构建三维N=2超对称高自旋规范理论与电流多重态

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本文探讨了N=2的超对称高自旋规范理论及其在三维空间的应用，特别是在保形超重力的背景下。作者Jessica Hutomo、Sergei M. Kuzenko和Daniel Ogburn聚焦于线性超多重谱系，这些谱系是N=2三维保形超重力的关键组成部分。他们构建了一种拓扑BF型项，这是一种在规范理论中常见的非平凡项，它展示了超对称的特性。研究的核心是引进共形高自旋规范超场，这些超场通过Chern-Simons类型的动作用于N=2超重力的线性化版本之上。这种构造增强了超对称理论的规范性和超魏尔不变性，即在任何保形平坦的超空间中保持不变，且涉及线性化的N=2超 Cotton张量的自旋推广，这是一个关键的超对称物理量。对于无质量的高旋转超多重子，在(1,1) anti-de Sitter (AdS)超空间中，作者提出了两个离壳的Lagrangian规范公式，这两个公式通过对偶变换相互关联。这种对偶关系使得理论的表述更为灵活，能够在不失去一致性的情况下处理不同描述。这些无质量理论的利用，使得在(1,1) AdS超空间中可以构建一致的高旋转超电流多重峰，这是大规模手性超多重模型的重要组成部分，提供了具体的超流多重态实例。文章还讨论了高旋转超多重子集的脱壳公式，这是一套数学工具，用于将物理问题从离壳形式转换到更为直观的物理表述，这对于理解和分析超对称理论的物理性质至关重要。这项工作深化了我们对N=2超对称理论的理解，尤其是在高自旋规范理论和超流多重态的构造方面，对于理论物理学家和相关领域的研究人员来说，具有重要的研究价值。

The T

αðnÞ

is said to be super-Weyl primary of dimension d

if its infinitesimal super-Weyl transformation law is

αðnÞ

¼ dσT

αðnÞ

: ð2:11Þ

As follows from (2.8), the super-Cotton tensor is super-

Weyl primary of dimension þ2. We now introduce several

types of primary superfields that will be important for our

subsequent consideration.

A symmetric rank-n spinor superfield G

αðnÞ

is called

longitudinal linear if it obeys the following first-order

constraint:

ðα

…α

nþ1

¼ 0; ð2:12Þ

which implies

þ 2nRÞG

αðnÞ

¼ 0: ð2:13Þ

If G

αðnÞ

is super-Weyl primary, then the constraint (2.12) is

consistent provided the dimension d

ðnÞ

and Uð1Þ

charge

ðnÞ

of G

αðnÞ

are related to each other as follows:

ðnÞ

¼ −

− q

ðnÞ

: ð2:14Þ

In the scalar case, n ¼ 0, the constraint (2.12) becomes the

condition of covariant chirality,

G ¼ 0. The dimension

and Uð1Þ

charge q

of any primary chiral scalar

superfield G are related as d

þ q

¼ 0, in accordance

with [12].

The longitudinal linear superfields form a ring. Given

two such superfields G

αðnÞ

and

αðmÞ

, their product

αðnþmÞ

≔ G

ðα

…α

nþ1

…α

nþm

is longitudinal linear. If

αðnÞ

and

αðmÞ

are super-Weyl primary superfields, their

product G

αðnþmÞ

is also super-Weyl primary.

Given a positive integer n, a symmetric rank-n spinor

superfield Γ

αðnÞ

is called transverse linear if it obeys the

first-order constraint

βα

…α

n−1

¼ 0;n≠ 0; ð2:15Þ

which implies

− 2ðn þ 2ÞRÞΓ

αðnÞ

¼ 0: ð2:16Þ

If Γ

αðnÞ

is super-Weyl primary, then the constraint (2.15) is

consistent provided the dimension d

ðnÞ

and Uð1Þ

charge

ðnÞ

of Γ

αðnÞ

are related to each other as follows:

ðnÞ

¼ 1 þ

− q

ðnÞ

: ð2:17Þ

In the n ¼ 0 case, the constraint (2.15) is not defined.

However its corollary (2.16) is perfectly consistent,

− 4RÞΓ ¼ 0; ð2:18Þ

and defines a covariantly linear scalar superfield Γ. The

dimension d

and Uð1Þ

charge q

of any primary linear

scalar Γ are related as d

þ q

¼ 1, in accordance

with [12].

In the case of 4D N ¼ 1 AdS supersymmetry, longi-

tudinal linear and transverse linear superfields were pio-

neered by Ivanov and Sorin [6] who studied the superfield

representations of the AdS isometry group OSpð1j4Þ. In the

framework of 4D N ¼ 1 conformal supergravity, primary

longitudinal linear and transverse linear supermultiplets

were introduced for the first time by Kugo and Uehara [38].

Such superfields were used in [7–9,18,29] for the descrip-

tion of off-shell massless gauge theories in four and three

dimensions.

The constraints (2.12) and (2.15) are solved in terms of

prepotentials Ψ

αðn−1Þ

and Φ

αðnþ1Þ

as follows:

αðnÞ

ðα

…α

; ð2:19aÞ

αðnÞ

ðβα

…α

: ð2:19bÞ

Provided the constraints (2.12) and (2.15) are the only

conditions imposed on G

αðnÞ

and Γ

αðnÞ

respectively, the

prepotentials Ψ

αðn−1Þ

and Φ

αðnþ1Þ

can be chosen to be

unconstrained complex, and are defined modulo gauge

transformations of the form

αðn−1Þ

ðα

…α

n−1

; ð2:20aÞ

αðnþ1Þ

ðγα

…α

nþ1

; ð2:20bÞ

with the gauge parameters ζ

αðn−2Þ

and ξ

αðnþ2Þ

being uncon-

strained. If the linear superfields G

αðnÞ

and Γ

αðnÞ

are super-

Weyl primary, then their prepotentials Ψ

αðn−1Þ

and Φ

αðnþ1Þ

can also be chosen to be super-Weyl primary.

