text: Here, µ and p are deﬁned as in (8) and (9), and α is the so-called interior penalty F F parameter, which is a positive constant independent of h and p. As usual, we require that α is suﬃciently large. Anticipating the result of Theorem 3.4, we state that α > | | α = 2 C (1 + λ C /M )2 will suﬃce, where λ = 1 + 1 + θ and C is the constant from 0 θ A A θ Lemma 2.4. ∈ The interior penalty hp-DG approximation of (1) is now stated as follows: ﬁnd u h,p V such that h,p ∀ ∈ (13) N(u ; v) = L(v) v V . h,p h,p · × We note that, in the linear case of A( ) = I, with I the d d identity matrix, and for particular choices of the parameters θ and α, the DG formulation (13) reduces to various well-known DG methods. Notable examples include the symmetric interior penalty − method for θ = 1 and α > α > 0 (cf. [1]), and the nonsymmetric interior penalty method 0 for θ = 1 and α > 0 (cf. [21]). Under suitable regularity conditions, one can show that (13) is a consistent approxi- mation of (1). ∈ ∩ Lemma 3.1 (Galerkin orthogonality). Assume that (1) has a strong solution u Hs(Ω) C0(Ω), s > 3/2. Then, − ∀ ∈ (14) N(u; v) N(u ; v) = 0 v V . h,p h,p ∈ | ∈ F Proof. Since u C0(Ω), we have that [[u]] = 0 strongly for all F . Moreover, F ∇ · ∇ ∇ ∈ h,0 since u satisﬁes (1a) almost everywhere, we have that A( u) u L2(Ω). From [6, ∇ ∇ | ∈ F Lemma 1.24], it then follows that [[A( u) u]] = 0 almost everywhere for all F . F ∈ h,0 Therefore, upon integration by parts, we ﬁnd that N(u; v) = L(v) for all v V , from h,p which we infer the stated result. (cid:90) (cid:90) (cid:88) For the analysis of the hp-DG approximation (13), we introduce the norms ||| ||| |∇ (cid:90)| | | ∈ v 2 := v 2 dx + σ [[v]] 2 ds, v V (h, p), ∈T K Γ K h,0,D h ||| ||| ||| ||| − {{|∇ |}} ∈ v 2 := v 2 + σ 1 v 2 ds, v V (h, p). + Γ h,0,D We note that these norms are equivalent on V for any α > 0. Indeed, by Lemma 2.4 h,p there exists a constant C such that ||| ||| ≤ ||| ||| ≤ − ||| ||| ∀ ∈ (15) v 2 v 2 (1 + Cα 1) v 2 v V . + h,p (cid:98) ∩ ∇ → Next, let X(Γ ) = Π ∈T L2(∂K Γ ) and deﬁne the trace operator : V (h, p) h,0,D K(cid:40)∈hT h∈,0F,D σ [X(Γ )]d such that,(cid:12)for K and F , h,0,D (cid:98) (cid:12) h h,K (cid:12) ∇ | | − ∈ F ∇ (( w) ) σ[[w]] if F , K F h,0 ( σ w) = ∇ | | − | − ∈ F K (( w) ) σn(w g ) if F . K F K D h,D 7 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Here,

text: µ

text: and

text: p

text: are

text: deﬁned

text: as

text: in

text: and

text: and

text: α

text: is

text: the

text: so-called

text: interior

text: penalty

text: F

text: F

text: parameter,

text: which

text: is

text: a

text: positive

text: constant

text: independent

text: of

text: h

text: and

text: p.

text: As

text: usual,

text: we

text: require

text: that

text: α

text: is

text: suﬃciently

text: large.

text: Anticipating

text: the

text: result

text: of

text: Theorem

text: we

text: state

text: that

text: α

text: α

text: C

text: λ

text: C

text: /M

text: will

text: suﬃce,

text: where

text: λ

text: θ

text: and

text: C

text: is

text: the

text: constant

text: from

text: θ

text: A

text: A

text: θ

text: Lemma

text: The

text: interior

text: penalty

text: hp-DG

text: approximation

text: of

text: is

text: now

text: stated

text: as

text: follows:

text: ﬁnd

text: u

text: h,p

text: V

text: such

text: that

text: h,p

text: N(u

text: v)

text: L(v)

text: v

text: V

text: h,p

text: h,p

text: We

text: note

text: that,

text: in

text: the

text: linear

text: case

text: of

text: A(

text: I,

text: with

text: I

text: the

text: d

text: d

text: identity

text: matrix,

text: and

text: for

text: particular

text: choices

text: of

text: the

text: parameters

text: θ

text: and

text: α,

text: the

text: DG

text: formulation

text: reduces

text: to

text: various

text: well-known

text: DG

text: methods.

text: Notable

text: examples

text: include

text: the

text: symmetric

text: interior

text: penalty

text: method

text: for

text: θ

text: and

text: α

text: α

text: (cf.

text: and

text: the

text: nonsymmetric

text: interior

text: penalty

text: method

text: for

text: θ

text: and

text: α

text: (cf.

text: Under

text: suitable

text: regularity

text: conditions,

text: one

text: can

text: show

text: that

text: is

text: a

text: consistent

text: approxi-

text: mation

text: of

text: Lemma

text: (Galerkin

text: orthogonality).

text: Assume

text: that

text: has

text: a

text: strong

text: solution

text: u

text: Hs(Ω)

text: s

text: Then,

text: N(u;

text: v)

text: N(u

text: v)

text: v

text: V

text: h,p

text: h,p

text: F

text: Proof.

text: Since

text: u

text: we

text: have

text: that

text: [[u]]

text: strongly

text: for

text: all

text: F

text: Moreover,

text: F

text: since

text: u

text: satisﬁes

text: almost

text: everywhere,

text: we

text: have

text: that

text: A(

text: u)

text: u

text: From

text: F

text: Lemma

text: it

text: then

text: follows

text: that

text: [[A(

text: u)

text: u]]

text: almost

text: everywhere

text: for

text: all

text: F

text: F

text: Therefore,

text: upon

text: integration

text: by

text: parts,

text: we

text: ﬁnd

text: that

text: N(u;

text: v)

text: L(v)

text: for

text: all

text: v

text: V

text: from

text: h,p

text: which

text: we

text: infer

text: the

text: stated

text: result.

text: For

text: the

text: analysis

text: of

text: the

text: hp-DG

text: approximation

text: we

text: introduce

text: the

text: norms

text: v

text: v

text: dx

text: σ

text: [[v]]

text: ds,

text: v

text: V

text: (h,

text: p),

text: ∈T

text: K

text: Γ

text: K

text: h

text: v

text: v

text: σ

text: v

text: ds,

text: v

text: V

text: (h,

text: p).

text: Γ

text: We

text: note

text: that

text: these

text: norms

text: are

text: equivalent

text: on

text: V

text: for

text: any

text: α

text: Indeed,

text: by

text: Lemma

text: h,p

text: there

text: exists

text: a

text: constant

text: C

text: such

text: that

text: v

text: v

text: Cα

text: v

text: v

text: V

text: h,p

text: Next,

text: let

text: X(Γ

text: Π

text: ∈T

text: Γ

text: and

text: deﬁne

text: the

text: trace

text: operator

text: V

text: (h,

text: p)

text: σ

text: [X(Γ

text: )]d

text: such

text: K

text: and

text: F

text: h

text: h,K

text: F

text: w)

text: σ[[w]]

text: if

text: F

text: K

text: F

text: σ

text: w)

text: F

text: K

text: w)

text: σn(w

text: g

text: if

text: F

text: K

text: F

text: K

text: D

text: h,D

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: {{ · }} · By the fact that [[ ]] = [[ ]](cid:88), we have the following useful identity: (cid:90) ∇ ∇ · ∇ N(w; v) = A( w) w v dx (cid:98) (cid:98) ∈T K h − (cid:90) {{ ∇ ∇ · − ∇ }} (16) A( σ w) σ w (θσ 1 v + [[v]]) ds Γ h,0,D − {{ ∇ ∇ · ∇ }} + θ σ 1 A( w) w v ds. Γ h,0,D Rewriting the semilinear form N according to (16) and using Assumption 1.1, we are able to prove the following two lemmata. ≡ Lemma 3.2 (Lipschitz continuity). There exists a constant C C (θ, C ) such that N N A − ≤ ||| − ||| ||| ||| ∀ ∈ (17) N(w ; v) N(w ; v) C w w v w , w , v V (h, p). 1 2 N 1 2 + + 1 2 |{{ · }}| ≤ {{| | | |}} ≤ {{| |}} {{| |}} Proof. Sta∈rting fromT (16) and(cid:88)usi(cid:90)ng that q1 q2 q1 q2 2 q1 q2 for all q , q [H1(Ω, )]d, we have that 1 2 h (cid:90) − ≤ | ∇ ∇ − ∇ ∇ | |∇ | N(w ; v) N(w ; v) A( w ) w A( (cid:0)w ) w v dx (cid:1) 1 2(cid:98) (cid:98) (cid:98) 1 (cid:98)1 2 2 ∈T K K h (cid:90) {{| ∇ ∇ − ∇ ∇ |}} | | − {{|∇ |}} | | + A( w ) w A( w ) w 2 θ σ 1 v + [[v]] ds σ 1 σ 1 σ 2 σ 2 Γ h,0,D | | − {{| ∇ ∇ − ∇ ∇ |}}{{|∇ |}} + 2 θ σ 1 A( w ) w A( w ) w v ds. 1 1 2 2 Γ h,0,D We use the Lipschitz condition (2) from(cid:90) Assumption 1.1 to bound each of these terms, (cid:88) yielding − (cid:90) ≤ |∇ − ∇ (cid:90)| |∇ | N(w ; v) N(w ; v) C w w v dx 1 2 A 1 2 ∈T K K h | | (cid:90) | − | {{|∇ |}} | − | | | + 2 θ C [[w w ]] v ds + C σ [[w w ]] [[v]] ds A 1 2 A 1 2 Γ Γ h,0,D h,0,D | | − {{|∇ − ∇ |}} {{|∇ |}} + 4 θ C σ 1 w w v ds. A 1 2 Γ h,0,D Upon application of the Cauchy-Schwarz inequality, we arrive at (17) with C = (2 + | | N 4 θ ) C . A ∈ − Lemma 3.3 (Strong monotonicity). Let θ [ 1, 1] and select α > α = 2 C(1 + | | 0 λ C /M )2, where λ = 1 + 1 + θ and C is the constant from Lemma 2.4. There θ A A θ ≡ exists a positive constant M M (M , α /α) such that N N A 0 − − − ≥ ||| − ||| ∀ ∈ (18) N(w ; w w ) N(w ; w w ) M w w 2 w , w V . 1 1 2 2 1 2 N 1 2 1 2 h,p 8 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: By

text: the

text: fact

text: that

label: equation

text: we

text: have

text: the

text: following

text: useful

text: identity:

text: N(w;

text: v)

text: A(

text: w)

text: w

text: v

text: dx

text: ∈T

text: K

text: h

text: A(

text: σ

text: w)

text: σ

text: w

text: (θσ

text: v

text: [[v]])

text: ds

text: Γ

text: θ

text: σ

text: A(

text: w)

text: w

text: v

text: ds.

