The topologically twisted index of \( \mathcal{N} \) = 4 superYangMills on T 2 × S2 and the elliptic genus
HJE
The topologically twisted index of superYangMills on
Junho Hong 0 1 2
James T. Liu 0 1 2
0 Ann Arbor , MI 481091040 , U.S.A
1 The University of Michigan , USA
2 Leinweber Center for Theoretical Physics, Randall Laboratory of Physics
We examine the topologically twisted index of N = 4 superYangMills with gauge group SU(N ) on T 2 S2, and demonstrate that it receives contributions from multiple sectors corresponding to the freely acting orbifolds T 2=Zm summing over these sectors, the index can be expressed as the elliptic genus of a twodimensional N = (0; 2) theory resulting from KaluzaKlein reduction on S2. This provides an alternate path to the `hightemperature' limit of the index, and con rms the connection to the rightmoving central charge of the N = (0; 2) theory.
Supersymmetric Gauge Theory; AdSCFT Correspondence; Conformal Field

T 2
S2
and the elliptic genus
Theory
1 Introduction
2
3
2.1
2.2
3.1
3.2
Multiple solutions to the BAEs
The topologically twisted index
The index as an elliptic genus
The topologically twisted index in the `hightemperature' limit
4.1
4.2
Expanding Zfm0;n0;r0g in the `hightemperature' limit
Examination of the determinant factor
4.3
The full index in the `hightemperature' limit
4.2.1
4.2.2
4.2.3
4.3.1
4.3.2
O(1) order determinant
First subexponential order determinant
B Invariance of det B under T and S transformations
C Proof that the map (3.34) is bijective
1
Introduction
Recent advances in supersymmetric eld theories have led to a new era of precision
holography through AdS/CFT. It has been driven on the eld theory side by the key developments
of rigid supersymmetry [1] and supersymmetric localization [2]. The former allows for a
systematic construction of supersymmetric eld theories on a curved background with
topological twists. The latter yields exact eld theory results reliable even at strong coupling
limit. Combining these two developments, now we can compute exact eld theory results
of various topologically twisted SCFTs on curved backgrounds, which can be explored in
{ 1 {
Sd 1 which computes the supersymmetric index
as well as Wilson loop observables in various representations of the gauge group.
In particular, a threedimensional topologically twisted index was introduced as the
supersymmetric index on S1
S2 with a topological twist on S2 [3].
When applied to
the ABJM theory [
4
], it has an interesting feature. In the largeN limit, the topologically
twisted index of the ABJM theory on S1 S2 matches the entropy of the dual asymptotically
AdS4 magnetic black hole, when it is extremized with respect to the chemical potentials [5].
This is regarded as the rst counting of the microstates of a supersymmetric asymptotically
AdS4 black hole.
Similarly, the fourdimensional topologically twisted index can be introduced as the
supersymmetric index on T 2
S2 with a topological twist on S2. In particular, we can
apply this to N = 4 SU(N ) superYangMills (SYM) with a similar goal in mind, namely
counting the microstates of the dual asymptotically AdS5 magnetic black string. This is
still an open problem, however, and here we review some of the unsolved issues in both the
eld theory and supergravity sides of the duality.
Field theory side.
The topologically twisted N = 4 SU(N ) SYM on T 2
S2 can be
constructed by equipping S2 with background gauge elds that couple to the SO(6)
Rsymmetry current, satisfying the conditions categorized in [6]. The explicit computation of
the topologically twisted index in the large N limit, however, has not yet been performed
unlike in the ABJM theory case. Instead, it has been investigated in the `hightemperature'
! 0+, where the modular parameter of the torus is given by
studied in [6, 8]. To be speci c, based on the well known duality between N = 4 SU(N )
SYM on R1;3 and Type IIB supergravity in AdS5
S5 background, we may expect that the
same eld theory on T 2
S2 with topological twists is holographically dual to Type IIB
supergravity in an asymptotically AdS5 magnetic black string background with conformal
boundary T 2
S2. The AdS3
S2 nearhorizon solution for the string is known and
numerical evidence suggests that it can be extended into a full solution [6]. However, a full
analytic supergravity solution with such asymptotic conditions has yet to be constructed.1
At this stage, we focus on the eld theory side by taking a closer look at the
topologically twisted index of N = 4 SU(N ) SYM on T 2
S2. As demonstrated in [7], the
hightemperature limit of the index, when extremized over the chemical potentials
a,
matches the rightmoving central charge of the N = (0; 2) SCFT associated with the AdS5
magnetic black string
< log Z
=
i
2
; a; na
=
cr(na);
!0+
6
2
(1.1)
1Asymptotically AdS5 black hole solutions with conformal boundary R
S3 have been constructed
in [9, 10]. Even in this case, however, matching its entropy with microstate counting in the largeN limit
of the dual eld theory has not yet been done due to various issues.
{ 2 {
HJEP07(218)
where fnag are integer magnetic charges satisfying P3
negative [6]. In a way, this is not surprising, as (1.1) is just the expected behavior in the
Cardy limit of the SCFT. Away from this limit, however, the index must transform as
a weak Jacobi form. This can be seen by KaluzaKlein reducing on S2, whereupon the
supersymmetric index on T 2 becomes the elliptic genus [11].
In this paper, we clarify the connection between the topologically twisted index of
N = 4, SU(N ) SYM on T 2
S2 and the elliptic genus. As constructed in [7], the index
can be computed using Je reyKirwan residues. The result is thus given in terms of a
sum over solutions to a set of algebraic equations, commonly referred to as the `Bethe
ansatz equations' (BAEs). In contrast to the S2
S1 index, where there is only a single
solution to the BAEs (up to permutations) [3, 5], here we nd multiple solutions, where
the `eigenvalues' are uniformly distributed over the T 2. Furthermore, the existence of these
multiple solutions is fundamental in order for the index to be an elliptic genus.
Once the index is understood as an elliptic genus, we revisit the hightemperature
! i0+, by performing the modular transformation
!
our results are left at the conjecture level, we reproduce the Cardy limit (1.1), where
1= . Although some of
cr(na) = 3(N 2
1)
1
n1n2n3
(n1n2 + n2n3 + n3n1)
;
(1.2)
in agreement with [7]. Since this expression is valid for arbitrary N , it also holds in the
largeN case with holographic dual. More generally, however, it would be interesting to
explore the largeN limit at arbitrary values of the modular parameter . Unfortunately,
this still appears to be a rather challenging problem, as the only expression we have for
the index at arbitrary N is given as a sum over sectors, each corresponding to a di erent
solution to the BAEs.
The outline of the paper is as follows. In section 2, we rst review the topologically
twisted index of N = 4 SYM on T 2
S2, then demonstrate that the BAEs admit multiple
solutions. In section 3, we connect the index to the elliptic genus and in particular
demonstrate that it transforms as a weak Jacobi form. Given this understanding of its modular
properties, we then revisit the hightemperature limit in section 4. Finally, we conclude
with some comments on the largeN limit in section 5.
