On S-duality of the Superconformal Index on Lens Spaces and 2d TQFT

0.34 MB·English

by  Luis F. Alday

On S-duality of the Superconformal Index on Lens
Spaces and 2d TQFT
Luis F. Alday, Mathew Bullimore and Martin Fluder
Mathematical Institute, University of Oxford,
24-29 St Giles’, Oxford OX1 3LB, United Kingdom
3
1
0
2

n
a
J

1 Abstract
3
We consider the 4d superconformal index for N = 2 gauge theories on S1 ×L(r,1),
]
h
where L(r,1) is a Lens space. We focus on a one-parameter slice of the three-
t
-
p dimensional fugacity space and in that sector we show S-duality. We do so by
e
h rewriting the index in a way that resembles a correlation function of a 2d TFT,
[
which however, we do not identify.

1
v
6
8
4
7
.
1
0
3
1
:
v
i
X
r
a
1 Introduction
Over the last few years we have seen a huge progress in understanding N = 2 super-
symmetric theories in four dimensions. Much of this progress was due to the realization
that we could systematically construct such four dimensional theories by compactifying the
six-dimensional theory living of N M5−branes (the putative N = (2,0) six-dimensional
theoryoftypeA )onaRiemannsurfacewithpunctures[1]. Furthermore, theproperties
N−1
oftheresulting4dtheoryareincorrespondencewiththepropertiesoftheRiemannsurface.
For instance, S-duality corresponds to different pant decompositions of the same Riemann
surface. Thissuggeststhatthephysicalquantitiesofthe4dtheoryshouldbecloselyrelated
to those of the theory on the two-dimensional Riemann surface. The first example of this
relation is the correspondence between the partition function on S4 and correlators in 2d
Liouville CFT [2].
Another powerful relation, more relevant to this paper, involves the super-conformal
index, or the super-symmetric partition function on S1 × S3. It was found in [3] that
invarianceoftheindexunderS-dualitytranslatesintoassociativityfortheoperatoralgebra
of a 2d TQFT. Furthermore, for a particular choice of the fugacities, this 2d theory was
identified as q−deformed 2d Yang-Mills in the zero area limit [4]. Extensions to other
choices of fugacities were studied in [5].
On general grounds, the relation between the super-conformal index and q−deformed
Yang-Mills can be understood as follows. We think of the partition function on S1×S3 as
arising from the compatification over Σ of an observable on the six dimensional N = (2,0)
theoryonS1×S3×Σ. Ontheotherhand,wecanalsocompactifythesix-dimensionaltheory
on S1 ×S3, leading to an observable in a two dimensional theory on the Riemann surface
Σ. If the 6d observable is “protected”, this leads to a relation between the observables
in the 4d and 2d theories. By analogous reasoning, considering the 6d theory on S4 ×Σ,
leads to the relation mentioned above, between the partition function on S4 and Liouville
correlators.
The above idea is hard to make precise, since little is known about the six-dimensional
theory. However, in the case of the super-conformal index, the presence of the S1 makes
thingssimpler: thesixdimensionaltheorycompactifiedoverS1 givesrisetofive-dimensional
N = 2Yang-Millstheory[6,7],sowecouldtrytounderstandthesuper-conformalindex/2d
TQFT duality by studying the 5d theory on S3 ×Σ. Indeed, in [8] the partition function
2
of the 5d theory on S3×Σ was computed and it was found to be identical to the partition
function of the 2d q-deformed Yang-Mills theory. It is tantalizing then to conjecture that
in general, the super-symmetric partition function of a 4d N = 2 theory on S1×M will be
3
given in terms of some 2d TQFT, given by the compactification of five-dimensional N = 2
Yang-Mills theory on M .
3
In this paper we make a step towards this direction by studying the super-symmetric
partition function on S1 ×L(r,1) = S1 ×S3/Z . This index was computed in [9], where
r
S−duality was checked up to a few orders in a fugacities expansion. The aim of this paper
is to prove S-duality (in a subspace of the fugacities space) by following an strategy similar