text: Γ

text: Rewriting

text: the

text: semilinear

text: form

text: N

text: according

text: to

text: and

text: using

text: Assumption

text: we

text: are

text: able

text: to

text: prove

text: the

text: following

text: two

text: lemmata.

text: Lemma

text: (Lipschitz

text: continuity).

text: There

text: exists

text: a

text: constant

text: C

text: C

text: (θ,

text: C

text: such

text: that

text: N

text: N

text: A

text: N(w

text: v)

text: N(w

text: v)

text: C

text: w

text: w

text: v

text: w

text: w

text: v

text: V

text: (h,

text: p).

text: N

text: Proof.

text: Sta∈rting

text: fromT

text: that

text: for

text: all

text: q

text: q

text: )]d,

text: we

text: have

text: that

text: h

text: N(w

text: v)

text: N(w

text: v)

text: A(

text: w

text: w

text: A(

text: w

text: v

text: dx

text: ∈T

text: K

text: K

text: h

text: A(

text: w

text: w

text: A(

text: w

text: w

text: θ

text: σ

text: v

text: [[v]]

text: ds

text: σ

text: σ

text: σ

text: σ

text: Γ

text: θ

text: σ

text: A(

text: w

text: w

text: A(

text: w

text: w

text: v

text: ds.

text: Γ

text: We

text: use

text: the

text: Lipschitz

text: condition

text: Assumption

text: to

text: bound

text: each

text: of

text: these

text: terms,

text: yielding

text: N(w

text: v)

text: N(w

text: v)

text: C

text: w

text: w

text: v

text: dx

text: A

text: ∈T

text: K

text: K

text: h

text: θ

text: C

text: [[w

text: w

text: v

text: ds

text: C

text: σ

text: [[w

text: w

text: [[v]]

text: ds

text: A

text: A

text: Γ

text: Γ

text: θ

text: C

text: σ

text: w

text: w

text: v

text: ds.

text: A

text: Γ

text: Upon

text: application

text: of

text: the

text: Cauchy-Schwarz

text: inequality,

text: we

text: arrive

text: at

text: with

text: C

text: N

text: θ

text: C

text: A

text: Lemma

text: (Strong

text: monotonicity).

text: Let

text: θ

text: and

text: select

text: α

text: α

text: λ

text: C

text: /M

text: where

text: λ

text: θ

text: and

text: C

text: is

text: the

text: constant

text: from

text: Lemma

text: There

text: θ

text: A

text: A

text: θ

text: exists

text: a

text: positive

text: constant

text: M

text: M

text: (M

text: α

text: /α)

text: such

text: that

text: N

text: N

text: A

text: N(w

text: w

text: w

text: N(w

text: w

text: w

text: M

text: w

text: w

text: w

text: w

text: V

text: N

text: h,p

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: for an| y| ≤(cid:15) > 0. Finally, for T4, using th(cid:90)e Lipschitz condition (2) together with the fact that θ 1, and subsequently applying Lemma 2.4, we obtain ≥ − | | −(cid:90){{|∇ |}} T θ C (cid:88)σ 1 w 2 ds 4 A Γ h,0,D ≥ − − |∇ | C C α 1 w 2 dx. A ∈T K K h ≥ Substituting the above bounds for T to T back into (19) and recalling that C M > 0, 1 4 A A we deduce that (cid:90) (cid:0) (cid:1) (cid:88) − − − N(w ; w w ) N(w ; w w ) 1 1 2 2 1 2 ≥ − − (cid:90) − |∇ − ∇ | M (2 + (cid:15) 1) λ C C α 1 w w 2 dx A θ A 1 2 ∈T K K h − | − | + (M λ C (cid:15)) σ [[w w ]] 2 ds, A θ A 1 2 Γ h,0,D | | (cid:16) (cid:17) (cid:88) (cid:90) where λ = 1 + 1 + θ . Upon selecting (cid:15) = M /(2 λ C ), we arrive at θ A θ A N(w ; w − w ) − N(w ; w − w ) ≥ M 1 −(cid:90)α0 |∇w − ∇w |2 dx 1 1 2 2 1 2 A 1 2 α ∈T K K h 1 | − | + M σ [[w w ]] 2 ds, A 1 2 2 Γ h,0,D − where α = 2 C (1 + λ C /M )2. Hence, we have proved (18) with M = M max(1, 1 0 θ A A N A 2 α /α). We conclude by noting that M > 0 whenever α > α . 0 N 0 With the aid of Lemma 3.2 and Lemma 3.3, we are now in the position to prove that ∈ the DG approximation (13) admits a unique solution u V . Necessary and suﬃcient h,p h,p conditions for existence and uniqueness are provided by Theorem A.1 in the Appendix. The following result is an immediate consequence. ∈ − Theorem 3.4 (Existence and uniqueness). Let θ [ 1, 1] and α > α = 2 C (1 + | | 0 λ C /M )2, where λ = 1 + 1 + θ and C is the constant from Lemma 2.4. Then, θ A A θ ∈ the DG approximation (13) has a unique solution u V . h,p h,p 4 A priori error analysis We begin by introducing the following hp-approximation results. ∈ T Lemma 4.1. Let K such that K = T (Kˆ ), where Kˆ is either the unit d-simplex h K or the unit d-hypercub≥e, and TK ∈is a CrK-diﬀeomorphism in compliance with Assump- tion 2.1(i). For sK 0, let v HsK(K) and d→eﬁne tK = min(rK, sK). Then, for pK = 1, 2, . . . , there exists a mapping πK : HsK(K) SpK(K) and a constant C indepen- dent of h , p and v such that: K K ≤ ≤ (i) for 0 k t , K − (cid:107) − (cid:107) ≤ hµK k (cid:107) (cid:107) v π (v) C K − v ; K Hk(K) ptK k HtK(K) K 10 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: for

text: an|

text: y|

text: Finally,

text: for

text: using

text: Lipschitz

text: condition

text: together

text: with

text: the

text: fact

text: that

text: θ

text: and

text: subsequently

text: applying

text: Lemma

text: we

text: obtain

text: T

text: θ

text: C

text: w

text: ds

text: A

text: Γ

text: C

text: C

text: α

text: w

text: dx.

text: A

text: ∈T

text: K

text: K

text: h

text: Substituting

text: the

text: above

text: bounds

text: for

text: T

text: to

text: T

text: back

text: into

text: and

text: recalling

text: that

text: C

text: M

text: A

text: A

text: we

text: deduce

text: that

text: N(w

text: w

text: w

text: N(w

text: w

text: w

text: M

text: λ

text: C

text: C

text: α

text: w

text: w

text: dx

text: A

text: θ

text: A

text: ∈T

text: K

text: K

text: h

text: (M

text: λ

text: C

text: σ

text: [[w

text: w

text: ds,

text: A

text: θ

text: A

text: Γ

text: where

text: λ

text: θ

text: Upon

text: selecting

text: M

text: λ

text: C

text: we

text: arrive

text: at

text: θ

text: A

text: θ

text: A

text: N(w

text: w

text: w

text: N(w

text: w

text: w

text: M

text: |∇w

text: ∇w

text: dx

text: A

text: α

text: ∈T

text: K

text: K

text: h

text: M

text: σ

text: [[w

text: w

text: ds,

text: A

text: Γ

text: where

text: α

text: C

text: λ

text: C

text: /M

text: Hence,

text: we

text: have

text: proved

text: with

text: M

text: M

text: θ

text: A

text: A

text: N

text: A

text: α

text: /α).

text: We

text: conclude

text: by

text: noting

text: that

text: M

text: whenever

text: α

text: α

text: N

text: With

text: the

text: aid

text: of

text: Lemma

text: and

text: Lemma

text: we

text: are

text: now

text: in

text: the

text: position

text: to

text: prove

text: that

text: the

text: DG

text: approximation

text: admits

text: a

text: unique

text: solution

text: u

text: V

text: Necessary

text: and

text: suﬃcient

text: h,p

text: h,p

text: conditions

text: for

text: existence

text: and

text: uniqueness

text: are

text: provided

text: by

text: Theorem

text: in

text: the

text: Appendix.

text: The

text: following

text: result

text: is

text: an

text: immediate

text: consequence.

text: Theorem

text: (Existence

text: and

text: uniqueness).

text: Let

text: θ

text: and

text: α

text: α

text: C

text: λ

text: C

text: /M

text: where

text: λ

text: θ

text: and

text: C

text: is

text: the

text: constant

text: from

text: Lemma

text: Then,

text: θ

text: A

text: A

text: θ

text: the

text: DG

text: approximation

text: has

text: a

text: unique

text: solution

text: u

text: V

text: h,p

text: h,p

text: A

text: priori

text: error

text: analysis

text: We

text: begin

text: by

text: introducing

text: the

text: following

text: hp-approximation

text: results.

text: T

text: Lemma

text: Let

text: K

text: such

text: that

text: K

text: T

text: (Kˆ

text: where

text: Kˆ

text: is

text: either

text: the

text: unit

text: d-simplex

text: h

text: K

text: or

text: the

text: unit

text: d-hypercub≥e,

text: and

text: TK

text: ∈is

text: a

text: CrK-diﬀeomorphism

text: in

text: compliance

text: with

text: Assump-

text: tion

text: For

text: sK

text: let

text: v

text: HsK(K)

text: and

text: d→eﬁne

text: tK

text: min(rK,

text: sK).