2
The topologically twisted index of N
= 4 SYM on T 2
S2
The topologically twisted index of N
= 4 SYM with gauge group SU(N ) was de ned
in [3, 7] as the supersymmetric index of the theory on T 2
S2 with a topological twist
on S2. The index depends on the modular parameter q = e2 i as well as avor chemical
potentials
a and magnetic uxes na, and may be written as [7]
Z( ; a; na) = A
X
1
I2BAEs det B
N
Y
3
Y
j6=k a=1
1(uj
1(uj
uk; )
uk +
a; )
1 na
;
where the prefactor A is given by
(2.1)
(2.2)
A = iN 1 ( )3(N 1) Y
1( a; ) (N 1)(1 na):
3
a=1
{ 3 {
The sum in (2.1) is over all solutions, I = fu0; u1; : : : ; uN 1; vg, of the `Bethe ansatz
the ui's sum to zero.
constrained to satisfy
While this Jacobian is explicitly constructed from N
1 of the N eigenvalues ui, it is easily
seen that it does not depend on which one is omitted because of the SU(N ) condition that
According to [7], the avor chemical potentials
a and the magnetic uxes na are
eiBj = 1
a; )
a; )
:
B
; BN )
; uN 1; v)
:
3
X
a=1
Note that the uj 's are also constrained to satisfy the SU(N ) condition PjN=01 uj = 0. In
up to sign and we will x these degrees of freedom later according to our purpose.
{ 4 {
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)
(2.8)
=N .
Here we exclude
a = 2 Z in order to avoid issues with the vanishing of 1(0; ) but we
do not necessarily assume
invariant under
a 2 R or 0 < < a < 2 . Instead, the twisted index (2.1) is
2.1
Multiple solutions to the BAEs
A solution to the BAEs, (2.3), was obtained in the `hightemperature' limit,
under the condition P3
a=1 a = 2 . It can be written as
! i0+, in [7]
uj = u
2 ~j;
v = (N + 1) ;
where u is a constant chosen to enforce the SU(N ) condition Pj uj = 0, and ~ =
While this solution was obtained in the hightemperature limit, it actually satis es the
BAEs for any
in the upper half plane and for arbitrary
a's satisfying the constraint
P3
a=1
Furthermore, we show below that (2.8) is in fact a special case of a larger set of BAE
solutions. The key observation is that the uj variables are doubly periodic, as they are
de ned on T 2, with periods uj ! uj + 2 and uj ! uj + 2
. Based on this periodicity,
the solution (2.8) then corresponds to the uj 's being evenly distributed along the thermal
circle. This de nes the torus T 2=ZN with modular parameter ~ =
=N . Then modular
invariance suggests that having the uj 's evenly distributed along the other cycle of the T 2
ought to yield another solution, this time with modular parameter ~ = N .
Taking this one step further, we expect that any uj 's evenly distributed over the torus
T 2 satisfy the BAEs, (2.3). In this case, the set of uj 's de nes a freely acting orbifold
T 2=Zm
Zn where fmg is the set of all positive divisors of N with N = mn. The
corresponding uj 's can be written explicitly as
Note that we have introduced a double index notation
uj
1 is a constant that, along with m and n, speci es the orbifold.
In order to prove that (2.9) indeed satis es the BAEs, we substitute it into (2.4), so
that the BAEs reduce to the claim that
(2.10)
(2.11)
(2.12)
!
:
(2.13)
eiv =! Y
3 m 1 n 1 1
Y
Y
We now use the double periodicity of 1, (A.5), to shift the product over ^j and k^ as
an appropriately chosen v.
determine v by choosing ^j0 = m
1 and k^0 = n
the identity
a + 2 ^j+mk^~ ;
Inserting this into the r.h.s. of (2.11) and using the constraint P3
a=1 a = 2 Z, then gives
eiv =! Y
3 m 1 n 1 1
Y
Y
In particular, the r.h.s. is now manifestly independent of ^j0 and k^0, thus demonstrating
that the full set of BAEs reduce to a single equation that can be consistently satis ed for
While this is su cient to demonstrate that (2.9) satis es the BAEs, we can explicitly
{ 5 {
1 in (2.12) with the upper sign to obtain
= ei[(N+1) +(n 1)m a]:
(2.14)
Inserting this into (2.13), taking the product over a and reducing the exponent then gives
v = (N + 1) , which is also compatible with the solution (2.8) of [7].
As a result, we have found multiple solutions to the BAEs, (2.9), labeled by three
integers m, n, and r such that N = mn and r = 0; : : : ; n
1. While we have not proven
that these are the complete set of solutions to the BAEs (up to permutations), we argue
below in section 3 that they are in fact complete based on modular covariance of the index.
We now compute the topologically twisted index for a particular sector labeled by fm; n; rg
by inserting the solution (2.9) into (2.1). Making this substitution gives
where the primes indicate that ^j1 = k^1 = 0 is to be omitted from the double product. The
product over ^j1 and k^1 can be shifted using (2.12) as follows:
m ^j2 1 n k^2 1
Y0
Y0
^j1= ^j2 k^1= k^2 1
As a result, we have
1 2 ^j1+k^1 ~ ;
m
a + 2 ^j1+k^1 ~ ;
m
= e imk^2 a Y0 Y0
The product of the theta functions can be simpli ed by using the product form of 1(u; )
given in (A.2). We nd
mY10 nY10 1 u + 2 ^j+mk^~ ;
Substituting these expressions and (2.2) into (2.17) then gives
Zfm;n;rg =
i
3 "
Y
det Bfm;n;rg a=1
it is convenient to maintain the original single index notation for the uj 's. Noting that (2.5)
singles out u0 as the constrained variable, the entries of the matrix are
uj ; a; )
g(u
u ; a; ) + g(u
u0; a; ); (2.21a)
uj ; a; )
g(u0
u ; a; ) + g(0; a; );
where ;
2 f1; 2; : : : ; N
1g. Here we have de ned
Since g(u; a; ) is an even function of u, we can derive the identities
g(u; a; )
log 1( a + u; ) 1( a
u; ) :
and
Bj;0 = N:
X
j=0
det B = N det
; BN 1)
; uN 1)
B ;
B0;
B ;0
B0;0
(2.22)
(2.23)
(2.24)
(2.25)
#
;
(2.26)
(2.27)
(2.28)
HJEP07(218)
Consequently, we have
Therefore it is enough to study the determinant of the (N
(N
1)square matrix
whose entries are given by (2.21a).
At this stage, we return to index pair notation given in (2.9) by
u ; a; )
!
then de ne the Gfunction as
; N
j=0
3
i X
a=1
log
1
a + 2
Accordingly, the sum in (2.21a) can be written in terms of index pair notation as
where
= n^j + k^ . Now changing the summation over ^j and k^ into a product within the
log and inserting (2.12) then gives
so that the sum in (2.21a) is in fact independent of which entry
is being considered.
Simplifying the product of theta functions within the log using (2.18), we get
where the prime denotes di erentiation with respect to the rst argument of 1. Finally,
inserting (2.30) into (2.21a), we can rewrite (2.21a) as
[Bfm;n;rg] ; =
!
h
IN 1 + B~fm;n;rg ;
i
(2.31)
where B~fm;n;rg is an (N
1)
1) square matrix with entries
[B~fm;n;rg] ; = Gfm;n;rg(^j ; k^ ; a; )
Then (2.24) leads to
det Bfm;n;rg = N
det(1 + B~fm;n;rg):
(2.33)
Finally, the contribution to the topologically twisted index from the sector labeled by
fm; n; rg is given by combining (2.20) with (2.33),
Zfm;n;rg( ; a; na) =
3 "
Y
a=1
"
3
a=1
#N 1
:
(2.34)
3
The index as an elliptic genus
As we have seen above, there are multiple solutions to the BAEs, each labeled by a set of
integers fm; n; rg, corresponding to the modding out of the original T 2 by a freely acting
Zm
Zn action. The sum over these multiple solutions I 2 BAEs in (2.1) is nontrivial,
and explicitly takes the form
Gfm;n;rg(^j; k^; a; ) = 2i X @ a log 1(m a; ~)
(2.30)
X ZfN=n;n;rg( ; a; na);
{ 8 {
(3.1)
where Zfm;n;rg is given in (2.34). In this section, we study this expression further for
arbitrary
and N . In particular, we show explicitly that the index is an elliptic genus,
which can be seen based on reduction over the S2 [11]. Here the sum in (3.1) is crucial to
ensure proper modular behavior of the index, since modular transformations permute the
individual sectors labeled by fm; n; rg.
