Spin（32）/ℤ2杂散弦和小弦理论的非几何真空资源-CSDN文库

Open

Access

166 浏览量 2020-03-24 02:46:01 上传评论收藏 596KB PDF 举报

资源推荐

资源详情

资源评论

JHEP11(2017)064

Published for SISSA by Springer

Received: September 8, 2017

Accepted: November 2, 2017

Published: November 13, 2017

Non-geometric vacua of the Spin(32)/Z

heterotic

string and little string theories

Anamar´ıa Font

a,b

and Christoph Mayrhofer

Departamento de F´ısica, Centro de F´ısica Te´orica y Computacional,

Facultad de Ciencias, Universidad Central de Venezuela,

A.P. 20513, Caracas 1020-A, Venezuela

Max-Planck-Institut f¨ur Gravitationsphysik, Albert-Einstein-Institut,

Am M¨uhlenberg 1, D-14476 Golm, Germany

Sommerfeld Center for Theoretical Physics,

Theresienstraße 37, 80333 M¨unchen, Germany

E-mail: afont@fisica.ciens.ucv.ve, christoph.mayrhofer@lmu.de

Abstract: We study a class of 6d N = (1, 0) non-geometric vacua of the Spin(32)/Z

heterotic string which can be understood as ﬁbrations of genus-two curves over a complex

one-dimensional base. The 6d N = (1, 0) theories living on the defects that arise when

the genus-two ﬁber degenerates at a point of the base are analyzed by dualizing to F-

theory on elliptic K3-ﬁbered non-compact Calabi-Yau threefolds. We consider all possible

degenerations of genus-two curves and systematically attempt to resolve the singularities

of the dual threefolds. As in the analogous non-geometric vacua of the E

× E

heterotic

string, we ﬁnd that many of the resulting dual threefolds contain singularities which do not

admit a crepant resolution. When the singularities can be resolved crepantly, we determine

the emerging eﬀective theories which turn out to be little string theories at a generic point

on their tensor branch. We also observe a form of duality in which theories living on

distinct defects are the same.

Keywords: Conformal Field Models in String Theory, F-Theory, String Duality, Super-

strings and Heterotic Strings

ArXiv ePrint: 1708.05428

Open Access,

 The Authors.

Article funded by SCOAP

https://doi.org/10.1007/JHEP11(2017)064

JHEP11(2017)064

monodromies in the duality group when going around points on the base where the moduli

become singular. Since among these monodromies there are transformations that invert

the torus volume, the compactiﬁcations are intrinsically non-geometric. The scheme of

ﬁbrations of T

moduli was later extended to heterotic strings where duality with F-theory

can be used to extract properties of the resulting non-geometric vacua [2]. Taking the base

to be complex one-dimensional leads to six-dimensional non-geometric heterotic vacua that

have received attention more recently [3–7].

In this paper, we will further examine six-dimensional N = (1, 0) non-geometric het-

erotic vacua described locally as T

ﬁbrations over a base B parametrized by t ∈ C. As in

recent works, we focus on conﬁgurations in which the heterotic gauge background is chosen

to have SU(2) structure so that the gauge group is broken to E

×E

or Spin(28)×SU(2)/Z

depending on whether one starts with the E

× E

or the Spin(32)/Z

heterotic string. In

this situation the heterotic moduli comprise one complex Wilson line together with the

complex structure and the complexiﬁed K¨ahler modulus of T

. The T-duality group act-

ing on the space spanned by these three moduli is O(2, 3, Z) [8]. Restricting this group

to SO

(2, 3, Z), the subgroup of order four which can be identiﬁed with Sp(4, Z), gives

an isomorphism between the heterotic moduli space and the moduli space of genus-two

curves [3, 4]. Hence, the non-geometric heterotic vacua can be deﬁned equivalently as

ﬁbrations of a genus-two Riemann surface over the base B. In general, a non-trivial holo-

morphic ﬁbration will degenerate at certain points on B and encircling them will induce an

Sp(4, Z) T-duality transformation on the moduli, thereby signalling the presence of defects,

dubbed T-fects in [9], such as NS5s or more exotic 5-branes. Now, the possible degenera-

tions of genus-two ﬁbers over the t-plane have been classiﬁed by Ogg, and Namikawa and

Ueno [10, 11]. Our objective is to continue the study, initiated in [5], of the six-dimensional

theories living on the T-fects corresponding to degenerations in the Namikawa-Ueno list.

This program is carried out by dualizing the conﬁguration to F-theory, where the T-fects

can be characterized geometrically.

