所有4D的Chern-Simons不变量，N=1个规范的超结构层次资源-CSDN文库

Open

Access

54 浏览量 2020-03-24 07:38:55 上传评论收藏 527KB PDF 举报

资源推荐

资源详情

资源评论

JHEP04(2017)103

Published for SISSA by Springer

Received: February 21, 2017

Accepted: April 8, 2017

Published: April 19, 2017

All Chern-Simons invariants of 4D, N = 1 gauged

superform hierarchies

Katrin Becker,

Melanie Becker,

William D. Linch III,

Stephen Randall

and

Daniel Robbins

George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy,

Texas A&M University,

College Station, TX 77843-4242, U.S.A.

Department of Physics, University of California,

Berkeley, CA 94720-7300, U.S.A.

Department of Physics, University at Albany,

Albany, NY 12222, U.S.A.

E-mail: kbecker@physics.tamu.edu, mbecker@physics.tamu.edu,

wdlinch3@gmail.com, srandall@berkeley.edu, dgrobbins@albany.edu

Abstract: We give a geometric description of supersymmetric gravity/(non-)abelian p-

form hierarchies in superspaces with 4D, N = 1 super-Poincar´e invariance. These hierar-

chies give rise to Chern-Simons-like invariants, such as those of the 5D, N = 1 gravipho-

ton and the eleven-dimensional 3-form but also generalizations such as Green-Schwarz-

like/BF -type couplings. Previous constructions based on prepotential superﬁelds are rein-

terpreted in terms of p-forms in superspace thereby elucidating the underlying geometry.

This vastly simpliﬁes the calculations of superspace ﬁeld-strengths, Bianchi identities, and

Chern-Simons invariants. Using this, we prove the validity of a recursive formula for the

conditions deﬁning these actions for any such tensor hierarchy. Solving it at quadratic

and cubic orders, we recover the known results for the BF -type and cubic Chern-Simons

actions. As an application, we compute the quartic invariant ∼ AdAdAdA + . . . relevant,

for example, to seven-dimensional supergravity compactiﬁcations.

Keywords: Superspaces, Chern-Simons Theories, Diﬀerential and Algebraic Geometry,

Supergravity Models

ArXiv ePrint: 1702.00799

Open Access,

 The Authors.

Article funded by SCOAP

doi:10.1007/JHEP04(2017)103

JHEP04(2017)103

are generalizations of the 1-form ﬁeld of Mawell theory: abelian gauge transformations

of such ﬁelds are exterior derivatives of (p − 1)-form gauge parameters and the exterior

derivative of such a potential is an invariant (p + 1)-form ﬁeld-strength. The presence

of such abelian p-forms is typical in extended supergravity and higher-dimensional super-

gravity theories: these theories have additional spinor degrees of freedom arising from a

larger gravitino, and the additional spin-0 and spin-1 degrees of freedom of the “gravi-p-

forms” are needed to balance this without introducing more spin-2. Well-known examples

include the graviphoton (+ scalar) of 4D, N = 2 supergravity and the 3-form of eleven-

dimensional supergravity.

Gravitational tensor hierarchies arise naturally, then, in extended supergravity and

higher-dimensional supergravity theories on product manifolds. More generally, the back-

ground spacetime can have the structure of a non-trivial bundle over Y , in which case the

mixed components of the graviton become the Kaluza-Klein gauge ﬁeld for the algebra of

diﬀeomorphisms on Y . The reduced components of the gravi-p-forms are charged under

this non-abelian gauge algebra: in addition to their usual abelian p-form transformation,

they transform as matter ﬁelds under Y diﬀeomorphisms.

The structure of the gravitational tensor hierarchy can be generalized by replacing the

collection of dimensionally reduced forms with a more general set not necessarily resulting

from any dimensional reduction. Maps between these new forms must be deﬁned to replace

the de Rham diﬀerential. Provided this is done in a manner compatible with the de Rham

complex of forms on X, there results an abelian tensor hierarchy of forms on X with values

in this new complex.

Abstracting further, the algebra of diﬀeomorphisms on Y can then be replaced with

a general non-abelian gauge algebra provided a representation is assigned to each degree

in the new hierarchy. The action of this algebra should satisfy certain “equivariance”

conditions with respect to the de Rham diﬀerential on X and the maps of the p-form

hierarchy. These conditions can be interpreted as gauging the abelian hierarchy with

respect to the new non-abelian gauge algebra; such hierarchies are referred to as “non-

abelian tensor hierarchies”.

It is important to emphasize that the p-forms of the hierarchy

transform linearly under the non-abelian group. Despite the terminology, the non-abelian

aspect of the gauge structure is only that of the gauge ﬁeld with the “tensors” transforming

as matter ﬁelds.