text: Then,

text: for

text: pK

text: there

text: exists

text: a

text: mapping

text: πK

text: HsK(K)

text: SpK(K)

text: and

text: a

text: constant

text: C

text: indepen-

text: dent

text: of

text: h

text: p

text: and

text: v

text: such

text: that:

text: K

text: K

text: (i)

text: for

text: k

text: t

text: K

text: hµK

text: k

text: v

text: π

text: (v)

text: C

text: K

text: v

text: K

text: Hk(K)

text: ptK

text: k

text: HtK(K)

text: K

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: 1 Optimal Random Access and Random Spectrum Sensing for an Energy Harvesting Cognitive Radio with and without Primary Feedback Leveraging Ahmed El Shaﬁe, Member, IEEE, Abstract— We consider a secondary user (SU) with energy and some delay-optimal properties are derived. Throughput harvesting capability. We design access schemes for the SU optimal energy allocation is investigated in [6] for energy which incorporate random spectrum sensing and random access, harvesting systems in a time-constrained slotted setting. In and which make use of the primary automatic repeat request [7], [8], minimization of the transmission completion time is (ARQ) feedback. We study two problem-formulations. In the ﬁrst problem-formulation, we characterize the stability region of considered in an energy harvesting system and the optimal the proposed schemes. The sensing and access probabilities are solution is obtained using a geometric framework. In [9], obtained such that the secondary throughput is maximized under energy harvesting transmitters with batteries of ﬁnite energy the constraints that both the primary and secondary queues are storage capacity are considered and the problem of throughput stable. Whereas in the second problem-formulation, the sensing maximization by a deadline is solved for a static channel. and access probabilities are obtained such that the secondary throughput is maximized under the stability of the primary queue The authors of [10] consider the scenario in which a set of and that the primary queueing delay is kept lower than a speciﬁed nodes shares a common channel. The PU has a rechargeable value needed to guarantee a certain quality of service (QoS) for battery and the SU is plugged to a reliable power supply. the primary user (PU). We consider spectrum sensing errors They obtain the maximum stable throughput region which and assume multipacket reception (MPR) capabilities. Numerical describes the maximum arrival rates that maintain the stability results show the enhanced performance of our proposed systems. of the network queues. In [11], the authors investigate the effects of network layer cooperation in a wireless three-node Index Terms—Cognitive radio, energy harvesting, queueing network with energy harvesting nodes and bursty trafﬁc. In delay. [12], Sultan investigated the optimal cognitive sensing and access policies for an SU with an energy queue. The analysis is I. INTRODUCTION based on Markov-decision process (MDP). In [13], the authors investigated the maximum stable throughput of a backlogged Cognitive radio technology provides an efﬁcient means of secondary terminal with energy harvesting capability. The SU utilizing the radio spectrum [2]. The basic idea is to allow randomly accesses the channel at the beginning of the time secondary users to access the spectrum while providing certain slot without employing any channel sensing. The secondary guaranteed quality of service (QoS) performance measures terminal can leverage the availability of primary feedback for the primary users (PUs). The secondary user (SU) is and exploit the multipacket reception (MPR) capability of the a battery-powered device in many practical situations and receivers to enhance its throughput. its operation, which involves spectrum sensing and access, In this paper, we develop spectrum sensing and transmission is accompanied by energy consumption. Consequently an methods for an energy harvesting SU. We leverage the primary energy-constrained SU must optimize its sensing and access automatic repeat request (ARQ) feedback for secondary ac- decisions to efﬁciently utilize the energy at its disposal. cess. Due to the broadcast nature of the wireless channel, this An emerging technology for energy-constrained terminals is feedback can be overheard and utilized by the secondary node energy harvesting which allows the terminal to gather energy assuming that it is unencrypted. The proposed protocols can from its environment. An overview of the different energy alleviate the negative impact of channel sensing because the harvesting technologies is provided in [3] and the references secondary access is on basis of the sensed primary state as therein. well as the overheard primary feedback. The problem with Data transmission by an energy harvester with a recharge- depending on spectrum sensing only is that sensing does able battery has got a lot of attention recently [4]–[13]. not inform the secondary terminal about its impact on the The optimal online policy for controlling admissions into the primary receiver. This issue has induced interest in utilizing the data buffer is derived in [4] using a dynamic programming feedback from the primary receiver to the primary transmitter framework. In [5], energy management policies which stabilize to optimize the secondary transmission strategies. For instance, the data queue are proposed for single-user communication in [14], the SU observes the ARQ feedback from the primary Part of this work has been presented in the 8th International Conference on receiver as it reﬂects the PU’s achieved packet rate. The SU’s Wireless and Mobile Computing, Networking and Communications (WiMob), objective is to maximize its throughput while guaranteeing 2012 [1] a certain packet rate for the PU. In [15], the authors use A. El Shaﬁe is with Wireless Intelligent Networks Center (WINC), Nile University, Giza, Egypt. e-mail: ahmed.salahelshaﬁe@gmail.com a partially observable Markov decision process (POMDP)

text: Optimal

label: title

text: Random

text: Access

text: and

text: Random

text: Spectrum

text: Sensing

text: for

text: an

text: Energy

text: Harvesting

text: Cognitive

text: Radio

text: with

text: and

text: without

text: Primary

text: Feedback

text: Leveraging

text: Ahmed

label: author

text: El

text: Shaﬁe,

text: Member,

text: IEEE,

text: Abstract—

text: We

label: abstract

text: consider

text: a

text: secondary

text: user

text: (SU)

text: with

text: energy

text: and

text: some

text: delay-optimal

text: properties

text: are

text: derived.

text: Throughput

text: harvesting

text: capability.

text: We

text: design

text: access

text: schemes

text: for

text: the

text: SU

text: optimal

text: energy

text: allocation

text: is

text: investigated

text: in

text: for

text: energy

text: which

text: incorporate

text: random

text: spectrum

text: sensing

text: and

text: random

text: access,

text: harvesting

text: systems

text: in

text: a

text: time-constrained

text: slotted

text: setting.

text: In

text: and

text: which

text: make

text: use

text: of

text: the

text: primary

text: automatic

text: repeat

text: request

text: minimization

text: of

text: the

text: transmission

text: completion

text: time

text: is

text: (ARQ)

text: feedback.

text: We

text: study

text: two

text: problem-formulations.

text: In

text: the

text: ﬁrst

text: problem-formulation,

text: we

text: characterize

text: the

text: stability

text: region

text: of

text: considered

text: in

text: an

text: energy

text: harvesting

text: system

text: and

text: the

text: optimal

text: the

text: proposed

text: schemes.

text: The

text: sensing

text: and

text: access

text: probabilities

text: are

text: solution

text: is

text: obtained

text: using

text: a

text: geometric

text: framework.

text: In

text: obtained

text: such

text: that

text: the

text: secondary

text: throughput

text: is

text: maximized

text: under

text: energy

text: harvesting

text: transmitters

text: with

text: batteries

text: of

text: ﬁnite

text: energy

text: the

text: constraints

text: that

text: both

text: the

text: primary

text: and

text: secondary

text: queues

text: are

text: storage

text: capacity

text: are

text: considered

text: and

text: the

text: problem

text: of

text: throughput

text: stable.

text: Whereas

text: in

text: the

text: second

text: problem-formulation,

text: the

text: sensing

text: maximization

text: by

text: a

text: deadline

text: is

text: solved

text: for

text: a

text: static

text: channel.

text: and

text: access

text: probabilities

text: are

text: obtained

text: such

text: that

text: the

text: secondary

text: throughput

text: is

text: maximized

text: under

text: the

text: stability

text: of

text: the

text: primary

text: queue

text: The

text: authors

text: of

text: consider

text: the

text: scenario

text: in

text: which

text: a

text: set

text: of

text: and

text: that

text: the

text: primary

text: queueing

text: delay

text: is

text: kept

text: lower

text: than

text: a

text: speciﬁed

text: nodes

text: shares

text: a

text: common

text: channel.

text: The

text: PU

text: has

text: a

text: rechargeable

text: value

text: needed

text: to

text: guarantee

text: a

text: certain

text: quality

text: of

text: service

text: (QoS)

text: for

text: battery

text: and

text: the

text: SU

text: is

text: plugged

text: to

text: a

text: reliable

text: power

text: supply.

text: the

text: primary

text: user

text: (PU).

text: We

text: consider

text: spectrum

text: sensing

text: errors

text: They

text: obtain

text: the

text: maximum

text: stable

text: throughput

text: region

text: which

text: and

text: assume

text: multipacket

text: reception

text: (MPR)

text: capabilities.

text: Numerical

text: describes

text: the

text: maximum

text: arrival

text: rates

text: that

text: maintain

text: the

text: stability

text: results

text: show

text: the

text: enhanced

text: performance

text: of

text: our

text: proposed

text: systems.

text: of

text: the

text: network

text: queues.

text: In

text: the

text: authors

text: investigate

text: the

text: effects

text: of

text: network

text: layer

text: cooperation

text: in

text: a

text: wireless

text: three-node

text: Index

text: Terms—Cognitive

text: radio,

text: energy

text: harvesting,

text: queueing

text: network

text: with

text: energy

text: harvesting

text: nodes

text: and

text: bursty

text: trafﬁc.

text: In

text: delay.

text: Sultan

text: investigated

text: the

text: optimal

text: cognitive

text: sensing

text: and

text: access

text: policies

text: for

text: an

text: SU

text: with

text: an

text: energy

text: queue.

text: The

text: analysis

text: is

text: I.

text: INTRODUCTION

text: based

text: on

text: Markov-decision

text: process

text: (MDP).

text: In

text: the

text: authors

text: investigated

text: the

text: maximum

text: stable

text: throughput

text: of

text: a

text: backlogged

text: Cognitive

text: radio

text: technology

text: provides

text: an

text: efﬁcient

text: means

text: of

text: secondary

text: terminal

text: with

text: energy

text: harvesting

text: capability.

text: The

text: SU

text: utilizing

text: the

text: radio

text: spectrum

text: The

text: basic

text: idea

text: is

text: to

text: allow

text: randomly

text: accesses

text: the

text: channel

text: at

text: the

text: beginning

text: of

text: the

text: time

text: secondary

text: users

text: to

text: access

text: the

text: spectrum

text: while

text: providing

text: certain

text: slot

text: without

text: employing

text: any

text: channel

text: sensing.

text: The

text: secondary

text: guaranteed

text: quality

text: of

text: service

text: (QoS)

text: performance

text: measures

text: terminal

text: can

text: leverage

text: the

text: availability

text: of

text: primary

text: feedback

text: for

text: the

text: primary

text: users

text: (PUs).

text: The

text: secondary

text: user

text: (SU)

text: is

text: and

text: exploit

text: the

text: multipacket

text: reception

text: (MPR)

text: capability

text: of

text: the

text: a

text: battery-powered

text: device

text: in

text: many

text: practical

text: situations

text: and

text: receivers

text: to

text: enhance

text: its

text: throughput.

text: its

text: operation,

text: which

text: involves

text: spectrum

text: sensing

text: and

text: access,

text: In

text: this

text: paper,

text: we

text: develop

text: spectrum

text: sensing

text: and

text: transmission

text: is

text: accompanied

text: by

text: energy

text: consumption.

text: Consequently

text: an

text: methods

text: for

text: an

text: energy

text: harvesting

text: SU.

text: We

text: leverage

text: the

text: primary

text: energy-constrained

text: SU

text: must

text: optimize

text: its

text: sensing

text: and

text: access

text: automatic

text: repeat

text: request

text: (ARQ)

text: feedback

text: for

text: secondary

text: ac-

text: decisions

text: to

text: efﬁciently

text: utilize

text: the

text: energy

text: at

text: its

text: disposal.

text: cess.

text: Due

text: to

text: the

text: broadcast

text: nature

text: of

text: the

text: wireless

text: channel,

text: this

text: An

text: emerging

text: technology

text: for

text: energy-constrained

text: terminals

text: is

text: feedback

text: can

text: be

text: overheard

text: and

text: utilized

text: by

text: the

text: secondary

text: node

text: energy

text: harvesting

text: which

text: allows

text: the

text: terminal

text: to

text: gather

text: energy

text: assuming

text: that

text: it

text: is

text: unencrypted.

text: The

text: proposed

text: protocols

text: can

text: from

text: its

text: environment.

