Asymptotic M5brane entropy from Sduality
X =
M5brane entropy from Sduality
Seok Kim 0 1
June Nahmgoong 0 1
Seoul National University 0 1
Theory, Supersymmetric Gauge Theory
0 1 Gwanakro , Gwanakgu, Seoul 08826 , Korea
1 Department of Physics and Astronomy & Center for Theoretical Physics
We study M5branes compacti ed on S1 from the D0D4 Witten index in the Coulomb phase. We rst show that the prepotential of this index is Sdual, up to a simple anomalous part. This is an extension of the wellknown Sduality of the 4d N = 4 theory to the 6d (2; 0) theory on nite T 2. Using this anomalous Sduality, we asymptotic free energy scales like N 3 when various temperaturelike parameters are large. This shows that the number of 5d KaluzaKlein proportional to N 3. We also compute some part of the asymptotic free energy from 6d chiral anomalies, which precisely agrees with our D0D4 calculus.
Sduality; Duality in Gauge Field Theories; Field Theories in Higher Dimensions; M

Asymptotic
1
3
1 Introduction 2
2.3 6d (2; 0) theories of DN and EN types
High temperature limit of the index
3.1
Tests with U(1) partition function
3.2 6d chiral anomalies on Omegadeformed R
4
T 2
Conclusions and remarks
Introduction
Strong/weakcoupling duality, or Sduality, exists in a number of quantum systems. In 4d
gauge theories, it takes the form of electromagnetic duality, which inverts the gauge
coupling and exchanges the roles of elementary charged particles and magnetic monopoles [1].
It is realized in the simplest manner in maximally supersymmetric YangMills (SYM)
theory [2]. In this case, the spectrum of dyons in the Coulomb phase was shown to
exhibit SL(2; Z) duality [3], providing a robust evidence of Sduality. Sduality in quantum
eld theories has also been a cornerstone of developing string dualities [4]. In both QFT
and string theory, Sduality provides valuable insights on the strongly coupled regions of
the systems.
Sduality of maximal SYM has many implications. In particular, this duality is related
to the existence of 6d superconformal eld theories called (2; 0) theories [5]. 4d maximal
SYM theories with gauge groups U(N ), SO(2N ), EN are realized by compactifying 6d (2; 0)
theories on small T 2. The SL(2; Z) duality originates from the modular transformation on
T 2. On one side, this relation highlights the farreaching implications of the 6d CFTs to
challenging lower dimensional systems. On the other hand, the 6d CFTs lack microscopic
de nitions, so that Sduality can provide useful clues to better understand the mysterious
6d CFTs. In this paper, we study the Sduality of the 6d (2; 0) theories compacti ed on
nite T 2, and use it to explore some interesting properties of these systems.
Our key observable of the 6d (2; 0) theory is the partition function of the D0D4 system.
More precisely, we study the Witten index of the quantum mechanical U(k) gauge theory
for k D0branes bound to N separated D4branes and fundamental open strings, and also
study their generating function for arbitrary k. In Mtheory, this system is made of N
M5branes wrapping S1. The D0D4 systems describe the 6d (2; 0) CFT on M5branes in
{ 1 {
the sectors with nonzero KaluzaKlein momenta. From the viewpoint of 5d
superYangMills theory on D4branes, this partition function is also known as Nekrasov's instanton
partition function [6]. Although the 5d YangMills description of its instanton solitons is
UV incomplete, the D0D4 system provides a UV complete description for computing the
instanton partition function. The results in our paper rely only on the UV complete D0D4
S1, where S1 is the temporal circle for the Witten index. However, with
D0branes (YangMills instantons) providing the KaluzaKlein (KK) modes of Mtheory,
there are evidences that this index (multiplied by a 5d perturbative part) is a partition
T 2. See [7, 8] for the AN 1 theories, and [9] for the
DN theories.
Regarding the D0D4 index as a 6d partition function on R
4
T 2, one can naturally
ask if it transforms in a simple manner under the Sduality acting on T 2. In this paper,
we establish the Sduality of the prepotential of this index,
nding a simple anomaly
of Sduality which we can naturally interpret with 6d chiral anomalies. Note that the
prepotential F is the leading coe cient of the free energy
log Z
mass parameter, explained in section 2. The anomaly of Sduality takes the
following form. The prepotential F can be divided into two parts, F (a; m) = FSdual(a; m)+
Fanom(m), where FSdual is related to its Sdual prepotential by a 6d generalization of the
Legendre transformation. (See section 2 for details.) Fanom is a simple function which
does not obey Sduality, thus named anomalous part. We nd Fanom in a closed form in
section 2, which in particular is independent of the Coulomb VEV a.
This
nding has two major implications. Firstly, similar result was found for the
prepotential F 4d of the 4d N = 2 theory [10], related to our prepotential by taking the
small T 2 limit. F 4d is also given by the sum FS4ddual(a; m) + Fa4ndom(m), where FS4ddual is self
Sdual. Since Fa4ndom is independent of a, F 4d is Sdual in the SeibergWitten theory, which
only sees a derivatives of F 4d. In our 6d uplift, F appearing in the SeibergWitten theory
also does not see Fanom(m) for the same reason.
Secondly, the partition function Z itself is a Witten index of the 6d theory on R4;1
S1.
So the full prepotential F = FSdual +Fanom including the a independent Fanom is physically
meaningful, as the leading part of the free energy
log Z when 1; 2 are small. At this
stage, we note our key discovery that Fanom contains a term proportional to N 3 in a suitabe
large N limit, to be explained in section 2. In particular, we further consider the limit in
which the KK modes on the circle become light. This amounts to taking the chemical
potential
conjugate to the KK momentum (D0brane charge) to be much smaller than
the inverseradius of S1. The small
limit is the strong coupling limit of the 5d
YangMills theory, or the limit in which the sixth circle decompacti es. This is the regime in
which 6d CFT physics should be visible. The prepotential in this limit can be computed
from our anomalous Sduality, since it relates the small
(strong coupling) regime to the
wellunderstood large
(weak coupling) regime. Fanom determines the small
limit of the
free energy, and makes it scale like N 3. We also show that the term in the asymptotic free
{ 2 {
energy proportional to N 3 is related to the chiral anomaly of the 6d (2; 0) theory, using
the methods of [11]. These
ndings show that the number of 5d KK
elds for D0brane
bound states grows like N 3, as we decompactify the system to 6d.
The rest of this paper is organized as follows. In sections 2.1 and 2.2, we develop the
anomalous Sduality of the prepotential and test it either by expanding F in the 5d N = 1
mass m, or by making the `Mstring' expansion [8]. In section 2.3, we discuss the 6d (2; 0)
theories of DN and EN types. In section 3, we study the high temperature free energy and
show that it scales like N 3 in a suitable large N limit. In section 3.1, we test our result
for U(1) theory. In section 3.2, we account for the imaginary part of the asymptotic free
energy from 6d chiral anomalies. Section 4 concludes with comments and future directions.
8 supersymmetry and U(N ) global symmetry. See, for instance, [7] for the details of this
system. Here, we shall only explain some basic aspects. The bosonic variables consist of
four Hermitian k
k matrices am
a _ , two complex k
N matrices q _ , ve Hermitian
k matrices 'I , and a quantum mechanical U(k) gauge eld At. Here, m = 1; 2; 3; 4 is
the vector index on R4 for the spatial worldvolume of the D4branes.
and _ are doublet
indices of SU(
2
)l and SU(
2
)r respectively, which form SO(
4
) rotation of R4. I = 1;
; 5
is the vector index on R5 transverse to the D4branes. When 'I are all diagonal matrices,
their eigenvalues are interpreted as D0brane positions transverse to D4branes. Similarly,
when am are all diagonal, their eigenvalues are interpreted as D0brane positions along
D4brane worldvolume. q _ represent internal degrees of freedom. The bosonic potential
energy is given by
1
2
i _
2
_
{ 3 {
of moduli space satisfying V = 0, or Di = 0, ['I ; am] = 0, 'I q _ = 0 and ['I ; 'J ] = 0.
The rst branch is obtained by taking q _ = 0, and am, 'I to be diagonal matrices. The
k sets of eigenvalues of (am; 'I ) represent the positions of k D0branes on R9, unbound
to the D4branes. The second branch is obtained by taking '
I = 0, and q _ , am to satisfy
Di = 0. After modding out by the U(k) gauge orbit, one can show that this branch is
described by 4N k real parameters. The two branches meet at '
I = 0, q _ = 0. Far away
from this intersection, each branch is described by a nonlinear sigma model (NLSM) on its
moduli space. We are interested in the second branch, describing 6d CFT on M5branes in
1
4
i
(2.1)
(2.2)
the sector with k units of KK momentum. The Witten index of the second branch can be
computed easily by deforming the system by a FayetIliopoulos (FI) parameter, shifting Di
in (2.2) by three constant i
. After this deformation, the rst branch becomes nonBPS,
since q _ = 0 cannot solve Di = 0 with i 6= 0. So the Witten index acquires contributions
only from the second branch.
One can understand the second branch from the low energy eld theory of D4branes,
the 5d maximal SYM theory. D0branes are realized in YangMills theory as instanton
solitons, classically described by nite energy stationary solutions of the following BPS
equation,
Fmn =
m; n; p; q = 1;
; 4 :
The nite energy solutions are labeled by the instanton number k, de ned by
1
2 mnpqFpq ;
k
know how to UV complete the full 5d SYM, the NLSM can be UV completed to the U(k)
separate the D4branes along a line, giving nonzero VEV to
5 only. In this setting,
we shall study the BPS bound states of the D0branes and the fundamental open strings
stretched along the
5 direction, suspended between a pair of D4branes. The bound states
preserve 4 Hermitian supercharges. In 6d (2; 0) theory, we compactify a spatial direction
on a circle with radius R0. The BPS states saturate the bound E
the energy, and P is the quantized momentum on S1 which is k in the D0D4 system. vi are
the N eigenvalues of the scalar
5, and qi's are the U(1)N electric charges in the Coulomb
branch, satisfying q1 +
+ qN = 0. From the 6d viewpoint, they are the selfdual strings
with charges qi coming from open M2branes, with P units of momenta on them. We also
de ne H
R0(E
viqi), which is the (dimensionless) energy on the selfdual strings.
RP0 + viqi, where E is
The 6d index is de ned by
Z( ; m; 1;2; v) = Tr h( 1)F e2 i H+2P e 2 i H 2 P e 1(J1+JR)+ 2(J2+JR)e2mJLe viqii : (2.6)
{ 4 {
(2.3)
(2.4)
Here, J1; J2 are two Cartans rotating the two 2planes of R4, JL; JR are the Cartans of
SO(5) unbroken by the VEV of 5. The measure is chosen so
that it commutes with 2 of the 4 Hermitian supercharges preserved by the BPS states, or
a complex supercharge Q and its conjugate Qy. See [7] for the details. One also
nds that
H P
2
fQ; Qyg. Since only the states saturating the BPS bound H
the index, Z is independent of . With H = P understood, the factor e2 i H+P
2
P contribute to
! e2 i P
weights the BPS states with the momentum P along the circle. So Z can be written as
where q
e2 i , and Z0
1 by de nition. Z can be computed in the weakly coupled
type IIA regime, in which D0branes are much heavier than the stretched fundamental
strings. Zk is computed as the nonperturbative Witten index of the D0D4 system with
xed k. Zpert comes from the zero modes at P = 0, the perturbative open string modes
on the D4branes. This factor can also be understood as coming from the perturbative
partition function of the 5d maximal SYM. Since we are in the weakly coupled regime,
Zpert can be computed unambiguously from the quadratic part of the YangMills theory.
