LATTICE CLASSIFICATION OF THE FOUR-DIMENSIONAL ...

KFKI-1937-WA

J. P. P. Z.

BALOG FORGÁCS VECSERNYÉS HORVÁTH

LATTICE CLASSIFICATION OF THE FOUR-DIMENSIONAL HETEROTIC STRINGS

Hungarian academy of Sciences CENTRAL RESEARCH INSTITUTE FOR PHYSICS BUDAPEST

КЛС1-19в7-40/А МОИ»

LATTICE CLASSIFICATION OF THE FOUR-DIHENSIOHAL HETEROTIC STRIN6S J. BALOG, P. FORGÁCS and P. VECSERNYÉS Central Research'Institute for Physic« H-1525 Budapest 114, P.O.B.49, Hungary Z. HORVÁTH* Institute for Theoretical Physics Loránd Eötvös University H-1088 Budapest, Hungary

HU ISSN 0368 5330

ABSTRACT jte proggse^jft lattice slicing procedureileadás»; to the classification of all four-dimensional chiral heterotic strings based on Conway and Sloane's 22-diMensional self-dual Euclidean lattices. By reversing this procedure it is possible to construct all these theories.

АННОТАЦИЯ Предлагается процесс разрезания решеток, приводящий к классификации всех моделей четырехмерных киральных гетеротических струн, основанных на 22-мерных самодвойственных евклидовых структурах Конвея и Слоана. Обращение этой проце­ дуры позволяет построить все теории такого типа.

KIVONAT Egy rács-szeletelésí módszert javaslunk amellyel az összes négy dimenziós chirális heterotikus hurmodell osztályozható Conway és Sloane 22 dimenziós önduális rácsai alapján. Az eljárás megfordításával lehetővé válik az összes ilyen tipusu elmélet megkonstruálása.

1

There has been a great deal of interest in the construction of four-dimensional, chiral consistent string theories. Presently, there are two successful methods, one is the fermionic construction [lj in

and the other is the с о variant lattice approach developed Í2,3l

based on Г*»5] • Sinje we shall follow the latter

approach we give a very brief description of it. The

10-2n

dimensional heterotic strings can be associated with an even self-dual Lorentsian lattice

| ^^2n*3n 8-n*

> J ? h e

l

e

f

t

l a

**

i c e

»

which is the maximal/Euclidean part, comes from the compactification of 16+2n bosonic coordinates. The right lattice is the momentum lattice of the right moving compactified bosons and bosonized fermions together with an

(8-n)

dimensional part which

corresponds to the bosonized aupereonformal ghost and space-time n

D

P n o u

n

fermions, through the mapping of ^ _ © 1 i i *° 8-n * ' • 8 in Ref. [з] a method was given to construct chiral models and n

some interesting examples were found, the problem of classifi­ cation and systematic construction of all chiral models was left open. In

[з] it has been suggested that there exists a huge

number of models /with 10 ^

as an upper limit/.

In this paper we show how to classify and construct all chiral models in 10-2n dimensionsс The classification is based t o

on mapping the I ig-on*8+2n Euclidean lattices

П ,

2

п

•

A

1

16:

1

*"

2n

dimensional self-dual

these lattices have been

enumerated by Conway and Sloane [_&,7J » The number of those lattices which are relevant to construct four dimensional models is only 68. One of our results ie that the left gauge groups can be only certain subgroups of the possible 68 groups associated with these lattices / I / . 2 2

•

In what follows we present the main idea behing our construe-

2

tion. As an illustration we rederive all the 10-dimensional heterotic strings [ß]

effortlessly and give a potentially inter­

esting four-dimensional nodal. The details of the classification will appear elsewhere \э\

•

Our basic observation is that the gauge groups of the consistent 10-diaensional heterotie strings constructed by the 1

covariant lattice method [2J correspond to the ainiaal /"roof / lattices of possible 16-diaeneional self-dual lattices listed in £б J . Indeed, consider the (16+в)-diaensional етеп Lorentsian «elfdual lattice 1 . 1 6

theory.

I де g

6

of a 10-diaensional beterotic string

has the following decomposition with respect

to the right lattice:

6

16

Г ,8 - (A?.o) и (Д »..) и (^i .») U (4c .«).

(i)

16

-16 Where /\*

denotes a set of vectors of the left lattice associ­

ated with vectors of the i conjugacy class of D«, the right lattice. / Д±

can be enpty for sose i./

The rules for addition of *he oonjugacy classes /üj are identical to those for i in D and also the mutual scalar products 8 and squares are the same mod l' and sod 2, respectively. /This follows from the self-duality of the whole lettice./ If all four conJugacy classes appear in (l) then it is easy to see that the 16-dimensional lattice Где constructed as

Г« - A*uA? i s odd self-dual. Vectors fros the minimal lattice of Где of

(2)

3

length squared 2 correspond to gauge bosons of the string theory since these states are in the Z ^ I

1 6

6

6

and i£\J ,o I sectors of

and I nß.Q » respectively. Conversely, given an odd self-dual lattice

|

, it is

l f e

easy to reconstruct the full lattice of the beterotie string. First one has to divide П

1

6

into Z\J

6

and Д ^

/corresponding

to the set of even and odd vectors respectively/. f/\^ ) dual of to * Q

6 Q

/the

Z\l / splits into four conjugacy classes with respect /this follows fros the self-duality of Г ^ Л

Combining

tbea with the corresponding conjugacy classes of the right lattice. Do»

(l)

i s

recovered.