text: An

text: overview

text: of

text: the

text: different

text: energy

text: alleviate

text: the

text: negative

text: impact

text: of

text: channel

text: sensing

text: because

text: the

text: harvesting

text: technologies

text: is

text: provided

text: in

text: and

text: the

text: references

text: secondary

text: access

text: is

text: on

text: basis

text: of

text: the

text: sensed

text: primary

text: state

text: as

text: therein.

text: well

text: as

text: the

text: overheard

text: primary

text: feedback.

text: The

text: problem

text: with

text: Data

text: transmission

text: by

text: an

text: energy

text: harvester

text: with

text: a

text: recharge-

text: depending

text: on

text: spectrum

text: sensing

text: only

text: is

text: that

text: sensing

text: does

text: able

text: battery

text: has

text: got

text: a

text: lot

text: of

text: attention

text: recently

text: not

text: inform

text: the

text: secondary

text: terminal

text: about

text: its

text: impact

text: on

text: the

text: The

text: optimal

text: online

text: policy

text: for

text: controlling

text: admissions

text: into

text: the

text: primary

text: receiver.

text: This

text: issue

text: has

text: induced

text: interest

text: in

text: utilizing

text: the

text: data

text: buffer

text: is

text: derived

text: in

text: using

text: a

text: dynamic

text: programming

text: feedback

text: from

text: the

text: primary

text: receiver

text: to

text: the

text: primary

text: transmitter

text: framework.

text: In

text: energy

text: management

text: policies

text: which

text: stabilize

text: to

text: optimize

text: the

text: secondary

text: transmission

text: strategies.

text: For

text: instance,

text: the

text: data

text: queue

text: are

text: proposed

text: for

text: single-user

text: communication

text: in

text: the

text: SU

text: observes

text: the

text: ARQ

text: feedback

text: from

text: the

text: primary

text: Part

text: of

text: this

text: work

text: has

text: been

text: presented

text: in

text: the

text: International

text: Conference

text: on

text: receiver

text: as

text: it

text: reﬂects

text: the

text: PU’s

text: achieved

text: packet

text: rate.

text: The

text: SU’s

text: Wireless

text: and

text: Mobile

text: Computing,

text: Networking

text: and

text: Communications

text: (WiMob),

text: objective

text: is

text: to

text: maximize

text: its

text: throughput

text: while

text: guaranteeing

text: a

text: certain

text: packet

text: rate

text: for

text: the

text: PU.

text: In

text: the

text: authors

text: use

text: A.

text: El

text: Shaﬁe

text: is

text: with

text: Wireless

text: Intelligent

text: Networks

text: Center

text: (WINC),

text: Nile

text: University,

text: Giza,

text: Egypt.

text: e-mail:

text: ahmed.salahelshaﬁe@gmail.com

text: a

text: partially

text: observable

text: Markov

text: decision

text: process

text: (POMDP)


text: Hyperka¨ hler Sigma Model and Field Theory on Gibbons-Hawking Spaces Anindya Dey1 and Andrew Neitzke2 1Theory Group, University of Texas at Austin 2Department of Mathematics, University of Texas at Austin Abstract We describe a novel deformation of the 3-dimensional sigma model with hyperka¨hler tar- N get, which arises naturally from the compactiﬁcation of a 4-dimensional = 2 theory on a hyperka¨hler circle bundle (Gibbons-Hawking space). We derive the condition for which the de- formed sigma model preserves 4 out of the 8 supercharges. We also study the contribution from a NUT center to the sigma model path integral, and ﬁnd that supersymmetry implies it is a holomor- phic section of a certain holomorphic line bundle over the hyperka¨hler target. We study explicitly the case where the original 4-dimensional theory is pure U(1) super Yang-Mills, and show that the contribution from a NUT center in this case is simply the Jacobi theta function. Contents 1 Introduction and main results 1 2 The deformed hyperka¨hler sigma model 6 2.1 Fields of the undeformed model . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Data for the deformed model . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Hyperka¨hler identities and their extensions . . . . . . . . . . . . . . . . . . . 7 2.4 Action of the deformed model . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.5 SUSY veriﬁcation: 1-fermion terms . . . . . . . . . . . . . . . . . . . . . . . . 12 2.6 Supersymmetry for boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3 U(1) theories compactiﬁed on S1 16 3.1 Bosonic U(1) gauge theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 N 3.2 = 2 U(1) gauge theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 N 3.2.1 Dimensional reduction from 6D = 1 super Yang-Mills . . . . . . . 19 3.2.2 4D action and SUSY transformation . . . . . . . . . . . . . . . . . . . 20 3.2.3 3D action and dualization . . . . . . . . . . . . . . . . . . . . . . . . . 20 × 3.3 U(1) SYM on R3 S1: hyperka¨hler sigma model picture . . . . . . . . . . . 22 1

text: Hyperka¨

font: MTRKEA+URWPalladioL-Roma

text: hler

text: Sigma

text: Model

text: and

text: Field

text: Theory

text: on

text: Gibbons-Hawking

text: Spaces

text: Anindya

text: and

text: Andrew

text: Group,

text: University

text: of

text: Texas

text: at

text: Austin

text: of

text: Mathematics,

text: University

text: of

text: Texas

text: at

text: Austin

text: Abstract

font: TQXHEG+URWPalladioL-Bold

text: We

text: describe

text: a

text: novel

text: deformation

text: of

text: the

text: sigma

text: model

text: with

text: hyperka¨hler

text: tar-

text: N

text: get,

text: which

text: arises

text: naturally

text: from

text: the

text: compactiﬁcation

text: of

text: a

text: theory

text: on

text: a

text: hyperka¨hler

text: circle

text: bundle

text: (Gibbons-Hawking

text: space).

text: We

text: derive

text: the

text: condition

text: for

text: which

text: the

text: de-

text: formed

text: sigma

text: model

text: preserves

text: out

text: of

text: the

text: supercharges.

text: We

text: also

text: study

text: the

text: contribution

text: from

text: a

text: NUT

text: center

text: to

text: the

text: sigma

text: model

text: path

text: integral,

text: and

text: ﬁnd

text: that

text: supersymmetry

text: implies

text: it

text: is

text: a

text: holomor-

text: phic

text: section

text: of

text: a

text: certain

text: holomorphic

text: line

text: bundle

text: over

text: the

text: hyperka¨hler

text: target.

text: We

text: study

text: explicitly

text: the

text: case

text: where

text: the

text: original

text: theory

text: is

text: pure

text: super

text: Yang-Mills,

text: and

text: show

text: that

text: the

text: contribution

text: from

text: a

text: NUT

text: center

text: in

text: this

text: case

text: is

text: simply

text: the

text: Jacobi

text: theta

text: function.

text: Contents

text: Introduction

text: and

text: main

text: results

text: The

text: deformed

text: hyperka¨hler

text: sigma

text: model

text: Fields

text: of

text: the

text: undeformed

text: model

text: Data

text: for

text: the

text: deformed

text: model

text: Hyperka¨hler

text: identities

text: and

text: their

text: extensions

text: Action

text: of

text: the

text: deformed

text: model

text: SUSY

text: veriﬁcation:

text: terms

text: Supersymmetry

text: for

text: boundaries

text: theories

text: compactiﬁed

text: on

text: Bosonic

text: gauge

text: theory

text: N

text: gauge

text: theory

text: N

text: Dimensional

text: reduction

text: from

text: super

text: Yang-Mills

text: action

text: and

text: SUSY

text: transformation

text: action

text: and

text: dualization

text: SYM

text: on

text: hyperka¨hler

text: sigma

text: model

text: picture


text: 4 U(1) theories on Gibbons-Hawking spaces 24 4.1 Bosonic U(1) gauge theory on Gibbons-Hawking space . . . . . . . . . . . . 24 N 4.2 = 2, U(1) gauge theory on Gibbons-Hawking space . . . . . . . . . . . . 26 N 4.2.1 Dimensional reduction from 6D = 1 SYM . . . . . . . . . . . . . . 26 4.2.2 4D action, SUSY and localization equations . . . . . . . . . . . . . . . 27 4.2.3 3D action: 4D instanton and Bogomolny equations . . . . . . . . . . 28 4.3 U(1) SYM on Gibbons-Hawking Space: hyperka¨hler sigma model picture . 30 5 The NUT centers 34 5.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 5.2 UV computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.3 Holomorphy of the boundary terms . . . . . . . . . . . . . . . . . . . . . . . 36 A U(1) SYM on NUT space as hyperka¨hler sigma model: rescaled version 37 B SUSY variation of the hyperka¨hler sigma model: 3-fermion and 5-fermion terms 40 N C 6D spinors and = 1 SYM in 6D 45 D 4D and 3D spinors 47 × E Killing spinor on TN T1,1 49 4 1 Introduction and main results It is a well-known principle that some aspects of quantum ﬁeld theories become eas- ier to understand when the theories are compactiﬁed to lower dimensions. This principle was exploited in particular in [Gaiotto:2008cd], where the wall-crossing phenomenon N in = 2 supersymmetric theories in four dimensions was studied by formulating the × theories on S1 R3, with S1 of ﬁxed radius R. A crucial input to that analysis was a good understanding of the constraints imposed by supersymmetry [Seiberg:1996nz, AlvarezGaume:1981hm]: they say that the IR Lagrangian of the compactiﬁed theory is (around a generic point of its moduli space) a sigma model into a hyperka¨hler manifold M M [R]. The metric of [R] typically depends in a highly nontrivial way on the parame- ter R, reﬂecting the fact that quantum corrections due to BPS particles of mass M scale as − e MR. In this paper we consider a different but related problem: we begin again with an N × = 2 theory in four dimensions, but rather than studying it on S1 R3, we take our spacetime to be a circle ﬁbration over R3, with isolated degenerate ﬁbers. Generically such a compactiﬁcation would not preserve any supersymmetry, at least without some modiﬁcation of the theory; however, we consider the special case where the spacetime X is actually hyperka¨hler (a Gibbons-Hawking space). Thus X has metric locally of the form 2

text: theories

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text: Gibbons-Hawking

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text: Gibbons-Hawking

text: space

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text: Dimensional

text: reduction

text: from

text: SYM

text: action,

text: SUSY

text: and

text: localization

text: equations

text: action:

text: instanton

text: and

text: Bogomolny

text: equations

text: SYM

text: on

text: Gibbons-Hawking

text: Space:

text: hyperka¨hler

text: sigma

text: model

text: picture

text: The

text: NUT

text: centers

text: Setup

text: UV

text: computation

text: Holomorphy

text: of

text: the

text: boundary

text: terms

text: A

text: SYM

text: on

text: NUT

text: space

text: as

text: hyperka¨hler

text: sigma

text: model:

text: rescaled

text: version

text: B

text: SUSY

text: variation

text: of

text: the

text: hyperka¨hler

text: sigma

text: model:

text: and

text: terms

text: N

text: C

text: spinors

text: and

text: SYM

text: in

text: D

text: and

text: spinors

text: E

text: Killing

text: spinor

text: on

text: TN

text: Introduction

text: and

text: main

text: results

text: It

text: is

text: a

text: well-known

text: principle

text: that

text: some

text: aspects

text: of

text: quantum

text: ﬁeld

text: theories

text: become

text: eas-

text: ier

text: to

text: understand

text: when

text: the

text: theories

text: are

text: compactiﬁed

text: to

text: lower

text: dimensions.