Although we compute Zpert and Zk in this special regime, we naturally expect the result
to be valid at general type IIA coupling, since this is a Witten index independent of the
continuous coupling.
Zk and Zpert are known for classical gauge groups. For U(N ), Zk is given by [7, 12{14]
Zk =
X
N
Y
Y sinh Eij(s)+m
2
+ sinh Eij(s) m
2
+
Yi;PiN=1 jYij=k i;j=1 s2Yi
2
sinh Eij(s) sinh Eij(s) 2 +
2
1
k=0
Z( ; m; 1;2; v) = Zpert(m; 1;2; v) X qkZk(m; 1;2; v)
(2.7)
The summation is made over N Young diagrams Yi with total number of boxes k, and
s runs over all boxes of the Young diagram Yi. hi(s) is the distance from s to the right
end of the Young diagram Yi, and vj (s) is the distance from s to the bottom end of the
Young diagram Yj . See [7] for the details. One often calls Zinst
P1
k=0 qkZk the instanton
partition function.
Zpert is given by [15, 16]
Zpert =
Y
2adj
" ~3( (v)+ ++m
2 i
j 2 1i ; 2 2i )~3( (v)+ + m
2 i
j 2 1i ; 2 2i ) # 2
1
~3( 2(vi) j 2 1i ; 2 2i )~3( (v)+2 +
2 i
j 2 1i ; 2 2i )
where ~3(zjw1; w2)
3(zj1; w1; w2) 3(1
zj1; w1; w2), and
N (zjw1;
; wN ) is the
Barnes' Gamma function. As noted in [16],
in the adjoint representation includes
Cartans,
= 0, for which ` 3(0j 2 1i ; 2 2i )' in the denominator would diverge. For these ,
{ 5 {
(2.8)
(2.9)
(2.10)
one replaces `~3(0jw1; w2)' factors by ~03(0jw1; w2)
lim[z ~3(zjw1; w2)]. See [16] for more
details. For t1
where prime here again means excluding the zero modes at n1 = n2 = 0 for the Cartans
= 0. The overall factor F is given for gauge group G by [15]
Zpert(v; 1;2; m) = e F P E
" 1 sinh m+ + sinh m
2
2
+
2
where 3 is the Barnes' zeta function. When t1; t2 < e (v) for all
2 adj, Zpert is
rewritten as
where P E[f (x; y; z;
)]
exp P1
n=1 n1 f (nx; ny; nz;
) , adj
P
2adj e (v), and r is
the rank of gauge group which is r = N for U(N ). The term 2r in P E comes from excluding
r fermionic zero modes for the Cartans.
One may multiply an alternative perturbative factor Zpert
e "0 [ZpUe(r1t)]N Z^pert to Zinst,
where [ZpUe(r1t)]N is the perturbative partition function for the N Cartans, Z^pert is de ned by
F =
=
i
2
i
2
X
2adj
X
(v)2adj
(v) + 2 + j1;
{ 6 {
are positive for positive roots and larger than m; 1;2. This expression will be useful when
studying Sduality from the Mstring viewpoint, in section 2.2. Zpert and Zpert are
different in subtle ways, which shall not a ect the studies of prepotential in this paper but
has implications on the Sduality of Z, which we comment on in section 2.2. (2.14) has
a more natural interpretation as the Witten index of charged Wbosons in the Coulomb
phase [7]. However, as an abstract partition function, Zpert is more natural as it is
manifestly Weylinvariant.
It will also be useful to know the simple structures of the Abelian partition function,
ZU(1) = ZpUe(r1t)ZiUns(t1). Firstly, the perturbative U(1) partition function can be written as
ZpUe(r1t) = e
= e
i(m2 2+)
after summing over all Young diagrams in (2.8).
Given Z = ZpertZinst, or Z = ZpertZinst, one can write this partition function as
Z = P E
f ( ; m; 1;2; v)
2 sinh 21 2 sinh 22
exp
n=1
"X1 1 f (n ; nm; n 1;2; nv) #
n 2 sinh n21 2 sinh n22
;
(2.16)
(2.17)
(2.18)
(2.19)
(2.20)
im2
4 jGj+
X
2adj
Fpert(v; m) =
Li3(e
(v))
Li3(e ( (v)+m))
Li3(e ( (v) m)) ;
1
2
1
2
or a similar expression for Z using f . The expression appearing in P E is called the single
particle index, containing all the information on the BPS bound states. The coe cients
of f in fugacity expansion are also called GopakumarVafa invariants [18, 19]. The factor
2 sinh 2112 sinh 22 comes from the centerofmass zero modes of the particle on R4, which
would have caused the path integral for Z to diverge at 1 = 2 = 0. So 1;2 also plays the
role of IR regulators. f ( ; m; 1;2; v) takes into account the relative degrees of freedom of
the bound state, in which 1;2 are just chemical potentials. In particular, 1;2 ! 0 limit is
smooth in f .
In this paper, we shall mostly discuss the limit 1; 2 ! 0. In this limit, one nds
Fpert(v; m)
from (2.18). F = Fpert + Finst is the prepotential. Finst can be obtained from (2.8) after
a straightforward but tedious calculation. Fpert can be obtained from (2.13), which is
given by
at e 1 < 1, e 2 < 1, by following the discussions till (2.13) for N = 1. The instanton part
HJEP12(07)
can be written as [17]
where Lis(x) = P1
a branch cut. The
n=1 xnns for jxj < 1, and can be continued to the complex x plane with
rst term coming from F will play no role in this paper. One way
of obtaining (2.20) is to rst take v; m to be purely imaginary, to guarantee convergence
of the sum in (2.13), and take the limit 1;2 ! 0 to obtain (2.20). Then, (2.20) can be
{ 7 {
analytically continued to complex v; m. One may alternatively start from Zpert and obtain
its prepotential,
Fpert =
m2
2
( ij +j + X
(v)) + X
>0
>0
+ is the set of positive roots. Here, from the identity
2Li3(e
(v))
Li3(e
(v) m
) + rFpUe(r1t) : (2.21)
Lin(e2 ix) + ( 1)n(e 2 ix) =
for all positive roots , and also Im( (v)) is chosen such that all Im( (v)
m) are within
the range (0; 2 ] for positive roots. Then one nds
Fpert
Fpert =
=
m2
2
X
>0
im2
2 j +j
(v)
(2 i)3
6
X
>0
B3
(v)
2 i
1
1 and Im(x) < 0, where Bn(x)'s are Bernoulli polynomials, one nds
3 x2 + 12 x. So at least in this setting, Fpert and Fpert di er only
2
by a trivial constant independent of v. The last constant will play no role in this paper.
It will be helpful to consider the prepotential of the U(1) theory separately. From (2.16)
and (2.17), the prepotential fU(1) = FpUe(r1t) + FiUns(t1) for the U(1) theory is given by
2Li3(qn) Li3(emqn) Li3(e mqn) +
2Li3(1) Li3(em) Li3(e m
) +
For studying the Sduality of this prepotential, it will be useful to make an expansion of
fU(1) in m. One rst nds that the instanton part is given by
2Li3(qn) Li3(emqn) Li3(e mqn) =
m2 X1 Li1(qn) 2 X
1
1
X
m2j+2
Li1 2j (qn)
n=1
= m2 log ( )+X
j;n;k=1 (2j +2)!
1
m2j+2
j=1 2j(2j +2)!
(E2j ( ) 1) ;
= m2 X1 log(1 qn) 2
1
X
m2j+2
k2j 1qnk = m2 log ( ) 2 X
where ( ) = Qn1=1(1
qn) = q 214 ( ) is the Euler function, and we used the identity
j=1 n=1 (2j +2)!
1
m2j+2 k2j 1qk
j;k=1 (2j +2)! 1 qk
1
2
n=1
k=1
X1 k2j 1qk
1
qk
=
4j
B2j (E2j ( )
1)
{ 8 {
for the Eisenstein series E2n( ). Bn are the Bernoulli numbers: B1 =
and so on. The perturbative prepotential can be expanded in m by using
1
6
at small z, with Hn = Pn
p=1 p1 . One nds
This will be useful later for understanding N fU(1), as a part of the U(N ) prepotential.
One can understand the chemical potentials from the viewpoint of the 4d e ective
action in the Coulomb branch. The dimensionless variables m; 1;2, v take the form of
m = RM ;
1;2 = R"1;2 ; v = Ra ;
where R is the radius of the temporal circle of R4 S1. M is the mass deformation parameter
of the 4d N = 2
YangMills theory, or the 5d N = 1 theory. (More precisely, M is 2
times the mass.) "1;2 are the Omega deformation parameters which have dimensions of
mass. a is the Coulomb VEV of the scalar eld
5
.
is identi ed as
1
30
1
2
1
42
3
4
1
4
= i
R
R0
;
{ 9 {
where R0 is the radius of the sixth circle. This is the inverse gauge coupling in 4d.
can be
complexi ed with a real part, given by the RR 1form holonomy of type IIA theory on S1.
The 4 dimensional limit of the partition function is obtained by taking R ! 0 with
xed ; M; "1;2; a. From (2.8), one
nds that all sinh functions of v; 1;2; m are replaced
by linear functions of a; "1;2; M , and the R dependences cancel between numerator and
denominator. As a result, the 4d limit Zk4d of the instanton partition function is given by
a rational function of M; "1;2; a of degree 0. This makes Zi4ndst and Fi4ndst to enjoy a simple
scaling property,
Zi4ndst( ; M; "1;2; a) = Zi4ndst( ; M; "1;2; a) ;
Fi4ndst( ; M; a) =
2Fi4ndst( ; M; a) : (2.34)
This will be used in section 2.1 to provide two interpretations of the 4d Sduality, and
extend one version to 6d. As for the perturbative part Fpert, one can use (2.29) to obtain
limR!0 Fpert. One nds
Fp4edrt =
X
2adj
M 2 log R
+
( (a) + M )2
4
3
4
(a)2
2
log (a)
log( (a) + M ) +
log( (a)
M )
( (a)
M )2
4
where the rst term independent of the Coulomb VEV is unphysical in the SeibergWitten
theory. The perturbative prepotential satis es the following pseudoscaling property,
Fp4edrt( M; a) =
2
Fp4edrt(M; a) + jGj 2
log
;
which is homogenous and degree 2 up to a Coulomb VEV independent shift.
Zinst or Finst are only known as q expansion when q
1, or
! i1. This is useful
when the `temperature' is much smaller than the KaluzKlein scale R10 , when the KK modes
or
are `heavy.' However, to study 6d SCFT, it is more interesting to explore the regime q ! 1,
! i0+, in which case the circle e ectively decompacti es. The two regimes are weakly
coupled and strongly coupled regimes, respectively. So if there is Sduality for the partition
function on R
4
T 2, it will be helpful to study the interesting decompactifying regime
from the wellunderstood region
! i1. Developing the Sduality of the prepotential F
is the goal of this section. (In section 2.2, we also comment on the Sduality of the full
partition function.)
2.1
Sduality and its anomaly
Following [10], we review the basic set up for studying the Sduality of 4 dimensional
prepotential, and extend it to the 6d theory on T 2.
The prepotential F of general 4d N = 2 gauge theory determines the e ective action
in the Coulomb branch. The magnetic dual description uses the dual Coulomb VEV aD(a)
and the dual prepotential FD(aD), de ned by the following Legendre transformation,
aD =
(2.37)
For theories with higher rank r > 1, a has many components, ai with i = 1;
; r.
Exwhose sum structures will not be explicitly shown to make the notations simpler. For
generic N = 2 theories, F; FD depend on other parameters like hypermultiplet masses and
the coupling constant (or the dynamically generated scale
instead of the coupling).
