text: ﬂat

text: connection

text: over

text: the

text: boundary

text: of

text: a

text: disk

text: ex-

text: tends

text: over

text: the

text: interior.

text: By

text: the

text: long

text: exact

text: sequence

text: d[g,g])

text: d[g|,g]

text: α

text: α

text: P+

text: is

text: injective.

text: On

text: the

text: other

text: hand

text: V

text: d[g,g])

text: vanishes,

text: since

text: X−

text: V

text: is

text: ho-

text: α

text: motopic

text: to

text: a

text: ﬁnite

text: set

text: of

text: points

text: and

text: the

text: Poincar´e

text: lemma

text: holds

text: for

text: cohomology

text: with

text: twisted

text: coeﬃcients.

text: Hence

text: d[g,g];

text: d[g,g])

text: surjects

text: onto

text: d[g|,g]

text: α

text: α

text: α

text: P−

text: Since

text: V

text: d[g,g])

text: and

text: V

text: d[g,g])

text: vanish,

text: d[g,g])

text: is

text: isomorphic

text: α

text: α

text: α

text: to

text: d[g|,g]

text: d[g|,g]

text: Finally

text: surjectivity

text: onto

text: d[g|,g]

text: follows

text: α

text: P−

text: α

text: P−

text: α

text: P+

text: from

text: the

text: exact

text: sequence.

text: In

text: order

text: to

text: understand

text: the

text: topology

text: of

text: L(Q)

text: we

text: use

text: an

text: explicit

text: description

text: in

text: terms

text: of

text: representations

text: of

text: the

text: extended

text: fundamental

text: group

text: as

text: in

text: Remark

text: Lemma

text: Suppose

text: that

text: Y

text: is

text: an

text: elementary

text: bundle

text: bordism

text: from

text: a

text: surface

text: X−

text: of

text: genus

text: g

text: to

text: a

text: surface

text: X

text: of

text: genus

text: g

text: Let

text: Q

text: be

text: a

text: G-bundle

text: over

text: Y

text: Then

text: in

text: the

text: of

text: Remark

text: the

text: M(P−)

text: M(P

text: is

text: the

text: set

text: of

text: pairs

text: of

text: equivalence

text: classes

text: [a

text: b

text: a

text: b

text: [a

text: b

text: a

text: Id]

text: g

text: g

text: satisfying

text: the

text: relation

text: g

text: [a

text: b

text: z.

text: In

text: particular,

text: the

text: projection

text: π

text: L(Q)

text: j

text: j

text: M(P−)

text: gives

text: L(Q)

text: the

text: structure

text: of

text: a

text: G

text: -bundle

text: over

text: M(P−),

text: whosevertical

text: tangent

text: space

text: is

text: the

text: associated

text: bundle

text: g

text: ker(Dπ)

text: [a

text: b

text: a

text: b

text: ξ]

text: g

text: [a

text: b

text: z

text: /G

text: g

text: g

text: j

text: j

text: Proof.

text: By

text: Lemma

text: the

text: moduli

text: space

text: M(Q)

text: is

text: a

text: Hom(π

text: (Y

text: Zss)

text: ﬁber

text: bundle

text: over

text: a

text: component

text: of

text: Hom(π

text: (Y

text: Ad(G))/

text: Ad(G).

text: Choose

text: generators

text: α

text: β

text: α

text: β

text: π

text: (X

text: [α

text: β

text: Id

text: j

text: j

text: such

text: that

text: the

text: maps

text: of

text: fundamental

text: groups

text: in

text: Lemma

text: (a)

text: realize

text: (a)

text: π

text: (Y

text: as

text: the

text: quotient

text: of

text: π

text: (X

text: by

text: the

text: subgroup

text: generated

text: by

text: β

text: and

text: (b)

text: π

text: (X−)