For example, consider the case N = 6, where the index (3.1) is a sum over the twelve
fm; n; rg = f1; 6; 0g; f1; 6; 1g; f1; 6; 2g; f1; 6; 3g; f1; 6; 4g; f1; 6; 5g;
with corresponding modular parameters
~ =
6
;
+ 1
6
;
+ 2
6
;
+ 3
6
;
+ 1
6
;
+ 2
6
;
+ 3
6
;
+ 4
6
;
+ 4
6
;
+ 5
6
Y
"
( )3
more involved. Consider, for example, the action of S on the f1; 6; 2g sector, with ~ = ( +
1)=6 , and then perform a SL(2; Z) transformation
2)=6. We rst take S: ~ ! ~0 = (2
~0 ! (2~0
1)=(3~0
1) to bring this into the form (2
+ 2)=3, corresponding to the
f2; 3; 2g sector. Of course, the detailed modular properties of the topologically twisted
index depends on how precisely the various building blocks of Zfm;n;rg transform.
Before considering the general case, we gain additional insight from the example of
N = 2. In this case, there are only three sectors, denoted by f1; 2; 0g, f1; 2; 1g and f2; 1; 0g.
The topologically twisted index is then given by the sum
where i = 2; 3; 4 correspond to the f2; 1; 0g, f1; 2; 1g and f1; 2; 0g sectors, respectively.
Then the modular properties of the index can be derived from those of the elliptic theta
functions, i.
Turning to the general case, for the index to be an elliptic genus, it must transform
as a weak Jacobi form of weight zero. Here it is worth recalling that, for a single chemical
potential, a Jacobi form of weight k and index m transforms according to
a^ + 2 for a single
a^. Since 1 picks up a minus sign
for every 2
shift, the numerator of (2.20) picks up a sign ( 1)(1 mN)(1 na^), while the
denominator is unchanged since the logarithmic derivatives of 1 are not sensitive to the
sign. As a result, we nd
Zfm;n;rg ! ( 1)(1 mN)(1 na^)Zfm;n;rg = ( 1)2ma^ ( 1)N(N m)(1 na^)Zfm;n;rg;
where we substituted in the index ma^ from (3.8). Writing N = mn then gives N (N
m2n(n
1), which is an even integer. Thus the second factor above is simply +1, and we
are left with Zfm;n;rg ! ( 1)2ma^ Zfm;n;rg, in agreement with (3.7a). Note that this result
, we rst consider the numerator factors in (2.34)
usis valid even if we only shift a single
a^
.
For the shift
a^ !
a^ + 2
ing (A.5). For 1( a^; ), we nd simply
where ya^ = ei a^ . For 1(m
a^; ~), we rst write
and use the relation q~n = e2 in~ = e2 irqm to obtain
1( a^ + 2 ; ) =
q 1=2ya^ 1 1( a^; );
1(m( a^ + 2
); ~) = 1(m
a^ + 2 (n~
r); ~) = ( 1)r+nq~ n2=2ya^ N 1(m
a^; ~); (3.11)
and indices
under the constraint P
a + 2 and
T :
!
a !
+ 1 and S :
a + 2
!
both N and na are even, and an integer otherwise.
It is straightforward to generalize this to the case of three chemical potentials, and we
verify below that the index (3.1) indeed transforms as a weak Jacobi form of weight zero
ma =
N 2
2
1
(1
na);
a a = 0. To do so, we rst consider the periodic shifts
for (3.7a), and next consider the modular transformations
1= for (3.7b). Note that the index ma is a halfinteger when
(3.8)
a !
(3.9)
m) =
(3.10)
(3.12)
(3.13)
(3.14)
(3.15)
HJEP07(218)
1(m( a^ + 2
); ~) = ( 1)n+r(n+1)q N=2ya^ N 1(m
a^; ~):
This demonstrates that the numerator picks up an overall factor
h
( 1)1 N(n+r(n+1))q(N2 1)=2yN2 1i1 na^
;
a^
under a shift of
. As above, the sign factor can be rewritten as
1
N (n + r(n + 1)) =
=
(N 2
(N 2
1) + N (n(m
1) + n2m(m
1)
1)
r(n + 1))
rmn(n + 1):
Since the last two terms in the nal expression are even, they do not contribute to the
overall sign, and we are left with
Zfnmum;ne;rrg ! ( 1)2ma^ q ma^ ya^
2ma^ Zfnmum;ne;rrg;
which is the expected result for a Jacobi form of index ma^ given by (3.8).
Since the numerator by itself transforms properly under the shift of
that the denominator must be inert under this shift. This is not entirely obvious, though,
as the logarithmic derivatives of 1 transform as
i;
a^; ~)
iN;
(3.16)
as can be seen directly from (3.10) and (3.12). The sum of logarithmic derivatives, however,
is invariant so long as we simultaneously shift another chemical potential, say
^b, by
2
,
since then these additional factors will cancel. Therefore the denominator is invariant under
this combined shift, and hence (3.13) extends to Zfm;n;rg itself. Note that this simultaneous
shift is in fact required to maintain the condition that the
a^'s sum to 2 Z.
3.2
Modular transformations
We now turn to the properties of the topologically twisted index under modular
transformations. Since a general transformation can be generated by a combination of T and S, it
is su cient for us to demonstrate the following properties:
These follow from the de nition (3.7b) for a Jacobi form of weight zero and indices ma for
the chemical potentials
T transformation
We begin with the T transformation. As indicated in (3.17a), we expect the partition
function to be invariant under T . Nevertheless, the individual sectors labeled by fm; n; rg
will get permuted, as in the N = 6 example shown in (3.4). We thus work one sector at a
time, and in particular consider the T transformation of Zfm;n;rg.
To proceed, we consider the expression (2.20), and observe that the numerator is built
from the combination
(3.18)
(3.19)
(3.20)
which transforms as a weak Jacobi form of weight
1 and index 1=2, as can be seen
from (A.4). For ( a; ), we have simply
However, the transformation is not as direct for (m
a; ~), since T : ~ ! ~ + m=n, which
is not a SL(2; Z) transformation on ~. In this case, it is more useful to note that
where r0 = r + m (mod n). Since
is invariant under integer shifts of the modular
parameter, we end up with
T : (m a; ~) !
fm0; n0; r0g = fm; n; r + m (mod n)g:
n
n
g
;
Gfm;n;rg(^j; k^; a; + 1) = Gfm0;n0;r0g(^j0; k^0; a; );
^j0 = ^j + k^ r + m
n
(mod m);
k^0 = k^:
S : ( a; ) !