The fundamental heterotic/F-theory duality relates the heterotic string compactiﬁed

on T

and F-theory compactiﬁed on an elliptically ﬁbered K3 surface [12–14]. At the

moduli level, the explicit map when there are no Wilson lines, i.e. the gauge group on the

heterotic side is unbroken, was found in [2, 15]. In the case at hand, when there is one

Wilson line breaking the gauge group to E

× E

or Spin(28) × SU(2)/Z

, the map from

the heterotic moduli to the dual K3 moduli was established lately in [3, 4, 16, 17]. This

map can be expressed in terms of Siegel modular forms of the genus-two curve encoding

the heterotic moduli. Thus, in F-theory the non-geometric heterotic vacua described as

genus-two ﬁbrations over a base correspond to speciﬁc K3 ﬁbrations over the same base.

Moreover, to preserve supersymmetry the total space of the F-theory ﬁbration must be a

Calabi-Yau — a threefold in the case that the base of the K3-ﬁbration is complex one-

dimensional. Since in F-theory there is a well-deﬁned geometric formalism to analyze

degenerations of the ﬁber along the base, the heterotic/F-theory duality enables us to infer

properties of the T-fects of non-geometric heterotic vacua.

The T-fects connected to the genus-two degenerations in the Namikawa-Ueno list [11]

were surveyed in [5] in the context of the E

× E

heterotic string. The purpose of the

– 2 –

JHEP11(2017)064

present work is to extend the analysis to the Spin(32)/Z

heterotic string. One motivation

is to check for the existence of dualities among defects observed in [5]. The ultimate

goal is to discover the main features of the theories living on the T-fects. The study of

such theories in the Spin(32)/Z

heterotic string actually started with the seminal treatise

of Witten [18] who showed that a heterotic 5-brane, or equivalently a small instanton,

supports a six-dimensional (1,0) supersymmetric gauge theory with group Sp(1) and 16

hypermultiplets in the fundamental representation. Already exploiting tools of F-theory,

the theories arising from Spin(32)/Z

small instantons sitting at ADE singularities in K3

were later analyzed in great detail in [19–21]. These generic theories were also derived from

the dual perspectives of type I D5-branes [22, 23], and type IIA conﬁgurations of D6, D8

and NS5-branes [24, 25]. In the Spin(32)/Z

heterotic string, F-theory methods were also

used early on in [26]. Various aspects of non-geometric vacua of the Spin(32)/Z

heterotic

string have been considered more recently in [2, 3, 6, 7].

In the following, we will present the results of a systematic study of Spin(32)/Z

heterotic T-fects associated to the genus-two degenerations in the Namikawa-Ueno clas-

siﬁcation [11]. As in the E

× E

case addressed in [5], we will apply the duality map

to every genus-two degeneration in the Namikawa-Ueno list in order to obtain the dual

F-theory background. Since this background turns out to have a elliptic ﬁbration with

a non-minimal singularity, we will attempt to turn the singularity into a minimal one by

performing a series of blow-ups in the base of the ﬁbration. When the resolution can be

accomplished we will determine the emerging smooth geometry. Introducing blow-ups is

equivalent to giving generic vevs to scalars in tensor multiplets of the 6d N = (1, 0) theory

on the defect, i.e. we move onto the tensor branch of the theory. Thus, knowing the smooth

geometry allows to deduce the gauge groups and matter content characterizing the IR limit,

valid in the tensor branch, of the theory living on the defect. Analogous techniques have

actually been employed in the recent classiﬁcation of SCFTs [27–31] and little string theo-

ries (LSTs) [32]. Actually, the theories that we obtain fall into known conﬁgurations whose

UV completions are conjectured to be LSTs [32, 33]. The theories indeed have a mass scale

and enjoy T-duality upon circle compactiﬁcation, both typical properties of LSTs [34].

Let us ﬁnally give an overview of the rest of the article. The six-dimensional non-

geometric heterotic vacua of interest are described in more detail in section 2. There

we recall the basics of heterotic compactiﬁcations on T

and review the formulation of

heterotic/F-theory duality in terms of a map between genus-two (sextic) curves and ellip-

tically ﬁbered K3 surfaces. In addition, we explain how the map connects degenerations of

sextics over a complex one-dimensional base, classiﬁed by Namikawa and Ueno, with de-

generations of K3 ﬁbered Calabi-Yau threefolds. In section 3 we ﬁrst sketch the procedure

to resolve singularities and then apply the method to local heterotic degenerations which

have a geometric description in some duality frame. We also discuss truly non-geometric

singularities that exhibit a kind of duality with geometric defects. In section 4 we catalog

all possible local heterotic degenerations admitting F-theory duals that can be resolved into

smooth Calabi-Yau threefolds. We conclude with further observations about the results.

Appendix A contains the resolutions of several models corresponding to small instantons

on ADE singularities.

– 3 –

JHEP11(2017)064

2 Non-geometric heterotic vacua and F-theory

This section is devoted to outlining the construction of the six-dimensional non-geometric

heterotic vacua studied in this paper. We will ﬁrst explain the structure of the vacua and

then discuss how to exploit F-theory/heterotic duality to analyze their properties.