text: as

text: the

text: subgroup

text: of

text: Y

text: generated

text: by

text: α

text: β

text: α

text: β

text: g

text: g


text: 10 KATRIN WEHRHEIM AND CHRIS WOODWARD ∈ (cid:48) ∈ (cid:48) with unique critical points y Y and y Y in each part, j(+1) j(+1) j(+1) j(+1) whose attaching cycles (for some choice of a metric) switch in the following (cid:48) (cid:48) sense: The attaching cycles of y and y in X and those of y and y in (cid:48) j (cid:48) j+1 j j j+1 X are disjoint, while in X − = X − the attaching cycle of y is homotopic j (cid:48) j 1 j 1 j (cid:48) to that of y , and the attaching cycle of y is homotopic to that of y ; and j+1 j+1 (cid:48) j analogously for the intersections of stable manifolds with X = X . More j+1 j+1 precisely, in this situation, critical point switch is the move from ([Y ]) (cid:48) (cid:48) (cid:48) i i=1,...,m to ([Y ]) (cid:48) with m = m, [Y ] = [Y ] for i < j and i > j + 1, and (cid:48) ◦ i (cid:48)i=1,...,m ◦ i i [Y ] [Y ] = [Y ] [Y ] as above. j j+1 j j+1 (c) A cylinder cancellation is the move ◦ ∪ from [Y ] = . . . [Y ] [Y ] . . . to [Y ] = . . . [Y Y ] . . . , j j+1 j j+1 ∈ { − } where for some j 1, . . . , m 1 one of the two consecutive elementary ◦ morphisms [Y ], [Y ] is cylindrical. Then the composition [Y ] [Y ] is an j j+1 j j+1 elementary morphism as well. More precisely, in this situation, critical point (cid:48) (cid:48) − cancellation is the move from ([Y ]) to ([Y ]) (cid:48) with m = m 1, i i=1,...,m i i=1,...,m (cid:48) (cid:48) ◦ (cid:48) [Y ] = [Y ], i < j, [Y ] = [Y ] [Y ], [Y ] = [Y ] i i j j j+1 i i+1 for i > j. A cylinder creation is the same move with the roles of ([Y ]) (cid:48) i i=1,...,m and ([Y ]) (cid:48) interchanged. i i=1,...,m Remark 2.2.10. (Stabilizations of Heegaard splittings versus Cerf moves) In dimen- sion n = 2 we can compare critical point creation to the stabilization of Heegaard splittings. A stabilization of a Heegaard splitting is obtained by connect sum with ∪ the standard Heegaard splitting of a sphere S3 = H− H into solid tori H−, H . ∪ + + More precisely, given a Heegaard splitting Y = Y− Y , its stabilization is ob- (cid:48) ∪X (cid:48)+ ta(cid:48)ined by pulling the Heeg∼aard splitting Y S3 = Y− Y−+ into×compress(cid:48)i(cid:48)o∪n bo(cid:48)(cid:48)dies Y± = Y± H± back to Y = Y S3. Equivalently, let [ 1, 1] X = Y− Y be + the decomposition of the cylindrical bordism consisting of two elementary bordisms, (cid:48) each carrying a Morse function with index 1 resp. 2. Then Y± is obtained from (cid:48)(cid:48) Y± by attaching Y± at X, since attaching a one-handle is equivalent to connected ∪ sum with a torus. Thus if the Heegaard splitting Y = Y− Y is induced by a + Cerf decomposition then the stabilization is obtained from a critical point creation. Conversely any critical point creation can be viewed as a connected sum as above, and so induces a stabilization of the corresponding Heegaard splitting. Theorem 2.2.11 (Connected Cerf theory). Let Y be a bordism as before of dimen- ≥ ◦ ◦ sion n + 1 3, and ﬁx a Cerf decomposition [Y ] = [Y ] . . . [Y ]. Then any other 1 m Cerf decomposition of [Y ] can be obtained up to equivalence from ([Y ]) by a i i=1,...,m ﬁnite sequence of Cerf moves.

text: KATRIN

text: WEHRHEIM

text: AND

text: CHRIS

text: WOODWARD

text: with

text: unique

text: critical

text: points

text: y

text: Y

text: and

text: y

text: Y

text: in

text: each

text: part,

text: whose

text: attaching

text: cycles

text: (for

text: some

text: choice

text: of

text: a

text: metric)

text: switch

text: in

text: the

text: following

text: sense:

text: The

text: attaching

text: cycles

text: of

text: y

text: and

text: y

text: in

text: X

text: and

text: those

text: of

text: y

text: and

text: y

text: in

text: j

text: j

text: j

text: X

text: are

text: disjoint,

text: while

text: in

text: X

text: X

text: the

text: attaching

text: cycle

text: of

text: y

text: is

text: homotopic

text: j

text: j

text: j

text: j

text: to

text: that

text: of

text: y

text: and

text: the

text: attaching

text: cycle

text: of

text: y

text: is

text: homotopic

text: to

text: that

text: of

text: y

text: and

text: j

text: analogously

text: for

text: the

text: intersections

text: of

text: stable

text: manifolds

text: with

text: X

text: X

text: More

text: precisely,

text: in

text: this

text: situation,

text: critical

text: point

text: switch

text: is

text: the

text: move

text: from

text: ([Y

text: i

text: to

text: ([Y

text: with

text: m

text: m,

text: [Y

text: [Y

text: for

text: i

text: j

text: and

text: i

text: j

text: and

text: i

text: i

text: i

text: [Y

text: [Y

text: [Y

text: [Y

text: as

text: above.

text: j

text: j

text: (c)

text: A

text: cylinder

text: cancellation

text: is

text: the

text: move

text: from

text: [Y

text: [Y

text: [Y

text: to

text: [Y

text: [Y

text: Y

text: j

text: j

text: where

text: for

text: some

text: j

text: m

text: one

text: of

text: the

text: two

text: consecutive

text: elementary

text: morphisms

text: [Y

text: [Y

text: is

text: cylindrical.

text: Then

text: the

text: composition

text: [Y

text: [Y

text: is

text: an

text: j

text: j

text: elementary

text: morphism

text: as

text: well.

text: More

text: precisely,

text: in

text: this

text: situation,

text: critical

text: point

text: cancellation

text: is

text: the

text: move

text: from

text: ([Y

text: to

text: ([Y

text: with

text: m

text: m

text: i

text: i

text: [Y

text: [Y

text: i

text: j,

text: [Y

text: [Y

text: [Y

text: [Y

text: [Y

text: i

text: i

text: j

text: j

text: i

text: for

text: i

text: j.

text: A

text: cylinder

text: creation

text: is

text: the

text: same

text: move

text: with

text: the

text: roles

text: of

text: ([Y

text: i

text: and

text: ([Y

text: interchanged.

text: i

text: Remark

text: (Stabilizations

text: of

text: Heegaard

text: splittings

text: versus

text: Cerf

text: moves)

text: In

text: dimen-

text: sion

text: n

text: we

text: can

text: compare

text: critical

text: point

text: creation

text: to

text: the

text: stabilization

text: of

text: Heegaard

text: splittings.

