B型高自旋引力：自由费米子与Gross-Neveu模型的双重理论

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"B型正式更高的自旋重力" 这篇研究文章深入探讨了高自旋引力的Type-B形式，这是一种在理论物理学中具有重要意义的领域，特别是在量子场论和弦理论中。作者Maxim Grigorieva和Evgeny Skvortsov通过详细的工作，提出了非线性的方程来描述这一理论，这些方程与自由费米子或Gross-Neveu模型具有等价性，但这种等价性取决于特定的边界条件。首先，高自旋引力是一种理论，它扩展了传统的引力理论，其中包含自旋更高的粒子，如自旋3或自旋4的粒子，而不仅仅是自旋2的引力子。Type-B形式则是在这种理论的一个特定框架下进行研究的，它提供了一种理解和处理高自旋场相互作用的新方法。在文章中，作者强调了这些非线性方程的来源：它们直接源于基本原理，即单轨迹运算符的共形场论（CFT）分区函数在任意背景下的规范不变性。这是一个重要的概念，因为CFT在理解AdS/CFT对应（反德西特空间/共形场论对应）中扮演关键角色，这是现代理论物理中的一个强大工具，用于连接量子场论和引力理论。此外，他们还得到了描述在自旋平坦背景上某些混合对称场传播的方程。混合对称场是指那些具有复杂对称性的场，它们不满足简单的Lorentz对称性，而是满足更复杂的对称群。在高自旋引力中，这样的场可能具有特殊的物理意义和应用。关键词包括AdS-CFT对应、高维场论、高自旋对称性和BRST量子化。AdS-CFT对应是将反德西特空间的引力问题与共形场论的量子问题联系起来的理论，是探索量子引力的重要途径。高维场论则涉及在多于四维的空间时间中研究物理现象。高自旋对称性是本文的核心概念，而BRST量子化是一种在量子场论中处理规范对称性的技术，对于理解和构造理论的规范不变量子版本至关重要。这篇研究工作深化了我们对高自旋引力的理解，提供了新的非线性方程，这些方程可以用来研究自由费米子和Gross-Neveu模型，并且在自旋平坦背景下描述混合对称场的传播。这对于进一步探索量子引力理论、强相互作用以及可能存在的高自旋粒子的性质具有重要意义。

JHEP05(2018)138

Let us consider a free Dirac fermion ψ and view the equations of motion

ψ(iγ · ∂

)ψ =⇒ iγ

∂

ψ = 0 (2.4)

as a physical state condition in the underlying quantum constrained system. The con-

straints are ﬁrst class and read

− ∂

· ∂

, iγ · ∂

, [iγ · ∂

, iγ · ∂

] = −2∂

· ∂

, (2.5)

where the brackets denote super-commutator, with the Grassmann degree being the stan-

dard Z

-grading in the Cliﬀord algebra of γ-matrices, so that in the above relations it is the

anti-commutator. The constraints are operators acting in the space of “wave functions”

with values in the Dirac spinor representation. Here and below we exploit the analogy

between free ﬁelds and ﬁrst-quantized particle models.

Now suppose we would like to add at least inﬁnitesimal sources for all operators (2.1),

i.e. couplings of the type

s,q

...a

...c

...a

...c

+ ∂

...a

...c

+ perm − traces) , (2.6)

where we also indicated the desired gauge invariance. Here, the sources h

s,q

have the same

algebraic properties as J

s,q

, i.e. they are traceless and have the Young symmetry depicted

in (2.1). Since most of J

s,q

are conserved tensors (save for (2.2)), the sources h

s,q

enjoy a

gauge symmetry.

At this stage the sources in (2.6) are inﬁnitesimal. If they are not, the currents J

s,q

are not conserved: the invariance of the eﬀective action W [h] can be restored upon adding

appropriate counterterms

ψ(h . . . h)ψ to S

and correcting the gauge symmetry δh by non-

linear terms. This also results in nonlinear deformation of the conservation condition for

the currents. As we explained in the Introduction, an elegant way to introduce sources

so that the total action S

+ S

remains gauge invariant even for non-inﬁnitesimal sources

was proposed in [26] in the case of scalar ﬁeld (see also [29] for a recent discussion of

curved backgrounds and nonlinear gauge symmetries). Now we are going to apply the

same strategy to the Dirac fermion.