An alternative possibility is that Д * are empty. In this case

6

end й J / o r ^ J /

Л* - Д * - Г where

г,

1 де

i 8

i t s e l f

lattice of Eg x Eg

B D

e v e n

6

6

1 6

eelf-duel lattice, i.e. the weight

or Spin (32) /Zg , corresponding to the

original heterotic strings [в J . So we have shown that the possible 10-dimensional heterotic strings are in one-to-one correspondence with the 16-dimsneional self-dual lattices of Conway and Sloane. A siailar sapping of the Lorentzian lattice onto an Euclidean self-dual one is also possible for lower dimensional theories. However, since the right lattice is sore complex in this case, the reconstruction procedure is no longer unique. Nevertheless, as we shall see, the possible gauge groups are still those associated with a minimal lattice from the list of Ref. [7] * or certain subgroups of these.

*

Before turning to the elassification of lower theories, lot ua describe tbo "«lieing" of Lorentsian latticea, • proeodnxo wo ahall frequently use in our analyaia. Consider » (k+l)-dimensional odd / even / self-dual lattice \ \

%

"**"

1

aetrle At-/» and cut it into two piece*:

ft,l -»

Гк,1-г © Г w

e

Where w £ ["kii-r ** ( * ) of ljj.1

6

Г^.1

f o r

(r íl) -

г

kf ,A')^f.A»)«fc»,.A.)

(ш)

will be an odd self-dual Lorentzian lattice« In the first case we have to find four eonjugacy classes 2

Д о ' £i??. Ду?» Л?? 1

2

3

i n

Г22 witb «•• group structure

10 гг ад. . Then we have to choose from the A ^ , A ^ ж

Х%

2

2

classes a pair of primitive vectors

(2t

1$

c o n d u g e c y

2tg) of length

2

mod 4. Hatching the corresponding scalar products we get

Гаг,, - [bf.hi) U(&? •»!• AL)vfo?*a.At) u

(12c

We have to repeat this glueing twice more. Then we obtain a 122:9

o d d

e e 3

i

- ~

d u a l

f r o m

build I 22:1A

Lorentzian lattice. Finally, we have to ' 22:9

we want all of the De one possibility

a n d

D

5

^

a

8 i m i l a r

«earner. Because

conjugacy classes to occur we have only

in this case.

Let us now see a concrete example. We choose the lattice Z

22

A f t e r

*" ' 22 *

glueing it with the X,

lattice

(9a) three

times in a particular way our self-dual Lorentzian lattice is generated by the following conjugacy classes:

(«) 9

(Af.A .);(Af,A?). The order of these elements are 4-,4,2,2, respectively, so they generate a

group, thus the whole lattice

I 2219 contains 64 eonjugacy classes. Л J

2

and Дь ' denote

the dual lattice of the 22-dimensional left and the 9-dinensional right lattice. The minimal lattice of Д

2

2

is the root lattice

of the D ^ K D , xD, xDj* Dj Lie-algebra. In addition to the root lattice Д ~

contains two other conjugacy classes from the

>

11

D

14

, l D

f

x D

l

w e ±

8

До •

nie Д ^

h t

3-Attlees

(ooooo) U (ovvw) U (vwoo)

(14)

lattice is generated by the following representatives

of the D£ weight lattice:

(ooo ooi ooi) , (ooo oio oio) , (oio 2 ° 2 ~ 2 ° 2 ) t (15) (ill 000 000) , (000 111 OOO)

t

(000 000 111 ) .

The representatives of the nontrivial conjugacy claeeee in (1?) are generated by the following vectorst

22

(Д ;

A ? ) - (e.e.o,s,o; | | 0 000 001)

( Д ^ Л ? ) a*

f

(o»o,o,s c | o | OOOOOO) , f

t

aC

2

(Да ; A»*) - (o,s,s e,ei ooo ooo oio ) , t

2

(Д^ ;

Д ^ ) - (o,c,e c c; 000 \ \ О \ \ о). f

f

Denoting the set of the even and odd vectors in • 22s9 ^ ^ о and Д ^ , respectively, the

lattice is an odd self-dual Lorentsian lattice. In order to get

12

our final even eelf-dual lattice from the Z^^

< 22 14

w e

e h o o 8 e

a

v e c t o r

eonjugacy class which has the following properties:

2

i/ ita length

ia 5» in auch a way that its left part ia of

2 length

p 3\

8 and its 9-dimensional part has length 1

ii/ its half ie in the ( Л ' )

lattice}

ill/ it has odd scalar producta with the elements of Д ~

.

She half of this vector can he a representative of the Д ' В

eonjugacy class because our requirements ensure the appropriate group structure and scalar product matchings with that of the Dв conjcgacy class to get an even self-dual Lorentzian lattice aat 5

Pan* - (А?.До )«(Д».^)о(д31, 5) (д^дП,д5 j . Л

и

Bequirenent i/ guarantees that we have zero mass fermions as well. The concrete fore of the Д '

l

representative is given by

(o,s,o,v,s ; | 0 0 0 0 | 0 0 - | )

•

Having constructed this lattice it is easy to find the massless fermion spectrum. Ve get the following multiplets of

2

l^o, (s • c v ) • f

(y, s • c ) (o,c • c,o) J , f

L

which are indeed ehiral, but only because of the r(l) factors.

13 Thus, the SO (28) gauge group only couples to a "shadow world" and only the SO (6) x U(l)

group couples to the low energy

feraions.

We are grateful to Prof. Z. Perjés for fruitful discussions«

14

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(2]

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Lattice classification of the four-dimensional heterotic strings

Kiadja a Központi Fizikai Kutató Intézet Fel«15s kiadói Szegó Károly Szakmai lektor: Perjéa Zoltán Nyelvi lektor: Hraskó Péter Példányszám 316 Törzsszám 87-30» KéazUlt a KPKI sokszorosító üzemében Felelős vezetó: Töreki Béláné Budapest, 1907. június hó