text: A

text: stabilization

text: of

text: a

text: Heegaard

text: splitting

text: is

text: obtained

text: by

text: connect

text: sum

text: with

text: the

text: standard

text: Heegaard

text: splitting

text: of

text: a

text: sphere

text: H−

text: H

text: into

text: solid

text: tori

text: H−,

text: H

text: More

text: precisely,

text: given

text: a

text: Heegaard

text: splitting

text: Y

text: Y−

text: Y

text: its

text: stabilization

text: is

text: ob-

text: ∪X

text: by

text: pulling

text: the

text: Heeg∼aard

text: splitting

text: Y

text: Y−

text: Y−+

text: Y±

text: Y±

text: H±

text: back

text: to

text: Y

text: Y

text: Equivalently,

text: let

text: X

text: Y−

text: Y

text: be

text: the

text: decomposition

text: of

text: the

text: cylindrical

text: bordism

text: consisting

text: of

text: two

text: elementary

text: bordisms,

text: each

text: carrying

text: a

text: Morse

text: function

text: with

text: index

text: resp.

text: Then

text: Y±

text: is

text: obtained

text: from

text: Y±

text: by

text: attaching

text: Y±

text: at

text: X,

text: since

text: attaching

text: a

text: one-handle

text: is

text: equivalent

text: to

text: connected

text: sum

text: with

text: a

text: torus.

text: Thus

text: if

text: the

text: Heegaard

text: splitting

text: Y

text: Y−

text: Y

text: is

text: induced

text: by

text: a

text: Cerf

text: decomposition

text: then

text: the

text: stabilization

text: is

text: obtained

text: from

text: a

text: critical

text: point

text: creation.

text: Conversely

text: any

text: critical

text: point

text: creation

text: can

text: be

text: viewed

text: as

text: a

text: connected

text: sum

text: as

text: above,

text: and

text: so

text: induces

text: a

text: stabilization

text: of

text: the

text: corresponding

text: Heegaard

text: splitting.

text: Theorem

text: (Connected

text: Cerf

text: theory).

text: Let

text: Y

text: be

text: a

text: bordism

text: as

text: before

text: of

text: dimen-

text: sion

text: n

text: and

text: ﬁx

text: a

text: Cerf

text: decomposition

text: [Y

text: [Y

text: [Y

text: Then

text: any

text: other

text: m

text: Cerf

text: decomposition

text: of

text: [Y

text: can

text: be

text: obtained

text: up

text: to

text: equivalence

text: from

text: ([Y

text: by

text: a

text: i

text: ﬁnite

text: sequence

text: of

text: Cerf

text: moves.