1 i 2a
e 4
Then since the above (^j; k^) ! (^j0; k^0) is a bijective map from Zm
and hence the denominator transforms in the expected manner as well. As a result, T
permutes the sectors without any additional factors, T : Zfm;n;rg ! Zfm0;n0;r0g. Finally,
since fm; n; rg ! fm0; n0; r0g is bijective, it is clear that the full partition function (3.1) is
indeed invariant under T transformations, (3.17a).
3.2.2
S transformation
We now turn to the S transformation, which takes
a !
a= along with
!
1= .
Once again, we start with the numerator. Since
(u; ) de ned in (3.18) is a weak Jacobi
form of weight
1 and index 1=2, we immediately have
where
with
The combination of (3.19) and (3.21) then demonstrates the simple transformation
T : Zfnmum;ne;rrg ! Zfnmum0;ner0;r0g;
as anticipated in (3.4).
To be complete, we must also investigate the T transformation on the denominator
of (2.20), which comes from the determinant of Bfm;n;rg. Here we use the double
periodicity (A.5) and the modular property (A.3b), to obtain the map
For (m a; ~), it is important to realize that S does not simply take ~ to
1=~. Instead,
we want to map ~ into a new ~0, at least up to a SL(2; Z) transformation. In particular,
we demand
where ~0 = (m0 + r0)=n0. The resulting SL(2; Z) transformation is given by
a =
r
g
;
c =
ad
bc = 1;
g
gcd(n; r);
(3.21)
(3.22)
(3.23)
Inserting this expression along with (3.27) into (2.20) then gives
S : Zfnmum;ne;rrg !
N 1e 2 i Pa ma 2a Znumer
fm0;n0;r0g;
with ma given in (3.8).
The extra factor of
N 1 is canceled by a similar factor arising from det B in the
denominator. For this determinant, we use the double periodicity (A.5) and the modular
property (A.3b), along with the requirement P
a a = 0 to obtain the map
with
Gfm;n;rg(^j; k^; a= ; 1= ) =
Gfm0;n0;r0g(^j0; k^0; a; )
^j0 =
k^0 =
g (k^ + dk^0)
n
g
n
^j +
r
g
^
k
mod
mod g;
N
g
:
In appendix C, we show that the above (^j; k^) ! (^j0; k^0) is a bijective map from Zm
Zn
to Zm0
Zn0 . Therefore, we get (see appendix B)
S : det Bfm;n;rg !
N 1 det Bfm0;n0;r0g;
which cancels the extra factor of N 1 in the numerator. As a result, S permutes the sectors
with a common factor, S : Zfm;n;rg ! e 2
i P
a ma 2a Zfm0;n0;r0g. Then since fm; n; rg !
fm0; n0; r0g is selfinverse and therefore bijective, the full partition function (3.1) transforms
under S transformation as (3.17b).
Finally, we wish to explain why the chemical potentials must sum to zero in order for
the index to be a proper modular form, in particular under the Stransformation: since S
takes
a to
a= , we must demand the simultaneous conditions
and ~0 takes the form
Here b and d are uniquely determined as the solution to (3.29) under the constraint for r0,
r0 < n0. Also note that we can make use of the simple relation c~0 + d = m0 =m, which
can be derived without explicit knowledge of b and d. Given (3.28), we then nd
S : (m a; ~) !
m a r
;
n
m
=
;
m0 a a~0 + b
c~0 + d c~0 + d
=
m
m0
e iN4 2a
(m0 a; ~0):
to satisfy the rst constraint given in (2.6) for both Z( ; a; na) and Z( 1= ; a= ; na),
which only makes sense when P3
of the degrees of freedom introduced in (2.7),
is not a serious restriction on the index.
a=1 a = 0. Of course, we can always use the second type
a !
a + 2 Z, to set P3
a=1 a = 0, so this
(3.30)
(3.31)
(3.32)
(3.33)
(3.34a)
(3.34b)
(3.35)
where
where
Given the construction of the index as a sum over sectors, (3.1), we now revisit the
`hightemperature' limit,
! 0+ with
= i =2 , rst investigated in [7] for the single sector
Zf1;N;0g. Note that, in what follows, we restrict to purely imaginary , corresponding to a
square torus, and real chemical potentials
a. In order to explore this limit, it is natural
to perform an S transformation (3.17b) assuming P
modular parameter has large imaginary part. In particular, we write
a
a = 0 so that the transformed
The partition function Z( 0; 0a; na) receives contributions from individual sectors
Zfm0;n0;r0g as we have seen in (3.1), and we generically expect only one or a handful of
sectors to dominate. To see this, we rst work on the expression for a xed sector, and
then look for the dominant contribution to the sum over sectors.
4.1
Expanding Zfm0;n0;r0g in the `hightemperature' limit
In order to expand Zfm0;n0;r0g, we rewrite (2.34) as
Zfm0;n0;r0g( 0; 0a; na) =
Q
a
( 0a; 0) (m0 0a; ~0) N 1 na
n0 det 1 + B~fm0;n0;r0g
h
The numerator can be easily treated using the asymptotic expression for , (A.9), as
( 0a; 0) =
i( 1)Da exp
4
i 2a +
i
(m0 0a; ~0) =
i( 1)Xa ei nr00 Xa(Xa+1) exp
da(1
da)
1 + O(e 22 min(da;1 da)) ; (4.4a)
iN
4
2
a +
i m0
n0
xa(1
xa)
1 + O(e 22 mn00 min(xa;1 xa)) ;
da
xa
a
2
n0 a
2
Note that these expressions break down if da = 0 or xa = 0, so from now on we assume
da's are not integer multiples of 1=n0 where this does not occur.
For the denominator, we rst examine the logarithmic derivative term in (4.3). So
long as we avoid the special cases xa = 0, the asymptotic expression (4.4b) is di erentiable
with respect to its rst argument, and we obtain
(4.6)
Since P
a a = 0 and we avoid special cases, we see that Xa must generically sum to either
1 or
2. Therefore (4.6) is in fact just
The remaining term, namely det(1 + B~fm0;n0;r0g), is more di cult to analyze. So for the
moment we leave it implicit. In this case, combining the numerator terms (4.4) with (4.6)
and taking into account the prefactor in (4.1) gives
where ' is a phase independent of , and the transformed quantities fm0; n0; r0g are given
Examination of the determinant factor
The asymptotic expression for the index, (4.7), is now complete up to the expansion of the
determinant. Unfortunately, its structure is rather intricate, and we have been unable to
nd a simple universal formula describing its asymptotics. The main issue is the observation
that the high temperature limit of log det(1 + B~fm0;n0;r0g) can be of either O(1) or O(1= ).
This term is relatively unimportant in the former case, but will contribute to the leading
order behavior in (4.7) in the latter case. However, which case the determinant factor is in
depends in a nonobvious manner on the chemical potentials
a and is not easily obtained.
We now proceed with a closer look at the matrix B~fm0;n0;r0g de ned in (2.32). To avoid
unnecessary notation, we will omit the universal arguments ( 0a; 0) = ( a= ; 1= ) and
occasionally the sector labels fm0; n0; r0g, in what follows. In this case, the B~ matrix entries
can be simply written as
[B~fm0;n0;r0g] ; = Gfm0;n0;r0g(^j0 ; k^0 )
;
^j0=0
Pm0 1 Pn0 1
k^0=0 Gfm0;n0;r0g(^j0; k^0)
(4.8)
where we have the index pair associations
! (^j0 ; k^0 ) and
! (^j0 ; k^0 ). At this stage it is
convenient to note that while this is originally an (N
1)
(N
1) square matrix, it can
be extended to an N
N square matrix by including the
= 0 and
= 0 entries. This
is equivalent to allowing ^j0 and k^0 to independently run over 0 : : : m0
1 and 0 : : : n0
1
without removing the (0; 0) pair. Since the rst column of B~ with entries [B~] ;0 vanishes
identically, however, the determinant det(1+ B~) can be viewed either as an (N
1) (N
1)
or an N
N determinant.