text: This

text: principle

text: was

text: exploited

text: in

text: particular

text: in

text: where

text: the

text: wall-crossing

text: phenomenon

text: N

text: in

text: supersymmetric

text: theories

text: in

text: four

text: dimensions

text: was

text: studied

text: by

text: formulating

text: the

text: theories

text: on

text: with

text: of

text: ﬁxed

text: radius

text: R.

text: A

text: crucial

text: input

text: to

text: that

text: analysis

text: was

text: a

text: good

text: understanding

text: of

text: the

text: constraints

text: imposed

text: by

text: supersymmetry

text: they

text: say

text: that

text: the

text: IR

text: Lagrangian

text: of

text: the

text: compactiﬁed

text: theory

text: is

text: (around

text: a

text: generic

text: point

text: of

text: its

text: moduli

text: space)

text: a

text: sigma

text: model

text: into

text: a

text: hyperka¨hler

text: manifold

text: M

text: M

text: [R].

text: The

text: metric

text: of

text: [R]

text: typically

text: depends

text: in

text: a

text: highly

text: nontrivial

text: way

text: on

text: the

text: parame-

text: ter

text: R,

text: reﬂecting

text: the

text: fact

text: that

text: quantum

text: corrections

text: due

text: to

text: BPS

text: particles

text: of

text: mass

text: M

text: scale

text: as

text: e

text: MR.

text: In

text: this

text: paper

text: we

text: consider

text: a

text: different

text: but

text: related

text: problem:

text: we

text: begin

text: again

text: with

text: an

text: N

text: theory

text: in

text: four

text: dimensions,

text: but

text: rather

text: than

text: studying

text: it

text: on

text: we

text: take

text: our

text: spacetime

text: to

text: be

text: a

text: circle

text: ﬁbration

text: over

text: with

text: isolated

text: degenerate

text: ﬁbers.

text: Generically

text: such

text: a

text: compactiﬁcation

text: would

text: not

text: preserve

text: any

text: supersymmetry,

text: at

text: least

text: without

text: some

text: modiﬁcation

text: of

text: the

text: theory;

text: however,

text: we

text: consider

text: the

text: special

text: case

text: where

text: the

text: spacetime

text: X

text: is

text: actually

text: hyperka¨hler

text: (a

text: Gibbons-Hawking

text: space).

text: Thus

text: X

text: has

text: metric

text: locally

text: of

text: the

text: form


text: L (cid:48) 4πδ ibp = ψE αζ¯αEeiEE(cid:48)∂N ϕjgij (2.41a) − (cid:48) ∂νϕieiEA(cid:48)γσναζ¯σEγαNβψA β (2.41b) 1 (cid:48) +2(cid:101)µνNGµνAiψE αζ¯αEeiEE(cid:48) (2.41c) where here and below, N denotes a unit normal to the boundary. Collecting all terms involving ζ¯E we have L L L 4πδ = 4πδ ∂ + 4πδ ibp (2.42a) (cid:48) = 4π∂iHψE αζ¯αEeiEE(cid:48) (2.42b) (cid:48) +ψE αζ¯αEeiEE(cid:48)∂N ϕjgij (2.42c) − (cid:48) ∂νϕieiEA(cid:48)γσναζ¯σEγαNβψA β (2.42d) 1 (cid:48) +2(cid:101)µνNGµνAiψE αζ¯αEeiEE(cid:48) (2.42e) The above equation can be rewritten as L (cid:48) L δ = eiEE(cid:48)ψE αζ¯σEδ σαi (2.43a) L 1 − 4πδ σαi = (4π∂iH + 2(cid:101)µνNGµνAi)δασ + (∂N ϕiδασ ∂νϕi(γνγN)σα). (2.43b) Using the condition that the ﬁelds are constant over the boundary, this variation re- duces to L (cid:48) 1 4πδ = ψE αζ¯σEeiEE(cid:48)(4π∂iH + 2(cid:101)µνNGµνAi)δασ (2.44a) L Evidently δ = 0 if 1 eiEE(cid:48)(∂iH + 8π (cid:101)µνNGµνAi) = 0. (2.45) After integrating over the boundary this condition gives eiEE(cid:48)(∂iQ + kAi) = 0, (2.46) 16 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: L

text: 4πδ

font: JVMVKC+PazoMath-Italic

text: ibp

font: DHXBIT+URWPalladioL-Ital

text: ψE

text: ϕjgij

text: β

text: where

text: here

text: and

text: below,

text: N

text: denotes

text: a

text: unit

text: normal

text: to

text: the

text: boundary.

text: Collecting

text: all

text: terms

text: involving

text: ζ¯E

text: we

text: have

text: L

text: L

text: L

text: 4πδ

text: 4πδ

text: 4πδ

text: ibp

text: +ψE

text: ϕjgij

text: β

text: The

text: above

text: equation

text: can

text: be

text: rewritten

text: as

text: L

text: L

text: δ

text: αζ¯σEδ

text: σαi

text: L

text: 4πδ

text: σαi

text: (∂N

text: ϕiδασ

text: ∂νϕi(γνγN)σα).

text: Using

text: the

text: condition

text: that

text: the

text: ﬁelds

text: are

text: constant

text: over

text: the

text: boundary,

text: this

text: variation

text: re-

text: duces

text: to

text: L

text: 4πδ

text: ψE

text: L

text: Evidently

text: δ

text: if

text: 8π

text: After

text: integrating

text: over

text: the

text: boundary

text: this

text: condition

text: gives

text: kAi)

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: § Spinor equation in Appendix : ζa = ζa (constant), 0 (4.22) Γ1Γ2Γ3Γ4ζ0a = ζ0a. The above equation ensures that exactly half of the original SUSY on ﬂat space is pre- served. From a 4D standpoint, the fermionic parameters generating the preserved SUSY on NUT space are constant chiral spinors. This is indeed the preserved supersymmetry for any Gibbons-Hawking space X. The dualized 3D theory on R3, which we shall discuss momentarily, is the best place to demonstrate this. 4.2.2 4D action, SUSY and localization equations The standard 4D SYM action on X can be obtained by dimensional reduction of the × 6D action discussed above. As in the case for R3 S1, we add the standard topological (cid:20) (cid:21) term and a boun(cid:90)dary term to the bosonic action. S(b4) =Im4πτ X d(cid:90)et[e] d4x 12 FmnFmn + ∂mA5∂m(cid:90)A5 − ∂mA0∂mA0 + 12 DabDab + iIm(cid:90) τ det[e] d(cid:0)4x F F˜mn + i θ(cid:1)m Θ ∧ F(4) (4.23) mn 4π X 8π2R S∞3 Sf =Im τ det[e] d4x −2iλ¯ aσ¯ mDmλa (m, n = 1, 2, 3, 4) (4) 4π X The SUSY transformation, generated by a chiral half of the supersymmetry parameters × on R3 S1, may be summarized as: − ˙ δAm = iζa,ασmαβλ¯ a˙ (m = 1, 2, 3, 4) β δA5 = ζα,aλα,a, δA0 = ζα,aλα,a −1 − δλa,α = 2 Fmn(σmn)αβζβ,a Dabζb,α (4.24) − δλ¯ αa˙ = iζa, α(σ¯ m)α˙ α∂mA5 + iζa, α(σ¯ m)α˙ α∂mA0 − − δDab = iζa,α(σm)αα˙ Dmλ¯ b,α˙ iζa,α(σm)αα˙ Dmλ¯ b,α˙ To ensure the convergence of the 4D path integral, one needs to consider the theory reduced from a Euclidean version of the 6D theory and this can be achieved by setting A = iAE, with AE real. 0 0 0 With this modiﬁcation, the localization equations give the following solutions for the bosonic ﬁelds: − F = (cid:63)(4) F A + iAE = a¯ (constant), (4.25) 5 0 − A iAE = a (constant) 5 0 28 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Spinor

text: equation

text: in

text: Appendix

text: ζa

text: ζa

text: (constant),

text: The

text: above

text: equation

text: ensures

text: that

text: exactly

text: half

text: of

text: the

text: original

text: SUSY

text: on

text: ﬂat

text: space

text: is

text: pre-

text: served.

text: From

text: a

text: standpoint,

text: the

text: fermionic

text: parameters

text: generating

text: the

text: preserved

text: SUSY

text: on

text: NUT

text: space

text: are

text: constant

text: chiral

text: spinors.

text: This

text: is

text: indeed

text: the

text: preserved

text: supersymmetry

text: for

text: any

text: Gibbons-Hawking

text: space

text: X.

text: The

text: dualized

text: theory

text: on

text: which

text: we

text: shall

text: discuss

text: momentarily,

text: is

text: the

text: best

text: place

text: to

text: demonstrate

text: this.

text: action,

text: SUSY

text: and

text: localization

text: equations

text: The

text: standard

text: SYM

text: action

text: on

text: X

text: can

text: be

text: obtained

text: by

text: dimensional

text: reduction

text: of

text: the

text: action

text: discussed

text: above.

text: As

text: in

text: the

text: case

text: for

text: we

text: add

text: the

text: standard

text: topological

text: term

text: and

text: a

text: term

text: to

text: the

text: bosonic

text: action.

text: X

text: FmnFmn

text: DabDab

text: τ

text: det[e]

text: F

text: F˜mn

text: i

text: Θ

text: mn

text: 4π

text: X

text: Sf

text: =Im

text: τ

text: det[e]

text: −2iλ¯

text: aσ¯

text: mDmλa

text: (m,

text: n

text: 4π

text: X

text: The

text: SUSY

text: transformation,

text: generated

text: by

text: a

text: chiral

text: half

text: of

text: the

text: supersymmetry

text: parameters

text: on

text: may

text: be

text: summarized

text: as:

text: δAm

text: iζa,ασmαβλ¯

text: a˙

text: (m

text: β

text: ζα,aλα,a,

text: ζα,aλα,a

text: −1

text: δλa,α

text: Fmn(σmn)αβζβ,a

text: Dabζb,α

text: δλ¯

text: αa˙

text: iζa,

text: α(σ¯

text: m)α˙

text: iζa,

text: α(σ¯

text: m)α˙

text: δDab

text: iζa,α(σm)αα˙

text: Dmλ¯

text: b,α˙

text: iζa,α(σm)αα˙

text: Dmλ¯

text: b,α˙

text: To

text: ensure

text: the

text: convergence

text: of

text: the

text: path

text: integral,

text: one

text: needs

text: to

text: consider

text: the

text: theory

text: reduced

text: from

text: a

text: Euclidean

text: version

text: of

text: the

text: theory

text: and

text: this

text: can

text: be

text: achieved

text: by

text: setting

text: A

text: iAE,

text: with

text: AE

text: real.

text: With

text: this

text: modiﬁcation,

text: the

text: localization

text: equations

text: give

text: the

text: following

text: solutions

text: for

text: the

text: bosonic

text: ﬁelds:

text: F

text: F

text: A

text: iAE

text: a¯

text: (constant),

text: A

text: iAE

text: a

text: (constant)