text: 5 (cid:80) (cid:104) − (cid:105) (cid:80) (a) (b) χ(τ) = T js((cid:126)q, τ)js( (cid:126)q, 0) has the well known form ± τ x x − (Jχ/2)2 sin2 k D ( τ)D (τ). For AFM (cid:126)k,σ x (cid:126)kσ (cid:126)k+q(cid:126)σ × SBs, this is given by χ(τ) = (J∆/2)2 cos2 k − (cid:126)k− x − [D− ↓(τ)D ↑(τ) + D ↑( τ)D− − ↓( τ) ∗ (cid:126)k−+q(cid:126) ∗ (cid:126)k − − (cid:126)k (cid:126)k q(cid:126) F−(cid:126)k( τ)F(cid:126)k+q(cid:126)(τ) F(cid:126)k−q(cid:126)( τ)F(cid:126)k(τ)]. These correla- tion functions are diagramatically shown in Fig. 3. FIG. 3: Feynman diagram representing the spin conductance Without disorders and interactions, the conductivity for (a) FM SBs and AFM SFs and (b) AFM SBs. Single wavy lines represent the normal propagators of APs, while consists of a delta function located at ω = 0 with the double wavy lines are the anomalous propagators. Dots are amplitude D, i.e., Drude weight. For FM SBs and AFM the vertexes. SFs, this is given b(cid:18)y (cid:19) (cid:88) inside the MI or from one electrode to the other elec- − Jχ 2 (cid:48) D = 2π sin2k n (ω ), (15) trode through the MI. This problem might require the 2 x X (cid:126)k treatment of nonequilibrium conditions including both (cid:126)k NM regions and a MI region. However, this treatment is so far only available at very low temperatures [22]. and for (cid:18)AFM(cid:19)SB we ﬁnd (cid:32) (cid:33) Here, instead we discuss the spin conductivity of bulk (cid:88) MIs. Combined with the spin injection discussed in the previous section, this should provide important insights − J∆ 2 J∆γ(cid:48) 2 (cid:48) D = π cos2k (cid:126)k n (sω ). (16) into the performance of MI/NM systems. 2 x ω B (cid:126)k We start from deriving the bulk spin current by the (cid:126)k,s=±1 (cid:126)k † − † equation of motion for Sz = a ↑a ↑ a ↓a ↓. Under (cid:126)r (cid:126)r (cid:126)r (cid:126)r (cid:126)r the MF approximation, the spin current on the bond ((cid:126)r, Figure 4 shows the numerical results for D for the FM † − † (cid:126)r+xˆ) is expressed as js((cid:126)r) = Jχ(ia ↑a ↑ ia ↓a ↓ + model (a) and the AFM model (b). For the AFM model, x 4 (cid:126)r (cid:126)r+xˆ (cid:126)r (cid:126)r+xˆ both SB and SF results are shown. At high tempera- H.c.) for the FM SB and the AFM SF cases and js((cid:126)r) = J∆(ia ↑a ↓ + ia ↓a ↑ + H.c.) for the AF SB caxse. In tures, D increases with decreasing temperature for all the 4 (cid:126)r (cid:126)r+xˆ (cid:126)r (cid:126)r+xˆ cases. But for SBs, D is cutoﬀ at the Bose condensation a momentum space, these spin current operators in the or magnetic ordering indicated by triangles because the small-(cid:126)q limit bec(cid:88)ome (cid:16) (cid:17) ﬂuctuating components decrease rapidly (the maximum of D is located slightly above T ). These D become zero c Jχ † − † at zero temperature, which is consistent with the results jxs((cid:126)q) = 2 sin kx a(cid:126)k↑a(cid:126)k+q(cid:126)↑ a(cid:126)k↓a(cid:126)k+q(cid:126)↓ (13) using the spin wave theory [23, 27]. Such a temperature (cid:126)k behavior is analogous to that of the uniform magnetic susceptibility, maximizing or diverging at T , as magnetic c for the FM S(cid:88)Bs and th(cid:16)e AFM SFs and (cid:17) systems become most susceptible against external (mag- netic) perturbations. By contrast, the spin conductance J∆ − † † does not diverge even for a FM. This is because the spin jxs((cid:126)q) = 2 i cos kx a−(cid:126)k+q(cid:126)↓a(cid:126)k↑ a(cid:126)k↑a−(cid:126)k−q(cid:126)↓ (14) current operator and the magnetic moment have the dif- (cid:126)k ferent momentum dependence or the form factor, see Eqs. (13) and (14), and only thermally populated SBs con- for the AFM SBs. tribute to the spin conductance. Within the MF approx- The spin conductivity σ(ω) is obtained by using the imation used here, AFM SFs behave as non-interacting standard Kubo formula [23]. For SB cases, care must fermions. Thus, the conductance shows the standard be- be taken for the condensed components of SBs, which havior due to the thermal broadening. At ﬁnite temper- represent the ordered moments not superconductivity atures, D decreases with increasing temperature roughly − [24]. While the condensed components could introduce proportional to (T/T ) 2 in all the cases. In fact, T is 0 0 novel spin superconductivity [25, 26], this requires proportional to the MF order parameter, χ or ∆, which an additional theoretical or experimental setup, the should also have the T dependence in principle. Since χ2 |(cid:104) · (cid:105)| planer spin anisotropy for FM or an external magnetic or ∆2 is related to S(cid:126) S(cid:126) (cid:48) , roughly proportional to − (cid:126)r (cid:126)r − ﬁeld for AFM, and the analyses on the dynamics of T 1, D is expected to be scaled as T 3 at much higher the condensed components, which is beyond the scope temperatures than the ordering temperature. of our study. Thus, we focus on the contribution When the eﬀect of disorder or the Gilbert dumping (cid:82) from the ﬂuctuating components to the spin conduc- [28] is to introduce a ﬁnite lifetime for APs, τ (qp qp tivity, which is given by the analytic continuation of stands for quasiparticle), σ(ω) becomes a Lorentzian as (cid:104) − (cid:105)| σ(iΩl) = →i 01/T dτeiΩlτ Tτjxs((cid:126)q, τ)jxs( (cid:126)q, 0) q(cid:126)→0/iΩl σ(ω) = Dτqp/π[(ωτqp)2 + 1], and the dc conductivity with iΩ ω + iδ. For FM SBs and AFM becomes ﬁnite. For the AFM SB case, there appears l SFs, the current-current correlation function additional continuum at ﬁnite frequencies extended up ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: (a)

font: default

text: (b)

text: χ(τ)

text: T

text: τ)js(

text: has

text: the

text: well

text: known

text: form

text: τ

text: x

text: x

text: k

text: D

text: τ)D

text: (τ).

text: For

text: AFM

text: x

text: SBs,

text: this

text: is

text: given

text: by

text: χ(τ)

text: k

text: x

text: [D−

text: ↓(τ)D

text: ↑(τ)

text: D

text: τ)D−

text: τ)

text: These

text: correla-

text: tion

text: functions

text: are

text: diagramatically

text: shown

text: in

text: Fig.

text: FIG.

text: Feynman

label: caption

text: diagram

text: representing

text: the

text: spin

text: conductance

text: Without

text: disorders

text: and

text: interactions,

text: the

text: conductivity

text: for

text: (a)

text: FM

text: SBs

text: and

text: AFM

text: SFs

text: and

text: (b)

text: AFM

text: SBs.

text: Single

text: wavy

text: lines

text: represent

text: the

text: normal

text: propagators

text: of

text: APs,

text: while

text: consists

text: of

text: a

text: delta

text: function

text: located

text: at

text: ω

text: with

text: the

text: double

text: wavy

text: lines

text: are

text: the

text: anomalous

text: propagators.

text: Dots

text: are

text: amplitude

text: D,

text: i.e.,

text: Drude

text: weight.

text: For

text: FM

text: SBs

text: and

text: AFM

text: the

text: vertexes.

text: SFs,

text: this

text: is

text: given

text: inside

text: the

text: MI

text: or

text: from

text: one

text: electrode

text: to

text: the

text: other

text: elec-

text: Jχ

text: D

text: 2π

text: n

text: (ω

text: trode

text: through

text: the

text: MI.

text: This

text: problem

text: might

text: require

text: the

text: x

text: X

text: treatment

text: of

text: nonequilibrium

text: conditions

text: including

text: both

text: NM

text: regions

text: and

text: a

text: MI

text: region.

text: However,

text: this

text: treatment

text: is

text: so

text: far

text: only

text: available

text: at

text: very

text: low

text: temperatures

text: and

text: for

text: we

text: ﬁnd

text: Here,

text: instead

text: we

text: discuss

text: the

text: spin

text: conductivity

text: of

text: bulk

text: MIs.

