Discrete Torsion

SUSY Gauge Theories

4.4 Discrete Torsion

defining cohomology of a singular space is ordinarily fairly hard, and to get something reasonable, one has to do so using "simplicial" rather than de-Rham cohomology.

What we find here is that string theory anticipates the cohomology of the resolved orbifold since the Hodge numbers the CFT computes correspond precisely to what one would have obtained by blowing up the orbifold. This seems to be a generic behavior of string theory on orbifolds.

this to higher genus surfaces. The effect of torsion is to change the transformation law of the ga'hb' twisted states under gahb, by addition of the phase (4.2), and this changes the notion of

r

invariance in the orbifold.

"Ordinary" orbifold CFT corresponds to no torsion, p = 0, but there are r - 1 closed string theories with torsion one can define. As we will see, while an orbifold theory without torsion has a "good" geometric interpretation, an orbifold theory with torsion does not.

4.4.1 Z2 x Z 2 Theory with Torsion

Before we go on to do the general case, let us return to the Z₂ x Z₂ example.

We recall that g-twisted states corresponded to 1, dz1 , dz1 , dz1 /\ dz1 . Now,

so that

r

invariant states are those transforming as ( -1) under h. These are precisely dz1 and dz1 , so that this time it is H¹•0(5_{9 )} and H⁰•1(5_{9 )} that survive the quotient.

Taking into the account the other two elements of

r

and the shifts in the cohomology labels, the H^1•0( 5_{9 )} orbifold with torsion has

hl,1 = 0, ^{h2,1 =} 3.

Generators of H²•1(M/f) correspond to deformations of complex structure of the orbifold. Complex structure of

C3 /f

is given in terms of

r

invariant monomials on

C3,

^modulo any relations between them. Here,

r

invariant monomials are Xi =

z[,

i

=

1, 2, 3, and y

=

z1z2z3 , which satisfy one relation:

By projecting onto ^Xi = const, the space contains three curves of singularities of the

form

y 2 CX XiXj.

The three elements of H²,1(M/f) can be thought of as deforming each curve of singu- larities (from the orbifold point of view, this seems as a natural interpretation, since the twisted states which give rise to the deformations are supported there. However, one should be cautious since the concept of locality when it comes to deformations of the complex structure is obscure.) Such a deformation could for example look like

This resolves each curve of singularities, however, one clearly has a conifold singularity left at the origin: upon adding the deformations the x1x2x3 term becomes irrelevant and can be neglected. What is left over is an equation of the conifold. So upon turning on the deformations present in string theory, we have found that we cannot completely resolve the singularities. From the mathematical standpoint there is no obstruction to deforming the conifold away via

however in string theory this deformation is absent, leaving a stable conifold singu- larity. This singularity is not a singularity in the CFT, unlike the conifold treated in [48], but is simply a region where stringy effects are large due to concentrated curvature. The conifold theory obtained above is smooth: it is a deformation of the orbifold CFT which does not have singularities, and the deformations we employed are not expected to introduce singularities. This manifests itself here precisely by the impossibility of turning on the offending deformation c. One final note: in the case of an orbifold without torsion, the spectrum we computed corresponded to the blowup of the orbifold. There, unlike in this case, no additional singularity at the intersection was found - basically the reason is that for a Z₂ x Z₂ orbifold, resolving singularities in codimension two automatically resolves the singularities at codimension three, as

can be easily seen torically.

4.4.2 General

Zm x

Zn Case

Take the orbifold group

r

= Zm x Zn to act as

g: (z1,z2,z3)---+ (z1,e2;izz,e_2;;z3), h: (z1,z2,z3)-+ (e2~iz1,zze_2~iz3).

Torsion depends only on the ratio of p and r in the eq.(4.2), so let q = gcd(p, r).

Using the formula ( 4.2),

for all

a, a',

b, b'. Let us denote

m =rs, n =rt, gcd(s, t)

=

1,

and put

g =

g~ and

Ti=

h~. From above, we see that a subgroup

generated by

g,

and h is completely unaffected by torsion. Thus in a spectrum of the

r

orbifold with p units of torsion, we will find a complete twisted sector of a C3

/f

orbifold.

This contribution is as follows. In the "ordinary" orbifold, the CFT spectrum agrees with the spectrum on the blowup of the orbifold. The cohomology of this orbifold can be computed by toric methods, with the following result:

• The C3

/f

orbifold has three curves of Zqs, Zqt, and Zq singularities. Blowing them up contributes qs - 1, qt - 1 and q - 1 to h^1•1. (Roughly, each singularity is of the form <C x C² /Z*. Blowing up replaces the singular curve by a chain of S²'s fibered over C, which contributes to h_{4 ,} and by the dual-of-the-dual to h 1,1 _)

- b 2rria 2rrib _.l.!c!...(qta+qsb)

• Since

fl

h : ( z1, z2, z_{3 )} ----+ ( e -q;- _{z 1,} e ---qt z2, e ^q2 st z_{3 ),} the origin is fixed by elements of the form

flhb,

where

0 <a, 0

<

b, qta

+

^qsb

<

q² st,

a simple counting shows that there are

H

^q2 st - qs - qt - q

+

2) such elements, and they contribute to h¹•1

: the fixed set is a point whose cohomology con- sists of constant functions contributing to H⁰•0 and this assignment gets shifted by 1 following (4.1). Poincare duality requires the same contribution to h²•2

,

which comes from the remainder of elements (a, b) fixing the origin which satisfy instead:

q² st

<

qta

+

qsb

<

2q² st.

It is easy to see that no other elements of

r

can contribute to the H¹•1 cohomology of the <C3

/f

orbifold. For, curves of singularities contribute to H¹•1 and H²•2 via 1 and dz /\ dz, and these are always invariant under all other elements of

r,

so either torsion is non-trivial, p

#

^{0 mod} r, and they are projected out, or torsion is trivial and they have already been accounted for. Thus,

h^1•1

=

-(q1 ²st

+

qs +qt+ q - 4).

2 Let us now turn to H²•1

• The contribution to these group elements can only come from the curves of singularities, from sectors twisted by ga, ha or gash-at, and so elements generating them are always of the form dz and dz. Consider, for example

ga twisted states, which propagate along 5₉ = (z_{1 ,} 0, 0). According to the above discussion, we are instructed to keep states that transform as

cap

^{under h, which is}

only possible if n =rand ap = ±1 mod r. The second equation is the statement that q = gcd(p, r) = 1, and if in addition n = r then, for every choice of sign, there is only one solution for a in Zn. The choice of sign in effect picks out one of the dz or dz's

as the "invariant" element of the fixed set cohomology group. Similar considerations for other elements can be used to show:

•If m = n = r where r = gcd(m,n), that is if

r

=Zr x Zn and if in addition q = gcd(r,p) = 1, then h²•1 = 3 = h¹•2

.

• In all other cases, h²•1 = O = h¹•2 .