to [4]: We uncover the structure behind S-duality by rewriting the index in a way that
resembles the correlation function of a 2d TFT. The structure of the index on Lens spaces
however, is much more complicated that the structure of the index for r = 1, and we are
not able to identify the relevant 2d TFT.
The organization of this paper is as follows. In section two we review the computation
of the index on S1×L(r,1). In section three we define the particular slice to be studied and
we show that for this slice the index can be written in a way that resembles a correlator of
a 2d TFT. Furthermore, we show that correlators of the putative 2d TFT satisfy crossing
symmetry, which translates into S-duality for the index computation. In section four we
end with some conclusions and open problems.
2 The Index on S1 × L(r,1)
The index is defined as [10]
I = Tr(−1)Fpj1+j2−rq−j1+j2−rtR+r (1)
where F is the fermion number and the trace is taken over the states of the theory on
S3 satisfying E − 2j − 2R + r = 0. E stands for the conformal dimension, (j ,j ) for
2 1 2
the Cartan generators of the SU(2) ⊗SU(2) isometry group and (R,r) for the Cartan
1 2
generators of the SU(2) ⊗U(1) R−symmetry.
R r
The index on S1×L(r,1) was computed in [9], to where we refer the reader for the de-
tails. The Lens space L(r,1) = S3/Z is defined as the orbifold of S3 : {(z ,z ) (cid:15) C2,|z |2+
r 1 2 1
|z |2 = 1} under the identification
2
3
(z ,z ) ∼ (e2πi/rz ,e−2πi/rz )
1 2 1 2
where SU(2) acts on (z ,z ) as a doublet. Z acts on the S1 fiber of the Hopf fibration,
1 1 2 r
denoted by S1 , i.e. the Z action is embedded into U(1) ⊂ SU(2) .
H r i 1
The orbifold theory has a set of vacua labeled by a non-trivial holonomy V along the
S1 , with Vr = 1. In this paper we will restrict to the case of A -type theories in which
H 1
case the holonomy can be taken of the form
V = diag(e2πim/r,e2πi(r−m)/r), m = 0,1,...,[r/2]. (2)
Different sectors are then labeled by the integer m.
The total index of a theory is constructed from the following building blocks, which
could be interpreted as three-point functions and propagators:
(cid:32) (cid:33)
(cid:88)(cid:88) 1
I(m1,m2,m3)(a ,a ,a ) = I(m1,m2,m3)exp g(tn,pn,qn,[m.s] )ans1ans2ans3
trif 1 2 3 0 n r 1 2 3
s n=1
(cid:32) (cid:33)
η(m)(a) = η(m)exp (cid:88) 1 (cid:0)f(tn,pn,qn;[2m] )a2n +f(tn,pn,qn,[−2m] )a−2n +f(tn,pn,qn;0)(cid:1)
0 n r r
n=1
where [x] denotes x mod r and we have introduced s = (s ,s ,s ), s = ± and m =
r 1 2 3 i
(m ,m ,m ). The zero-point contributions are given by
1 2 3
(cid:18)√t(cid:19)−41(cid:80)s([m.s]r−[m.s]2r/r) (cid:18)√t(cid:19)[2m]r−[2m]2r/r
I(m1,m2,m3) = , η(m) = (3)
0 pq 0 pq
Finally, the one letter contributions are given by
pm + qr−m (cid:16) √ √ (cid:17)
1−pr 1−qr
f(t,p,q;m) = pq(1+1/ t)−1− t +δ (4)
m,0
1−pq
pm + qr−m (cid:16) pq (cid:17)
g(t,p,q;m) = 1−pr 1−qr t1/4 − (5)
1−pq t1/4
The total index is then obtained by ”gluing” such contributions. For instance, the ”four-
point function” is obtained by joining two three-point functions with the corresponding
propagators
4
[r/2](cid:90)
(cid:88)
[dz]mI(m1,m2,m)(a ,a ,a)η(m)(a)I(m,m3,m4)(a,a ,a ) (6)
trif 1 2 trif 3 4
m=0
where the measure is given by
(cid:40)
2−a2−a−2da [2m] = 0
[da]m = 4π a r (7)
1 da [2m] (cid:54)= 0
4π a r
S−duality implies that the four-point function (6) is symmetric under the interchange
of any two pairs (a ,m ) ↔ (a ,m ). This was verified to a few orders in the fugacities
i i j j
expansion in [9]. In order for this crossing property to work, one needs to sum over all
intermediate values of m. The aim of this paper is to understand such a property as arising
from a 2d TQFT computation, as it was done for the r = 1 case in [3] and subsequent
papers.
3 A particular slice and 2d TFT picture
3.1 A particular slice
In [5] a particular limit, denoted as the ”Macdonald Index”, was studied. This limit
corresponds to p = 0, general q,t and is characterized by its enhanced supersymmetry.