text: Combined

text: with

text: the

text: spin

text: injection

text: discussed

text: in

text: the

text: previous

text: section,

text: this

text: should

text: provide

text: important

text: insights

text: J∆

text: D

text: π

text: n

text: (sω

text: into

text: the

text: performance

text: of

text: MI/NM

text: systems.

text: x

text: ω

text: B

text: We

text: start

text: from

text: deriving

text: the

text: bulk

text: spin

text: current

text: by

text: the

text: equation

text: of

text: motion

text: for

text: Sz

text: a

text: ↑a

text: a

text: ↓a

text: Under

text: the

text: MF

text: approximation,

text: the

text: spin

text: current

text: on

text: the

text: bond

text: Figure

text: shows

text: the

text: numerical

text: results

text: for

text: D

text: for

text: the

text: FM

text: is

text: expressed

text: as

text: Jχ(ia

text: ↑a

text: ia

text: ↓a

text: model

text: (a)

text: and

text: the

text: AFM

text: model

text: (b).

text: For

text: the

text: AFM

text: model,

text: x

text: both

text: SB

text: and

text: SF

text: results

text: are

text: shown.

text: At

text: high

text: tempera-

text: H.c.)

text: for

text: the

text: FM

text: SB

text: and

text: the

text: AFM

text: SF

text: cases

text: and

text: J∆(ia

text: ↑a

text: ia

text: ↓a

text: H.c.)

text: for

text: the

text: AF

text: SB

text: caxse.

text: In

text: tures,

text: D

text: increases

text: with

text: decreasing

text: temperature

text: for

text: all

text: the

text: cases.

text: But

text: for

text: SBs,

text: D

text: is

text: cutoﬀ

text: at

text: the

text: Bose

text: condensation

text: a

text: momentum

text: space,

text: these

text: spin

text: current

text: operators

text: in

text: the

text: or

text: magnetic

text: ordering

text: indicated

text: by

text: triangles

text: because

text: the

text: limit

text: ﬂuctuating

text: components

text: decrease

text: rapidly

text: (the

text: maximum

text: of

text: D

text: is

text: located

text: slightly

text: above

text: T

text: These

text: D

text: become

text: zero

text: c

text: Jχ

text: at

text: zero

text: temperature,

text: which

text: is

text: consistent

text: with

text: the

text: results

text: sin

text: kx

text: using

text: the

text: spin

text: wave

text: theory

text: Such

text: a

text: temperature

text: behavior

text: is

text: analogous

text: to

text: that

text: of

text: the

text: uniform

text: magnetic

text: susceptibility,

text: maximizing

text: or

text: diverging

text: at

text: T

text: as

text: magnetic

text: c

text: for

text: the

text: FM

text: and

text: AFM

text: SFs

text: and

text: systems

text: become

text: most

text: susceptible

text: against

text: external

text: (mag-

text: netic)

text: perturbations.

text: By

text: contrast,

text: the

text: spin

text: conductance

text: J∆

text: does

text: not

text: diverge

text: even

text: for

text: a

text: FM.

text: This

text: is

text: because

text: the

text: spin

text: i

text: cos

text: kx

text: current

text: operator

text: and

text: the

text: magnetic

text: moment

text: have

text: the

text: dif-

text: ferent

text: momentum

text: dependence

text: or

text: the

text: form

text: factor,

text: see

text: Eqs.

text: and

text: and

text: only

text: thermally

text: populated

text: SBs

text: con-

text: for

text: the

text: AFM

text: SBs.

text: tribute

text: to

text: the

text: spin

text: conductance.

text: Within

text: the

text: MF

text: approx-

text: The

text: spin

text: conductivity

text: σ(ω)

text: is

text: obtained

text: by

text: using

text: the

text: imation

text: used

text: here,

text: AFM

text: SFs

text: behave

text: as

text: non-interacting

text: standard

text: Kubo

text: formula

text: For

text: SB

text: cases,

text: care

text: must

text: fermions.

text: Thus,

text: the

text: conductance

text: shows

text: the

text: standard

text: be-

text: be

text: taken

text: for

text: the

text: condensed

text: components

text: of

text: SBs,

text: which

text: havior

text: due

text: to

text: the

text: thermal

text: broadening.

text: At

text: ﬁnite

text: temper-

text: represent

text: the

text: ordered

text: moments

text: not

text: superconductivity

text: atures,

text: D

text: decreases

text: with

text: increasing

text: temperature

text: roughly

text: While

text: the

text: condensed

text: components

text: could

text: introduce

text: proportional

text: to

text: (T/T

text: in

text: all

text: the

text: cases.

text: In

text: fact,

text: T

text: is

text: novel

text: spin

text: superconductivity

text: this

text: requires

text: proportional

text: to

text: the

text: MF

text: order

text: parameter,

text: χ

text: or

text: which

text: an

text: additional

text: theoretical

text: or

text: experimental

text: setup,

text: the

text: should

text: also

text: have

text: the

text: T

text: dependence

text: in

text: principle.

text: Since

text: χ2

text: planer

text: spin

text: anisotropy

text: for

text: FM

text: or

text: an

text: external

text: magnetic

text: or

text: ∆2

text: is

text: related

text: to

text: roughly

text: proportional

text: to

text: ﬁeld

text: for

text: AFM,

text: and

text: the

text: analyses

text: on

text: the

text: dynamics

text: of

text: T

text: D

text: is

text: expected

text: to

text: be

text: scaled

text: as

text: T

text: at

text: much

text: higher

text: the

text: condensed

text: components,

text: which

text: is

text: beyond

text: the

text: scope

text: temperatures

text: than

text: the

text: ordering

text: temperature.