Taking the logarithmic derivative of (u; ) and using the asymptotic expansion (A.9)
gives the hightemperature expansion of G
temperature limit of the matrix B~fm0;n0;r0g with entries (4.8). We keep the O(1) and the
, which is necessary to study the
highrst subexponential term, G(^j0; k^0) = G0(^j0; k^0) + Gexp(^j0; k^0) +
, where
1 + Da(k^0=n0) + Da( k^0=n0) ;
G0(^j0; k^0) =
G
exp(^j0; k^0) =
3
X
X
a=1 =
and we have de ned
where the determinant on the left is that of an N
N matrix, while that on the right is of
an n0
n0 matrix.
since B~0f1;1;0g = 0, so that
At this point, we are still left with the n0 n0 determinant to evaluate. However, there
is an important special case corresponding to fm0; n0; r0g = fN; 1; 0g. This case is trivial
det(1 + B~0fN;1;0g) = 1:
The situation is more complicated when n0 6= 1. While we do not have a proof,
numerical evidence indicates that the O(1) order determinant only takes on two possibilities,
depending on the chemical potentials:
da(x)
2
a + x (mod 1);
Da(x)
2
a + x :
one will dominate, depending on the relative magnitudes of da( k^0=n0).
Note that, while Gexp is a sum of twelve exponentially small terms, generically only a single
Given the asymptotic form of G(^j0; k^0), the B~ matrix can be expanded into the sum
of an O(1) matrix and a subexponential one, B~ = B~0 + B~exp. If det(1 + B~0) 6= 0, then we
are essentially done, as it will not contribute at the O(1= ) order in the hightemperature
limit. However, if this vanishes, the subexponential contribution becomes important. We
thus consider the O(1) order determinant rst, before turning to the subexponential one.
4.2.1
O(1) order determinant
For the B~0 matrix, we note that its entries are built from Gfm0;n0;r0g(^j0; k^0), where here we
0
have restored the sector labels fm0; n0; r0g. However, examination of (4.9a) demonstrates
that it is actually independent of m0 and r0 as well as the index ^j0. As a result, we can
write the matrix expression
where U is the m0
m0 square matrix whose entries are all unity. Since U has only one
nonvanishing eigenvalue equal to m0, we then see that
det(1 + B~0fm0;n0;r0g) = det(1 + B~0f1;n0;0g);
~0
Bfm0;n0;r0g =
1
m0
U
~0
det(1 + B~0fm0;n0;r0g) = 0 or n02:
(4.9a)
(4.9b)
(4.10)
(4.11)
(4.12)
(4.13)
(4.14)
HJEP07(218)
~0 = 0, while the gray regions correspond to nontrivial 1+B~0, but still with vanishing determinant.
B
The determinant evaluates to n02 in the unshaded regions. The yellow triange corresponds to the
region 0 < d1
d2
In order to investigate where the determinant vanishes, we take da =
a=2 (mod 1) and
assume none of them are integer multiples of 1=n0 as in (4.5a). Furthermore, without loss
of generality, we let Pa da = 1, which follows from the requirement P
a
a = 0. (The other
a da = 2, can be mapped to this one by taking into account the invariance
For small values of n0, the regions in (d1; d2) parameter space where the determinant
vanishes are shown in
gure 1. Here the (n0
1)
(n0
identically in the black regions. For prime n0, this appears to be the only places where the
determinant vanishes, while for composite n0 there are additional regions with vanishing
determinant but with nontrivial 1 + B~0, represented by the gray regions.
1) matrix 1 + B~0f1;n0;0g vanishes
For general n0, consider that the B~0 matrix is obtained from
G0(^j0; k^0) = <>>0;
8
>
>
>
>
>>1;
>:>2;
1; da2 > min(k^0) and (da1
da1 < min(k^0) and (da2
da2 < min(k^0) < da3 < max(k^0);
da3 < min(k^0);
min(k^0))(da3
min(k^0))(da3
max(k^0)) < 0;
max(k^0)) < 0;
(4.15)
which is a direct consequence of (4.9a). Here da's are ordered as 0 < da1
1 and we have de ned min(k^0) and max(k^0) as the min and max of fk^0=n0; 1
da2
da3 <
k^0=n0g,
respectively. In particular, note that G0(^j0; k^0) =
0;k^0 where
l0
n0
l0 + 1
n0
;
< da2 < da1 + da2 <
with l0 = 0; 1;
;
(4.16)
n0
1
2
;
which corresponds to the black regions in gure 1. Inserting G0(^j0; k^0) =
then explains why 1 + B~0 vanishes identically in these regions. However, the resulting B~0
0;k^0 into (4.8)
matrix outside of the black regions is rather di cult to work with. Nevertheless, for prime
n0, we conjecture based on numerical evidence that det(1 + B~0) = n02 everywhere outside of
the black regions speci ed by (4.16). The case for composite n0 is clearly more complicated,
as can be seen from the gure.
First subexponential order determinant
ponentially suppressed contributions to B~. For prime n0, we can derive
Whenever the O(1) order determinant vanishes, it becomes necessary to examine the
ex1 + B~f1;n0;r0g = exp 2 i
f1;n0;r0g(da) r0
n0
f1;n0;r0g(da)
C( f1;n0;r0g(da));
(4.17)
at leading order whenever we are in the black regions speci ed by (4.16). Here C( ) is an
(n0
1)
(n0
1)square matrix de ned by
[C( )] ; = 2 ; +
; +
;n0
;
;n0
;
; (n0 )
;
(4.18)
and
f1;n0;r0g(da) and
f1;n0;r0g(da) are given by
f1;n0;r0g(da) = <da3
f1;n0;r0g(da) =
8
(n0
l0 + 1;
n0n0 1 ;
n
l0;
:min da2
nl00 ; da3
l0 = 0;
we have det C( ) = n02 for any
= 1;
; n0 1. Combining this result with the conjecture
for the O(1) behavior made above, we nd for prime n0 (excluding the special case)
det(1 + B~f1;n0;r0g) = n02 exp 2 i(n0
1)
f1;n0;r0g(da) r0
n0
f1;n0;r0g(da)
;
(4.22)
Note that here we are excluding the special case da2
nl00 = da3
For prime n0, we can prove det C( ) = n02 for any
= 1; : : : ; n0
1. In particular, we
can rst show det C(1) = n02 by mathematical induction. Then since
with a permutation
( ) =
(mod n0) and the corresponding permutation matrix
C( ) = ~ 1C(1)~ ;
[~ ] ; =
( ); ;
(4.19a)
(4.19b)
(4.20)
(4.21)
at leading order, where we set
regions speci ed by (4.16).
f1;n0;r0g(da) to vanish outside of the black
We now have all the components needed to work out the high temperature expansion
of the index in the fm; n; rg sector provided the corresponding m0 is unity and n0 is prime.
For such a sector, substituting (4.22) into (4.7) yields
da)
and cf1;n0;r0g(da) > 0 is a positive function away from
special values of the chemical potentials da.
