The Untwisted Stabilizer in Simple Current Extensions

A method is presented to compute the order of the untwisted stabilizer of a simple current orbit, as well as some results about the properties of the resolved fields in a simple current extension.

a r X i v :h e p -t h /9611194v 1 23 N o v 1996The Untwisted Stabilizer in Simple Current Extensions Peter Bantay Mathematics Department University of California at Santa Cruz Abstract

A method is presented to compute the order of the untwisted stabilizer of a simple current orbit, as well as some results about the properties of the resolved fields in a simple current extension.

In a recent paper[1],Fuchs,Schellekens and Schweigert presented an Ansatz to describe the modular properties of a CFT obtained by simple current extensions (for a review see[2]).An important role is played in their Ansatz by the so-called untwisted stabilizer,a subgroup of the ordinary stabilizer de?ned by some cohomological properties.In general it is quite di?cult to determine the untwisted stabilizer,as it is related to the tranformation properties of the genus one holomorphic one-point blocks of the simple currents under the mapping class group action,and the latter data are not readily available.

The purpose of this note is to present a procedure that allows under suitable circumstances the determination of the order of the untwisted stabilizer solely from the knowledge of the SL(2,Z)representation on the space of genus one characters of the original theory.The idea is to exploit the Frobenius-Schur indicator introduced for CFTs in[6].

We shall not go into the details of the FSS Ansatz[1],let’s just recall the basic setting.We are given some CFT and a group G of integral spin simple currents,and we would like to construct the SL(2,Z)representation of the new CFT obtained by extending the original one with the simple currents in G.The ?rst thing to do is of course to determine the primaries of the extended theory. The scheme is as follows:

1.First we keep only those primaries p of the original theory which have zero

monodromy charge with respect to all the simple currents in G,resulting in

a set I G0.

2.In the next step we identify those primaries from I G0that lie on the same

G-orbit,i.e.for which there exists a simple current in G transforming one into the other.

3.We split the so obtained orbits into several new primaries.Naively one

would think that each orbit[p]should be split into|S p|new ones,where S p={α∈G|αp=p}is the stabilizer of the orbit[p],but it turns out that this so-called?xed point resolution1is governed by a subgroup U p of the full stabilizer S p,the so-called untwisted stabilizer[1].The actual de?nition

A method is presented to compute the order of the untwisted stabilizer of a simple current orbit, as well as some results about the properties of the resolved fields in a simple current extension.

of U p involves the consideration of the space of genus one holomorphic one-point blocks for the simple currents in G,thus it is in general a di?cult problem to determine U p.

After having performed the above three steps,we are led to the following description of the primary?elds of the extended theory:they are in one-to-one correspondence with pairs(p,ψ),where p is a G-orbit of I G0,andψis a linear character of U p,i.e.an element of the dual group?U p.As it cannot lead to confusion,we shall also denote by p any representative of the orbit p.

Now that we know how to describe the primaries,we can formulate the FSS Ansatz describing the SL(2,Z)representation of the resolved theory.For the exponentiated conformal weightsωp=exp(2π??p)of the primaries-i.e.the eigenvalues of the T-matrix-we have

ω(p,ψ)=ωp,(1)

while for the S-matrix the FSS Ansatz reads[1]

|G|

S(p,ψ),(q,χ)=e p e q

2(3) is the square root of the index of U p in S p,which is known to be an integer on general grounds,and S pq(α)is the matrix element of the mapping class S acting on the space of genus one holomorphic one-point blocks of the simple current α,in the canonical basis of[3].In particular,S pq(0):=S pq is just the ordinary S-matrix of the original theory we started with.It may be shown[1,3]that the above Ansatz leads to a consitent SL(2,Z)representation for the extended theory,e.g.it satis?es the relations of[4].

Our aim is to determine the order of U p,or what is the same,the e p-s from Eq.(3).The?rst thing to note is that,while to determine the actual matrix elements of S through Eq.(2)we need to know all of the matrix elements S pq(α) -which are in general hard to compute-,upon summing overχ∈U q in Eq.(2), a standard argument of character theory yields

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A method is presented to compute the order of the untwisted stabilizer of a simple current orbit, as well as some results about the properties of the resolved fields in a simple current extension.

χ∈?U q S(p,ψ),(q,χ)=

e p

|S p|

.(6)

Substituting Eq.

(2)

into

Verlinde’s

formula

[5]

N

r

pq

= s S ps S qs¯S rs

e p e q

1

ω2

r

,(9)

to arrive at

ν(p,ψ)=

e p

2

q r,s N p rs S qr S0s

ω2s

|S p|

α∈G

Z[p,α],(12)

then because the lhs.of Eq.(10)can only take the values±1and0,we get the following alternative:

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A method is presented to compute the order of the untwisted stabilizer of a simple current orbit, as well as some results about the properties of the resolved fields in a simple current extension.

1.Z G(p)=0.In this case the resolved?eld(p,ψ)is complex,and we get no

information on e p from Eq.(10).

2.Z G(p)=0.In this case the resolved?eld(p,ψ)is either real or pseudo-real

according to the sign of Z G(p),moreover

1

e p=

1

e q

A method is presented to compute the order of the untwisted stabilizer of a simple current orbit, as well as some results about the properties of the resolved fields in a simple current extension.

Acknowledgement:It is a pleasure to acknowledge discussions with Geo?Mason and Christoph Schweigert.

References

[1]J.Fuchs,A.N.Schellekens and C.Schweigert,Nucl.Phys.B473(1996),323.

[2]A.N.Schellekens and S.Yankielowicz,Int.J.Mod.Phys.A5(1990),2903.

[3]P.Bantay:Simple Current Extensions and Mapping Class Group

Representations,hep-th/9611124.

[4]P.Bantay and P.Vecsernyes:Mapping Class Group Representations and

Generalized Verlinde Formula,hep-th/9506186.

[5]E.Verlinde,Nucl.Phys.B300(1988),360.

[6]P.Bantay:The Frobenius-Schur Indicator in Conformal Field Theory,

hep-th/9610192.

Supported partially by OTKA T016251

e-mail:bantay@hal9000.elte.hu

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