text: of

text: our

text: study.

text: Thus,

text: we

text: focus

text: on

text: the

text: contribution

text: When

text: the

text: eﬀect

text: of

text: disorder

text: or

text: the

text: Gilbert

text: dumping

text: from

text: the

text: ﬂuctuating

text: components

text: to

text: the

text: spin

text: conduc-

text: is

text: to

text: introduce

text: a

text: ﬁnite

text: lifetime

text: for

text: APs,

text: τ

text: (qp

text: qp

text: tivity,

text: which

text: is

text: given

text: by

text: the

text: analytic

text: continuation

text: of

text: stands

text: for

text: quasiparticle),

text: σ(ω)

text: becomes

text: a

text: Lorentzian

text: as

text: σ(iΩl)

text: →i

text: dτeiΩlτ

text: τ)jxs(

text: σ(ω)

text: and

text: the

text: dc

text: conductivity

text: with

text: iΩ

text: ω

text: iδ.

text: For

text: FM

text: SBs

text: and

text: AFM

text: becomes

text: ﬁnite.

text: For

text: the

text: AFM

text: SB

text: case,

text: there

text: appears

text: l

text: SFs,

text: the

text: current-current

text: correlation

text: function

text: additional

text: continuum

text: at

text: ﬁnite

text: frequencies

text: extended

text: up

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 7 it is reasonable that only the Drude feature is detectable IV. CONCLUSION in the GHz range. In Ref. [16], Wu et al. report that the voltage response due to the PM SSE is proportional − ∼ to T α with α 3–4. Our results indicate that a part of To summarize, we have presented rather simple de- this temperature dependence is from the intrinsic tem- scriptions to understand the spin current injection at a perature dependence of the spin conductivity in a PMI paramagnetic insulator/normal metal interface and the and the spin injection at the interface. However, de- spin current transport inside a paramagnetic or magnetic tailed analyses for this experiment require even more in- insulator based on the Schwinger boson and Schwinger formation, such as the temperature dependence of order fermion mean-ﬁeld theories. Both the spin current injec- parameters, the lifetime of APs due to disorder, particle- tion js and spin current transport D show rather uni- int particle interactions, and the interactions with phonons. versal behaviors irrespective of the sign of the magnetic As related subjects, there have appeared some the- coupling, FM or AFM, and methodology: js depends int oretical proposals to detect novel electronic states of on temperature rather weakly above magnetic transition spin liquid systems by interfacing with conventional sys- temperature, while it decreases rapidly with decreasing tems. References [22, 31] proposed a similar experimental temperature below the transition temperature. Simi- setup, the injection of spin current into a MI, to detect larly, D decreases with increasing temperature above the spinon or magnon excitation spectra inside the insula- transition temperature, and rapidly decreases with de- tor. Their focus is the voltage dependence of the spin creasing temperature below the transition temperature. injection through a clean interface at low temperatures. These ﬁndings suggest that there is an optimal temper- On the other hand, our focus is rather the temperature ature window for the performance of a magnetic insula- dependence of the spin injection through a rough inter- tor/normal metal junction around the magnetic transi- face where momentum transfer is not conserved. Thus, it tion temperature of the constituent magnetic insulator. should be experimentally more easily accessible. Refer- Such a device would potentially function even above the ence [32] considered a junction consisting of a spin liquid magnetic transition temperature. The current results insulator sandwiched by two FMs. They proposed that support the conjecture of Ref. [15]—the enhancement a spinon Fermi surface could induce a characteristic os- of V around T is mainly due to the spin-current ISHE N cillatory behavior in the magnetic coupling between the transport by thermal magnons inside an AFM spacer— two FMs as a function of the thickness of the spin liquid and suggest that a similar mechanism contributes to the spacer. In our case, a magnetic coupling between NM and enhancement of V around T of a FM insulator used ISHE C PMI is rather dynamical, the injection of spin current in for the spin current generator without a spacer [14]. It or out of PMI. Further, we focused on the universal be- would be also interesting to analyze nonequilibrium sit- havior of PMI described by both bosonic and fermionic uations in which a magnetic insulator is sandwiched by APs, which do not necessarily have a Fermi surface. two normal metals [19] with diﬀerent spin accumulations In contrast to metal- or semiconductor-based mag- and temperatures. Our approach would provide some netic interfaces as widely discussed in the literature [19], useful insights into the phenomena reported in Ref. [16]. MI/NM interfaces do not necessarily accompany the Finally, further experimental tests for the frequency- charge accumulation inside a magnetic region when in- dependent spin conductance, as also predicted in Ref. j(cid:80)ecting or extracting spins. Instead, when the interface [23], would be desirable, but experimental techniques re- spin accumulation exists, the Lagrange multiplier or the main to be developed. local chemical potential, enforcing the local constraint † a a = 2S, may have to be ﬁxed accordingly using Acknowledgments σ (cid:126)rσ (cid:126)rσ techniques developed for inhomogeneous magnetic sys- tems [33, 34]. One of the interesting directions is com- bining techniques developed in Refs. [19, 33, 34] and ap- S.O. thanks A. Bhattacharya, S. Takahashi, N. Na- plying them to nanostructures with the spin injector, de- gaosa, S. Maekawa, D. Xiao, D. Hou, Z. Qiu and E. tector, and a magnetic/PM insulator; similar structures Saitoh for fruitful discussions. The research by S.O. is are originally proposed in Ref. [35]. Such nanostructures supported by the U.S. Department of Energy, Oﬃce of might be useful to measure the spin diﬀusion length in Science, Basic Energy Sciences, Materials Sciences and magnetic/PM insulators. Engineering Division. [1] Concepts in Spin Electronics, edited by S. Maekawa (Ox- [5] S. Murakami, N. Nagaosa, and S. C. Zhang, Science 301, ford University Press, 2006). 1348 (2003). [2] M. I. Dyakonov and V. I. Perel, Phys. Lett. A35, 459 [6] J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jung- (1971). wirth, and A. H. MacDonald, Phys. Rev. Lett. 92, 126603 [3] J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999). (2004). [4] S. Zhang, Phys. Rev. Lett. 85, 393 (2000). [7] E. Saitoh, M. Ueda, H. Miyajima, G. Tatara, App. Phys. ##LTLine## ##LTLine## ##LTLine##