We expect that this expression continues to
hold for arbitrary values of fm; n; rg, although we have been unable to obtain a general
expression for the determinant factor
fm0;n0;r0g(da) apart from the above case.
4.2.3
The N = 2 and 3 cases
We now give a couple of examples supporting the results (4.22) and (4.23). For notational
convenience, here we set 0 < d1
d2
d3 < 1 without loss of generality and therefore the
domain in (d1; d2) parameter space shrinks down to the yellow triangle in
gure 1.
For the N = 2 case, we have a total of three sectors, labeled by fm0; n0; r0g = f2; 1; 0g,
f1; 2; 0g and f1; 2; 1g. The determinant in the f2; 1; 0g sector is trivial as seen in (4.13), so
we focus on the f1; 2; r0g case. From (4.8) and (4.9a), we have
det(1 + B~f1;2;r0g) =
1 + 2 P3
a=1 Da(1=2)
up to higher order terms. Due to the constraint P
restricted as
exp
Gf1;2;r0g(0; 1)
and therefore (4.24) leads to (\ " denotes the nonvanishing leading order)
det(1 + B~f1;2;r0g)
When d3 > 1=2, we use the expansion (4.9b) to obtain
exp
Gf1;2;r0g(0; 1)
2 exp 2 i
d3
1=2
where we used the fact that
minfda(1=2); 1
da(1=2)g = d3(1=2) = d3
(d3 > 1=2);
1
2
;
8
<4;
:
r0
2
(4.25)
(4.26)
(4.27)
(4.28)
which is valid for d3 > 1=2. Consequently, we have
which is consistent with (4.22). As a result, the N = 2 index is given by (4.23) with
(4.29)
(4.30)
(4.31)
(4.32)
(4.33)
(4.34)
(4.35)
(4.36)
(4.37)
and therefore we have
det(1 + B~f1;3;r0g)
(9;
d2 < 1=3 < d3 < 2=3;
2Gexp(0; 1)2 + 5Gexp(0; 1)Gexp(0; 2) + 2Gexp(0; 2)2; otherwise:
result is independent of k^0, and is given by
We can pull out the leading order behavior of Gexp(0; k^0) with k^0 = 1; 2 from (4.9b). The
exp
Gf1;3;r0g(0; k^0)
8
d2 < 13 ; d3 > 23 ;
; d2 > 13 ; d3 < 23 ;
where we made use of
minfda(1=3); 1
da(1=3); da( 1=3); 1
da( 1=3)g
G
exp(0; 1) $ G
exp(0; 2) ;
We now turn to the N = 3 index. Here there are four sectors, given by fm0; n0; r0g =
f3; 1; 0g and f1; 3; r0g with r0 = 0; 1; 2. Since the determinant in the f3; 1; 0g sector is trivial
as seen in (4.13), we focus on the f1; 3; r0g case. From (4.8) and (4.9a), we have
up to higher order terms, where we have suppressed the f1; 3; r0g subscript from G(^j0; k^0).
Due to the constraint P
1, the sum Pa(Da(1=3) + Da(2=3)) is restricted as
3
a=1
X(Da(1=3) + Da(2=3)) =
(
1; d2 < 1=3 < d3 < 2=3;
0;
otherwise;
det(1 + B~f1;3;r0g)
9
2
3
;
1
3
;
=
(
d3(1=3) = d3
d2( 1=3) = d2
d2 < 1=3; d3 > 2=3;
; d2 > 1=3; d3 < 2=3:
8>1;
>
<exp h4 i r30
>>:exp h4 i 2r0
3
d3 2=3 i
d2 1=3 i
d2 < 1=3; d3 < 2=3;
d2 < 1=3; d3 > 2=3;
; d2 > 1=3; d3 < 2=3;
which is consistent with (4.22). As a result, the N = 3 index is given by (4.23) with
f3;1;0g = 0;
f1;3;r0g = max(0; d2
1=3; d3
2=3):
4.3
After the above examination of the individual fm; n; rg sectors, we now return to the full
index, (3.1), in the high temperature limit. From (4.23), we expect the leading behavior
of each individual sector Zfm;n;rg to scale exponentially in 1= . Thus, the sectors with the
largest positive coe cient of 1=
will dominate the full index, and the other sectors will
be exponentially suppressed. As a result, we are left with identifying the dominant sectors
and their contribution to the index. Note that the degeneracy, if any, of the dominant
sectors does not contribute to the leading order expansion of the full index.
The high temperature limit was investigated in [7], where the BAE were solved in the
0;
constraint P3
and the result is
As discussed in [7], this is to be extremized with respect to the potentials, da under the
a=1 da = 1. This can be performed by the method of Lagrange multipliers,
log Zf1;N;0g da
=
cr(na) +
;
where the extremum values, da, are given by
da =
na(na
2
1)
;
1
(n1n2 + n2n3 + n3n1);
and cr(na) is the rightmoving central charge of the 2d N = (0; 2) SCFT arising from the
KK compacti cation of the topologically twisted N = 4 SYM over S2 [7]. Here we have
assumed that two of na's are negative so cr(na) given in (1.2) is positive.
Note that the lefthand side of (4.40) only corresponds to a single sector of the full
index. Nevertheless, this connection to the rightmoving central charge suggests that the
f1; N; 0g sector is a dominant one, so that
6
2
log Z da
=
cr(na) +
;
where Z is the full index, and is indeed the only physically relevant quantity to connect
to the central charge. Note that, if log Z is truly dominated by log Zf1;N;0g, then da can
be considered not just an extremum of log Zf1;N;0g, but the full index as well. Hence the
identi cation of the central charge with the extremized index in the `hightemperature'
limit [7] remains valid in the presence of multiple BAE solutions.
Of course, it is still necessary to demonstrate that the f1; N; 0g sector is a dominant
one. To do so, we must show that
(4.39)
(4.40)
(4.41)
(4.42)
at leading order in 1= for any fm; n; rg. This inequality can be written explicitly by
inserting (4.40) into (4.23):
3
X(1
a=1
na)
xa(1
xa)
n02
da(1
da)
2(n0 1)
N 2
fm0;n0;r0g(da);
(4.43)
which originate from the determinant of 1 + B~.
for any fm0; n0; r0g. The di culty in proving this inequality lies in the fm0;n0;r0g(da) factors
fm0;n0;r0g(da) factors depend in a complicated manner on the
extremized potentials da. However, they are always nonnegative and in fact vanish in the
white regions of gure 1. In this case the claim (4.43) reduces to
da(1
da)
0;
(4.44)
for any integers na with the constraints Pa na = 2 and two of them being negative. Note
that the latter is necessary for the 2d SCFT arising from the KK compacti cation to have a
positive rightmoving central charge [6]. Here we prove this reduced claim under the same
constraints, but without the integer condition. To begin with, note that the map (4.40) is
in fact invertible between
fna : X na = 2; two of them are negativeg $ fda : X da = 1; (1=2 d1 d2)2 > d1d2g;
a
for any da within the domain given in (4.45). Now we de ne f (d1; d2) as the l.h.s. of the
above inequality. Then within the subdomain of xed bn0dac, where @daf is well de ned,
we can consider the extremum of f which satis es
) (1
2
da)2
xa(1
xa)
n02
da(1
da) +
1
1
2da
1
2xa
n0
(1
2da) = k; (4.49)
with the inverse map
Hence, using
the above claim can be rewritten equivalently as
na =
2da(2da
1)
1
4(d1d2 + d2d3 + d3d1)
:
(1
na)(1
2da) =
Q3a=1(1
na) > 0;
3
X
a=1 1
1
2da
xa(1
xa)
n02
da(1
da)
0;
(4.45)
(4.46)
(4.47)
(4.48)
where k is some constant independent of a. At this extremum, the determinant of the
Hessian is given by
16k2
(1
2d1)(1
2d2)(1
2d3)
< 0;
(4.50)
so it is in fact a saddle point. Note that we have used da1
da2 < 1=2 < da3 , ordered as
before, which is valid in the domain given in (4.45). This implies the minimum of f within
the subdomain of xed bn0dac must stay on its boundary. If one investigates the values of f
on this boundary, it is straightforward (though tedious) to check that f is minimized where
xa1 ! 0+; xa2 ! 0+; xa3 ! 1
xa1 ! 0+; xa2 ! 1 ; xa3 ! 1
for
for
3
X
a=1
3
X
a=1
bn0dac = n0
bn0dac = n0
1;
2:
For both cases, we have
f !