For 4d N = 2 theory, the prepotential F 4d (to be distinguished with the 6d
prepotential F which we shall consider later) depends on the microscopic coupling constant
and the adjoint hypermultiplet mass M . The prepotential can be divided into the classical,
perturbative, and instanton contributions,
F 4d = Fcl( ; a) + Fp4edrt(a; M ) + Fi4ndst( ; a; M )
Fcl( ; a) + f 4d( ; a; M )
(2.38)
(2.35)
(2.36)
HJEP12(07)
D =
1
. For the 4d N = 2 theory, FD4d is de ned by
where Fcl( ; a) =
i a2, and Fp4edrt. f 4d
Fp4edrt + Fi4ndst is the quantum prepotential. To
study self Sdual theories, it is convenient to de ne FD4d as a function of the dual coupling
FD4d( D; aD; M ) = L[F 4d]( ; a; M ) = F 4d( ; a; M )
( ; a; M ) :
(2.39)
a
Then, self Sduality exists if FD4d and F 4d are same function, FD4d( ; a; M ) = F 4d( ; a; M ).
This Sduality has been tested in detail in [10]. More precisely, it was found that
F 4d( ; a; M ) = FS4ddual( ; a; M ) + Fa4ndom( ; M ) ;
where FS4ddual satis es
FS4ddual( D; aD; M ) = FS4ddual( ; a; M )
a
and Fa4ndom is an anomalous part of Sduality, depending on ; M but is independent of the
Coulomb VEV a [10]. Since the Coulomb branch e ective action is obtained by taking
a derivatives of F 4d, F 4d and FS4ddual are identical in the SeibergWitten theory. This
establishes the Sduality of the 4d N = 2 theory in the Coulomb branch e ective action.
Let us rephrase the 4d Sduality in a way that is suitable for 6d extension. Fi4ndst satis es
the scaling property (2.34). Combining the perturbative part, one nds
(2.40)
Applying this to F 4d( D; aD; m), one obtains
F 4d( ; a; M ) =
2
F 4d( ; a; M ) + jGj 2
log
:
F 4d( D; aD= ; M= ) =
2F 4d( D; aD; M )
So the left hand side of (2.41) can be written as
FS4ddual( D; aD; M ) =
2FS4ddual( D; aD= ; M= )
+ jGjM 2
2
log
+ 2Fa4ndom( D; M= )
Fa4ndom( D; M ) :
(2.42)
(2.43)
(2.44)
rM2 2 log M + M 2(
rewritten as
Let us consider the structure of Fa4ndom. Since the prepotential has mass dimension 2, one
may think that its M dependence is simply M 2. However, the perturbative part (2.35)
shows that there is a term rM2 2 log M in F 4d which scales in an odd manner. In the
computational framework of [10], which we shall explain below in our 6d version, FS4ddual is
by construction taken to be a series expansion in M 2. This means that the odd term
rM2 2 log M should have been put in Fa4ndom. Therefore, had one been doing the
calculation of [10] using (2.35) as the perturbative part, one would have found that Fa4ndom =
), where (
) only depends on . Using this structure, (2.44) can be
FS4ddual( D; aD; M ) =
2FS4ddual( D; aD= ; M= ) + (jGj
r)
log :
(2.45)
(2.46)
(2.47)
(2.48)
(2.49)
(2.50)
The nal result holds for complex . Similar property holds for 1;2
This makes the appearance of M to be more natural on the left hand side of (2.47).
R"1;2, i.e. 1D;2 = 1;2 .
Secondly, let us discuss how a should transform. In 4d, we already stated that
naturally appears on the left hand side of (2.41). For simplicity, let us discuss these variables
in the limit of large Coulomb VEV, v
Ra
1, a
m. The second term can be ignored
in this limit, yielding the semiclassical result aD =
a. In this limit, we shall discuss
what is the natural Sdual variable using the Abelian 6d (2; 0) theory. In 4d, aD = a is a
2F~S4ddual( D; aD= ; M= ) = F~S4ddual( ; a; M )
a
instead of (2.41). To summarize, by trivially rede ning FS4ddual and Fa4ndom by the last term
of (2.46), one can reformulate the standard Sduality (2.41) as (2.47). Only (2.47) will
naturally generalize to the Sduality on R
4
T 2.
Now we seek for the Sduality of the 6d prepotential. Note that in 4d, (2.41) and (2.47)
are equivalent by making a minor rede nition of Fa4ndom, using (2.42). In 6d, a property
like (2.42) does not hold. Before making a quantitative study of the 6d Sduality, we rst
explain that (2.47) is more natural in 6d. To discuss the 6d prepotential, it is convenient
to work with the dimensionless parameters v; m; 1;2.
Firstly, in the 6d theory compacti ed on T 2, the complex mass parameter m is simply
the holonomy of the background gauge eld for SU(
2
)L global symmetry, along the two
sides of T 2. Then after making an Sduality of the torus, exchanging two sides of T 2, one
naturally expects mD = m . Let us brie y review this by taking a rectangular torus, for
simplicity. In this case, the complex structure
of the torus is purely imaginary.
is
related to the two radii of T 2 by
= i
R
R0
;
where R0 is the radius of the circle which compacti es the 6d theory to 5d SYM, and
R is the radius of another circle which compacti es the 5d theory to 4d. The Sduality
transformation exchanges R $ R0. So the dual complex structure is D = i R0 =
R
More precisely, Sduality rotates the torus by 90 degrees on a plane. It also transforms the
1
.
two SU(
2
)L holonomies along the two circles. Let Re(M ) be the holonomy on the circle
with radius R, and Im(M ) that on the circle with radius R0. Under Sduality, one
nds
Re(MD) = Im(M ), Im(MD) =
Re(M ). So one nds MD =
iM . In F , M appears in
the dimensionless combination m
RM , which transforms as
So de ning
one nds that F~S4ddual satis es
F~S4ddual( ; a; M ) = F 4d
S dual( ; a; M )
jGj
2
r
aD = a +
natural aspect of Sduality being electromagnetic duality. Also, it makes sense to multiply
a by a complex number , since a is a complex variable living on a plane. However, in
6d CFT on T 2, a lives on a cylinder. The real part of a is the VEV of the real scalar in
the 6d selfdual tensor multiplet, which is noncompact. On the other hand, the imaginary
part of a comes from the holonomy of the 2form tensor eld B on T 2, implying that it
is a periodic variable. So it does not good make sense to rotate a living on a cylinder by
complex . More precisely, the 6d scalar
and the 5d scalar a are related by a
R0 . So
one nds
a
R0( + iB12) ;
(2.51)
HJEP12(07)
where 1 and 2 denote two directions of T 2. Thus, v = Ra
the dimensionful variables, This requires one to use aD
under R $ R0, meaning that it makes more sense to set vD
RR0( + iB12) is invariant
v in the limit v
a as the dual variable, instead of
aD
a. This does not rotate the variable a by a complex number, so makes better sense
in 6d. Incidently, we have already found the alternative (but equivalent) statement (2.47)
of Sduality which uses aD as the dual variable, instead of aD. Note that the usage of
aD = a + 2 i
uplift, it is natural and consistent to regard vD
M . Thus, in the 6d
So it appears natural to seek for a 6d generalization of (2.47) rather than (2.41). This
is what we shall establish in the rest of this section. Namely, we shall nd that the 6d
prepotential is divided into two,
F = FSdual( ; v; m) + Fanom( ; m)
where v = Ra, m = RM , and Fanom is independent of the Coulomb VEV. FSdual satis es
2FSdual
= FSdual( ; v; m)
v
We have some freedom to choose Fanom, by adding/subtracting v independent Sdual
expressions to Fanom, FSdual. We shall explain that one can choose Fanom as
Fanom = N fU(1)( ; m) +
N 3
N
288
m4E2( )
where fU(1) is the U(1) prepotential (2.25). The rst term N fU(1) comes from the N 6d
Abelian tensor multiplets in U(1)N , which has their own Sduality anomaly. The second
term of Fanom is one of the key
ndings of this paper, which comes from the charged part
of the partition function. After replacing m = M R, and multiplying R12 to the above Fanom
to get to the conventionally normalized prepotential (as noted in footnote 1), one can take
the 4d limit of Fanom. The second term proportional to N 3
N vanishes in the 4d limit
R ! 0, as it is proportional to M 4R2.
1Here, one may wonder that f appearing on the right hand side should have been R2f . However, we
shall de ne the prepotential as the coe cient of the dimensionless
that is conventional in the SeibergWitten theory, making f dimensionless. Namely, fours in 6d is related
to the conventionally normalized prepotential by fours = R2fconventional.
1 ,
(2.52)
(2.53)
(2.54)
With the motivations and results given, we now properly set up the calculation and
show the claims made above. As in 4d, we decompose the 6d prepotential as
F ( ; v; m) = Fcl + Fpert + Finst
Fcl + f ;
(2.55)
where Fcl
i v2. The prepotential is Sdual if it satis es
1
2
F
D =
= F ( ; v; m)
( ; v; m) :
(2.56)
We rst study the structures of this equation, before showing that it is satis ed by our
FSdual. Firstly, replacing F by Fcl, one can check that Sduality trivially holds at the
classical level:
2Fcl( D; vD) =
2
v
where vD is replaced by its classical value vD = v (formally at f = 0). Now we
subtract (2.56) by (2.57) to nd the following condition for the quantum prepotential f :
2
f
1
; v +
= f ( ; v; m) +
( ; v; m)
1
4 i
We are going to study the last equation. Note again that the e ective action in the
Coulomb branch only contains v derivatives of F , or f . Thus, in SeibergWitten theory,
f is ambiguous by addition of v independent functions, possibly depending on
and m.
However, the Sduality requirement (2.58) is sensitive to the value of f , including the v
independent part. So when one tries to establish the Sduality of the Coulomb branch
e ective action, one should have in mind that one may have to add suitable Coulomb VEV
independent terms to f computed microscopically from Z.
Following [10], we shall establish the Sduality (2.58) and its anomaly (2.54) by
expanding f in the mass m when it is small enough. We shall still get an exact statement (2.54),
which we check for certain orders in m. One should however have in mind that the exact
statement (2.54) may be valid only within a
nite region of m; v in the complex planes.
In section 2.2, we shed more lights on the exactness of (2.54), by making an Mstring
expansion [8].
As studied in the 4d limit [10], there is a natural way of achieving the Sduality
requirement (2.58). This is to require that f is expanded in quasimodular forms of suitable
weights. To precisely explain its meaning, we rst expand f in m as
f ( ; v; m) =
1
n=1
X m2nfn( ; v) :
This series makes sense as follows. Firstly, the m ! 0 limit exhibits enhanced maximal
supersymmetry. So at m = 0, the classical prepotential Fcl =
i v2 acquires no quantum
corrections, meaning that f vanishes at m = 0. Also, the prepotential is an even function
(2.57)
(2.58)
(2.59)
of m, which restrict the expansion as above.2 Then, following [10], we require that fn
is a quasimodular form of weight 2n
2, which means the following. Quasimodular
forms are polynomials of the rst three Eisenstein series E2, E4, E6, where each series has
weight 2; 4; 6 respectively under Sduality in the following sense:
6
i
E2( 1= ) = 2
E2 +
; E4( 1= ) = 4E4( ) ; E6( 1= ) = 6E6( ) :
(2.60)
More concretely, they are given by
E2 = 1
1
24 X
n=1
nqn
1
qn
; E4( ) = 1 + 240 X
; E6 = 1
1
n=1
n q
3 n
1
qn
504 X
1
n=1
n q
5 n
1
qn
: (2.61)
Higher Eisenstein series E2n are polynomials of E4; E6 with weight 2n. To study the
quasimodular property, it is helpful to decompose their dependence on
into the dependence
through E2 and the dependence through E4; E6. We thus write fn( ; v; E2( )), where the
dependence through E2 is explicitly shown. A weight 2n 2 quasimodular form fn satis es
where
6i . In terms of f , this is equivalent to
fn( 1= ; v; E2( 1= )) = 2n 2
fn( ; v; E2( ) + ) ;
2
f
1
; v;
m
; E2( 1= )
= f ( ; v; m; E2( ) + ) :
We now investigate how quasimodularity is related to the Sduality (2.58). One can
make (2.63) to be equivalent to (2.58) by specifying the E2 dependence of f , which we now
turn to.