text: it

text: is

text: reasonable

text: that

text: only

text: the

text: Drude

text: feature

text: is

text: detectable

text: IV.

label: section

text: CONCLUSION

text: in

text: the

text: GHz

text: range.

text: In

text: Ref.

text: Wu

text: et

text: al.

text: report

text: that

text: the

text: voltage

text: response

text: due

text: to

text: the

text: PM

text: SSE

text: is

text: proportional

text: to

text: T

text: α

text: with

text: α

text: Our

text: results

text: indicate

text: that

text: a

text: part

text: of

text: To

text: summarize,

text: we

text: have

text: presented

text: rather

text: simple

text: de-

text: this

text: temperature

text: dependence

text: is

text: from

text: the

text: intrinsic

text: tem-

text: scriptions

text: to

text: understand

text: the

text: spin

text: current

text: injection

text: at

text: a

text: perature

text: dependence

text: of

text: the

text: spin

text: conductivity

text: in

text: a

text: PMI

text: paramagnetic

text: insulator/normal

text: metal

text: interface

text: and

text: the

text: and

text: the

text: spin

text: injection

text: at

text: the

text: interface.

text: However,

text: de-

text: spin

text: current

text: transport

text: inside

text: a

text: paramagnetic

text: or

text: magnetic

text: tailed

text: analyses

text: for

text: this

text: experiment

text: require

text: even

text: more

text: in-

text: insulator

text: based

text: on

text: the

text: Schwinger

text: boson

text: and

text: Schwinger

text: formation,

text: such

text: as

text: the

text: temperature

text: dependence

text: of

text: order

text: fermion

text: mean-ﬁeld

text: theories.

text: Both

text: the

text: spin

text: current

text: injec-

text: parameters,

text: the

text: lifetime

text: of

text: APs

text: due

text: to

text: disorder,

text: particle-

text: tion

text: js

text: and

text: spin

text: current

text: transport

text: D

text: show

text: rather

text: uni-

text: int

text: particle

text: interactions,

text: and

text: the

text: interactions

text: with

text: phonons.

text: versal

text: behaviors

text: irrespective

text: of

text: the

text: sign

text: of

text: the

text: magnetic

text: As

text: related

text: subjects,

text: there

text: have

text: appeared

text: some

text: the-

text: coupling,

text: FM

text: or

text: AFM,

text: and

text: methodology:

text: js

text: depends

text: int

text: oretical

text: proposals

text: to

text: detect

text: novel

text: electronic

text: states

text: of

text: on

text: temperature

text: rather

text: weakly

text: above

text: magnetic

text: transition

text: spin

text: liquid

text: systems

text: by

text: interfacing

text: with

text: conventional

text: sys-

text: temperature,

text: while

text: it

text: decreases

text: rapidly

text: with

text: decreasing

text: tems.

text: References

text: proposed

text: a

text: similar

text: experimental

text: temperature

text: below

text: the

text: transition

text: temperature.

text: Simi-

text: setup,

text: the

text: injection

text: of

text: spin

text: current

text: into

text: a

text: MI,

text: to

text: detect

text: larly,

text: D

text: decreases

text: with

text: increasing

text: temperature

text: above

text: the

text: spinon

text: or

text: magnon

text: excitation

text: spectra

text: inside

text: the

text: insula-

text: transition

text: temperature,

text: and

text: rapidly

text: decreases

text: with

text: de-

text: tor.

text: Their

text: focus

text: is

text: the

text: voltage

text: dependence

text: of

text: the

text: spin

text: creasing

text: temperature

text: below

text: the

text: transition

text: temperature.

text: injection

text: through

text: a

text: clean

text: interface

text: at

text: low

text: temperatures.

text: These

text: ﬁndings

text: suggest

text: that

text: there

text: is

text: an

text: optimal

text: temper-

text: On

text: the

text: other

text: hand,

text: our

text: focus

text: is

text: rather

text: the

text: temperature

text: ature

text: window

text: for

text: the

text: performance

text: of

text: a

text: magnetic

text: insula-

text: dependence

text: of

text: the

text: spin

text: injection

text: through

text: a

text: rough

text: inter-

text: tor/normal

text: metal

text: junction

text: around

text: the

text: magnetic

text: transi-

text: face

text: where

text: momentum

text: transfer

text: is

text: not

text: conserved.

text: Thus,

text: it

text: tion

text: temperature

text: of

text: the

text: constituent

text: magnetic

text: insulator.

text: should

text: be

text: experimentally

text: more

text: easily

text: accessible.

text: Refer-

text: Such

text: a

text: device

text: would

text: potentially

text: function

text: even

text: above

text: the

text: ence

text: considered

text: a

text: junction

text: consisting

text: of

text: a

text: spin

text: liquid

text: magnetic

text: transition

text: temperature.