The crucial step is to realize that all derivatives of

ψ that are hidden in J

s,q

can be

integrated by parts and the coupling can be written in the following suggestive form

= i

ψ f(γ, ∂

|x)ψ , (2.7)

where f is a function of γ and ∂, whose Taylor coeﬃcient can depend on x:

f(γ, ∂

|x) =

s,q

...a

...c

(x)γ

...c

∂

. . . .∂

. (2.8)

Here f

s,q

are not irreducible tensors anymore. These coeﬃcients are related to the source

s,q

, but it makes sense not to work at this stage with the operators that are quasi-

primary and to extend the base of local bilinear operators to the one covered by generating

– 6 –

JHEP05(2018)138

functions (2.3). It is clear that S

encodes the usual lower spin couplings like Yukawa,

electro-magnetic and gravitational ones as well as couplings to higher spin sources. In

what follows we refer to f as to background ﬁelds to stress that they are not required to

be inﬁnitesimal.

The full action of the conformal Dirac ﬁeld coupled to all sources can be written as

S = S

+ S

ψ(iγ · ∂

+ if)ψ , (2.9)

where it is useful to absorb the kinetic term into F = iγ · ∂

+ if.

To describe sources in terms of the constrained system (2.5), we should also allow for

a deformation h = h(γ, ∂

|x) of the −∂

constraint, which can altogether be packed into

H = −∂

+ h. In these terms the equations of motion for the Dirac ﬁeld are identiﬁed as

physical state conditions:

F ψ = (iγ · ∂

+ if)ψ = 0 , Hψ = (−∂

+ h)ψ = 0 . (2.10)

If we insist that the constraint algebra is unchanged we ﬁnd

[F, F ] = 2H ⇐⇒ 2[γ · ∂

, f] + [f, f] = −2h , (2.11)

so that, as expected, h is determined by f and H does not introduce any new indepen-

dent sources.

There is a natural gauge symmetry which acts on both ψ and F and leaves the La-

grangian (2.9) unchanged. Indeed, consider

δF = [, F ] + {α, F } + {β, H}, δψ = ψ − αψ − F βψ , (2.12)

where the three gauge parameters , α and β are functions of the same type as F , i.e.

they depend on γ, ∂

and x. The reality of the action implies 

†

= −, α

†

= α and

†

= β. The -symmetry is a usual gauge transformation, while α and β are responsible

for reparameterization of the constraints, which we discuss in the next section.

To study gauge symmetry for F , H it is convenient to work in terms of symbols

rather than operators. To this end we introduce variables θ

, p

associated with γ

and

∂

, respectively. The algebra of operators is then the tensor product of Weyl algebra and

Cliﬀord algebra. Seen as a ?-product algebra of symbols it can be identiﬁed as an algebra

freely generated by x

, p

, θ

modulo the following relations:

, p

]

= δ

, [θ

, θ

]

= 2η

, (2.13)

where [A, B]

= A ? B − (−1)

|A||B|

B ? A denotes ?-product super-commutator, with the

Grassmann degree assigned according to |x

| = |p

| = 0 and |θ

| = 1. The linearized gauge

symmetries of f read as

δf = −(p · ∂

− θ · ∂

) + 2α(p · θ) + 2β(p · p) + . . . , (2.14)

where we dropped the terms with derivatives of α and β.

Note that the usual minimal coupling of fermion to gravity, |e|

ψγ

(∂

+ω

a,b

[γ

, γ

])ψ, is certainly a

part of (2.7) upon appropriate identiﬁcation of ﬁelds, see also the comment below about general covariance.

The Taylor coeﬃcients of f should be understood as f

...a

...m

, i.e. as tensors of the ﬁber Lorentz group

O(d − 1, 1) and transforming as coeﬃcients of diﬀerential operators in the m’s. In the present paper we do

not dwell on the global geometry issues.

– 7 –

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