Let us rst try to nd the desired E2 dependence, by requiring both (2.58) and (2.63).
By applying (2.63) to f ( 1 ; vD; m ; E2( 1= )), one obtains
; E2( 1= )
= f
; v +
; m; E2( ) +
;
(2.64)
(2.62)
(2.63)
2
(2.66)
For the sake of completeness, we repeat the logics presented in [10] and expand it to make
a proof. In fact, we shall make a stronger claim than needed. Namely, we need to nd
2Strictly speaking, there is a term rm2 2 log m in the perturbative part, which is easiest to see from the
4d limit (2.35). However, we shall expand fSdual as (2.59), while the term rm2 2 log m is moved to Fanom.
1
; v +
where again recall that
6i . Combining this with (2.58), one obtains
f
; v +
; m; E2( ) +
= f ( ; v; m; E2( )) +
( ; v; m; E2( ))
: (2.65)
We want to make this equation to hold, by specifying a particular E2 dependence of f .
[10] showed that the desired E2 dependence is
the E2 dependence of f which guarantees (2.65) only at
=
show that (2.66) guarantees (2.65) for arbitrary independent parameter , and then set
6i . However, we shall
= 6i later.
As a warmup, we follow [10] to make a series expansion of the left hand side of (2.65)
in small , to see how (2.66) guarantees (2.65) at low orders. One nds that
(l:h:s:) = f +
12
v~ = v +
; E~2 = E2 + ; f~ = f ( ; v~; E~2) :
=
which follows from (2.66). If (2.65) holds for general , its rst derivative would yield
So at 0 and 1 orders, one nds that it agrees with the right hand side if (2.66) is met.
Now assuming (2.66), we consider whether (2.65) is satis ed in full generality. To this
(l:h:s:) =
(r:h:s:) =
1
24
where for simplicity, we de ned
2
1
24
(2.67)
(2.68)
(2.69)
(2.70)
(2.71)
= 6i .
which one can show by using (2.68). On the other hand, (2.70) together with the O( 0)
component of (2.65) is equivalent to (2.65), since the O( 0) component is the only
information lost by taking
derivative. However, we have already shown around (2.67) that
the O( 0) component of (2.65) is satis ed. Therefore, showing (2.70) will be equivalent to
showing (2.65). So will show (2.70) by assuming (2.66). We take
at xed v; E2. Again using (2.66), one obtains
@@fv is zero at a particular value of , (2.71) guarantees that it is zero at di erent
values of . Since we already checked around (2.67) that (2.65) is true up to O( 1), we
= 0. This establishes
that (2.66) implies (2.70), and in turn that (2.66) implies (2.65). Finally, we insert
To summarize till here, (2.65) holds if f satis es (2.66). But (2.65) and (2.63) implies
the Sduality relation (2.58). Therefore, Sduality requirement (2.58) is satis ed if f
satis es the quasimodular property (2.63) and the modular anomaly equation (2.66). In the
rest of this subsection, we shall discuss the last two equations.
Following and extending [10], we show that the prepotential f obeys the two
properties (2.63), (2.66), up to an anomalous part which is independent of the Coulomb VEV v.
fn by
for n
for m = 1;
Again following [10], our strategy is to rst nd a prepotential fSdual in a series of m2
which satis es both (2.63) and (2.66). Then we show that f
fSdual is independent of v.
We expand fSdual like (2.59), fSdual = P1
n=1 m2nfn( ; v). (2.66) is given in terms of
1 n 1
24
(2.72)
; n
1, one can integrate the right hand side of (2.72) with E2 to get fn, up to
an integration constant independent of E2. The integration constant is a polynomial of E4
and E6 with modular weight 2n
2, whose coe cients depend only on v. These integration
constants depending on v can be xed once we know a few low order coe cients of f in q
expansion. Also, to start the recursive construction, the rst coe cient f1 at m2 should be
known. It will turn out that this can be also xed by the known perturbative part fpert [10].
This way, one can recursively generate the coe cients of fSdual from (2.63), (2.66) and the
knowledge of the few low order coe cients of f in q expansion. We emphasize here that
our purpose of making a recursive construction of fSdual is to show that the Coulomb VEV
dependent part of f is Sdual. Therefore, while xing the integration constants and f1 in
fSdual by using the low order q expansion coe cients of f , it su ces to use f up to the
addition of any convenient expression independent of v. So for technical reasons, we shall
t these integration constants and f1 by comparing fSdual with
f ( ; v; m)
N fU(1)( ; m)
(2.73)
rather than f itself. Note that N fU(1) is the prepotential contribution from U(1)N Cartan
part, coming from D0branes bound to D4branes but unbound to Wbosons which see
v. One reason for comparing with f
N fU(1) is that fU(1) does not admit a power series
expansion in m2 like (2.59). The Sduality anomaly of N fU(1) can be calculated separately
from (2.31).
With these understood, we start the recursive contruction by determining f1. This can
be xed solely from the perturbative part of (f
N fU(1))pert [10]. Namely, when instantons
are bound to Wbosons, there are fermion zero modes which provide at least a factor of
m4 in f . This means that m2 term f1 should come from the perturbative part only. This
fact can also be straightforwardly checked from the microscopic calculus. So one nds
f1 = (f
N fU(1))pert
where
is the set of roots of U(N ).
One can then compute f2 using (2.72) at n = 2,
m2
=
2
1 X Li1(e (v)) =
2
1 X log(1
2
e (v)) ;
(2.74)
One can integrate it with E2, to obtain
(1
e (v)) :
f2 =
=
2
1
(2.75)
(2.76)
where
( ) is given for
ej by
For a given , there are 2(N
2) elements of ( ). Using this, one nds
=
=
f2 =
=
2
2
E2( )
(v)), the rst term can be rewritten so that
There is no integration constant at weight 2. To proceed, we study the properties of the
U(N ) roots.
consists of vectors of the form ei
N orthogonal unit vectors. takes following possible values,
ej , i 6= j, i; j = 1;
; N , where ei are
X
2
=
2 ( )
1
3
X
i6=j6=k6=i
at O(m4) order.
2
1 e (v) 1 e (v)
1
1 e (v) + X
1 e (v) 1 e (v)
1
1 e (v) 5
4
(1 e (v))2
X 21(Ne (1v)) + X
(1 e (v))(1 e (v)) 5
2
X
2
2 ( )
X
2 ( )
1
2
X
i6=j6=k6=i
2
3
X
2
X
Here, one can simplify the second term by using
X
2
1
1
e (v) =
2
1 X 1 =
2
N (N
2
1 1 1
Also, using (1 ex)(1 ey) + (1 e x)(1 ey x) + (1 ex y)(1 e y) = 1, one nds
(1 e (v))(1 e (v)) = X
X
i6=j k6=i;j (1 evi vj )(1 evi vk ) (1 evi vj )(1 evk vj )
+
[(i; j; k)+(j; k; i)+(k; i; j)] =
=
N (N
1)(N
2) ;
(2.80)
where at the second step we symmetrized the summand by making a cyclic permutation
of i; j; k. This simpli es the third term. One thus nds
E2( )
24
"
2
f2 =
N #
1
3
(2.77)
(2.78)
3
HJEP12(07)
3
Before proceeding to higher order coe cients fn with n
3, let us rst discuss f2
that we computed by requiring Sduality of fSdual. Note that at m4, we have obtained an
all order result in the instanton expansion, coming from E2( ) = 1
24q
72q2
96q3
168q4
. So from the microscopic instanton calculus, one can expand f ( ; v; m) in small
m, and we can compare f and fSdual at m4 order. We nd that
(f
N fU(1)) fSdual m4
=
which we checked till q2 order for general N , and till q3 for N = 2; 3. Therefore, we
nd that the microscopic prepotential is compatible with Sduality at m4 order, up to the
addition of an `anomalous' term on the right hand side independent of the Coulomb VEV.
One can make further recursive calculations of fn for n
3, using (2.72), and test the
consistency of fSdual with our microscopic f . The next recursion relation of (2.72) is
Knowing f1; f2, one can integrate (2.83) to obtain
E2( )2
The integration constant c3(v) can be determined by expanding f3 in q, and comparing
the q0 order with the perturbative contribution (f
N fU(1))pert at m6 order. One obtains
c3(v) =
1
2880
X Li 3(e (v))
2
1
288
2
X Li 2(e (v))2 +
1
576
X
2
X
2 ( )
Li0(e (v))Li 2(e (v)) :
Inserting this c3(v) in (2.84), one can further study the higher order coe cients of f3 in q
expansion, against the microscopic result f . We nd that
(f
N fU(1)) fSdual m6
= 0 ;
which we checked till q2 order for general N , and till q3 order for N = 2; 3.
Integrating (2.72) to get higher fn's, the integration constants take the following form,
fn( ; v)
X
4a+6b=2n 2; a 0; b 0
ca;b(v)E4( )aE6( )b :
More concretely, one would get
f4
c0;1E6 ; f5
c2;0E42 ; f6
c1;1E4E6 ; f7
c3;0E43 + c0;2E62 ; f8
c2;1E42E6 ;
and so on. To x the coe cients cp;q(v), one should use some low order data of f
If there are k + 1 independent cp;q's, one should use up to k instanton coe cients of
(2.82)
(2.83)
3
5
(2.84)
(2.85)
(2.86)
(2.87)
(2.88)
N fU(1).
It is again important to understand the set ( ) for ADE, which we explain now.
For DN = SO(2N ), the 2N 2 2N roots in
are given by e
i ej , where i; j = 1;
; N
and i < j. Elements of ( ) are given for various
by
= ei
ej : ( ) = fk 6= i; j : ei
= ei + ej : ( ) = fk 6= i; j : ei
For E6, the number of roots is j j = 72. 40 roots take the form of
ej where i 6= j
and i; j = 1;
; 5, from the SO(10) subalgebra. Additional 32 roots take the form of
1
2
( e1
e5
e6
e7 + e8), where the total number of
signs is even. The structure
of ( ) is given for various
as follows. Firstly, when
= ei
ej , then
( ) = fk 6= i; j : ei
e
i
)
To proceed, we classify the roots
depending on their norm with . The possibilities are
(1) :
(
2
) :
( ) =
s1
s6 =
(s1e1 +
s6e6 + e7
e8) ;
(2.127)
where
means that all possible signs are allowed in the 32 spinorial elements. Thus, one
nds 12 + 8 = 20 elements of ( ) in this case. Similarly, for
= ei + ej , one nds
( ) = fk 6= i; j : ei
ek; ej
ekg [
(ei + ej +
)
where
s20 (s1e1 +
means the same. So again, one nds j ( )j = 12 + 8 = 20. For
=
e
i
one can do a similar analysis. Finally,
can be one of the 32 spinorial elements,
+ s5e5
e6
e7 + e8) with s0;
; s5 =
1 and s1
s5 = 1. Then,
( ) = fs0(siei + sj ej )g [ f
s0(siei + sj ej )g ;
with j ( )j = 16 + 16 = 32. When
one nds
so j ( )j = 5C2 + 5C2 = 20. For E7, j j = 126. 60 roots take the form of
i; j = 1;
; 6, from SO(12) subalgebra. Additional 64 roots take the form of
e6
For instance, for
e7 + e8), with total number of
signs being even. Finally, 2 more roots are given
e8). When
=
e
i
ej , ( ) takes the same structure as that shown for E6.