2.1 Heterotic vacua in 8 and 6 dimensions

The starting point is the compactiﬁcation of the heterotic string on a torus T

. The

emerging eight-dimensional theory contains moduli ﬁelds encoding the geometric and gauge

bundle data. The geometric moduli consist of the complexiﬁed K¨ahler modulus, ρ =

B + ω ∧ ¯ω, with B the Kalb-Ramond two-form and ω the holomorphic one-form of the

torus which follows from the metric on T

, and the complex structure modulus given by

τ =

ω/

ω, where a and b are the two generators of the non-trivial one-cycles of the

torus. Furthermore, from the gauge bundle data we have 16 complex Wilson line moduli

from the Cartan generators of the non-Abelian gauge group of the heterotic string, i.e.

+ i

, I = 1, . . . , 16. In the following, we restrict ourselves to background

gauge bundles which only have SU(2) structure, so it will break E

× E

down to E

× E

or Spin(32)/Z

to Spin(28) × SU(2)/Z

. With this choice there is only a single complex

modulus, called β in the following, whose real and imaginary parts are given by the Wilson

line of the SU(2) Cartan around the one-cycles of the T

as deﬁned above. It is well known

that the three complex parameters ρ, τ and β live on the heterotic moduli space [8]

het

= O(2; R) × O(3; R)\O(2, 3; R)/O(2, 3; Z ) , (2.1)

where O(2; R) × O(3; R)\O(2, 3; R) is the local moduli space of the T

compactiﬁcation and

O(2, 3; Z) the duality group which identiﬁes physically equivalent theories.

Having the eight-dimensional moduli from the torus compactiﬁcation we construct,

in the next step, six-dimensional vacua by letting the moduli ﬁelds vary along two real,

or one complex, dimension. Since we allow in this construction for identiﬁcations of the

moduli under the duality around paths of non-trivial homotopy, it is very cumbersome

to work directly with the moduli ﬁelds. To circumvent this diﬃculty we use, like in F-

theory, a geometric object which has (almost) the same moduli space as the ﬁelds we want

to describe. The variation of the ﬁelds becomes then a ﬁbration of the object along the

complex one-dimensional base. In our case, the geometriﬁcation is done via a genus-two

curve such that we end up with a genus-two ﬁbration.

Since the description of the heterotic moduli space in terms of genus-two curves will

be crucial in the following, we brieﬂy review it. To every point of the moduli space char-

acterized by ρ, τ and β, there is an associated genus-two curve Σ whose period matrix Ω

belongs to H

deﬁned by

(

Ω =

τ β

β ρ



det Im(Ω) > 0, Im(ρ) > 0

)

. (2.2)

The four independent one-cycles of Σ can be chosen to span a canonical homology basis,

and b

with i, j = 1, 2, such that intersection form is symplectic. The elements of

– 4 –

剩余34页未读，继续阅读

评论收藏

内容反馈

weixin_38733333

粉丝: 4
资源: 922

Spin（32）/ℤ2杂散弦和小弦理论的非几何真空

扩展的黎曼几何II：局部杂散双场理论

2d（0,2）F-理论真空中的Green-Schwarz机理和几何异常关系

SO（32）杂散字符串理论的逼真的三代模型

SO（32）杂散弦理论中的规范与磁通量耦合

非几何杂散弦和具有增强规范组的对偶F理论

SPIN32F0.zip

MM32SPIN2x(p) 库函数和例程.rar

异质超重力中的Spin（7）压实和1 / 4-BPS真空

MM32SPIN0x(n) 库函数和例程.rar

MM32SPIN05PF数据中文手册.rar

Seekfree_MM32SPIN27PS_Opensource_Library_逐飞_mm32spin27ps_

MM32 SPIN MCU 电机 FOC 驱动应用指导手册

MM32SPIN0x(q) 库函数和例程.rar

灵动微MM32SPIN系列软件

MindMotion.MM32SPIN0x_DFP.1.0.7_packs_MM32_

量子威尔逊表面和拓扑相互作用

MM32SPIN422C驱动.rar

29.SPIN32F0 DRV.zip

MM32支持的烧录器（中文版）

基于MM32SPIN080C 无感方波BLDC驱动篇应用笔记

spin和promela简介

windows下spin和xspin的安装

Spin_AD ESR/EPR模拟软件 顺磁-自由基信号模拟

MM32SPIN06_RCC.txt

SPIN_BASIC.rar_SPIN_SPIN算法源程序_spin basic_spin算法

G 2-非几何M理论背景的结构和量化

基于小弦理论的四个维度的N = 2超对称QCD的强子

Monte Carlo studies of mixed spin-1 and spin-3/2 Ising model on simple cubic lattice

最新资源

Spin_AD ESR/EPR模拟软件顺磁-自由基信号模拟