3
X
a=1
da(1
1
2da
da)
na
1
na
0;
which proves (4.48) and thereby the claim (4.43) in the white regions of gure 1 where
fm0;n0;r0g(da) vanishes.
4.3.2
The N = 2 and 3 cases
Of course, we are left to deal with the regions where
fm0;n0;r0g(da) is strictly positive. In
this case, the inequality (4.43) is stronger than the reduced claim (4.44), and the above
proof no longer applies. In the absence of a general expression for
verify (4.43) for N = 2 and 3, and leave the general case for N
we set 0 < d1
d2
d3 < 1 without loss of generality as in 4.2.3.
fm0;n0;r0g(da), we only
4 as a conjecture. Here
For N = 2, it su ces to prove the inequality (4.43) for fm0; n0; r0g = f1; 2; r0g. Inserting
fx1; x2; x3g = f2d1; 2d2; 2d3
1g and
f1;2;r0g(da) = d3
1=2 into (4.43) then reduces the
claim to
(n3
1)(n1n2
1)
0;
which is true since n2
n1
1 and n3
4. Hence the claim is proven for N = 2.
For N = 3, it su ces to prove the inequality (4.43) for fm0; n0; r0g = f1; 3; r0g. Inserting
fx1; x2; x3g = <
f1;3;r0g(da) =
8>>f3d1; 3d2; 3da
f3d1; 3d2; 3da
>>:f3d1; 3d2
(
d3
d2
into (4.43) and examining the resulting expression then proves the claim for N = 3.
1 ;
g
2 ;
g
(d2 < 1=3; d3 < 2=3);
(d2 < 1=3; d3 > 2=3);
g
1; 3da
1 ; (d2 > 1=3; d3 < 2=3);
2
3
1
3
; (d2 < 1=3; d3 > 2=3);
; (d2 > 1=3; d3 < 2=3);
(4.51)
(4.52)
(4.53)
HJEP07(218)
(4.54)
(4.55a)
(4.55b)
Our main observation is that the BAEs for the topologically twisted index for N = 4,
SU(N ) SYM on T 2
S2 have multiple solutions labeled by three integers m, n, and r such
that N = mn and r = 0; : : : ; n
1. Modular covariance of the index is only achieved after
summing over a complete set of these solutions. Taking this into account, we veri ed that
the index gives the elliptic genus of the (0; 2) theory [11, 12], which transforms as a weak
Jacobi form of weight zero. Based on this observation, we expect that the BAEs for general
supersymmetric indices where there is a T 2 factor will similarly admit multiple solutions.
This is equivalent to having multiple saddle points in the matrix integrals that arise from
localization of the path integral.
Multiple solutions of the BAEs, however, make it rather di cult to compute the index
explicitly. This is because we have to sum over all possible contributions to get the full
index (3.1). We conjecture that the contribution from a single sector, namely Zf1;N;0g,
will dominate in the `hightemperature' limit when extremized with respect to the
avor
chemical potentials
a, giving the result (4.41), which connects the index to the central
charge of the (0,2) theory. However, we have been unable to demonstrate this in full
generality because of the di culty in computing det Bfm0;n0;r0g in this limit.
This connection between the hightemperature limit of the topologically twisted index
and the rightmoving central charge, (1.2), was derived with the assumption that cr(na) > 0.
On the holographic side, positivity of the central charge is necessary for a good AdS3
supergravity solution to exist. However, it may be interesting to explore the case when
S2
a single magnetic charge na is negative, corresponding to cr(na) < 0 after extremization.
While the holographic dual is not obviously wellde ned, the
eld theory may still be
interesting on its own. In this situation, the f1; N; 0g sector may no longer dominate, and
additional sectors will have to be considered as well.
We were initially drawn to the topologically twisted T 2
S2 index because of our
interest in its largeN limit. This limit, however, is somewhat delicate, as the sum over
sectors involves the modular parameter ~ = (m
+ r)=n with N = mn. The di erent
sectors then have =~ ranging from (= )=N ! 0 to N =
! 1 for xed
limit. Similar to the hightemperature limit, we may expect O(N 2) contributions to arise
from the =~ ! 0 sectors, and in particular the f1; N; 0g sector. However, for nite modular
parameter , the
nal result ought to remain a weak Jacobi form of weight zero, as the
in the largeN
Cardy limit would not yet have been taken.
Assuming progress can be made with the largeN limit, this would allow us to
investigate the partition function for microstate counting of the dual magnetic black string, in
analogy with the AdS4 black hole story of [5]. However, an analytic supergravity solution
has not yet been constructed. (See [13] for a singular magnetic string and [6] for a
numerical solution.) So in order to complete the picture, it would be worth obtaining such a
solution that interpolates from an AdS3
S2 nearhorizon geometry [6] to asymptotic AdS5
with conformal boundary T 2
S2. If such an analytic solution can be found, an interesting
follow up would be to compare the log N term in the index with the corresponding
oneloop supergravity result. (See [14{16] for recent work on the topologically twisted index
for ABJM theory.) This would, however, require a more careful computation of log det B
than what we considered above, and hence may remain an open challenge.
Acknowledgments
The motivation to explore the largeN limit of the topologically twisted index on T 2
S2
arose out of conversations with L. Pando Zayas. We wish to thank S. M. Hosseini, F.
Larsen, L. Pando Zayas and V. Rathee for enlightening discussions and N. Bobev for
interesting comments. This work was supported in part by the US Department of Energy
under Grant No. DESC0007859.
A
Elliptic functions
Let q = e2 i and x = eiu. Then the Dedekind eta function is given by
1 1
1
n= 1
1
n= 1
1
x 21 ) Y(1
n=1
1(x; q) = 1(u; ) = iq 8 (x 2
x 1qn)
= i
X ( 1)nxn+ 12 q 12 (n+ 12 )2:
The Jacobi theta function 1 is given by
(A.1)
(A.2)
(A.3a)
(A.3b)
(A.4a)
(A.4b)
(A.5)
(A.6)
These elliptic functions satisfy the following modular properties
( + 1) = ei =12 ( );
1(u; + 1) = ei =4 1(u; );
( 1= ) = p i ( );
1(u= ; 1= ) = i
p i eiu2=4
1(u; ):
These modular properties, (A.3), can be extended to general SL(2; Z) transformations
u
1
;
=
c + d ( );
= 3pc + d e 4 i(ccu2+d) 1(u; );
where is a 24th root of unity.
