3d N=4超保形场论中希格斯分支算子的一维理论与拓扑相关函数

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"希格斯分支算子的一维理论" 这篇学术论文探讨了3d N = 4 $$\mathcal{N} = 4$$超保形场论中的半BPS（Bogomol'nyi-Prasad-Sommerfield）局部算子的相关函数。这些算子与超对称定位技术密切相关，该技术被用来精确计算特定类型的物理过程。研究重点在于一种特殊的算子——希格斯分支算子的扭曲线性组合，它们可以在三维空间的任意一条线上任意位置插入。希格斯分支算子是基于超多重标量构建的，这些算子形成一个一维的非交换算子代数，其相关函数具有拓扑性质。这意味着它们的2点和3点函数可以通过这个1d拓扑代数的性质直接推导出来。为了进行这样的计算，研究者们将3d超保形场论映射到一个圆形三球上，并利用了一个不属于$$\mathcal{N} = 4$$超对称代数的特定超荷进行超对称局部化。这种策略简化了问题，使得可以构建一个简单的模型来处理1d拓扑代数的相关函数。这个模型实际上是一个与矩阵模型耦合的一维高斯理论，可以视作拓扑规范量子力学的规范固定版本。这种模型的引入使得非保形理论，包括具有实质量和Fayet-Iliopoulos参数的理论，也能得到扩展。此外，论文还讨论了与库仑分支相关的1d拓扑代数，这为理解更广泛的物理现象提供了基础。论文的作者包括Mykola Dedushenko、Silviu S. Pufu和Ran Yacoby，他们分别来自普林斯顿大学的Joseph Henry Laboratories、加州理工学院的Walter Burke Institute for Theoretical Physics和魏茨曼科学研究所的物理学与天文学系。该论文发表于JHEP03(2018)138，并在2018年3月23日发布，是开放获取的，可供广大研究者查阅。这篇研究深入研究了3d N = 4超保形场论中的特殊算子和它们的相互作用，通过超对称定位和拓扑代数的方法，揭示了希格斯分支算子的非平凡性质，对于理解和计算这类理论的复杂物理过程有着重要的意义。

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JHEP03(2018)138

The algebra (2.24) will be central in our work, so let us describe it in more detail. Let

us denote the generators by J

(`)

αβ

, R

, and Q

)

for su(2|1)

and J

(r)

αβ

, R

, and Q

)

for

su(2|1)

. Abstractly, the algebra obeyed by J

(`)

αβ

, R

, and Q

)

(`)

, J

(`)

] = i

ijk

(`)

, [J

(`)

αβ

, Q

)

] =



αγ

)

+ ε

βγ

)



, (2.25)

, Q

)

] = ±Q

)

, {Q

)

, Q

−

)

} = −



(`)

αβ



, (2.26)

where

(`)

αβ

≡

−(J

(`)

+ iJ

(`)

) J

(`)

− iJ

(`)

. (2.27)

The generators of su(2|1)

obey the same relations with ` → r.

To be more concrete, let us explain how these generators act on the various operators

in the theory. We will take this opportunity to set up some of the notation we will use later.

2.3.1 Action of S

isometries

The commutators with S

isometries act on a gauge-invariant operator O as the Lie

derivative

(`)

, O] = −L

O, [J

(r)

, O] = −L

(2.28)

with respect to the Killing vectors v

and v

that obey the su(2) algebra, [v

, v

] = iε

ijk

and similarly for v

In an explicit description where the three-sphere of radius r is embedded in R

via

+ X

= r

(2.29)

we can use the parameterization

+ iX

= r cos θe

iτ

, X

+ iX

= r sin θe

iϕ

(2.30)

in terms of the coordinates θ ∈ [0,

], τ, ϕ ∈ [−π, π]. In this parameterization, the Killing

vectors in (2.28) are

(−cos(τ + ϕ)∂

− tan(θ) sin(τ + ϕ)∂

+ cot(θ) sin(τ + ϕ)∂

) ,

(sin(τ + ϕ)∂

− tan(θ) cos(τ + ϕ)∂

+ cot(θ) cos(τ + ϕ)∂

) ,

(∂

+ ∂

) ,

(cos(τ −ϕ)∂

+ tan(θ) sin(τ −ϕ)∂

+ cot(θ) sin(τ −ϕ)∂

) ,

(−sin(τ − ϕ)∂

+ tan(θ) cos(τ −ϕ)∂

+ cot(θ) cos(τ −ϕ)∂

) ,

(∂

− ∂

) .