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: 1 ∂ V ∂ ζE + µ ζE = 0, µ 2 V 1 ∂ V ∂ ζ¯E + µ ζ¯E = 0, (4.44) µ 2 V ⇒ R = f (ϕ0) = . 2V (cid:48) One can also read off qA(cid:48) directly by comparing the fermionic actions in the two de- B i scriptions: (cid:90) (cid:90) √ (cid:18) (cid:19) − (cid:48) (cid:48) (cid:48) Sf = 2i Im τR R3 d3x Vλ¯ aγi∂iλa = R3 d3x ψ¯αA(cid:48)γµβα(∂µψA β + qBA(cid:48)0∂µVψB β) (4.45) (cid:48) R 1/2 0 (cid:48) (cid:54) qA(cid:48) = − , qA(cid:48) = 0 (i = 0) B 0 V 0 1/2 B i (cid:48) Therefore, the effective q˜A (cid:48) that appears in the extended hyperka¨hler identity (2.7) 0B will be given as (cid:18) (cid:19) (cid:48) R 1 0 q˜A(cid:48) = (4.46) B 0 V 0 0 (cid:48) Given the explicit forms of q˜A(cid:48) and f (ϕ0), one obtains the following nontrivial com- B 0 ponents of the connection in the extended hyperka¨hler identity (2.7): Γφ = 0, Γφ¯ = − R 0φ 0φ¯ V R Γy Γy¯ = 0, = 0y 0y¯ 2V Γφ = 0, Γφ¯ = 0 (4.47) φ0 φ¯0 Γy Γy¯ = 0, = 0 y0 y¯0 ∀ Γk = 0 k. 00 Now we can check whether this connection obeys the constraints arising from the vanishing of 1-fermion terms in the SUSY variation of the action, (2.9)-(2.10), which we derived for a general deformed hyperka¨hler sigma model which preserves some super- symmetry. Note that the 3-fermion constraint (2.19) is satisﬁed trivially in this case. For U(1) SYM on X, the non-trivial part of the ﬁrst constraint, namely for j = 0 and arbitrary i, assumes the(cid:16)particular form (cid:17) − − elEE(cid:48) ∂ig0l + ∂0gil ∂lgi0 (Γik0 + Γ0ki)gkl + Fil = 0 (4.48) 33 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: V

text: ζE

text: µ

text: ζE

text: µ

text: V

text: V

text: ζ¯E

text: µ

text: ζ¯E

text: µ

text: V

text: R

text: f

text: One

text: can

text: also

text: read

text: off

text: directly

text: by

text: comparing

text: the

text: fermionic

text: actions

text: in

text: the

text: two

text: de-

text: B

text: i

text: scriptions:

text: Sf

text: Im

text: τR

text: Vλ¯

text: aγi∂iλa

text: β

text: β)

text: R

text: (i

text: B

text: V

text: B

text: i

text: Therefore,

text: the

text: effective

text: q˜A

text: that

text: appears

text: in

text: the

text: extended

text: hyperka¨hler

text: identity

text: will

text: be

text: given

text: as

text: R

text: B

text: V

text: Given

text: the

text: explicit

text: forms

text: of

text: and

text: f

text: one

text: obtains

text: the

text: following

text: nontrivial

text: com-

text: B

text: ponents

text: of

text: the

text: connection

text: in

text: the

text: extended

text: hyperka¨hler

text: identity

text: Γφ

text: Γφ¯

text: R

text: 0φ

text: 0φ¯

text: V

text: R

text: Γy

text: Γy¯

text: Γφ

text: Γφ¯

text: φ0

text: φ¯0

text: Γy

text: Γy¯

text: Γk

text: k.

text: Now

text: we

text: can

text: check

text: whether

text: this

text: connection

text: obeys

text: the

text: constraints

text: arising

text: from

text: the

text: vanishing

text: of

text: terms

text: in

text: the

text: SUSY

text: variation

text: of

text: the

text: action,

text: which

text: we

text: derived

text: for

text: a

text: general

text: deformed

text: hyperka¨hler

text: sigma

text: model

text: which

text: preserves

text: some

text: super-

text: symmetry.

text: Note

text: that

text: the

text: constraint

text: is

text: satisﬁed

text: trivially

text: in

text: this

text: case.

text: For

text: SYM

text: on

text: X,

text: the

text: non-trivial

text: part

text: of

text: the

text: ﬁrst

text: constraint,

text: namely

text: for

text: j

text: and

text: arbitrary

text: i,

text: assumes

text: form

text: Fil

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: From the structure of the intertwiners speciﬁed in (4.41) and the nonzero components of Γk in (4.47), it is clear that there are only two nontrivial components one needs to check, ij namely for l = y¯, i = y, and l = φ¯, i = φ. In the ﬁrst case, we have − − ∂yg0y¯ + ∂0gyy¯ ∂y¯gy0 (Γky0 + Γ0ky)gky¯ + Fyy¯ = ∂0gyy¯ + Fyy¯ (4.49) 4π − 8iπ = − = 0. Im τ (τ τ¯) In the second case , the constraint is satisﬁed trivially − − ∂φg0φ¯ + ∂0gφφ¯ ∂φ¯ gφ0 (Γkφ0 + Γ0kφ)gkφ¯ + Fφφ¯ (4.50) = 0 since each of the terms in the equation is individually zero. The connection derived in (4.47) for the hyperka¨hler sigma model which arises from the circle compactiﬁcation of U(1) SYM on X space therefore obeys the ﬁrst SUSY con- straint (2.9). Finally, the second SUSY constraint (2.10) can be written as − e (cid:48)(Γk Γk ) = 0 (4.51) k2E i0 0i ⇒ − − ∀ = Γy Γy = 0, Γφ Γφ = 0 i (4.52) i0 0i i0 0i which is trivially satisﬁed in this case, since all the relevant components of the connec- tion vanish. In appendix , we consider the sigma model again after rescaling the adjoint scalar so × that the metric looks closer to the one obtained via compactiﬁcation on R3 S1, with an effective radius R = R/V. ef f 5 The NUT centers 5.1 Setup So far we have described the local physics of the 3-dimensional sigma model which N one obtains by starting with the pure U(1) = 2 gauge theory in four dimensions and dimensionally reducing on a Gibbons-Hawking space GH. Now suppose we consider the actual compactiﬁed theory as opposed to the naive dimensional reduction. On general (cid:28) grounds we would expect that the local physics of this theory at energy scales E V/R (cid:28) (cid:107) (cid:107) and E dV can be described by the same ﬁelds which appear in the dimensionally reduced theory. In fact, here we can say more: since the four-dimensional theory is free 34 ##LTLine## ##LTLine##

text: From

text: the

text: structure

text: of

text: the

text: intertwiners

text: speciﬁed

text: in

text: and

text: the

text: nonzero

text: components

text: of

text: Γk

font: YEKKPP+PazoMath

text: in

text: it

text: is

text: clear

text: that

text: there

text: are

text: only

text: two

text: nontrivial

text: components

text: one

text: needs

text: to

text: check,

text: ij

text: namely

text: for

text: l

text: y¯,

text: i

text: y,

text: and

text: l

text: φ¯,

text: i

text: φ.

text: In

text: the

text: ﬁrst

text: case,

text: we

text: have

text: Fyy¯

text: Fyy¯

text: 4π

text: Im

text: τ

text: (τ

text: τ¯)

text: In

text: the

text: second

text: case

text: the

text: constraint

text: is

text: satisﬁed

text: trivially

text: ∂φg0φ¯

text: ∂0gφφ¯

text: ∂φ¯

text: Fφφ¯

text: since

text: each

text: of

text: the

text: terms

text: in

text: the

text: equation

text: is

text: individually

text: zero.

text: The

text: connection

text: derived

text: in

text: for

text: the

text: hyperka¨hler

text: sigma

text: model

text: which

text: arises

text: from

text: the

text: circle

text: compactiﬁcation

text: of

text: SYM

text: on

text: X

text: space

text: therefore

text: obeys

text: the

text: ﬁrst

text: SUSY

text: con-

text: straint

text: Finally,

text: the

text: second

text: SUSY

text: constraint

text: can

text: be

text: written

text: as

text: e

text: Γk

text: Γy

text: Γy

text: Γφ

text: Γφ

text: i

text: which

text: is

text: trivially

text: satisﬁed

text: in

text: this

text: case,

text: since

text: all

text: the

text: relevant

text: components

text: of

text: the

text: connec-

text: tion

text: vanish.

text: In

text: appendix

text: we

text: consider

text: the

text: sigma

text: model

text: again

text: after

text: rescaling

text: the

text: adjoint

text: scalar

text: so

text: that

text: the

text: metric

text: looks

text: closer

text: to

text: the

text: one

text: obtained

text: via

text: compactiﬁcation

text: on

text: with

text: an

text: effective

text: radius

text: R

text: R/V.

text: ef

text: f

text: The

text: NUT

text: centers

text: Setup

text: So

text: far

text: we

text: have

text: described

text: the

text: local

text: physics

text: of

text: the

text: sigma

text: model

text: which

text: N

text: one

text: obtains

text: by

text: starting

text: with

text: the

text: pure

text: gauge

text: theory

text: in

text: four

text: dimensions

text: and

text: dimensionally

text: reducing

text: on

text: a

text: Gibbons-Hawking

text: space

text: GH.

text: Now

text: suppose

text: we

text: consider

text: the

text: actual

text: compactiﬁed

text: theory

text: as

text: opposed

text: to

text: the

text: naive

text: dimensional

text: reduction.

text: On

text: general

text: grounds

text: we

text: would

text: expect

text: that

text: the

text: local

text: physics

text: of

text: this

text: theory

text: at

text: energy

text: scales

text: E

text: V/R

text: and

text: E

text: dV

text: can

text: be

text: described

text: by

text: the

text: same

text: ﬁelds

text: which

text: appear

text: in

text: the

text: dimensionally

text: reduced

text: theory.