Given two linear superfields G

αðnþ1Þ

and Γ

αðnÞ

such that

their Uð1Þ

charges are constrained by q

ðnþ1Þ

þ q

ðnÞ

¼ −1,

we can define a gauge-invariant and super-Weyl-invariant

BF term

ðnÞ

3j4

zEΨ

αðnÞ

¼ −ð−1Þ

3j4

zEΦ

αðnþ1Þ

; ð2:21Þ

where the superspace integration measure is d

3j4

z ≔

θd

θ and E

−1

≔ BerðE

Þ.

In the n ¼ 0 case, the prepotential solution (2.19b) is

still valid. The prepotential Φ

can be chosen to be

N ¼ 2 SUPERSYMMETRIC HIGHER SPIN GAUGE … PHYS. REV. D 98, 125004 (2018)

125004-5

unconstrained complex provided the constraint (2.18) is the

only condition imposed on Γ. However, if we are dealing

with a real linear superfield,

− 4RÞL ¼ 0;

L ¼ L; ð2:22Þ

then the constraints are solved [12] in terms of an uncon-

strained real prepotential V,

L ¼ i

V ¼ V; ð2:23Þ

which is defined modulo gauge transformations of the form

δV ¼ λ þ

λ; J λ ¼ 0;

λ ¼ 0: ð2:24Þ

If L is super-Weyl primary, then Eq. (2.17) tells us that the

dimension of L is þ1. In this case it is consistent to

consider the gauge prepotential V to be inert under the

super-Weyl transformations [12], δ

V ¼ 0.

Let us assume that the background curved superspace

allows the existence of a real transverse linear superfield

αðnÞ

βα

…α

n−1

¼ 0; D

βα

…α

n−1

¼ 0: ð2:25Þ

Then it is automatically conserved,

βγ

βγα

…α

n−2

¼ 0: ð2:26Þ

in accordance with (2.4b). The super-Cotton tensor W

αβ

an example of such supermultiplets. If W

αðnÞ

is super-Weyl

primary, then its dimension is equal to ð1 þ n=2Þ,in

accordance with (2.17). As will be shown in the next

subsection, a solution to (2.25) in terms of an unconstrained

prepotential exists for every conformally flat superspace.

C. Conformal gauge superfields

Let n be a positive integer. A real symmetric rank-n

spinor superfield H

αðnÞ

is said to be a conformal gauge

supermultiplet if (i) it is super-Weyl primary of dimension

ð−n=2Þ,

αðnÞ

¼ −

σH

αðnÞ

; ð2:27Þ

and (ii) it is defined modulo gauge transformations of

the form

αðnÞ

ðα

…α

− ð−1Þ

ðα

…α

; ð2:28Þ

with the gauge parameter λ

αðn−1Þ

being unconstrained

complex. The dimension of H

αðnÞ

in (2.27) is uniquely

fixed by requiring the longitudinal linear superfield g

αðnÞ

ðα

…α

in the right-hand side of (2.28) to be super-

Weyl primary. Indeed, the gauge parameter g

αðnÞ

must be

neutral with respect to the R-symmetry group Uð1Þ

since

αðnÞ

is real, and then the dimension of g

αðnÞ

is equal to

ð−n=2Þ, in accordance with (2.14).

Starting with H

αðnÞ

one can construct its real descendant

αðnÞ

ðHÞ¼AH

αðnÞ

, where A is a linear differential

operator involving D

, the torsion superfields and their

covariant derivatives, with the following properties:

(1) W

αðnÞ

is super-Weyl primary of dimension

ð1 þ n=2Þ,

αðnÞ



1 þ



σW

αðnÞ

: ð2:29Þ

(2) The gauge variation of W

αðnÞ

vanishes if the super-

space is conformally flat,

αðnÞ

¼ OðW

ð2Þ

Þ; ð2:30Þ

where W

ð2Þ

is the super-Cotton tensor (2.7).

(3) W

αðnÞ

is divergenceless if the superspace is con-

formally flat,

βαðn−1Þ

¼ OðW

ð2Þ

Þ;

βαðn−1Þ

¼ OðW

ð2Þ

Þ: ð2:31Þ

Here OðW

ð2Þ

Þ stands for contributions containing

the super-Cotton tensor and its covariant derivatives.

In general, W

αðnÞ

ðHÞ is uniquely defined modulo a

normalization and contributions involving the super-

Cotton tensor (2.7). The primary superfields W

αðnÞ

ðHÞ

may be interpreted as linearized higher-spin super-Cotton

tensors in supergravity; see also the next subsection.

D. Linearized conformal higher-spin supergravity

in conformally flat backgrounds

So far the supergravity background has been completely

arbitrary. In the remainder of this section we assume that

the background curved superspace M

3j4

is conformally

flat,

αβ

¼ 0: ð2:32Þ

Then each primary superfield W

αðnÞ

ðHÞ, with n ¼ 1; 2; …,

is gauge invariant,

αðnÞ

¼ 0; ð2:33Þ

and obeys the conservation equations (2.25). These proper-

ties and the super-Weyl transformation laws (2.27) and

(2.29) imply that the action

The super-Weyl transformation of the superspace density is

E ¼ −σE.

HUTOMO, KUZENKO, and OGBURN PHYS. REV. D 98, 125004 (2018)

125004-6

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