Motivated by applications to supergravity compactiﬁcations, the deﬁning conditions

of such hierarchies were reformulated in [13] and interpreted as coming either from closure

of the algebra of abelian and non-abelian gauge transformations or from the requirement

that there exist enough gauge-covariant ﬁeld-strengths. Assuming both of these conditions,

the resulting structure is that of a double chain complex that extends the superspace de

Rham complex and is equivariant under the action of the non-abelian gauge group. Being

a diﬀerential chain complex in superspace, such theories naturally deﬁne supersymmetric

characteristic classes, provided the appropriate traces are supplied. In particular, it is

The conditions deﬁning such a general non-abelian tensor hierarchy were formulated in [8] in an attempt

to construct six-dimensional superconformally invariant gauge theories (see also [9, 10]). The mathematical

structure of these models was investigated further in [11, 12].

– 2 –

JHEP04(2017)103

possible to deﬁne the analogs of Maxwell/Yang-Mills invariants, higher Chern/Pontryagin

classes, and Chern-Simons/BF -type invariants.

To better understand the four-dimensional phenomenology of such theories [4, 24, 56],

we ﬁrst embedded the abelian [14] and later the non-abelian [13] tensor hierarchies into

ﬂat, 4D, N = 1 superspace and explicitly constructed their manifestly supersymmetric

invariants. The ﬁeld-strengths of such hierarchies satisfy their Bianchi identities identically

and can be used to construct the usual Maxwell-type actions. The superspace diﬀerential

operators involved in the Bianchi identities turn out to be the adjoints of those appearing

in the gauge transformations of the super-p-form potentials. Because of this, the potentials

and ﬁeld-strengths can be used to construct a quadratic BF -type invariant. By an abuse

of language, we will refer to this as a quadratic Chern-Simons-type invariant for reasons

that will hopefully become clear if they are not already.

In order to construct cubic and higher-order Chern-Simons-like actions (including the

dimensional reductions of actual abelian Chern-Simons invariants), what is needed is a

set of composite superﬁelds constructed from the ﬁeld-strengths that satisfy the same con-

straints (i.e. Bianchi identities) as the ﬁeld-strengths themselves. Such a construction of the

Chern-Simons terms suﬃces since the inhomogeneous part of the p-form transformations

is abelian. Finding this set of composite superﬁelds and checking the constraints requires

considerable eﬀort in the prepotential formulation since the constraints are not linear in

superspace derivatives. Indeed, it is not clear a priori that such a set of composites exists

even in the abelian version of the hierarchy. In the non-abelian case the required interplay

between hierarchy identities, gauge-covariant superspace D-algebra identities, and Bianchi

identities seems miraculous.

This work originated in the desire to understand this “miracle” and to obviate the

cumbersome calculus of the prepotential formalism by reinterpreting it in superspace

diﬀerential-geometric terms. In such a formulation, the ﬁeld-strengths are speciﬁc Lorentz-

irreducible parts of super-(p+1)-forms [15]. The complicated Bianchi identities they satisfy

are relations “descendant” from the condition that the superforms be closed (or exact by

the Poincar´e lemma). Since the superspace de Rham operator is a graded derivation, the

wedge product of closed forms is closed and the miraculous cancellations of the prepo-

tential formalism would just be descendants of this trivial fact. Finally, it was imagined

that the Chern-Simons-like actions would simply be the integral of the (pullbacks of the)

higher-dimensional super-Chern-Simons form.

As it turns out, this interpretation is overly-simplistic for two reasons. The ﬁrst is

that the closed superforms that deﬁne irreducible representations of the super-Poincar´e

algebra do not form a ring under multiplication: to get an irreducible supermultiplet from

a superform, many parts must be set to zero as conventional constraints (similarly to what

is done to the torsion in superspace supergravity theories). On the other hand, when two

lower-degree forms are wedged together they will generally give contributions violating

these conditions and ruin the picture sketched above. This problem can be circumvented

They are given explicitly in terms of oﬀ-shell “prepotential” superﬁelds. Such a (ﬁnite) set of prepo-

tentials exists only because we have chosen to work in a superspace admitting no more than four real su-

percharges.

– 3 –

JHEP04(2017)103

by deﬁning an improved form in the same cohomology class that satisﬁes the original

conditions on the irreducible superform.

The second complication is that the Chern-Simons action is deﬁned by a form that

is not closed whereas the “ectoplasm” method used to construct supersymmetric actions

speciﬁcally requires the use of closed forms [20, 21]. Fortunately, it is known how to handle

this situation [16, 17]: in addition to the Chern-Simons form C constructed by wedging

superform potentials and ﬁeld-strengths, one constructs a second, inequivalent form K that

is both manifestly gauge invariant and satisﬁes dK = dC. (That such a form exists is a

phenomenon called “Weil triviality” [19], cf. section 4.1.) This gives a closed superform

J = C − K which, in turn, deﬁnes the superspace completion of the component Chern-

Simons action by the ectoplasm procedure (cf. section 4).