text: In

text: fact,

text: here

text: we

text: can

text: say

text: more:

text: since

text: the

text: four-dimensional

text: theory

text: is

text: free

text: ##LTLine##

text: ##LTLine##


text: L − (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) 4πδ = iqiEA(cid:48)ψF σζ¯σFeiFF(cid:48)ψ¯E(cid:48)αγµβα(∂µψA β + qBA(cid:48)j∂µϕjψB β) (B.1a) (cid:48) (cid:48) (cid:48) + iψ¯αA(cid:48)γµβα∂µ(qiAE(cid:48)ψF σζ¯σFeiFF(cid:48)ψE β) (B.1b) − (cid:48) (cid:48) (cid:48) iψ¯αA(cid:48)γµβα∂jqBA(cid:48)iψE σζ¯σEeEj E(cid:48)∂µϕiψB β (B.1c) − (cid:48) (cid:48) (cid:48) iψ¯αA(cid:48)γµβαqBA(cid:48)i∂µ(ψE σζ¯σEeiEE(cid:48))ψB β (B.1d) (cid:48) (cid:48) (cid:48) (cid:48) + iψ¯αA(cid:48)γµβαqBA(cid:48)i∂µϕiqBjE(cid:48)ψF σζ¯σFeFj F(cid:48)ψE β (B.1e) (cid:48) (cid:48) (cid:48) (cid:48) + iΩA(cid:48)B(cid:48)C(cid:48)D(cid:48)ψαA γµβα∂νϕieiBEγσνβζ¯σEψδC γµδωψ¯D ω (B.1f) L L (cid:48) Relabeling indices this becomes δ = iψ¯A(cid:48)αζ¯σEδ EA ασ where L (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) δ A ασ = + qA (cid:48)ψF σei (cid:48)γµα(∂ ψE β + qE(cid:48) ∂ ϕjψB β) (B.2a) E iE EF β µ B j µ − (cid:48) (cid:48) (cid:48) γµα∂ (qA (cid:48)ψF σei (cid:48)ψE β) (B.2b) β µ iE EF (cid:48) (cid:48) (cid:48) + γµβα∂jqBA(cid:48)iψE σeEj E(cid:48)∂µϕiψB β (B.2c) (cid:48) (cid:48) (cid:48) + γµαqA(cid:48) ∂ (ψE σei (cid:48))ψB β (B.2d) β B i µ EE − (cid:48) (cid:48) (cid:48) (cid:48) γµαqA(cid:48) ∂ ϕiqB (cid:48)ψF σej (cid:48)ψE β (B.2e) β B i µ jE EF (cid:48) (cid:48) (cid:48) (cid:48) + ΩA(cid:48) (cid:48) (cid:48)ψE γµτ∂ ϕieB γνσβψC γαδ (B.2f) E B C τ β ν iE δ µ which expands out to L (cid:48) (cid:48) (cid:48) (cid:48) δ A ασ = + qA (cid:48)ψF σei (cid:48)γµα∂ ψE β (B.3a) E iE EF β µ (cid:48) (cid:48) (cid:48) (cid:48) + qA (cid:48)ψF σei (cid:48)γµαqE(cid:48) ∂ ϕjψB β (B.3b) iE EF β B j µ − (cid:48) (cid:48) (cid:48) γµβα∂µϕj∂jqiAE(cid:48)ψF σeiEF(cid:48)ψE β (B.3c) − (cid:48) (cid:48) (cid:48) γµαqA (cid:48)∂ ψF σei (cid:48)ψE β (B.3d) β iE µ EF − (cid:48) (cid:48) (cid:48) γµβαqiAE(cid:48)ψF σ∂µϕj∂jeiEF(cid:48)ψE β (B.3e) − (cid:48) (cid:48) (cid:48) γµαqA (cid:48)ψF σei (cid:48)∂ ψE β (B.3f) β iE EF µ (cid:48) (cid:48) (cid:48) + γµβα∂jqBA(cid:48)iψE σeEj E(cid:48)∂µϕiψB β (B.3g) (cid:48) (cid:48) (cid:48) + γµαqA(cid:48) ∂ ψE σei (cid:48)ψB β (B.3h) β B i µ EE (cid:48) (cid:48) (cid:48) + γµβαqBA(cid:48)iψE σ∂µϕj∂jeiEE(cid:48)ψB β (B.3i) − (cid:48) (cid:48) (cid:48) (cid:48) γµαqA(cid:48) ∂ ϕiqB (cid:48)ψF σej (cid:48)ψE β (B.3j) β B i µ jE EF (cid:48) (cid:48) (cid:48) (cid:48) + ΩA(cid:48) (cid:48) (cid:48)ψE γµτ∂ ϕieB γνσβψC γαδ (B.3k) E B C τ β ν iE δ µ 42

text: L

text: 4πδ

text: β

text: β)

text: β)

text: σζ¯σEeEj

text: β

text: β

text: σζ¯σFeFj

text: β

text: γµβα∂νϕieiBEγσνβζ¯σEψδC

text: γµδωψ¯D

text: ω

text: L

text: L

text: Relabeling

text: indices

text: this

text: becomes

text: δ

text: EA

text: ασ

text: where

text: L

text: δ

text: A

text: ασ

text: qA

text: σei

text: ψE

text: β

text: ϕjψB

text: β)

text: E

text: iE

text: EF

text: β

text: µ

text: B

text: j

text: µ

text: γµα∂

text: (qA

text: σei

text: β)

text: β

text: µ

text: iE

text: EF

text: σeEj

text: β

text: (ψE

text: σei

text: β

text: β

text: B

text: i

text: µ

text: EE

text: ϕiqB

text: σej

text: β

text: β

text: B

text: i

text: µ

text: jE

text: EF

text: γµτ∂

text: ϕieB

text: γνσβψC

text: γαδ

text: E

text: B

text: C

text: τ

text: β

text: ν

text: iE

text: δ

text: µ

text: which

text: expands

text: out

text: to

text: L

text: δ

text: A

text: ασ

text: qA

text: σei

text: ψE

text: β

text: E

text: iE

text: EF

text: β

text: µ

text: qA

text: σei

text: ϕjψB

text: β

text: iE

text: EF

text: β

text: B

text: j

text: µ

text: β

text: γµαqA

text: ψF

text: σei

text: β

text: β

text: iE

text: µ

text: EF

text: β

text: γµαqA

text: σei

text: ψE

text: β

text: β

text: iE

text: EF

text: µ

text: σeEj

text: β

text: ψE

text: σei

text: β

text: β

text: B

text: i

text: µ

text: EE

text: β

text: ϕiqB

text: σej

text: β

text: β

text: B

text: i

text: µ

text: jE

text: EF

text: γµτ∂

text: ϕieB

text: γνσβψC

text: γαδ

text: E

text: B

text: C

text: τ

text: β

text: ν

text: iE

text: δ

text: µ


text: (cid:20) (cid:21) The Lagrangian of the 6D theory is L 1 1 1 6 = g2 2 FMNFMN + ψ¯aΓM∂Mψa + 2 DabDab (C.1) 6 − where we choose the following metric on the ﬂat space ηMN = ( 1, 1, 1, 1, 1, 1). The fermionic ﬁeld ψa is a symplectic Majorana-Weyl spinor which transforms as a doublet of the SU(2) R-symmetry. A SO(1, 5) spinor obeys the Weyl condition and is conjugate to self but does not obey the standard ”Majorana” condition. However, when combined with the SU(2) symmetry, one can have a modiﬁed reality condition on these spinors - R the ”symplectic Majorana” condition. Γ7ψa = ψa (Weyl condition) − ≡ (C.2) ψa, TC6 (cid:101)ab = (ψb )†iΓ0 ψ¯b (Symplectic Majorana condition) − − where Γ7 is the chirality matrix in 6D deﬁned as Γ7 = Γ0Γ1Γ2Γ3Γ4Γ5. C is the 6D − − − − 6 charge conjugation obeying C Γµ(C ) 1 = (Γµ)T. 6 6 SUSY transformation The action in equation (C.1) is invariant under the following SUSY transformation rules: 1 − − − δAM = (ζ¯aΓMψa ψ¯aΓMζa) = ψ¯aΓMζa (since ζ¯aΓMψa = ψ¯aΓMζa) 2 δψa = −1 FMNΓMNζa − Dabζb, δψ¯a = 1ζ¯aΓMNFMN − Dabζ¯b (C.3) 2 2 δDab = ζ¯aΓM∂Mψb + ζ¯bΓM∂Mψa Note that the SUSY parameter ζa is a Grassman-o×dd sy×mplectic Majorana-Weyl spinor and a solution of the Killing spinor equation on R3 S1 T1,1 : ⇒ ∂Mζa = 0 = ζa = constant (C.4) − In equation (C.3), we used that ψ¯aΓMζa = ζ¯aΓMψa which follows from the general relation involving 6D spinors − ψ¯aΓN1 N2...Nnζa = ( 1)nζ¯aΓNn Nn−1...N1ψa (C.5) The SUSY variation for ψ¯a can be obtained as follows: − i − i δψ¯a =δψa†iΓ0 = 2 FMN(ΓMNζa)†Γ0 = 2 FMNζa†(ΓMN)†Γ0 − 1 1 = FMNζ¯aΓNM (using ΓMN†Γ0 = Γ0ΓMN) = FMNζ¯aΓMN 2 2 46 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: The

text: Lagrangian

text: of

text: the

text: theory

text: is

text: L

text: FMNFMN

text: ψ¯aΓM∂Mψa

text: DabDab

text: where

text: we

text: choose

text: the

text: following

text: metric

text: on

text: the

text: ﬂat

text: space

text: ηMN

text: The

text: fermionic

text: ﬁeld

text: ψa

text: is

text: a

text: symplectic

text: Majorana-Weyl

text: spinor

text: which

text: transforms

text: as

text: a

text: doublet

text: of

text: the

text: R-symmetry.

text: A

text: spinor

text: obeys

text: the

text: Weyl

text: condition

text: and

text: is

text: conjugate

text: to

text: self

text: but

text: does

text: not

text: obey

text: the

text: standard

text: ”Majorana”

text: condition.

text: However,

text: when

text: combined

text: with

text: the

text: symmetry,

text: one

text: can

text: have

text: a

text: modiﬁed

text: reality

text: condition

text: on

text: these

text: spinors

text: R

text: the

text: ”symplectic

text: Majorana”

text: condition.

text: Γ7ψa

text: ψa

text: (Weyl

text: condition)

text: ψa,

text: (ψb

text: ψ¯b

text: (Symplectic

text: Majorana

text: condition)

text: where

text: Γ7

text: is

text: the

text: chirality

text: matrix

text: in

text: deﬁned

text: as

text: Γ7

text: C

text: is

text: the

text: charge

text: conjugation

text: obeying

text: C

text: Γµ(C

text: (Γµ)T.

