N=4 SYM的费米与玻色子质量变形：AdS-CFT对偶与超重力解析

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本文探讨了N=4超级杨-米尔斯(N $$ \mathcal{N} $$ = 4 SYM)理论中的费米子和玻色子质量变形，以及它们在阿贝尔-卡斯卡拉德(Abelian-Karcher)对偶框架下的物理效应。作者研究了一种非超对称的费米子质量变形，这种变形在超弦理论的背景下，特别关注了其在阿达-塞弗特空间(AdS)几何中的对应物——D3膜(Coulomb分支)上的影响。通过对RR和NS-NS两种形式势的背向反应进行分析，研究者发现这些势能与费米子的质量紧密相关，它们在Coulomb分支的潜在能量中表现为D3膜的玻色子质量矩阵。利用表示论和超重力方法，作者揭示了费米子质量参数完全决定了这个矩阵的迹部分，即矩阵元素的总和。然而，矩阵的迹less部分（即无迹分量）则被揭示为非规范化模式，意味着它们对于理论的整体行为有重要的但不同于迹部分的作用。本文解决了一个先前的理论争议，即在超重力流的描述中，玻色子质量矩阵的迹被认为是维数为(g s N)^(1/4)的严格激发，而其非零值被认为对应于非严格的流。这一结果表明，在玻色子和费米子质量平方和不相等的情况下，渐近AdS流可能无法提供完整的理论描述，因为它只能处理质量相等的系统，这与弱耦合理论的预期一致，即仅当理论具有共形紫外稳定点时，才可能出现这样的对偶关系。这项工作深化了我们对N=4 SYM的量子场论和引力双关的理解，强调了费米子和玻色子质量的相互作用及其在超对称破缺过程中的关键角色。它不仅提供了对量子场论与引力理论之间关系的新见解，还促进了对超重力和弦理论中对称性破缺机制的进一步探索。

JHEP05(2016)149

brane. Therefore, the trace of the boson mass matrix that one reads oﬀ from the bulk will

always be equal to the trace of the square of the fermion mass matrix.

This, in turn, indicates that in the presence of fermion masses, the stringy operator is

not dual to the sum of the squares of the boson masses, but to the diﬀerence between it and

the sum of the squares of the fermion masses. Mass deformations of the N = 4 theory where

the supertrace of the square of the masses is zero can therefore be described holographically

by asymptotically-AdS supergravity solutions [7–16]. However, to describe theories where

this supertrace is nonzero, one has to turn on “stringy” non-normalizable modes that

correspond to dimension-(g

1/4

operators, which will destroy the AdS asymptotics.

To see this we begin by considering the backreaction of the three-form ﬁeld strengths

corresponding to fermion mass deformations on the metric, the dilaton and the four-form

potential, which has been done explicitly for several particular choices of masses [4, 17].

This backreaction can give several terms that modify the action of a probe D3 brane,

giving rise to a Coulomb-branch potential that is quadratic in the fermion masses and that

transforms either in the 1 or in the 20

of SO(6). Furthermore, one can independently

turn on non-normalizable modes in the 20

of SO(6) that correspond to deforming the

Lagrangian with traceless boson bilinears, and that can also give rise to a Coulomb-branch

potential. Since all these terms behave asymptotically as r

−2

and transform in the same

SO(6) representation, disentangling the contributions of the non-normalizable modes from

the terms coming from the backreaction of the three-forms can be quite nontrivial. For

example, in equation (62) in [1], the Coulomb-branch potential appears to contain both

contributions in the 1 and in the 20

of SO(6) coming from the backreaction of the fermion

mass tensor T

ijk

, and to have no non-normalizable contribution.

We will show that the backreaction of the modes dual to the fermion masses can only

source terms in the D3 brane Coulomb-branch potential that are singlets under SO(6), and

hence the Coulomb-branch potential terms that transform in the 20

of SO(6) can only

come from non-normalizable L = 2 (traceless) modes that one has to turn on separately

from the fermion masses. Since the singlet term in the Coulomb-branch potential is the

supergravity incarnation of the trace of the boson mass matrix, our result implies that in

the bulk this boson mass trace is completely determined by the fermion masses: the sum

of the squares of the boson masses will always be equal to the sum of the squares of the

fermion masses.

Our calculation establishes that asymptotically-AdS

solutions can only be dual to

theories in which the sum of the squares of the boson masses is the same as the sum of the

squares of the fermion masses. Theories where these quantities are not equal cannot by

described holographically by such solutions.

From a ﬁeld theory perspective this interpretation is very natural: the solutions that

are asymptotically AdS

can only be dual to ﬁeld theories that have a UV conformal ﬁxed

point, and therefore their masses and coupling constants should not run logarithmically in

the UV (their beta-functions should be zero). At one loop this cannot happen unless the

sum of the squares of the boson masses is equal to the sum of the squares of the fermion

masses [18], which reduces the degree of divergence in the corresponding Feynman diagram

– 3 –

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