没有合适的资源?快使用搜索试试~ 我知道了~
最近通过以下公式计算了在流形ℳg,p上的三维N = 2 $$ \ mathcal {N} = 2 $$理论的分配函数,该流形是在闭合黎曼曲面Σg上度为p的S 1束。 超对称定位。 在本文中,我们使用全息M-理论对偶,在一类振动规理论中,在大N处计算这些分区函数。 我们提供了具有这样的三维圆束作为共形边界的超重力块对偶。 这些配置是N = 2 $$ \ mathcal {N} = 2 $$最小规范超重力的解决方案,并且属于Taub-NUT-AdS和Taub-Bolt-AdS类型,保留了1/4的超对称性。 我们讨论了将这些解决方案推广到M理论的条件,并通过全息重归一化计算了壳上作用。 我们表明,通过双重超保形场理论的分配函数的大N极限,可以正确地再现Bolt解的上升条件和壳上作用。 特别是,最近我们发现,Σg×S 1 =ℳg,0分区函数与AdS4黑洞的熵匹配,而S 3≅,0.1的自由能匹配,这是我们形式主义的特例,并且 我们评论他们之间的关系。
资源推荐
资源详情
资源评论
JHEP05(2018)116
Published for SISSA by Springer
Received: March 28, 2018
Accepted: May 5, 2018
Published: May 17, 2018
Partition functions on 3d circle bundles and their
gravity duals
Chiara Toldo
a,b
and Brian Willett
b
a
Columbia University in the City of New York, Pupin Hall,
538 West 120th Street, New York City, NY 10027, U.S.A.
b
Kavli Institute for Theoretical Physics, University of California,
Santa Barbara, CA 93106, U.S.A.
E-mail: ctoldo@kitp.ucsb.edu, bwillett@kitp.ucsb.edu
Abstract: The partition function of a three-dimensional N = 2 theory on the manifold
M
g,p
, an S
1
bundle of degree p over a closed Riemann surface Σ
g
, was recently computed
via supersymmetric localization. In this paper, we compute these partition functions at
large N in a class of quiver gauge theories with holographic M-theory duals. We provide the
supergravity bulk dual having as conformal boundary such three-dimensional circle bundles.
These configurations are solutions to N = 2 minimal gauged supergravity and pertain to
the class of Taub-NUT-AdS and Taub-Bolt-AdS preserving 1/4 of the supersymmetries.
We discuss the conditions for the uplift of these solutions to M-theory, and compute the
on-shell action via holographic renormalization. We show that the uplift condition and
on-shell action for the Bolt solutions are correctly reproduced by the large N limit of the
partition function of the dual superconformal field theory. In particular, the Σ
g
×S
1
∼
=
M
g,0
partition function, which was recently shown to match the entropy of AdS
4
black holes,
and the S
3
∼
=
M
0,1
free energy, occur as special cases of our formalism, and we comment
on relations between them.
Keywords: AdS-CFT Correspondence, Supergravity Models, Supersymmetric Gauge
Theory
ArXiv ePrint: 1712.08861
Open Access,
c
The Authors.
Article funded by SCOAP
3
.
https://doi.org/10.1007/JHEP05(2018)116
JHEP05(2018)116
Contents
1 Introduction 1
2 M
g,p
partition function at large N 4
2.1 Supersymmetric background on M
g,p
4
2.2 Computation of M
g,p
partition function 6
2.3 Large N computation 10
3 The supergravity dual 17
3.1 Minimal N = 2 gauged supergravity 17
3.2 NUTs and Bolts 19
3.3 Supersymmetry properties 21
3.4 Moduli space of solutions 23
3.5 Uplift to 11d 25
3.6 On-shell action via holographic renormalization 27
4 Holographic comparison for ABJM 29
4.1 Truncating to minimal supergravity 29
4.2 Holographic comparison 31
4.3 NUTs and Bolts for S
3
31
5 General quivers 33
5.1 The M
g,p
partition function for general quivers 33
5.2 Fractional R-charges and the R-symmetry background 36
5.3 Example: V
5,2
theory 41
5.4 Minimal supergravity for general quivers and the universal twist 43
6 Comparison with S
3
partition function 45
7 Discussion 48
A Explicit Killing spinors 49
B Moduli space of gravity solutions 52
C Details of partition function calculations 53
C.1 Contributions to twisted superpotential 55
C.2 Contributions to log Z
M
g,p
59
– i –
JHEP05(2018)116
1 Introduction
Recently, there has been much progress in performing exact, nonperturbative computa-
tions for superconformal field theories (SCFTs) on curved manifolds via the technique of
supersymmetric localization (see the review [1] and references therein). Such methods
have greatly developed in the past several years, providing a tool to study a wide variety of
SCFTs in various dimensions and backgrounds, leading to non-trivial tests of holography
and other known dualities.