= ei + ej , one nds ( ) = fk 6= i; jjei
ek; ej
ekg [ f 21 (ei + ej
)
g
= s20 (s1e1 +
+ s6e6
e7 + e8), with s1
s6 = 1,
( ) = fs0(siei + sj ej )g [ f
s0(siei + sj ej )g [ fs0(e8
e7);
+ s0(e7
e8)g (2.126)
with j ( )j = 6C2 + 6C2 + 2 = 32. Finally, when
= e7
e8, one nds
(2.121)
(2.123)
(2.124)
=
(2.125)
e
i
1
2
( e1
ej ,
jGj
c2
N 2
N
1
1
2N 2
2N
N
2
248
30
with j ( )j = 24 + 32 = 56. Other cases with roots of the form
s8e8) with s1
s8 = 1, one nds
with j ( )j = 32. The case with
the form of
e
ej , i; j = 1;
= e8
e7 is similar. For E8, j j = 240. 112 roots take
; 8, from SO(16) subalgebra. Additional 128 roots take
e8) with number of
signs being even, forming the SO(16) spinor
that can be checked with all ( ) we listed above is that, if
with j ( )j = 8C2 + 8C2 = 56. Including the SU(N ) case studied in section 2.1, one nds
4, where c2 is the dual Coxeter number. See table 1. Another useful fact
2
( ), then
) = 1. So at given , one nds
( ) = fsiei + sj ej g [ f
(siei + sj ej )g
2
( ), since
X
2 ( )
f ; =
(
X
2 ( )
f ;
2
for any expression f ; .
By following the analysis for the U(N ) case, till (2.1), one nds
2
f2 =
E2
96 44 X Li 1(e (v))
r) +
X
2
X
=
Now we use the identity (2.129) to rewrite the last term in the parenthesis as
1
1
(1 e (v))(1 e (v)) + (1 e (v))(1 e (v) (v)) + (1 e (v) (v))(1 e (v)) :
2 X
3
2
X
2 ( )
2
with renaming
1, (2.131) becomes 23 P
On the second term, we relabeled
into
in the rst sum, and then took
( ) as labeling the elements of (
). The third term is simply the second term
1 1 1
. Using the identity (1 ex)(1 ey) + (1 e x)(1 ey x) + (1 ex y)(1 e y) =
2
P
E2( )
24
E2( )
24
2
2
2 ( )
1 = 43 (jGj
where at the last step we used the identity jGj = r(c2 + 1) for simplylaced Lie algebra.
Namely, (2.132) implies
f2 contains E2( ) = 1
, so makes a prediction on the instanton corrections.
For G = SO(2N ), one can compare this against microscopic instanton calculus for the
= 1 theory [9].
We compared the two results at 1 instanton level for SO(8).
f2jq1 = fSdualjm4q1 =
where Li 1(x) = (1 xx)2 . On the other hand, the single instanton partition function Z1 for
the SO(2N ) theory can be obtained by starting from the Witten index for the quantum
mechanics describing an O4 plane, 2 D0branes and 2N D4branes (in the covering space).
The index is a complicated residue sum. One should further subtract the contributions
from D0branes unbound to D4O4, which was explained in [9]. Following this procedure,
we checked that
fSdual
(fSO(8)
4fU(1))
m4q1
=
c2jGj
12
:
One can continue to generate higher order fn's, and also the microscopic instanton calculus
for general DN at higher order in q, and compare them. Here we simply conjecture
f ( ; v; m) = fSdual( ; v; m) + rfU(1)( ; m) +
288
c2jGj m4E2( )
for all G = SU(N ); SO(2N ); EN , where r is the rank of G. For G = SU(N ), we have tested
it extensively in section 2.1, after adding one free tensor multiplet to make it U(N ). For
G = SO(2N ), we tested it till m4, q1 order only at N = 4, but in principle one can do all
the calculus of section 2.1, following the methods of [9]. For EN , this is just a prediction
by assuming Sduality and 5d perturbative results. The last term proportional to c2jGj
will be further tested in section 3, from the 6d chiral anomaly of SO(5) Rsymmetry.
3
High temperature limit of the index
In this section, we compute the asymptotic form of the prepotential at strong coupling, or
high `temperature'
! i0. This is the limit in which the compacti cation radius R0 of
the sixth circle becomes large, or equivalently in which D0branes become light. The key
technique of computation will be the anomalous Sduality that we developed in section 2.
Our convention is that the strong coupling theory of our interest is the `Sdualized'
theory. So we take D ! i0+, and
=
1D ! i1. Recall fSdual satis es
(2.133)
(2.134)
(2.135)
fSdual( D; vD; mD) = fSdual( ; v; m) +
(3.1)
1
4 i
where D =
1 , vD = v + 2 1i @@fv , mD = m . We replaced fSdual by f when it appears with
v derivatives, since fanom is independent of v. Inserting fSdual = f
fanom, one nds that
2f ( D; vD; mD) = f ( ; v; m) +
fanom( ; m) :
(3.2)
1
4 i
fanom( ; m) =
log( i)
N m2
2
i D +
6
+
N 3m4
48 i
:
(3.3)
Inserting this in (3.2), one obtains
f ( D; vD; mD) =
2f ( ; v; mD)+
1
4 i 3
N m2D
2
log( i)+
6
i
+
i
6
+
The limit
! i1 on the right hand side has to be understood with care, since
mD scales with . Also, we should study how v scales with
! i1, at xed vD.
Had v; m not scaled with , one would have naively expected that the instanton corrections
in f would have been suppressed at q
1, so that we could replace f on the right hand
side by fpert. Let us check when this is correct. This expectation is correct if Fk(v; m) does
not scale to be larger than qk. From (2.8), Fk scales like Fk
For this factor to be smaller than qk, one should require jRe( mD)j <
e kNm at
Re(m)
2Ni . Let us take
to be purely imaginary for convenience (although most of our nal results are valid for
complex ). Then, Fk can be ignored if
jIm(mD)j <
2
N
:
When Im(mD) reaches
N
2 , we encounter a phase transition, beyond which one should
make a new q expansion on the right hand side. The correct nature of this phase transition
will be commented on later. To make the simplest calculus at D ! i0+, we take mD to
satisfy (3.5).
Let us also discuss how v should scale at xed vD. We shall rst assume that v is nite
at nite vD, and then show that it is consistent with ignoring finst. If finst can be ignored,
then the relation between v and vD can be simpli ed as
v = vD
2 i
(3.5)
(3.6)
(3.7)
(3.8)
(3.9)
12 (log( ex))2 with the branch cut at ex 2 (1; 1).
(m + (v) + log( 1))2 ;
tribution, from the 1
Re(m)
1. Then,
where we used Li2(ex)+Li2(e x) =
So one can approximate
(v) m
2
6
)
1
2
Li2(e v
)
Li2(e v m
) :
Since we assume that v is nite, the rst term not containing m yields a subleading
con! 0 factor in (3.6). To be de nite, we take Im(mD) < 0 so that
Li2(e (v)+m) =
Li2(e
(log( 1) + (v) + m)2
where
Li2(e
(v) m) can be ignored at Re(m)
1. We ignored all the terms that vanish
after summing over , or are subleading in the 1 ! 0 limit. Expanding the square on the
right hand side, the term proportional to m2 = m2D 2 will vanish upon summing over .
The next term proportional to m (v) will be the nonzero leading term. One obtains
v
vDi
mD X
4 i
2
v = vDi
2 i
N mD (P v)i ;
at
mD X
2
2
Also fpert itself is given by
where we used
X
2
ej )
(3.11)
HJEP12(07)
Here, P is the N
N projection to SU(N ). Decomposing v = vU(1) + vSU(N), one nds that
vU(1) = (vU(1))D ; vSU(N)
1
1 + N2miD
(vSU(N))D
(3.10)
(3.12)
(3.14)
(3.15)
(3.16)
X
2adj
fpert =
Li3(e (v))
Li3(e (v) m
)
f ( D; vD; mD) !
2f ( ; v; mD) +
v =
N mD(vSU(N))i =
(vSU(N))iD :
(3.13)
where we used Li3(ex)
x3
6
2
ix2 +
32x if the real part of x is positive and large.
Therefore, the asymptotic prepotential is given by
12 fpert(v; mD). In this case, N2m3D
12
one obtains
In particular, one nds that the Coulomb VEV vD does not appear in the asymptotic
limit. This is natural since the Coulomb VEV is a dimensionful parameter, which should
not be visible in the large momentum limit. This is a result for
When 0 < Im(mD) < 2N , all the analysis above is same except the step of approximating
2
N
< Im(mD) < 0.
is replaced by
N2m3D . Combining the two cases,
i
iN mD
2
2
2
iN mD
2
iN mD
2
4#
;
3At mD =
that v can scale with . At mD =
2Ni , one nds that vSU(N) diverges. In this case, one has to approximate (3.6) by assuming
2Ni , we checked for N = 2; 3 that vSU(N) scales like p , which grows
large but is much smaller than m =
mD. Due to this fact, v does not a ect the asymptotic free energy,
and our nal result for fasymp below will be reliable even at mD =
2 X ei
i6=j
N mD
1 + N2miD
12
3
+
2Ni .
1
12
X
2adj
2
+
m3 =
N 2 3m3D
12
iN m2D
12
+
+
1
2
X
6
X
5
(0; 1)
(N; 0)
(N; 1)
Mls2
2πRIIB
M ! 2NπRls2IgIsB
12 fpert(v; mD) provides subleading contribution in
so that one nds
where
Im(mD) < 2 , respectively. Finally, when Im(mD) = 0,
N
+
iN mD
2
4#
;
i
where the superscript (0) means vanishing imaginary part of mD. At this stage, we note
that fasymp at Im(mD) 6= 0 can be written as the following holomorphic function with a
branch cut,
fasymp =
2Li4(1)
Li4(eNmD )
Li4(e NmD ) :
This expression will be helpful later.
We rst investigate fasymp for purely imaginary mD
ix, at
2N < x < 2N . One nds
fasymp =
3
i "
3N
N x 2
2
2
N x 3
2
+
N x 4#
2
:
The partition function undergoes a phase transition at x = 0, from certain perturbative
particles being massless at mD
x = 0. One may wonder how fasymp behaves beyond
N
2 . At x =
2 , one nds from the Sdual picture that finst cannot be ignored,
N
(e Nme2 i )k
O(1) at m =
mD !
2 i . This means that particles
with nonzero instanton number become light at these points. One can get some insights
on these nonperturbative massless particles.
To see this, it is helpful to recall the type IIB 5brane web realization of the 5d N = 1
system. More precisely, we realize the `Sdualized' setting at
! i1, using weakly coupled
type IIB string theory. The brane web rst consists of N D5branes and 1 NS5brane, all
(3.17)
(3.18)
(3.19)
HJEP12(07)
extended on 01234 directions, transverse to 789, and forming a web on the x5x6 plane. One
makes a twisted compacti cation (x5; x6)
(x5 + 2 RIIB; x6 + M `s2). The D5branes wrap
x5 direction, and form a web with the NS5brane extended along x6, as shown in gure 1.