text: SUSY

text: transformation

text: The

text: action

text: in

text: equation

text: is

text: invariant

text: under

text: the

text: following

text: SUSY

text: transformation

text: rules:

text: δAM

text: (ζ¯aΓMψa

text: ψ¯aΓMζa)

text: ψ¯aΓMζa

text: (since

text: ζ¯aΓMψa

text: ψ¯aΓMζa)

text: δψa

text: −1

text: FMNΓMNζa

text: Dabζb,

text: δψ¯a

text: Dabζ¯b

text: δDab

text: ζ¯aΓM∂Mψb

text: ζ¯bΓM∂Mψa

text: Note

text: that

text: the

text: SUSY

text: parameter

text: ζa

text: is

text: a

text: Grassman-o×dd

text: sy×mplectic

text: Majorana-Weyl

text: spinor

text: and

text: a

text: solution

text: of

text: the

text: Killing

text: spinor

text: equation

text: on

text: ∂Mζa

text: ζa

text: constant

text: In

text: equation

text: we

text: used

text: that

text: ψ¯aΓMζa

text: ζ¯aΓMψa

text: which

text: follows

text: from

text: the

text: general

text: relation

text: involving

text: spinors

text: ψ¯aΓN1

text: The

text: SUSY

text: variation

text: for

text: ψ¯a

text: can

text: be

text: obtained

text: as

text: follows:

text: i

text: i

text: δψ¯a

text: =δψa†iΓ0

text: FMNζ¯aΓNM

text: (using

text: ΓMN†Γ0

text: FMNζ¯aΓMN

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:90) (cid:90) (cid:104) (cid:105) (cid:90) (cid:0) (cid:1) × d4x ψ¯aΓm∂mψa = × d4x λ¯ †aσ¯ m∂mλa + λ†aσm∂mλ¯ a R3 S1 R3 S1 (D.4) − = (cid:18)× (cid:19)d4x 2iλ¯ aσ¯ m∂mλa (cid:18) (cid:19) R3 S1 (cid:18) (cid:19) Similarly, the SUSY parameter ζa = ζζ1aa can be decomposed as ζ1a = ζ0a,α , ζ2a = (cid:0) 2 (cid:1) 0 so that the SUSY transformation can also be written in terms of the 4D spinors. For ¯a ζ α˙ − − example, δAm = ψ¯aΓmζa = i ζ¯aσ¯ mλa ζaσmλ¯ a . The remaining rules can be derived similarly. Now consider further dimensional reduction of 4D spinors to 3D spinors. If (x , x , x , x ) denote local coordinates on the manifold M , then we treat x as the 1 2 3 4 4 2 circle direction. The basic rules for writing the Lagrangian and the SUSY transformation for the ﬁelds can be summarized as follows: → (cid:54) (σm)αβ˙ (γi)αβ = (γi)βα = (σ1, σ3, i) (m = 2) → (cid:54) (σ¯ m)α˙ β (γi)αβ = (γi)βα = (σ1, σ3, i) (m = 2) (D.5) → → (σ2)αβ˙ i(cid:101)αβ, (σ¯ 2)α˙ β i(cid:101)αβ → → λa,α λa,α, λ¯ a λ¯ a α˙ α where we normalize the antisymmetric tensor (cid:101)αβ as (cid:101)12 = (cid:101)21 = 1. The matrices − − γi := (γi)αβ = ( σ3, σ1, σ2) obey the following identities { } γi, γj = δij1 (Clifford algebra) (D.6) γiγj = δij1 + i(cid:101)ijkγk (Product Identity) (cid:90) (cid:0) (cid:1) (cid:90) The fermionic Lagrangian on dimensional reduction can therefore be rewritten as − − × d4x 2iλ¯ aσ¯ m∂mλa = 2i d3x λ¯ αa(γi)αβ∂iλaβ (D.7) R3 S1 R3 Similarly, the SUSY transformation rules can be rewritten in terms of the 3D spinors, using the above rules. × E Killing spinor on TN T1,1 4 × The metric on the space TN T1,1 is 4 R2 − − ds2 = V(r)dr2 + V(r)r2(dθ2 + sin2 θdφ2) + (dχ cos θdφ)2 + du2 dv2 (E.1) V 49 ##LTLine##

text: ψ¯aΓm∂mψa

text: λ¯

text: †aσ¯

text: m∂mλa

text: λ†aσm∂mλ¯

text: a

font: BEGFVD+PazoMathBlackboardBold

text: 2iλ¯

text: aσ¯

text: m∂mλa

text: Similarly,

text: the

text: SUSY

text: parameter

text: ζa

text: can

text: be

text: decomposed

text: as

text: so

text: that

text: the

text: SUSY

text: transformation

text: can

text: also

text: be

text: written

text: in

text: terms

text: of

text: the

text: spinors.

text: For

text: ¯a

text: ζ

text: α˙

text: example,

text: δAm

text: ψ¯aΓmζa

text: i

text: ζ¯aσ¯

text: mλa

text: ζaσmλ¯

text: a

text: The

text: remaining

text: rules

text: can

text: be

text: derived

text: similarly.

text: Now

text: consider

text: further

text: dimensional

text: reduction

text: of

text: spinors

text: to

text: spinors.

text: If

text: (x

text: x

text: x

text: x

text: denote

text: local

text: coordinates

text: on

text: the

text: manifold

text: M

text: then

text: we

text: treat

text: x

text: as

text: the

text: circle

text: direction.

text: The

text: basic

text: rules

text: for

text: writing

text: the

text: Lagrangian

text: and

text: the

text: SUSY

text: transformation

text: for

text: the

text: ﬁelds

text: can

text: be

text: summarized

text: as

text: follows:

text: (σm)αβ˙

text: (γi)αβ

text: (γi)βα

text: i)

text: (m

text: (σ¯

text: m)α˙

text: β

text: (γi)αβ

text: (γi)βα

text: i)

text: (m

text: (σ2)αβ˙

text: (σ¯

text: 2)α˙

text: β

text: λa,α

text: λa,α,

text: λ¯

text: a

text: λ¯

text: a

text: α˙

text: α

text: where

text: we

text: normalize

text: the

text: antisymmetric

text: tensor

text: as

text: The

text: matrices

text: γi

text: (γi)αβ

text: obey

text: the

text: following

text: identities

text: γi,

text: γj

text: (Clifford

text: algebra)

text: γiγj

text: (Product

text: Identity)

text: The

text: fermionic

text: Lagrangian

text: on

text: dimensional

text: reduction

text: can

text: therefore

text: be

text: rewritten

text: as

text: 2iλ¯

text: aσ¯

text: m∂mλa

text: λ¯

text: αa(γi)αβ∂iλaβ

text: Similarly,

text: the

text: SUSY

text: transformation

text: rules

text: can

text: be

text: rewritten

text: in

text: terms

text: of

text: the

text: spinors,

text: using

text: the

text: above

text: rules.

text: E

text: Killing

text: spinor

text: on

text: TN

text: The

text: metric

text: on

text: the

text: space

text: TN

text: is

text: (dχ

text: cos

text: V

text: ##LTLine##


text: With this projection condition imposed, the Killing spinor equation is clearly the same × × as that for R3 S1 T1,1 written in spherical polar coordinates on R3 for which we know the Killing spinor to be simply a constant spinor. Therefore, for a particular choice of vier- × beins (compatible with Cartesian coordinates on R3), the Killing spinor on TN T1,1 is 4 simply a constant spinor obeying the above projection condition. The above computation also shows that constancy and covariant-constancy are equivalent for any spinor on the × manifold TN T1,1 if the spinor obeys the projection condition. 4 Dimensional reduction of the Killing spinor equation On dimensionally reducing the theory along the S1 ﬁber, we demand that ∂ ζ = 0 — χ automatically true for the chiral spinor which is a solution of the Killing spinor equation D ζ = 0. The spinors generating the supersymmetry transformations in the dimension- χ ally reduced theory are therefore expected to be given by the remaining components the Killing spinor equation, viz. Dmζ = 0, with m = r, θ, φ. We will now express this equation in terms of the three-dimensional covariant derivative. We will be interested in reducing the theory to ﬂat 3D space, with the metric ds2 = dr2 + r2(dθ2 + sin2 θdφ2) (E.21) Choosing e1 = dr, e2 = rdθ, e3 = r sin θdφ, the spin connections for the above metric are − Ω 12 = dθ, (E.22) − Ω 13 = sin θ dφ, (E.23) − Ω 23 = cos θ dφ. (E.24) Now, one can easily check that the equation for the Killing spinor Dmζ = 0 on ﬂat 3D space is identical to the dimensionally reduced Killing spinor equation on the Taub-NUT space for a spinor satisfying the constraint E.20. The supersymmetry transformations for the 3D action are therefore generated by constant chiral spinors, ∂iζ = 0, (E.25) where xi are the Cartesian coordinates on R3. 52

text: With

text: this

text: projection

text: condition

text: imposed,

text: the

text: Killing

text: spinor

text: equation

text: is

text: clearly

text: the

text: same

text: as

text: that

text: for

text: written

text: in

text: spherical

text: polar

text: coordinates

text: on

text: for

text: which

text: we

text: know

text: the

text: Killing

text: spinor

text: to

text: be

text: simply

text: a

text: constant

text: spinor.

text: Therefore,

text: for

text: a

text: particular

text: choice

text: of

text: vier-

text: beins

text: (compatible

text: with

text: Cartesian

text: coordinates

text: on

text: the

text: Killing

text: spinor

text: on

text: TN

text: is

text: simply

text: a

text: constant

text: spinor

text: obeying

text: the

text: above

text: projection

text: condition.

text: The

text: above

text: computation

text: also

text: shows

text: that

text: constancy

text: and

text: covariant-constancy

text: are

text: equivalent

text: for

text: any

text: spinor

text: on

text: the

text: manifold

text: TN

text: if

text: the

text: spinor

text: obeys

text: the

text: projection

text: condition.

text: Dimensional

text: reduction

text: of

text: the

text: Killing

text: spinor

text: equation

text: On

text: dimensionally

text: reducing

text: the

text: theory

text: along

text: the

text: ﬁber,

text: we

text: demand

text: that

text: ζ

text: χ

text: automatically

text: true

text: for

text: the

text: chiral

text: spinor

text: which

text: is

text: a

text: solution

text: of

text: the

text: Killing

text: spinor

text: equation

text: D

text: ζ

text: The

text: spinors

text: generating

text: the

text: supersymmetry

text: transformations

text: in

text: the

text: dimension-

text: χ

text: ally

text: reduced

text: theory

text: are

text: therefore

text: expected

text: to

text: be

text: given

text: by

text: the

text: remaining

text: components

text: the

text: Killing

text: spinor

text: equation,

text: viz.

text: Dmζ

text: with

text: m

text: r,

text: θ,

text: φ.

text: We

text: will

text: now

text: express

text: this

text: equation

text: in

text: terms

text: of

text: the

text: three-dimensional

text: covariant

text: derivative.

text: We

text: will

text: be

text: interested

text: in

text: reducing

text: the

text: theory

text: to

text: ﬂat

text: space,

text: with

text: the

text: metric

text: Choosing

text: dr,

text: rdθ,

text: r

text: sin

text: θdφ,

text: the

text: spin

text: connections

text: for

text: the

text: above

text: metric

text: are

text: Ω

text: dθ,

text: Ω

text: sin

text: θ

text: dφ,

text: Ω

text: cos

text: θ

text: dφ.

text: Now,

text: one

text: can

text: easily

text: check

text: that

text: the

text: equation

text: for

text: the

text: Killing

text: spinor

text: Dmζ

text: on

text: ﬂat

text: space

text: is

text: identical

text: to

text: the

text: dimensionally

text: reduced

text: Killing

text: spinor

text: equation

text: on

text: the

text: Taub-NUT

text: space

text: for

text: a

text: spinor

text: satisfying

text: the

text: constraint

text: The

text: supersymmetry

text: transformations

text: for

text: the

text: action

text: are

text: therefore

text: generated

text: by

text: constant

text: chiral

text: spinors,

text: ∂iζ

text: where

text: xi

text: are

text: the

text: Cartesian

text: coordinates

text: on