Since only states with j +j −r = 0 will contribute to the index in this limit, we expect
1 2
a simplification. This can also be seen as follows. The building blocks of the index can
be written as products of elliptic gamma functions of the form Γ(z,p,q), where z is some
expression depending on the a and the fugacities. The following identity
i
∞
1 (cid:89)
Γ(z,0,q) = , (z;q) = (1−zqi) (8)
(z;q)
i=0
Then implies that the index can be written in terms of simpler functions. Indeed, in this
limit we find the following expressions
5
(cid:89) 1
I(m1,m2,m3)(a ,a ,a ) = I(m1,m2,m3) √ (9)
trif 1 2 3 0 ( tas1as2as3q[−m.s]r;qr)
s 1 2 3
(cid:18) −a2 (cid:19)δm,0 (cid:89)
η(m)(a) = (qr;qr)(t;qr) (a2sq[−2ms]r;qr)(a2stq[−2ms]r;qr) (10)
(1−a2)2
s=±
The zero-point contributions will be discussed momentarily. Mimicking [3], we define the
rescaled structure constants
(cid:113)
Cˆ(m1,m2,m3)(a1,a2,a3) = η(m1)(a1)η(m2)(a2)η(m3)(a3)It(rmif1,m2,m3)(a1,a2,a3) (11)
Note that the zero-point contributions are subtle in this limit, since they are proportional
toptosomepower. Itcanbecheckedthatthispowerisalwaysbiggerthanorequaltozero.
Furthermore, quite remarkably, this power is exactly zero (and so the contribution does
not vanish in the p → 0 limit), provided m ,m and m satisfy a selection rule. Namely,
1 2 3
for fixed m and m , m should run between |m −m | and min(|r−m −m |,m +m ).
1 2 3 1 2 1 2 1 2
Note that this agrees with the selection rules for SU(2) affine algebra at level r.
In what follows, we will make a further simplification in the space of fugacities, and we
will consider the limit t = qr. Note that this reduces to the ’Schur’ limit for r = 1 and so
the 2d-TFT we are after should reduce to q−deformed 2d YM in the zero area limit [4].
3.2 2d TFT interpretation
We would like to interpret the rescaled structure constants as the three-point correlation
functions of some 2d- TFT. Let us start by defining
1−qr
C(m1,m2,m3)(a ,a ,a ) ≡ Cˆ(m1,m2,m3)(a ,a ,a ) (12)
1 2 3 (qr;qr) 1 2 3
and let us study this object for cases of increasing difficulty. The simplest case corresponds
to m = m = m = 0. One can explicitly check that in this case the structure constants
1 2 3
coincide with that of the r = 1 case, up to a rescaling q → qr. Using the results of [4] we
can immediately write
6
∞
(cid:88) χ (a )χ (a )χ (a )
C(0,0,0)(a ,a ,a ) = (cid:96) 1 (cid:96) 2 (cid:96) 3 (13)
1 2 3
|(cid:96)|
qr
(cid:96)=1
where we introduced the Schur polynomials and the q−deformed dimension
a(cid:96) −a−(cid:96) q−(cid:96)/2 −q(cid:96)/2
χ (a) = , |(cid:96)| = (14)
(cid:96) a−1/a q q−1/2 −q1/2
The next case, allowed by the selection rules, corresponds to m = 0 and m = m = m,
1 2 3
with 0 < m < r/2. Note that in this case the three-point function is not symmetric under
the interchange a ↔ 1/a or a ↔ 1/a . Hence, we are not able to expand it purely in
2 2 3 3
terms of Schur polynomials. However, we can expand it in terms of Schur polynomials for
a :
1
C(0,m,m)(a ,a ,a ) = (cid:0)N(m)(a )N(m)(a )(cid:1)1/2(cid:88) χ(cid:96)(a1)U(cid:96)(m)(a2,a3) (15)
1 2 3 2 3
|(cid:96)|
qr
(cid:96)
forsomefunctionsU(m)(a ,a ). Forfutureconvenience,wehavepulledoutan(cid:96)−independent
(cid:96) 2 3
normalization factor, with
a2qr−2m +a−2q2m
N(m)(a) = 1− (16)
1+qr
The functions U(m)(a ,a ) possess some remarkable features. Computing them to several
(cid:96) 2 3
orders in a q−expansion we find that we can write them as
U(m)(a,b) = 1, U(m)(a,b) = U(m)(a)U(m)(b)+U(m)(a)U(m)(b) (17)
1 (cid:96) (cid:96),1 (cid:96),1 (cid:96),2 (cid:96),2
(m) (m)
whereU (a)andU (a)satisfyorthonormalitypropertieswithrespecttothe”measure”
(cid:96),1 (cid:96),2
N(m)(a):
(cid:73)
1 da
N(m)(a)U(m)(a)U(m)(a) = δ δ (18)
2πi a (cid:96),i (cid:96)(cid:48),j (cid:96)(cid:96)(cid:48) ij
Assuming that U(m)(a) contain only terms of the form a−(cid:96),...,a(cid:96), any solution of (18) leads
(cid:96),i
to the correct values for U(m)(a,b)! However, such solutions are not unique. We found it
(cid:96)
7