(2.31)

– 12 –

JHEP03(2018)138

We will make signiﬁcant use of the parameterization (2.30) in the remainder of this paper.

The metric in these coordinates is

) = r

(dθ

+ cos

(θ)dτ

+ sin

(θ)dϕ

). (2.32)

Coordinates θ and ϕ parametrize a disk with the metric ds

) = r

(dθ

+ sin

(θ)dϕ

where θ is the radial coordinate of the disk. The sphere metric then becomes:

) = ds

) + w

dτ

, w = r cos θ, (2.33)

which manifests S

as a U(1)-ﬁbration over D

, with the ﬁbers being “warped” by w and

shrinking to zero size at the boundary of the disk.

2.3.2 Action of R-symmetries

The action of R

and R

on the ﬁelds of the previous section depends on the precise

embedding of u(1)

and u(1)

into su(2)

⊕su(2)

given in terms of the matrices h and

in (2.12) as follows. Let us ﬁrst deﬁne the operators

, R

˙a

(2.34)

where R

and R

˙a

are the generators of su(2)

and su(2)

respectively. In our conventions,

we then have

= R

+ R

, R

= R

− R

(2.35)

This equation provides an identiﬁcation of R

and R

with linear combination of the Cartan

elements R

and R

of the R-symmetry of the superconformal algebra. In terms of their

action on ﬁelds, it is suﬃcient to describe how they act on su(2)

and su(2)

fundamental

operators. We have

, O

] =

, [R

, O

˙a

] =

˙a

(2.36)

with a straightforward generalization to operators with multiple su(2)

⊕ su(2)

indices.

For instance, [R

, O

] =

−

. The action of R

and R

operators can then be inferred from simply combining (2.35) and (2.36).

2.3.3 Action of supersymmetries

The action of the odd generators of su(2|1)

⊕ su(2|1)

on operators in general multiplets

can be quite complicated. As mentioned above, on the vector and hypermultiplet operators

their action is just a particular subset of the transformation rules (2.4)–(2.9). The precise

correspondence between the various ξ

a ˙a

obeying (2.13) and the supercharges Q

)

and

)

is given in (C.9) and (C.21).

Note that in our conventions h

= h

, and similarly for

– 13 –

JHEP03(2018)138

2.4 Central extension of non-conformal supersymmetry algebra

The discussion in section 2.3 was restricted to the case of vanishing mass and FI parameters.

Introducing these parameters amounts to central extensions of the algebra (2.24), as we

will now describe.

It is not hard to see, using Jacobi identity, that one cannot introduce central charges

in (anti-)commutators between left and right algebras, so one can only separately centrally

extend su(2|1)

and su (2|1)

. Each of these algebras admits only one non-trivial central

extension, so in total we have two central charges. We denote the centrally extended

algebras with a tilde, so the supersymmetry algebra of our theories is

su(2|1)

⊕

su(2|1)

Denoting central charges of the left and right subalgebras by Z

and Z

respectively, the

only place where they appear are the following anti-commutators:

)

, Q

−

)

} = −



(`)

αβ

+ ε

αβ



)

, Q

−

)

} = −



(r)

αβ

+ ε

αβ



(2.37)

Physically, the central charges Z

and Z

correspond to turning on real masses and FI

parameters. We turn on masses by coupling to background vectormultiplets in the Cartan

of the ﬂavor symmetry, as explained in section 2. The only components of these background

multiplets which are non-zero are

˙a

(Φ

b.g.

)

˙a

= −(Φ

b.g.

)

and h

b.g.

)

, as explained

in (2.16).

The supersymmetry algebra has a gauge transformation on the right, as written in

eq. (2.18), with the gauge parameter Λ of (2.19). In gauge theories, dynamical gauge ﬁelds

force us to consider only operators which are not charged under the corresponding gauge

symmetry. For such operators, the gauge transformation in the SUSY algebra vanishes.