In addition, 1 is quasidoubly periodic with (p; q 2 Z)
1(u + 2 (p + q ); ) = ( 1)p+qe ique i q2
1(u; ):
In the text, we have introduced the weak Jacobi form of weight 1 and index 1=2,
(u; )
(This is the squareroot of the unique weak Jacobi form of weight
2 and integer index 1,
sometimes denoted ' 2;1.) This can be expanded for =
1 (ie jqj
1), with the result
(u; )
Note that this expansion breaks down for integer `. We can also rewrite this expansion as
(u; )
0 =
(mod 1) = 1
;
`0 =
=
1:
u2
2
2
B
Invariance of det B under T and S transformations
Here we demonstrate that det B transforms according to (3.26) and (3.35) under T and S
transformations, respectively. We rst note that the eigenvalues for the BAE solution
denoted by fm; n; rg are canonically ordered according to (2.25). The key step here is then to
order the eigenvalues for the BAE solution denoted by fm0; n0; r0g di erently, according to
fm; n; rg sector : un^j+k^
fm0; n0; r0g sector : un^j+k^
!
!
Note that (^j; k^) ! (^j0; k^0) is a bijective map from Zm
S transformation cases so the above ordering for fm0; n0; r0g sector is valid. Furthermore,
it does not a ect the determinant of the B matrix as the determinant does not depend on
eigenvalue ordering.
Now we prove, with respect to the above ordering,
Bfm;n;rg( a; + 1) = Bfm0;n0;r0g( a; );
Bfm;n;rg( a= ; 1= ) =
N 1
Bfm0;n0;r0g( a; );
(A.7)
(A.8)
(A.9)
(A.10)
(B.1a)
(B.1b)
(B.2a)
(B.2b)
(B.3a)
(B.3b)
as unity for l 2 f1;
su ces to show
which automatically yields (3.26) and (3.35) respectively. Note that fm0; n0; r0g are di erent
for T and S cases. From (2.21), the (l; N ) entries of the l.h.s. and the r.h.s. are the same
; N g. In order to prove that the remaining entries also match, it
k^0; a; + 1) = Gfm0;n0;r0g(^j0
k^0; a= ; 1= ) =
for any ^j; ^j0 2 Zm and k^; k^0 2 Zn. Note that these are not trivial from (3.24) or (3.33) but
can be proved based on those relations and the following properties of the Gfunction:
Gfm;n;rg(^j + m; k^; a; ) = Gfm;n;rg(^j; k^; a; );
l:h:s: = Gfm;n;rg
^j0 + r
; fk^
k^0; ng;
a
;
Proof of (B.3a).
l:h:s: = Gfm;n;rg
= Gfm0;n0;r0g
= Gfm0;n0;r0g
= r:h:s:
(
^j + k^
^j0 + r
^j0 + r
$ ^
k
$ ^
k
m + r
k^0 % ; m
k^0 %
; m
Here fA; Bg denotes A mod B (0
the 3rd lines. The 2nd line comes from (3.24).
Proof of (B.3b).
; fk^
k^0; ng; a; + 1
m + r
; m
; fk^
k^0; ng; a;
; m ; k^
k^0; a;
A < B). Note that (B.4) has been used in the 1st and
!
)
1
!
N
g
; g ;
(B.4b)
!
(B.5)
Gfm0;n0;r0g
Gfm0;n0;r0g
n (^j
g
n
n
k^ + d
n
k^0 + d
r (k^
n
g
^j +
g
^j +
r
g
^ N
g
n (^j
N
g
r
g
^ N
g
r (k^
!
^j0) +
; a;
; g
n
g ^j0 +
r
g k^0; g
N
n
g ^j0 +
r
g k^0; g
N
; g ;
; a;
!
= r:h:s:
(B.6)
Note that (B.4) has been used in the 1st and the 4th lines. The 2nd line comes from (3.24)
followed by the identity M fA; Bg = fM A; M Bg.
C
Proof that the map (3.34) is bijective
First we prove that (3.34) is onetoone, i.e.
^j10 = ^j20 & k^10 = k^20
)
^j1 = ^j2 & k^1 = k^2:
(C.1)
To begin with, note that (3.34a) implies
which means k^1 = k^2 in fact. Combined with this fact, (3.34b) implies
Zn0 (m0 = g; n0 = N=g). To begin with, recall that we have
Next we prove that (3.34) is onto, i.e. there exists (^j; k^) 2 Zm
Zn satisfying (3.34)
)
)
g
n
( b) +
(d) = 1:
Then for any given (^j0; k^0) 2 Zm0
Zn0 , we have
This can be rewritten as
g
bk^0 +
r ^j0 +
g
= k^0:
n ^j0
r
dk^0
k^0 =
n (
bk^0 +
r ^j0 + r
$ dk^0
ng ^j0 %
; m
n ^j0; n ;
N )
:
As in appendix B, fA; Bg denotes A mod B (0
check that
A < B). Now it is straightforward to
^j =
k^ =
r ^j0 + r
$ dk^0
ng ^j0 %
)
; m
2 Zm;
n ^j0; n
2 Zn;
(C.2)
(C.3)
(C.4)
(C.5)
(C.6)
(C.7a)
(C.7b)
truly satisfy (3.34), so (3.34) is onto.
Open Access.
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
[1] G. Festuccia and N. Seiberg, Rigid supersymmetric theories in curved superspace, JHEP 06
(2011) 114 [arXiv:1105.0689] [INSPIRE].
[2] V. Pestun, Localization of gauge theory on a foursphere and supersymmetric Wilson loops,
Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].
[3] F. Benini and A. Za aroni, A topologically twisted index for threedimensional
supersymmetric theories, JHEP 07 (2015) 127 [arXiv:1504.03698] [INSPIRE].
[arXiv:0806.1218] [INSPIRE].
localization, JHEP 05 (2016) 054 [arXiv:1511.04085] [INSPIRE].
JHEP 06 (2013) 005 [arXiv:1302.4451] [INSPIRE].
and black strings in AdS5, JHEP 04 (2017) 014 [arXiv:1611.09374] [INSPIRE].
and a no go theorem, Int. J. Mod. Phys. A 16 (2001) 822 [hepth/0007018] [INSPIRE].
[hepth/0401042] [INSPIRE].
048 [hepth/0401129] [INSPIRE].
arXiv:1504.04355 [INSPIRE].
S2 and elliptic genus,
S2 and supersymmetric
localization, JHEP 03 (2014) 040 [arXiv:1311.2430] [INSPIRE].
large N in localization and the dual oneloop quantum supergravity, JHEP 01 (2018) 026
[arXiv:1707.04197] [INSPIRE].
[4] O. Aharony , O. Bergman , D.L. Ja eris and J. Maldacena , N = 6 superconformal ChernSimonsmatter theories, M2branes and their gravity duals , JHEP 10 ( 2008 ) 091 [5] F. Benini , K. Hristov and A. Za aroni, Black hole microstates in AdS4 from supersymmetric [6] F. Benini and N. Bobev , Twodimensional SCFTs from wrapped branes and cextremization , [9] J.B. Gutowski and H.S. Reall , Supersymmetric AdS5 black holes , JHEP 02 ( 2004 ) 006 [10] J.B. Gutowski and H.S. Reall , General supersymmetric AdS5 black holes , JHEP 04 ( 2004 ) [11] M. Honda and Y. Yoshida , Supersymmetric index on T 2 [12] C. Closset and I. Shamir , The N = 1 chiral multiplet on T 2