convenient to choose the following basis
U(m)(aqm) = χ (a) (19)
(cid:96),1 (cid:96)
a1−(cid:96)((1−a2+2(cid:96))(qr −q(cid:96)r)+a2(q((cid:96)+1)r −1)+a2(cid:96)(1−q((cid:96)+1)r))
U(m)(aqm) = −i (20)
(cid:96),2 (cid:112)
(1−q((cid:96)−1)r)(1−q((cid:96)+1)r)(a2 −1)(a2qr −1)
from which the orthogonality conditions can be checked explicitly. The above definitions
work for 0 < m < r/2. For the special cases m = 0,r/2 we obtain
(0) (r/2)
U (a) = U (a) = χ (a) (21)
(cid:96),1 (cid:96),1 (cid:96)
(0) (r/2)
U (a) = U (a) = 0 (22)
(cid:96),2 (cid:96),2
The doubling of the base functions, from χ (a) to U(0)(a) plus U(0)(a), when m (cid:54)= 0,r/2,
(cid:96) (cid:96),1 (cid:96),2
is expected, since for these cases, the gauge group is broken to the maximal torus.
Finally, let us mention that the orthonormality conditions (18) imply
1 (cid:73) da(cid:48)
N(m)(a(cid:48))U(m)(a,a(cid:48))U(m)(a(cid:48),b) = U(m)(a,b) (23)
2πi a(cid:48) (cid:96) (cid:96) (cid:96)
Let us now focus on the generic case in which m ,m ,m (cid:54)= 0. For the case r = 1, it
1 2 3
is convenient to expand the structure constants in terms of Schur polynomials. In other
words, given C(a ,a ,a ) we consider
1 2 3 r=1
(cid:90)
(cid:89)
C (a ,a ,a ) = [da ]χ (a )χ (a(cid:48))C(a(cid:48),a(cid:48),a(cid:48)) (24)
(cid:96)1,(cid:96)2,(cid:96)3 1 2 3 r=1 i (cid:96)i i (cid:96)i i 1 2 3 r=1
i
This action is diagonal in the index (cid:96) (see [11] for a physical explanation of this fact), which
leads to (13) and manifests the structure behind S-duality. For the case r > 1, it is then
natural to follow the same procedure with the replacement
χ (a)χ (a(cid:48)) → (cid:0)N(m)(a)N(m)(a(cid:48))(cid:1)1/2U(m)(a,a(cid:48)) (25)
(cid:96) (cid:96) (cid:96)
Hence, given a general three-point function C(m1,m2,m3)(a ,a ,a ), we can consider the
1 2 3
transformed one
8
(cid:90)
C(m1,m2,m3)(a ,a ,a ) = (cid:89)[da(cid:48)](cid:0)N(mi)(a )N(mi)(a(cid:48))(cid:1)1/2N(mi)(a ,a(cid:48))C(m1,m2,m3)(a(cid:48),a(cid:48),a(cid:48))
(cid:96)1,(cid:96)2,(cid:96)3 1 2 3 i i i (cid:96)i i i 1 2 3
i
(26)
Note that for the case m = 0, we obtain the previously mentioned results, since
(cid:0)N(m)(a)N(m)(a(cid:48))(cid:1)1/2U(m)(a,a(cid:48)) = χ (a)χ (a(cid:48)). Quite surprisingly, we find that this action
(cid:96) (cid:96) (cid:96)
is again diagonal in the index (cid:96)! Furthermore, the structure constant can be written as the
sum of these components
(cid:88)
C(m1,m2,m3)(a ,a ,a ) = C(m1,m2,m3)(a ,a ,a ) (27)
1 2 3 (cid:96) 1 2 3
(cid:96)
We define the three-point correlation functions of our putative 2d TFT as
(cid:88)
(cid:104)V(m1)(a )V(m2)(a )V(m3)(a )(cid:105) = (cid:104)V(m1)(a )V(m2)(a )V(m3)(a )(cid:105) (28)
1 2 3 1 2 3 (cid:96)
(cid:32) (cid:33)−1/2
3
(cid:89)
(cid:104)V(m1)(a )V(m2)(a )V(m3)(a )(cid:105) = N(mi)(a ) C(m1,m2,m3)(a ,a ,a ) (29)
1 2 3 (cid:96) i (cid:96) 1 2 3
i=1
The normalization factor, is such that in the 2d TFT the gluing is done with the measure
factor N(m)(a) = 1 − a2qr−2m+a−2q2m. Note that the m dependence of the measure factor
1+qr
can be absorbed by taking a → aqm. Furthermore, note that for q → 1, the measure factor
reduces to the usual SU(2) Haar measure.
Even though the correlation functions of the 2d TFT do not factorize into functions of
the a , the orthonormality properties (18) imply the following form
i
2
(cid:88)
(cid:104)V(m1)(a )V(m2)(a )V(m3)(a )(cid:105) = c(R)(q)f(R)(q)U(m1)(a )U(m2)(a )U(m3)(a ) (30)
1 2 3 (cid:96) (cid:96),ijk (cid:96),i 1 (cid:96),j 2 (cid:96),k 3
i,j,k=1
where we have stressed the fact that the functions c(R)(q) and f(R)(q) will depend on q.
(cid:96),ijk
Besides the explicit dependence on the m in the functions U(mi)(a ), the value of f(R)(q)
i (cid:96),i i (cid:96),ijk
can change as we jump from one ”region” to another. Here R runs over the possible regions
9
and the values of the m fix the region we are at. Let’s for simplicity consider the case in
i
which r is odd and assume 0 < m ≤ m ≤ m < r/2. In this case:
1 2 3