The twisted compacti cation guarantees that the open strings with tension F1 = 2 1 0
(where 0 = `s2) suspended between D5branes across the web have mass 2M . D1branes
wrapping x5 ending on NS5brane are identi ed as YangMills instantons. Unit instanton's
mass is given by D1 2 RIIB = 22 R0IgIBs , which should be identi ed with R10 in our M5brane
setting. So one obtains 2 RIIB = 2
R00gs . On the other hand,
= gis , which should be identi ed in our M5brane setting as i RR0 . So one nds gs = RR0 .
These provide the relations between the parameters RIIB, gs and the M5brane parameters
is given at zero axion by
R; R0. The slope of the (N; 1) 5brane is
between the two ends of the segment on gure 1. We stated above that
x6 = M 0, so
xx56 = gs1N , where
x5 and
x6 are the distances
one nds
x5 = N M gs 0.
In this setting, the segment of (N; 1) 5brane shrinks at M = 0. Here, the perturbative
hypermultiplet particle becomes massless, corresponding to the fundamental strings
connecting D5branes across the NS5brane. This causes the socalled
op phase transition.
The singular term proportional to jxj3 in (3.19) is caused by fpert in the Sdual setting,
from the particles becoming massless at M = 0. So we conclude that the cusp / jxj3 is
due to the op transition.
As one increases positive M , the next transition happens when the (N; 1) brane
segment goes around the circle in x5 direction, as shown on the bottomright side of gure 1.
This happens at
x5 = N M gs 0 = 2 RIIB,
So one nds that the transition happens at
N M gs 0 = 2 RIIB =
2 0gs =
R0
N
;
(3.20)
(3.21)
precisely when Finst cannot be ignored. Across x =
segment shrinks. So across this value, another transition happens, with the D1brane
segment extended along the shrinking segment being massless.
As one continues to change M , transitions due to nonperturbative massless particles
N
will happen at x = 2 n with n being integers. At n = 1; 2;
; N
1, the nature of this
transition is hard to study. This is because the massless particles are nontrivial bound
states of D1branes. Also, studying the
! i1 approximations around x = 0, not all
massless particles were responsible for the cusp at x = 0. So it will be important to know
which types of massless particles contribute to the cusp of fasymp at x = 2 n
N . However,
if n is a multiple of N , one
nds from the 5brane web diagram that the transition is an
SL(2; Z) transformation of the transition at x = 0, so that the same type of cusp will
N
2 , i.e. m =
2Ni , the N D5brane
happen. Indeed this has to be the case, since x
periodicity of the instanton partition function.
x + 2 (or mD
mD + 2 i) is the
Interestingly, if one takes the holomorphic extension (3.18) within jIm(mD)j < 2N to
the whole region of mD, one gets a de nite prediction on fasymp as a function of real x,
and also on the nature of phase transitions at all n. Plotting (3.18) for the entire real
x, one nds gure 2. fasymp(x) for 2Nn < x < 2 (n+1) is given by simply translating the
N
function in the range 0 < x < 2N by 2 n
N . This means that all the cusp structures are
completely the same at all n, at least in fasymp(x). It will be interesting to understand
how the nonperturbative massless particles cause the same cusp in (3.18). Also, in (3.18)
or in
gure 2, fasymp has a shorter period x
reduced period has to do with multiplewrapping of M5branes on S1, analogous to the
x + 2N . It will be interesting to see if the
mutiplewinding fundamental strings [28].
Now we study fasymp for purely real mD. The asymptotic free energy is given by
log Z
1 2
3 1 2 D
is proportional to N 3. Namely, one nds that the single particle index f ( ; 1; 2; m; v) in
the limit 1;2 ! 0,
! i0+ is given by
X1 1
n=1
n
f (n ; 1;2 = 0; nm; nv) !
3 1 2
N 3m4
16
N m2
4
;
(3.22)
(3.23)
where we dropped the D subscripts. This shows that the microscopic entropy (with minus
sign for fermions) of light D0branes bound to N D4branes exhibit large number of bound
states proportional to N 3. The second term proportional to N clearly comes from N free
tensor multiplets, as this comes from the Sdualization of N fU(1). One can understand
that the rst term proportional to N 3m4 is a remnant of the cancelation between bosonic
and fermionic states in the index, since this term vanishes at m = 0. It will be interesting
to guess what kind of index f ( ; 1;2 = 0; m; v) would exhibit the above behavior in the
high temperature limit. In particular, having the analytic expression (3.18) given in terms
of Li4 functions, with chemical potentials multiplied by N , it will be interesting to seek for
an interpretation using multiplewrapping of M5branes, or instanton partons [29].
Finally, we comment that one can obtain the asymptotic free energy at D ! i0+ for
all ADE theories, starting from (2.135) and following the analysis of this section. To make
a similar calculation, one also needs to know the perturbative partition function, and the
range of Im(mD) in which the instanton correction finst can be ignored on the right hand
side. The perturbative prepotential is straightforward for all ADE. As for the instanton
part, we should know when Fkqk is much smaller than 1 at q ! 0 for large real part of
mD. The leading behavior of Fk for large real m can be easily inferred, by knowing
the correct parameter scalings between the 5d N = 1 theory and the pure N = 1 theory.
Namely, one nds
Fkq
k
ekc2mqk ;
where c2 is the dual Coxeter number of the gauge group G. This is because the pure 5d
N = 1 theory is obtained by taking the limit m ! 1, q ! 0, with
This means that one can ignore the instanton part in the region
following the analysis for the U(N ) case, the asymptotic free energies of ADE theories are
ec2mq held
xed.
2
c2 < Im(mD) < 2c2 . By
3
r
imD
2
2
2jGj
3
+ (c2jGj + r)
4#
;
(3.24)
(3.25)
(3.26)
Im(mD) < 2c2 , and
3
3
r
imD
2
imD
2
2
+ (c2jGj + r)
imD
2
imD
2
4#
where
Tests with U(1) partition function
We provide a small consistency check of fasymp for the U(1) case. By this exercise, one can
also get better intuitions on the true nature of the approximations and phase transitions,
which perhaps may be a bit obscure in our Sduality based approach.
In the previous Sduality based approach, we rst took 1; 2 ! 0 limit of the partition
function, to focus on the prepotential only. Then using the Sduality, we extracted out
the
! 0 asymptotics of the prepotential, where q = e2 i
= e , at
nite m and N .
We reconsider the same limits directly with the U(1) instanton partition function. The
instanton partition function is given by
Zinst = exp
"X1 1 sinh n(m
2
n=1
n sinh n21 sinh n22 1
e n
e n
#
exp
" 4
X1 sinh2 n2m
n3
e n
1
e n
#
(3.27)
in the 1; 2 ! 0 limit. Now we take the
! 0 limit at xed m. This is somewhat tricky
at real m, which we also take to be positive. This is because the above formula is valid for
m <
when m is real. Physically, this is because the partition function Z has poles at
m = n for all positive integers n. So with
xed real m, one would hit many poles as one
takes the
! 0 limit. To deal with this situation more easily, we rst continue m to be
purely imaginary, m = ix, and continue back later to complex m.
Inserting m = ix and taking
! 0 limit, one obtains
Z
exp
= exp
4
X1 sin2 nx #
2
= exp
1
1 2
Li4(eix) + Li4(e ix)
"
1
1
X
The nal expression can be continued to complex x. Here, we use the property
where 0
Re(x) < 1 for Im(x)
0. Bn(x) are the Bernoulli polynomials, given by
Lin(e2 ix) + ( 1)nLin(e 2 ix) =
n!
Bn(x) ;
text
e
t
1
1
X Bn(x)
In particular, one nds B4(x) = 310
1 + 30x2
60x3 + 30x4 , so that
Li4(eix)+Li4(e ix) =
B4(x=2 ) =
for 0
x < 2 . This leads to the asymptotic formula
24
log Z
2 4
3 1 2
2 4
90
x 2
2
2 4
3
2
x 2
2
2
x 3
2
+
x 4
2
x 3
2
+
x 4
2
(3.28)
(3.29)
(3.30)
(3.31)
(3.32)
for 0
x < 2 , which is in complete agreement with the Sdualitybased result, (3.19),
upon inserting
and N = 1. When
2
< x
0, we use a di erent identity of
Li4 function to nd a similar expression, with the sign of the O(x3) term ipped. This also
shows that the continuation (3.18) beyond
2 < x < 2 by the Li4 functions to complex
x is indeed correct.
3.2
6d chiral anomalies on Omegadeformed R
4
T 2
In this subsection, we shall discuss the connection between the Sduality anomaly and the
6d chiral anomalies of global symmetries. In particular, we shall independently compute
some part of our asymptotic free energy fasymp based on chiral anomaly only. However,
let us start by giving a general comment, on why one should naturally expect Sduality
anomaly of the partition function if the system has chiral anomaly.
Consider a partition function of even dimensional chiral theories on T 2, like 2d theories
on T 2 or our system on R
4
T 2. For a global symmetry, one turns on a background gauge
eld A. In particular, let us turn on the
at connection of A on T 2. We shall only be
interested in Abelian
at connections, characterized by the commuting holonomies along
the two circles of T 2. Large gauge transformations would have made both holonomies to
be periodic, had there been no chiral anomalies. For simplicity, let us take a rectangular
torus with two radii r1, r2, respectively. Then the large gauge transformations would
have given the periods A1
A1 + r11 and A2
A2 + r12 .
With matter
elds having
integral charge q of this global symmetry, the modes of these elds would have frequencies
(!1; !2) = ( nr11 + qA1; nr22 + qA2) on T 2, with integral n1; n2, which is invariant under the
periodic shifts of A1, A2. This is a consequence of these gauge symmetries. However, in
quantum observables like the partition function on T 2, these large gauge transformations
may fail to be symmetries for theories with chiral anomalies. This is because one has to
regularize the path integral over these modes, by regarding one of the two directions as
temporal circle [30]. By this procedure, one of the two holonomies A1; A2 fail to be periodic
in the partition functions. This is precisely what happen for the 2d elliptic genera [31]. We
expect that similar things will happen to 6d chiral theories on R
4
T 2, but we cannot make
this expectation more precise here. We shall simply assume the failure of double periodicity
of background holonomies due to chiral anomalies, and then explain that it forces the
partition function to have Sduality anomaly, as we found in section 2 by nonzero Fanom.
Let us write the background holonomies as a complex number m. Had a free energy
F ( ; m) on T 2 been exactly Sdual, then its exact Sduality F ( 1= ; m= ) = F ( ; m)
means that m has double period. This is because if the right hand side has period in one
direction, say F ( ; m) = F ( ; m + 1), the left hand side forces F ( 1 ; m ) = F ( 1 ; m + 1 ),
and thus F ( ; m) = F ( ; m
), contradicting the obstruction of double periodicity from
chiral anomaly. This comment applies to our 6d partition functions. So we naturally expect
Sduality anomaly.
With these motivations in mind, rather than trying to elaborate on it, we shall make
a concretely calculation which shows that a particular term in our asymptotic high
temperature free energy dictated by Fanom can be computed using 6d chiral anomaly only.