In particular, recently these techniques have been successfully applied to the compu-
tation of the partition function of three-dimensional superconformal Chern-Simons-matter
theories on Σ
g
×S
1
in presence of background magnetic flux for the R- and flavor symmetries
through the Riemann surface, Σ
g
[2–5]. By performing a partial topological twist [6] on Σ
g
,
one obtains the so-called “topologically twisted Witten index” [3]. This was shown in [7] to
reproduce in the large N limit
1
the macroscopic entropy of supersymmetric magnetic AdS
4
black holes in theories of 4d FI-gauged supergravity. These black hole configurations, first
found in [8], consist of M2-branes wrapped around Σ
g
, and thus they implement the partial
topological twist for the QFT describing the low-energy dynamics of the M2-branes.
This recent success led to several extensions and developments. First of all, the en-
tropy matching was performed on more general supergravity backgrounds, including dyonic
black holes [9], black hole configurations arising from massive IIA supergravity trunca-
tions [10, 11] and solutions with hyperbolic horizon [12]. Moreover, unexpected relations
were discovered between the topologically twisted index on S
2
×S
1
and the corresponding
partition function on S
3
in the large N limit [13]. Specifically, the computation of the
twisted index involves, as an intermediate step, the computation of the twisted superpo-
tential, or Bethe potential, as a function of the flavor fugacities [3]. Then this quantity was
shown to coincide, for a suitable mapping of parameters, with the large N limit of the S
3
partition function of the same N = 2 theory [13]. Given that the partition function on S
3
of three-dimensional superconformal theories, such as the ABJM theory [14], has been ex-
tensively studied, and has connections to entanglement entropy and the F -theorem [15–17],
it is natural to ask if such a correspondence has a deeper meaning.
In parallel with these developments, a new class of partition functions for general 3d
N = 2 gauge theories was computed in [18] utilizing a three-dimensional uplift of the 2d
A-model [19]. These partition functions are defined on the manifold M
g,p
, a U(1) bundle
of Chern degree p ∈ Z over a Riemann surface Σ
g
,
S
1
p
→ M
g,p
→ Σ
g
, (1.1)
where g ∈ Z
≥0
denotes the genus of the Riemann surface. This set of manifolds includes
in particular the three-sphere S
3
and the product spaces Σ
g
× S
1
M
0,1
' S
3
, M
g,0
' Σ
g
× S
1
. (1.2)
1
Upon a suitable extremization of the index with respect to the fugacities, which corresponds to the
attractor mechanism in the gravity side.
– 1 –
JHEP05(2018)116
Thus, these partition functions include both the topologically twisted index of [2–5] and
the round S
3
partition function of [20–22] as special cases. This then provides a natural
framework to address the relation between the topologically twisted index and S
3
partition
function. At the same time, constructing explicit supergravity backgrounds whose bound-
ary is such a circle bundle and computing their renormalized on-shell action provides a
viable holographic check for these field theory computations.
In more detail, the partition function on M
g,p
can be computed by a sum over super-
symmetric “Bethe vacua,” [23],
Z
M
g,p
(m
i
, s
i
) =
X
I∈S
BE
F
I
(m
i
)
p
H
I
(m
i
)
g−1
Π
I
i
(m
i
)
s
i
, (1.3)
where the index I runs over the set S
BE
of vacua of the theory. Here m
i
and s
i
are,
respectively, real masses and fluxes for background flavor symmetry gauge fields, and F
I
,
H
I
, and Π
I
i
are certain functions appearing in the 3d uplift of the A-model, described in
section 2 below. We will argue that, for a class of quiver gauge theories with holographically
dual M-theory descriptions, in the large N limit this sum can be approximated by a single
dominant term, I
dom
, and we find the result:
log Z
M
g,p
(m
i
) ≈ p log F
I
dom
(m
i
) + (g − 1) log H
I
dom
(m
i
) + s
i
log Π
I
dom
i
(m
i
) , (1.4)
leading to a very simple dependence on the geometric and flux parameters. We find the par-
tition function exhibits the expected N
3/2
scaling, and reproduce and generalize the results
of [4, 7] in the case p = 0. However, we find that a large N solution exists only under certain
conditions on the mass and flux parameters. In the case of M
0,1
= S
3
, these conditions
differ from those under which previous large N computations of the S
3
partition function
were carried out, e.g., in [15], and we comment on this discrepancy in section 4 below.
We reproduce this result holographically, by providing supergravity backgrounds hav-
ing boundary M
g,p
in the framework of minimal N = 2 U(1) gauged supergravity. Such
solutions can be embedded locally in 11d on 7-dimensional Sasaki-Einstein manifolds. We
construct Euclidean regular solutions which preserve 1/4 of the supersymmetries and have
appropriately quantized magnetic flux. Starting from the analysis of [24–26], we find that
the boundary can be filled with multiple gravity configurations, with different topology. In
particular, for the boundary S
3
case we can have regular “NUT” solutions, with topology
R
4
, and for S
3
/Z
p
one finds mildly singular NUT/Z
p
solutions. On the other hand, for
general M
g,p
we find regular “Bolt” solutions, with topology O(−p) → Σ
g
. The different
topology has non-trivial consequences for the uplift of these solutions. Indeed, while there
are no requirements for the NUT solution to lift to M-theory, the Bolt uplifts to eleven
dimensions only for certain values of g and p, depending on the geometrical properties of
the internal Sasaki-Einstein 7-manifold. Interestingly, the same constraints are recovered
in the field theory computation by setting all fluxes equal, thus reproducing the universal
twist which corresponds to minimal gauged supergravity.