 I m +m +m = r
 1 2 2

R = II m +m = m (31)
1 2 3


 III other cases
This division into regions is of course expected, due to the presence of [m.s] in (9), namely,
r
within a region we have a ’continuous dependence’ on the m , but the expression ’jumps’
i
when we cross to a different region. The normalization factor c(R) has been pulled out in
(R)
order to have f = 1 and is given by
(cid:96)=1,111
1 1−qr
c(I)(q) = c(II)(q) = √ , c(III)(q) = √ (32)
1+qr 1+qr
The structure constants have certain symmetry properties under permutation of the
(R) (I)
holonomies m . This implies certain symmetries among the functions f , namely f
i (cid:96),ijk (cid:96),ijk
(III) (II)
and f are invariant under permutation of i,j,k and f is invariant under the in-
(cid:96),ijk (cid:96),ijk
terchange of i and j. Furthermore, up to a very high order in the q−expansion, we have
checked the additional symmetries
(II) (II) (II) (III) (III) (III) (III)
f = f = f = f = f = f = f (33)
(cid:96),111 (cid:96),122 (cid:96),212 (cid:96),111 (cid:96),122 (cid:96),212 (cid:96),221
(II) (II) (II) (III) (III) (III)
f = f = f = f = f = f = 0 (34)
(cid:96),112 (cid:96),121 (cid:96),211 (cid:96),112 (cid:96),121 (cid:96),211
All in all, at each order (cid:96) and for generic r, the structure constants depend on eight
functions of the fugacity q (for small values of r not all the regions are present). Let us
introduce the following notation
10