Despite these complications to the na¨ıve geometrization of the prepotential hierarchy,

we will ﬁnd that it is possible to give a recursive formula for the composite superﬁelds

deﬁning the Chern-Simons invariants of any non-abelian tensor hierachy of the type deﬁned

in reference [13]. Furthermore, it is possible to solve these recursion relations to obtain all

of the superspace invariants explicitly. We demonstrate this in detail by reproducing the

cubic invariant found in [13] and deriving a new quartic invariant. The former was recently

used (in conjunction with a superspace Hitchin functional) to derive the scalar potential

of eleven-dimensional supergravity in backgrounds admitting a (not necessarily closed) G

structure [23]. The quartic invariant would be a main ingredient in a similar analysis for

seven-dimensional supergravity backgrounds [22].

Outline. We begin in section 2 with a review of the non-abelian tensor hierarchy in

the prepotential formulation and use this to describe the problem of constructing Chern-

Simons-like gauge-invariant superspace actions. This pre-geometrical description is refor-

mulated in terms of super-p-forms in section 3 where the composite superﬁelds appearing

in the construction of the (secondary) characteristic classes are interpreted as products

of closed superforms. In section 4 we relate these composite superforms to supersymmet-

ric invariants by way of “ectoplasm”. This method takes as input a closed superform of

spacetime degree four and returns a chiral superspace integral. As mentioned above, the

product of closed irreducible superforms is not a closed irreducible superform. In sec-

tion 4.1, we construct a gauge-invariant composite superform with which we modify the

original composite superform to obtain a closed, irreducible, composite superform. Ap-

plying the ectoplasm method, this corrected composite form gives the Chern-Simons-like

superspace action. Since the construction is in terms of superforms, the result is manifestly

supersymmetric and gauge-invariant by the same logic as that for bosonic Chern-Simons

invariants. In section 5, we use this technology to derive and solve a recursion formula

for all Chern-Simons-like actions for any non-abelian tensor hierarchy. We summarize our

conclusions in section 6.

By “irreducible” we will always mean as a representation of the super-Poincar´e algebra. In particular,

a form can be both composite (i.e. constructed by wedging non-composite forms) and irreducible (i.e. it

satisﬁes the same constraints as the non-composite form of the same degree).

– 4 –

剩余30页未读，继续阅读

评论收藏

内容反馈

weixin_38686187

粉丝: 8
资源: 965

所有4D的Chern-Simons不变量，N = 1个规范的超结构层次

最新资源

所有4D的Chern-Simons不变量，N = 1个规范的超结构层次

Chern-Simons Theory, Knots and Moduli Spaces of Connections

N = 4 $$ \ mathcal {N} = 4 $$ Chern-Simons-matter理论中的量子1/2 BPS Wilson循环

希格斯相中的Bose-Fermi Chern-Simons对偶

Chern-Simons超引力的量子理论

N = 2边界Chern-Simons理论的量规和超对称不变性

4D，N = 1超空间中的Chern-Simons动作及其测量

超对称Chern-Simons-matter模型中重整化组的改进和对称性的动态破坏

N = 4 $$ \ mathcal {N} = 4 $$ Chern-Simons-matter理论中大量的1/2 BPS Wilson循环

三流形中的扁平连接和经典的Chern-Simons不变式

与3D重力耦合的粒子的有效Chern-Simons作用

在弱耦合下探测N = 4 Chern-Simons物论中的Wilson环

N = 2 Chern-Simons-matter模型的超势的重归一化组改进

具有SO和USp量规组的Chern-Simons物质对偶

异质超电势的有限变形：全纯Chern-Simons和L∞代数

4D中的Abelian张量层次结构，N = 1个超空间

精确的Chern-Simons /拓扑字符串对偶

Chern-Simons理论中的拓扑界面和AdS3 / CFT2对应关系

Poisson-Lie T-对偶作为Chern-Simons理论的边界现象

最小N = 4 $$ \ mathcal {N} = 4 $$ Chern-Simons问题理论的指数和对偶

Chern-Simons物质理论的矩阵模型超出了球形极限

N = 2超对称Chern-Simons理论的超对称解和Borel奇点

从11 + 3维流形上的Chern-Simons修正重力的起源

SIM（1）超空间中的Chern–Simons理论

最小N = 4 $$ \ mathcal {N} = 4 $$ Chern-Simons-matter理论的完全分解

Abelian Chern-Simons理论中的路径积分不变量

大N重整化组以3d N $$ \ mathcal {N} $$ = 1 Chern-Simons-Matter理论流动

Chern-Simons-fermion理论中的无序算子

最新资源