2
2
This holds provided that the reduction on Y
7
does not contain Betti vector multiplets in its spectrum.
There are additional subtleties in the quantization condition of the fluxes of the Betti vectors that we do
not treat here.
– 2 –
JHEP05(2018)116
The computation of the on-shell action of these two distinct bulk solutions is obtained
via standard techniques of holographic renormalization. The resulting on-shell action for
the NUT configuration coincides with the free energy of the corresponding theory on S
3
.
The renormalized on-shell action for Bolt solutions is instead of the form
I
Bolt±
=
√
2πN
3/2
12
s
Vol(S
7
)
Vol(Y
7
)
(4(1 − g) ∓ p) , (1.5)
with the additional constraint
± p + 2(g − 1) = 0 (mod I(Y
7
)) , (1.6)
where I(Y
7
) is the Fano index of the internal 7-manifold. In particular, for g = 0 we retrieve
the results of [25].
We are able to show that, for this solution in minimal gauged supergravity, the on-shell
action in the gravity side matches with the partition function of the corresponding field
theory (1.4)
− log Z
M
g,p
= I
Bolt
, (1.7)
as expected. In case of trivial fibration, p = 0, our formulas (1.4) and (1.5) find agreement
with those of [27]. In this particular case the on-shell action of the Euclidean solution
coincides with the entropy of supersymmetric 1/4 BPS black holes with constant scalars
and higher genus horizon.
3
Along with the matching with the Bolt solutions, we study the relation between the
S
3
partition function as computed by [15] and the result we obtain for the M
g,p
partition
function, in light of the result of [13]. In particular, we elaborate on how the interesting
relation between the extremal value of the twisted superpotential and the large N partition
function on S
3
, discovered in [13], fits in our framework by relating these both to the
partition function on the lens space S
3
/Z
2
.
The main text of the paper is organized as follows: in section 2 we provide the details
of the computation of the large N partition function of a class of N = 2 3d quiver gauge
theories on M
g,p
, focusing on the example of the ABJM theory. In section 3 we describe
Euclidean minimal gauged supergravity solution whose boundary is M
g,p
, and examine
their supersymmetry properties, along with their moduli space for regularity. Moreover,
we compute the on-shell action via holographic renormalization. In section 4 we show the
matching between the renormalized on-shell action of the Bolt solutions and the partition
function of the dual field theory for the ABJM theory. In section 5, we consider more
general quiver gauge theories, including the V
5,2
theory, and describe the truncation to
minimal supergravity for these theories, obtaining a generalization of the universal twist
of [27]. In section 6, we discuss the relation between the twisted superpotential and S
3
partition function observed by [13], and relate these to the lens space partition function.
Finally in section 7, we discuss some open issues and future directions. Several appendices
complete this paper, and they are devoted to the construction of the explicit Killing spinor
3
See also the recent analysis of [28, 29] for further computations regarding the equivalence between
renormalized on-shell action and BPS black hole entropy.
– 3 –
剩余67页未读,继续阅读
资源评论
weixin_38683895
- 粉丝: 6
- 资源: 899
上传资源 快速赚钱
- 我的内容管理 展开
- 我的资源 快来上传第一个资源
- 我的收益 登录查看自己的收益
- 我的积分 登录查看自己的积分
- 我的C币 登录后查看C币余额
- 我的收藏
- 我的下载
- 下载帮助
最新资源
- 这是 HIC-Yolov5 的存储库.zip
- 这只是另一个 YOLO V2 实现 在 jupyter 笔记本中训练您自己的数据集!.zip
- PicGo 是一个用于快速上传图片并获取图片 URL 链接的工具
- uniapp vue3 自定义下拉刷新组件pullRefresh,带释放刷新状态、更新时间、加载动画
- WINDOWS 2003邮箱服务器搭建
- 距离-IoU 损失更快、更好的边界框回归学习 (AAAI 2020).zip
- 该项目是运行在RK3588平台上的Yolo多线程推理demo,已适配读取视频文件和摄像头信号,demo采用Yolov8n模型进行文件推理,最高推理帧率可达100帧,秒 .zip
- 该项目使用 YOLOv8 通过用户友好的界面执行医学图像的分类、检测和分割等任务 .zip
- AI's prompts
- 该存储库将演示如何使用 OpenVINO 运行时 API 部署官方 YOLOv7 预训练模型.zip
资源上传下载、课程学习等过程中有任何疑问或建议,欢迎提出宝贵意见哦~我们会及时处理!
点击此处反馈
安全验证
文档复制为VIP权益,开通VIP直接复制
信息提交成功