Let us rst explain the anomalies of the 6d (2; 0) theory of AN 1 type. More precisely,
we shall consider the anomaly of the interacting AN 1 type theory times a decoupled free
selfdual tensor multiplet theory. This corresponds to the system of N M5branes including
the decoupled centerofmass multiplet. The anomaly polynomial 8form is given by
(3.33)
(3.34)
(3.35)
(3.36)
1
4
1
4
1
8
1
8
1
16
The Pontryagin classes are de ned by
p1 =
1
2(
2
)2 trR2 ;
p2 =
1
trR4 + (trR2)2 :
Here, traces are acting on either 6
6 matrices for SO(5; 1) tangent bundle T , or 5
matrices for SO(5) normal bundle N . Taking their curvatures to be R and F , respectively,
one nds
I8 = N I8(1) + N (N 2
1) p2(N )
24
where I8(1) is the anomaly of the single M5brane theory, or one free (2; 0) tensor multiplet,
1
48
I8(1) =
p2(T ) +
(p1(T )
p1(N ))2 :
N
48
+
1
4
(
2
)4I8 =
trF 4 + (trF 2)2 + trR4
(trR2)2 +
(trR2
trF 2)2
trF 4 + (trF 2)2
:
1
8
1
4
We shall restrict F to a Cartan part. In particular, since we shall be taking the Omega
backgrounds to be small, the Cartan for SU(
2
)R will have much smaller background
eld
than SU(
2
)L, from +
m. So we shall only turn on the background eld for the Cartan in
whose components are F ab =
F ba with a; b = 1;
SO(5), corresponding to our N = 1 mass m. F is a 5
; 5. The component corresponding to
5 matrixvalued 2form,
the Cartan of SU(
2
)L is obtained by keeping F 12 =
F 21 =
F 34 = F 43
F only. With
this restriction, one nds tr(F 2) !
4F 2, tr(F 4) ! 4F 4. Inserting these, the SO(5; 1) and
U(1)
SU(2)L anomalies are given by
(
2
)4I8 ! 24
F 4 +
N
48 2
1 F 2trR2 +
1
4
trR4
1
8
(trR2)2 :
(3.37)
Only the rst term N243 F 4 will be relevant for the computations below.
Our goal is to compute some part of the asymptotic free energy at high temperature
D ! 0, using 6d chiral anomalies. Recall that we found
Se =
log Z !
fasympt =
1 2
i
24 3 1 2 D
N 3m4
4 2N m2 +
(3.38)
where
stands for the m3 term which exists when m has imaginary component. The
m3 term will not be of our interest in this subsection. We obtained this expression at
1;2
1 and
D ! 0, where D
used purely imaginary
D with
4 ( + i) is the same D used before. Often, we
= i, but we keep real
in this subsection to see a
clear relation to chiral anomalies. For a reason to be explained below, we would like
to study the asymptotic free energy when all the chemical potentials 1;2; m are purely
imaginary. So inserting i 1;2, im (with real 1;2; m) in the places of 1;2; m in (3.38), one
imaginary part of the e ective action,
obtains Se
N 3m4 + 4 2N m2 + O(m3) . In this setting, we focus on the
Im(Se ) =
12 1 2 (1 + 2)
N 3m4 + 4 2N m2 + O(m3) ;
(3.39)
and compute it from 6d chiral anomalies. Especially, we shall compute part of Im(Se )
from the 5d e ective action approach for the 6d theory on small temporal circle. 6d chiral
anomaly determines a special class of terms in the 5d e ective action. It turns out that,
knowing the terms determined by anomaly, one can only compute the term proportional
to m4. So we shall pay attention to the rst term
Im(Se )
m4
N 3m4
12 1 2 (1 + 2) :
(3.40)
in (3.38).
We shall argue below that this term is completely dictated by 6d chiral anomaly, and then
we recompute this term using chiral anomaly only. This will provide another strong test
of our
ndings from the D0D4 calculus. Then, since one naturally expects that
supersymmetrization of (3.40) is holomorphic in D, one can reconstruct the term
iN3m4
24 3 1 2 D
We shall consider the 6d anomaly from the viewpoint of 5d e ective action, obtained
by compacti cation on a small circle of circumference
1, and discuss our asymptotic
free energy fasympt on R41;2
circle, the partition function is an index of the form
T 2 in this setting. On T 2, regarding one circle as the temporal
Z( ; v; m; 1;2) = Tr h( 1)F e 2 (H i P )eP2a=1 a(Ja+JR)e2mJL e viqi i :
(3.41)
Real 1;2; m is consistent with the conventions for the partition function presented at the
beginning of section 2. In this setting, the chemical potentials 1;2; m will twist the
translation on the temporal circle in a way that the twisted time evolution is not unitary (simply
HJEP12(07)
because the factors in the trace are not unitary transformations). This would cause a
complex deformation of the Euclidean action by twisting with chemical potentials.4 For a
technical reason, it will be convenient to keep these twistings to preserve the reality of the
action. So we replace
eP2a=1 a(Ja+JR)e2mJL e viqi ! ei P2a=1 a(Ja+JR)e2imJL e iviqi ;
(3.42)
which will make real twists of the Euclidean action. This is equivalent to the insertions of
i 1;2; im around (3.40). The factor e 2 H demands us to consider a 6d Euclidean theory
whose temporal coordinate y satis es periodicity y
T 2. Another circle factor is labeled by x, which we take to have periodicity x
x + 2 .
y + 2
. This forms a circle of the
De ning D = 4 ( + i), one obtains
e 2 (H i P )
e2 i D H +2P e 2 i D 2
H P = e 2 Im DH+2 iRe DP :
So D is the complex structure of T 2. This torus is endowed with the metric
ds2(T 2) = (dx
dy)2 + dy2
(3.43)
(3.44)
and periods (x; y)
(x + 2 ; y)
metric of R
4
T 2 is given by
(x + 2 ; y + 2 ). Including the chemical potential a, the
ds2(R4
T 2) =
X
a=1;2
dza
2i a
2
zady
+ (dx
dy)2 + dy2 ;
(3.45)
SU(
2
)L. Also, H 2 P
where za are complex coordinates of C
2
R
4 with charges Ja[zb] =
ab. Finally, the
chemical potential m is realized as the background gauge eld A = 2m dy for U(1)
fQ; Qg, where Q is a supercharge preserved by the index. So Z is
4Strictly speaking, Lagrangian formulation is not known in 6d. So when we refer to a Lagrangian
description, we mean a 5d Lagrangian after reducing on a small circle. See also comments in [11] concerning
the conversion between twistings and background gauge elds in the presence of anomalies.
Following [11] (see also [32]), we shall make a KK reduction on the small circle along
y, for small inversetemperature
1. To this end, one rewrites the background in the
form of
ds2 = e2 (dy + a)2 + hij dxidxj ;
5 is the 5d metric, e2 = 1+ 2 + 42 P
a 2ajzaj2 is the dilaton, and
a =
1
dx
2 ajzaj2 d a
(3.46)
(3.47)
gauge eld and A6 is the 5d scalar. So one nds A6 = 2m and A =
A6a.
is the graviphoton eld, where za = jzajei a . The 6d background gauge eld A for U(1)
SU(
2
)L is also rewritten in the form A = A6(dy + a) + A, where A is the 5d background
If the 6d theory compacti ed on a small circle has no 5d massless modes, one can
express the thermal partition function in terms of a 5d local e ective eld theory of
background
elds, where the 5d derivative expansion corresponds to a
series expansion. As
noted in [11], with massless modes in 5d, there could be nonlocal part of the e ective action
which is smooth in the
! 0 limit. In our case, the nonlocal part comes from the 5d
perturbative maximal SYM. There is additional di culty in using the derivative expansion
in our setting, since some of our background elds are proportional to
1, which may spoil
the orderings provided by the derivative expansion. So it appears tricky to directly employ
the formalism of [11, 32].
However, one can study the imaginary part (3.39) of our asymptotic free energy using
the 5d approach. The imaginary part can be computed completely by knowing the 5d
ChernSimons like terms. To explain this, note rst that we have been careful to set all
our background
elds to be real, e.g. by setting our chemical potentials to be imaginary.
With real background
elds turned on, suppose that we rst reduce the 6d theory on a
small circle to a general 5d Lorentzian spacetime. Then the 5d e ective action is real,
since Hermiticity is not broken in the Lorentzian theory. Now we Wickrotate the `time'
direction in this 5d setting. Since all background elds are real, the only possible step which
may cause complex e ective action is the Wick rotation to Euclidean 5d space. Here, note
that we are seeking for an e ective action of the vectors a; A; ! (spin connection), tensor
hij , and scalars A6, . To compute the imaginary part, one can focus on the local terms.
This is because the nonlocal terms come from the determinant of 5d maximal SYM whose
elds are covariantized by real background
elds, which is real. Among the local terms
obtained from scalar Lagrangian density, we should seek for terms containing the tensor
ijklm to obtain imaginary contribution after Wick rotation. It should be contracted with
antisymmetric tensors formed by the background
elds.
There are many possibilities,
arranged in derivative expansion. For instance, there could be complicated terms like
da ^ dA ^ d f ( ; A6), and so on.
Although there are many terms, let us comment that there can be gauge invariant
terms and gauge noninvariant terms in the imaginary action. The latter class should exist
because the 5d e ective action should realize 6d chiral anomalies. The coe cients of the
terms in the latter class are thus completely determined by known 6d anomalies [11, 32].
Among the gauge invariant terms, there can be action coming from gauge invariant
Lagrangian density, like the term that we illustrated in the last paragraph. Finally, there may
be ChernSimons terms in which Lagrangian densities are not gauge invariant but their
integrals are. So the imaginary action takes the following structure,
(3.48)
SCS = SC(1S) +SC(2S) +SGI
SC(1S) =
SC(2S) =
i 1 Z
3
iDr1 Z
96 2
i 2 Z
i 3 Z
Z
a^da^da+
A^dA^da+
a^R^R+i 4
A^dA^dA+
A6a^da^da+4A63A^da^da+6A62A^dA^da+4A6A^dA^dA +
4
;
HJEP12(07)
where r1 = 4 is the radius of the small sixth circle with circumference 2 . SC(1S) consists
of the gauge invariant ChernSimons terms. SC(2S) is part of the gauge noninvariant
ChernSimons terms that comes from U(1)
SO(5)R normal bundle anomaly in 6d,
namely the rst term
N243 F 4 of (3.37). Anomaly matching xes D = N 3, as well as the
relative coe cients as shown on the second line.5 The omitted terms
ChernSimons terms containing !, which we do not need here. The omitted terms in SC(2S)
can all be computed from mixed anomalies and gravitational anomalies of (3.37), which
we do not work out here as we shall not need them. Finally, SGI is the action containing
ijklm associated with gauge invariant Lagrangian density, e.g. da ^ da ^ d f ( ; A6), dA ^
dA ^ d[(da)ij (dA)ij ]g( ; A6), and so on. One point we emphasize is that SGI can come in
in nite series of derivative expansion, while SC(1S) and SC(2S) consist of nite number of terms
in SC(1S) are other
and can be completely classi ed.
The imaginary terms have rich possibilities. Here we consider the terms which are
nonzero with our background, and also the leading terms in small 1;2, proportional to
11 2 . A6 = 2m is constant in our background. Also, A =
graviphoton. Plugging in these values, one obtains
A6a is constant times the
(A6)n ijklm(rank 5 antisymmetric tensor of a; ; !; h) :
(3.49)
The parenthesis consists of the elds reduced from 6d metric (3.45). Note that, after
plugging in constant A6 and A =
the remaining
elds. This is because the only possible gauge noninvariant terms SC(2S),
A6a, all terms should be formally gauge invariant in
completely dictated by anomaly, also become gauge invariant like A46a ^ da ^ da with
constant A6.
Now we note the fact that, in the 6d metric, all za coordinates of R4 are multiplied
by a. So in the parenthesis of (3.49), the only za's not associated with a are derivatives.
So one makes a formal derivative expansion of this term, assigning the `mass dimensions'
[a] = 0, [ ] = 0, [h] = 0, [!] = 1. The lowest order term comes in two derivatives, and
is proportional to a ^ da ^ da. There can be no other gaugeinvariant terms at this order.