For background gauge ﬁelds, this is not so. We can have operators which are charged

under the corresponding global symmetry (which would be gauged if the gauge ﬁeld were

dynamical), and for them, such gauge transformations in the algebra will generate central

charges. A simple computation, using the expression (2.19) for Λ, shows that:

+ Z

) = i(Φ

b.g.

)

= i bm ,

− Z

) = 0 . (2.38)

Here bm = diag(m

), where m

are real masses for hypers q

, I being the ﬂavor index.

Analogously, FI parameters correspond to background twisted vectormultiplets in the

Cartan of the gauge group. They similarly generate central charges with:

+ Z

) = 0 ,

− Z

) = i

ζ. (2.39)

Here,

ζ = ζ

acts non-trivially only on operators charged under the topological symmetry,

where ζ

is the FI parameter and t

is the corresponding topological charge. Examples of

such operators are monopole operators.

– 14 –

JHEP03(2018)138

3 Cohomology in SCFTs

Our aim in this section and the next is to describe a procedure that generalizes the coho-

mological truncation of [4, 7, 8] from N = 4 SCFTs to the more general non-conformal

N = 4 theories on S

that were described in section 2. The construction of [4, 7, 8] was

based on identifying two supercharges Q

and Q

in N = 4 SCFTs on R

, such that

the OPE restricted to their cohomology gives a certain quantization of the Higgs branch.

It was also possible to ﬁnd another pair of supercharges, Q

and Q

, whose cohomology

similarly leads to a quantization of the Coulomb branch, though this second possibility was

not explored in detail. We will generalize both cases to non-conformal theories on S

, but,

just as in [4, 7, 8], our main focus will also be the Higgs branch.

We will ﬁnd that local operators in the cohomology, both for Q

and for Q

, can

only be inserted along a great circle S

⊂ S

The circle is the ﬁxed point locus of the

U(1) isometry that appears in the anti-commutator {Q

, Q

} or {Q

, Q

}. In the case of

, the operators that can be inserted on S

will be referred to as “twisted Higgs branch

operators”, because, as we will see, they are in 1-to-1 correpsondence with Higgs branch

chiral ring operators. Similarly, operators in Q

cohomology will be referred to as “twisted

Coulomb branch operators”.

In section 3.1, we start by reviewing the construction of [4, 7, 8] in ﬂat space, and

then in section 3.2 we translate this construction to S

. In section 4 we describe the

generalization of this cohomology directly based on the su(2|1)

⊕ su(2|1)

algebra.

3.1 SCFT in ﬂat space

Consider theories living on a three-dimensional Euclidean space R

with the standard

coordinates ~x = (x

, x

). The bosonic subalgebra of the osp(4|4) superconformal alge-

bra is so(4) ⊕ sp(4), where the generators of sp(4) are rotations M

αβ

, translations P

and

special conformal transformations K

, and the generators of the so (4)

∼

su(2)

⊕su(2)

R-

symmetry are denoted by R

and

˙a

and act on the Higgs and Coulomb branches, respec-

tively. The fermionic generators are Q

αa ˙a

and S

αa ˙a

, denoting Poincar´e and conformal super-

charges, respectively. The detailed description of this algebra can be found in appendix C.1.

Deﬁne the two supercharges Q

and Q

= Q

, Q

= Q

. (3.1)

In (3.1), r is some arbitrary parameter with dimensions of length.

The supercharges Q

1,2

are nilpotent, i.e., (Q

)

= (Q

)

= 0, and their anticommutator is given by

Z =

, Q

} = −M

. (3.2)

Whether we consider the cohomology of Q

or Q

, the above equation implies that it

can be represented by elements from the Z = 0 subspace. In order to satisfy Z = 0, local

operators with zero

charge can only be inserted at the ﬁxed point locus of the M

rotation, i.e., at the line x

= x

= 0.

There is some freedom in choosing Q

H,C

, which corresponds precisely to the choice of great circle on S

When we map (3.1) to S

, we will interpret r as the radius of the sphere.

– 15 –

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