text: The

text: current

text: results

text: insulator

text: sandwiched

text: by

text: two

text: FMs.

text: They

text: proposed

text: that

text: support

text: the

text: conjecture

text: of

text: Ref.

text: enhancement

text: a

text: spinon

text: Fermi

text: surface

text: could

text: induce

text: a

text: characteristic

text: os-

text: of

text: V

text: around

text: T

text: is

text: mainly

text: due

text: to

text: the

text: spin-current

text: ISHE

text: N

text: cillatory

text: behavior

text: in

text: the

text: magnetic

text: coupling

text: between

text: the

text: transport

text: by

text: thermal

text: magnons

text: inside

text: an

text: AFM

text: spacer—

text: two

text: FMs

text: as

text: a

text: function

text: of

text: the

text: thickness

text: of

text: the

text: spin

text: liquid

text: and

text: suggest

text: that

text: a

text: similar

text: mechanism

text: contributes

text: to

text: the

text: spacer.

text: In

text: our

text: case,

text: a

text: magnetic

text: coupling

text: between

text: NM

text: and

text: enhancement

text: of

text: V

text: around

text: T

text: of

text: a

text: FM

text: insulator

text: used

text: ISHE

text: C

text: PMI

text: is

text: rather

text: dynamical,

text: the

text: injection

text: of

text: spin

text: current

text: in

text: for

text: the

text: spin

text: current

text: generator

text: without

text: a

text: spacer

text: It

text: or

text: out

text: of

text: PMI.

text: Further,

text: we

text: focused

text: on

text: the

text: universal

text: be-

text: would

text: be

text: also

text: interesting

text: to

text: analyze

text: nonequilibrium

text: sit-

text: havior

text: of

text: PMI

text: described

text: by

text: both

text: bosonic

text: and

text: fermionic

text: uations

text: in

text: which

text: a

text: magnetic

text: insulator

text: is

text: sandwiched

text: by

text: APs,

text: which

text: do

text: not

text: necessarily

text: have

text: a

text: Fermi

text: surface.

text: two

text: normal

text: metals

text: with

text: diﬀerent

text: spin

text: accumulations

text: In

text: contrast

text: to

text: metal-

text: or

text: semiconductor-based

text: mag-

text: and

text: temperatures.

text: Our

text: approach

text: would

text: provide

text: some

text: netic

text: interfaces

text: as

text: widely

text: discussed

text: in

text: the

text: literature

text: useful

text: insights

text: into

text: the

text: phenomena

text: reported

text: in

text: Ref.

text: MI/NM

text: interfaces

text: do

text: not

text: necessarily

text: accompany

text: the

text: Finally,

text: further

text: experimental

text: tests

text: for

text: the

text: frequency-

text: charge

text: accumulation

text: inside

text: a

text: magnetic

text: region

text: when

text: in-

text: dependent

text: spin

text: conductance,

text: as

text: also

text: predicted

text: in

text: Ref.

text: or

text: extracting

text: spins.

text: Instead,

text: when

text: the

text: interface

text: would

text: be

text: desirable,

text: but

text: experimental

text: techniques

text: re-

text: spin

text: accumulation

text: exists,

text: the

text: Lagrange

text: multiplier

text: or

text: the

text: main

text: to

text: be

text: developed.

text: local

text: chemical

text: potential,

text: enforcing

text: the

text: local

text: constraint

text: a

text: a

text: may

text: have

text: to

text: be

text: ﬁxed

text: accordingly

text: using

text: Acknowledgments

text: σ

text: techniques

text: developed

text: for

text: inhomogeneous

text: magnetic

text: sys-

text: tems

text: One

text: of

text: the

text: interesting

text: directions

text: is

text: com-

text: bining

text: techniques

text: developed

text: in

text: Refs.

text: and

text: ap-

text: S.O.

text: thanks

text: A.

text: Bhattacharya,

text: S.

text: Takahashi,

text: N.

text: Na-

text: plying

text: them

text: to

text: nanostructures

text: with

text: the

text: spin

text: injector,

text: de-

text: gaosa,

text: S.

text: Maekawa,

text: D.

text: Xiao,

text: D.

text: Hou,

text: Z.

text: Qiu

text: and

text: E.

text: tector,

text: and

text: a

text: magnetic/PM

text: insulator;

text: similar

text: structures

text: Saitoh

text: for

text: fruitful

text: discussions.

text: The

text: research

text: by

text: S.O.

text: is

text: are

text: originally

text: proposed

text: in

text: Ref.

text: Such

text: nanostructures

text: supported

text: by

text: the

text: U.S.

text: Department

text: of

text: Energy,

text: Oﬃce

text: of

text: might

text: be

text: useful

text: to

text: measure

text: the

text: spin

text: diﬀusion

text: length

text: in

text: Science,

text: Basic

text: Energy

text: Sciences,

text: Materials

text: Sciences

text: and

text: magnetic/PM

text: insulators.

text: Engineering

text: Division.

text: Concepts

text: in

text: Spin

text: Electronics,

text: edited

text: by

text: S.

text: Maekawa

text: (Ox-

text: S.

text: Murakami,

text: N.

text: Nagaosa,

text: and

text: S.

text: C.

text: Zhang,

text: Science

text: ford

text: University

text: Press,

text: M.

text: I.

text: Dyakonov

text: and

text: V.

text: I.

text: Perel,

text: Phys.

text: Lett.

text: J.

text: Sinova,

text: D.

text: Culcer,

text: Q.

text: Niu,

text: N.

text: A.

text: Sinitsyn,

text: T.

text: Jung-

text: wirth,

text: and

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