This term indeed yields the desired
1
1 2
scaling. Firstly, the integral dxd2z1d2z1 can be
scaled into (
1 22 )2 times a measure depending on aza . Also, two derivatives in a ^ da ^ da
5Following [11], we show the form of the action with constant value of A6, taking into account the
covariant anomaly rather than the consistent anomaly. This is su cient for our calculus of the free energy.
can also be scaled with a , yielding another overall factor 1 22 . za in the remaining integral
appear in the combination aza , including the integration variable, so is independent of a.
So this term yields the right scaling
consider those terms that reduce to
11 2 . Therefore, to compute (3.40), we only need to
(A6)na ^ da ^ da
(3.50)
upon plugging in our background. This implies that one does not have to consider SGI
of (3.48), since they are associated with local Lagrangian density and cannot provide terms
So we only consider SC(1S) and SC(2S) of (3.48). Unlike the coe cients of SC(2S), coe cients
of SC(1S) cannot be determined with our limited knowledge of the 6d theory. So even after
restricting our interest to the imaginary part (3.39) of the e ective action, we cannot
compute them all due to our ignorance on these coe cients. Since the second term of
shall not need the mixed anomaly contributions in SC(2S) coming from the term
SC(1S) is quadratic in A, we cannot compute the O(m2) term of (3.39). This is why we
F 2trR2
in (3.37), which will also yield a contribution at O(m2), since knowing them is incomplete
to compute the whole O(m2) contributions. Also, the O(m3) term cannot be computed
since we do not know
4
. However, the ChernSimons terms that are quartic in A and
A6 are completely dictated by 6d anomalies, as shown on the second line of (3.48). Note
that quartic ChernSimons term is allowed precisely because we allow gauge noninvariant
ChernSimons term, to match 6d anomalies which are fourth order in the elds. Thus, we
can compute (3.40) from SC(2S) of (3.48). Note also that, for imaginary chemical potentials,
we have found earlier in this section that fasymp undergoes phase transitions due to massless
particles. This only changes O(m3) or lower order terms, so that the m4 order that we are
going to compute is una ected.
We also note in passing that, we can turn the logic around and use our D0D4 results
to constrain the 5d e ective action. Namely, we know from our D0D4 calculus the O(m2)
and O(m3) coe cients of Im(fasymp), and also the vanishing of the O(m0) coe cient. This
knowledge can be used to constrain 1
; 2
; 3
of (3.48). This information may be useful
for studying other high temperature partition functions of the 6d (2; 0) theories.
Coming back to the computation of (3.40), we plug A =
A6a and A6 = 2m into SC(2S)
iN 3(A6)4r1 Z
96 2
a ^ da ^ da :
To compute this, one should evaluate the graviphoton ChernSimons term,
Z
Z
1 + 2 +
4 2ajzaj2
2
3
dx) ^ 2
4 12 2 4dx1 ^ dy1 ^ dx2 ^ dy2 (3.52)
(3.51)
(3.53)
where za
R dx = 2 , R dxadya =
R d(ra2), (3.52) becomes
xa + iya, with x1; y1; x2; y2 being the Cartesian coordinates of R4. Since
64 3
1 2 Z 1
2
0
1+ 2 + 4 2ara2 3 =
2
4 3
2 Z 1
1 2
(1+ 2 +X +Y )3 =
2 3
2
(1+ 2) 1 2
;
96 2
a ^ da ^ da =
i
N 3
3 27 3
16m4
4
2 3
2
(1 + 2) 1 2
N 3m4
i
12 1 2 (1 + 2) ; (3.54)
where we plugged in r1 = 4 . This precisely agrees with (3.40), based on D0D4 calculus.
Finally, let us comment that the same calculation can be done to test some part
of (3.25) for all ADE theories. For ADE, (3.25) yields the imaginary part
simply by changing the coe cient N 3 ! c2jGj + r from (3.25). On the other hand, the
anomaly polynomial (3.33) is replaced by the following polynomial
Im(Se )
(c2jGj + r)m4
12 1 2 (1 + 2) ;
I8 = rI8(1) + c2jGj 24
p2(N )
(3.55)
(3.56)
N 3 by c2jGj + r, completely reproducing (3.55).
4
Conclusions and remarks
for ADE. Again after restricting SO(5)R to U(1)
replaced by c2jGj+r F 4. So the calculations of this subsection can be done by replacing all
SU(
2
)L, the term N243 F 4 of (3.37) is
In this paper, we explored Sduality of the prepotential of the 6d (2; 0) theories compacti ed
on T 2, on the Coulomb branch. We found evidences of Sduality and its anomaly. Using
this result, we computed the asymptotic free energy of this system compacti ed on S1 (in
the index version), when the Omega background parameters 1;2 and the chemical potential
conjugate to the KK momentum are small. The asymptotic free energy is proportional
to N 3 in a suitable large N limit, showing that the light KK
elds exhibit the N 3 degrees
of freedom. After suitably complexifying the chemical potentials, we showed that the
imaginary part of the free energy proportional to N 3 is completely reproduced from the 6d
chiral anomaly of the SO(5) Rsymmetry. Most results are generalized to the ADE class
of (2; 0) theories.
In the literatures, the N 3 scalings of various observables of 6d (2; 0) theory have been
found, using various approaches. Thermal entropy of black M5branes [33] or various other
quantities are computed from the gravity dual. Chiral anomalies are computed from the
anomaly in ow mechanism [34]. The supersymmetric Casimir energy on S5 was computed
from the superconformal index [35{40]. Perhaps among these, the mysteries of 6d CFT
may be most directly addressed from the thermal partition function calculus of [33]. So
it would be desirable to have a microscopic view of this phenomenon by directly counting
states of the 6d CFTs. As far as we are aware of, such a direct account for N 3 scaling
of states has not been available from a microscopic quantum calculus yet. Our studies
show the N 3 scalings of the microscopically counted degrees of freedom. More precisely,
we compacti ed the 6d SCFT on S1, so N 3 degrees of freedom are absent at low energy.
However, at high temperature compared to the inverseradius of the circle, we expect the
6d CFT physics to be visible, hopefully in our F . One subtlety is that fermionic states
are counted with minus sign in the index, so there may be cancelation between bosons and
fermions. Even after this possible cancelation, we nd that the uncanceled free energy still
exhibits N 3 scaling, which proves that the 6d CFT has N 3 degrees of freedom. We have
provided an alternative study of the asymptotic free energy based on 6d chiral anomalies,
which completely agrees with our D0D4 calculus.
Our studies based on D0D4 system also shows that the light D0brane particles are
responsible for the UV enhancement of degrees of freedom. Since D0branes are the key
objects which construct Mtheory at strong coupling limit of the type IIA strings, it is
natural to see that they are also responsible for the N 3 degrees of freedom of the 6d (2; 0)
theory. It will be interesting to better understand the single particle index f ( ; 1;2; m; v)
which yields this behavior. In particular, conjectures on instanton partons [29] may be
addressed in more detail.
The Coulomb branch partition function on R
4
T 2 was used as building blocks of
interesting CFT indices in the symmetric phase. We comment that our asymptotic free
energy proportional to N 3 does not appear in these symmetric phase indices. Let us explain
this with the 6d superconformal index, and the DLCQ index.
Firstly, it has been proposed that the D0D4 partition function, or more precisely
this partition function multiplied by the 5d perturbative part, is a building block for the
6d superconformal indices [35{38] on S5
S1. So one might wonder whether our nding
log Z / 1 2
N3m4
(with D = 2i ) at high temperature has implications to the supercofonrmal
index. One can immediately see that the answer is negative. For this discussion, the
relevant formula is presented in [38], which uses the product of 3 copies of Coulomb branch
partition functions on R
4 T 2 as the integrand. The angular momentum chemical potentials
of U(1)2
SO(6) on S5 are labeled by three numbers a1; a2; a3 satisfying a1 + a2 + a3 = 0.
In this setting, the 3 sets of Omega deformation parameters are given by ( 1; 2) = (a2
a1; a3
a1), (a3
a2; a1
a2), (a1
a3; a2
a3) respectively. Since the asymptotic formula
for Z is obtained in the limit of small 1; 2, one can study the superconformal index in the
limit of small a1; a2; a3. In this limit, the most divergent part in 1;2 is given by
log ZS5 S1
N 3m4
1
(a2 a1)(a3 a1) (a3 a2)(a1 a2) (a1 a3)(a2 a3)
(4.1)
It is an identity that the sum in the square bracket vanishes, so that the leading asymptotic
part proportional to N 3 vanishes on S5
S1. So our fasymp has no implication to the
superconformal index. However, study of the subleading part O( 1;2)0 will be interesting,
along the lines of our section 2.2. We hope to come back to this problem in the near future.
Secondly, the M5brane theory compacti ed on a lightlike circle can be studied using
the D0D4 quantum mechanics [41, 42]. Its index at DLCQ momentum k can be computed
by integrating the D0D4 index in the Coulomb branch suitably with the Coulomb VEV
v, as explained in [7]. So one nds (again with D = 2i
N 3m4
24 1 2
! 0)
:
(4.2)
ZDLCQ
exp
Here, unlike the partition function on R
T 2, where we have notion of multiparticles
so that log Z itself is meaningful as the singe particle index, the DLCQ index is de ned
with a con ning harmonic potential on R4 [7]. Thus, the exponent cannot be physically
meaningful separately. Also, the de nition of ZDLCQ is such that + = 1+ 2 has to be real
2
and bigger than other fugacities, as e + < 1 plays the role of main convergence parameter.
So one has to set 1 2 > 0. This implies that ZDLCQ does not exhibit exponential growth,
but is rather highly suppressed at small , presumably due to boson/fermion cancelation.
From these observations on the superconformal index and the DLCQ index, one
realizes that ZR4 T 2 contains interesting dynamical information which may be wiped out in
other observables.
Omega deformed partition functions can also be used to study 6d (1; 0) superconformal
eld theories. In fact, for many 6d (1; 0) systems, the index on R
4
T 2 is known in the
`selfdual string expansion,' similar to the Mstring expansion explained in our section 2.2.
The coe cients like Z(ni) of section 2.2 are elliptic genera of 2d CFTs for the 6d selfdual
strings in the tensor branch. Those elliptic genera have been studied for various 6d (1; 0)
theories [43{48]. The Sduality anomaly and the high temperature asymptotic free energies
could be studied using the approaches explored in this paper. This may be an interesting
approach to explore the rich physics of 6d CFTs and their compacti cations to 5d/4d.
It would also be interesting to further study the Sduality of the full index of the (2; 0)
theory, based on some ideas sketched in our section 2.2. Following [26], we nd it interesting
to study the Wilson/'t Hooft line defects uplifted to 6d surface operators. Sdualities of
other defect operators should also be interesting.
3
Finally, one may ask if a suitable M2brane partition function on R
S1 can exhibit
N 2 scaling, where is the Omega deformation parameter. Although this scaling has been
microscopically computed from the S3 partition function, or the entanglement entropy,
perhaps better physical intuitions can be obtained by directly accounting for where such
degrees of freedom come from, like we did for 6d SCFTs on S1 from D0branes
(instanton solitons).
Acknowledgments
We thank Prarit Agarwal, Joonho Kim, Kimyeong Lee, Jaemo Park, Jaewon Song, Shuichi
Yokoyama for helpful discussions, and especially HeeCheol Kim for many inspiring
discussions and comments. We also thank Joonho Kim for helping us with the SO(8) instanton
calculus. This work is supported in part by NRF Grant 2015R1A2A2A01003124 (SK, JN),
and Hyundai Motor Chung MongKoo Foundation (JN).
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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