Asymptotic M5-brane entropy from S-duality

Journal of High Energy Physics, Dec 2017

We study M5-branes compactified on S 1 from the D0-D4 Witten index in the Coulomb phase. We first show that the prepotential of this index is S-dual, up to a simple anomalous part. This is an extension of the well-known S-duality of the 4d \( \mathcal{N}=4 \) theory to the 6d (2, 0) theory on finite T 2. Using this anomalous S-duality, we find that the asymptotic free energy scales like N 3 when various temperature-like parameters are large. This shows that the number of 5d Kaluza-Klein fields for light D0-brane bound states is proportional to N 3. We also compute some part of the asymptotic free energy from 6d chiral anomalies, which precisely agrees with our D0-D4 calculus.

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Asymptotic M5-brane entropy from S-duality

X = M5-brane entropy from S-duality Seok Kim 0 1 June Nahmgoong 0 1 Seoul National University 0 1 Theory, Supersymmetric Gauge Theory 0 1 Gwanak-ro , Gwanak-gu, Seoul 08826 , Korea 1 Department of Physics and Astronomy & Center for Theoretical Physics We study M5-branes compacti ed on S1 from the D0-D4 Witten index in the Coulomb phase. We rst show that the prepotential of this index is S-dual, up to a simple anomalous part. This is an extension of the well-known S-duality of the 4d N = 4 theory to the 6d (2; 0) theory on nite T 2. Using this anomalous S-duality, we asymptotic free energy scales like N 3 when various temperature-like parameters are large. This shows that the number of 5d Kaluza-Klein proportional to N 3. We also compute some part of the asymptotic free energy from 6d chiral anomalies, which precisely agrees with our D0-D4 calculus. S-duality; 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 Omega-deformed R 4 T 2 Conclusions and remarks Introduction Strong/weak-coupling duality, or S-duality, 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 Yang-Mills (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 S-duality. S-duality in quantum eld theories has also been a cornerstone of developing string dualities [4]. In both QFT and string theory, S-duality provides valuable insights on the strongly coupled regions of the systems. S-duality 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 far-reaching implications of the 6d CFTs to challenging lower dimensional systems. On the other hand, the 6d CFTs lack microscopic de nitions, so that S-duality can provide useful clues to better understand the mysterious 6d CFTs. In this paper, we study the S-duality 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 D0-D4 system. More precisely, we study the Witten index of the quantum mechanical U(k) gauge theory for k D0-branes bound to N separated D4-branes and fundamental open strings, and also study their generating function for arbitrary k. In M-theory, this system is made of N M5-branes wrapping S1. The D0-D4 systems describe the 6d (2; 0) CFT on M5-branes in { 1 { the sectors with nonzero Kaluza-Klein momenta. From the viewpoint of 5d super-YangMills theory on D4-branes, this partition function is also known as Nekrasov's instanton partition function [6]. Although the 5d Yang-Mills description of its instanton solitons is UV incomplete, the D0-D4 system provides a UV complete description for computing the instanton partition function. The results in our paper rely only on the UV complete D0-D4 S1, where S1 is the temporal circle for the Witten index. However, with D0-branes (Yang-Mills instantons) providing the Kaluza-Klein (KK) modes of M-theory, 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 D0-D4 index as a 6d partition function on R 4 T 2, one can naturally ask if it transforms in a simple manner under the S-duality acting on T 2. In this paper, we establish the S-duality of the prepotential of this index, nding a simple anomaly of S-duality 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 S-duality takes the following form. The prepotential F can be divided into two parts, F (a; m) = FS-dual(a; m)+ Fanom(m), where FS-dual is related to its S-dual prepotential by a 6d generalization of the Legendre transformation. (See section 2 for details.) Fanom is a simple function which does not obey S-duality, 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 FS4-ddual(a; m) + Fa4ndom(m), where FS4-ddual is self S-dual. Since Fa4ndom is independent of a, F 4d is S-dual in the Seiberg-Witten theory, which only sees a derivatives of F 4d. In our 6d uplift, F appearing in the Seiberg-Witten 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 = FS-dual +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 (D0-brane charge) to be much smaller than the inverse-radius 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 S-duality, since it relates the small (strong coupling) regime to the well-understood 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 D0-brane 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 S-duality of the prepotential and test it either by expanding F in the 5d N = 1 mass m, or by making the `M-string' 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 D4-branes. 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 D4-branes. When 'I are all diagonal matrices, their eigenvalues are interpreted as D0-brane positions transverse to D4-branes. Similarly, when am are all diagonal, their eigenvalues are interpreted as D0-brane positions along D4-brane 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 D0-branes on R9, unbound to the D4-branes. 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 M5-branes 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 Fayet-Iliopoulos (FI) parameter, shifting Di in (2.2) by three constant i . After this deformation, the rst branch becomes non-BPS, 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 D4-branes, the 5d maximal SYM theory. D0-branes are realized in Yang-Mills 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 D4-branes along a line, giving nonzero VEV to 5 only. In this setting, we shall study the BPS bound states of the D0-branes and the fundamental open strings stretched along the 5 direction, suspended between a pair of D4-branes. 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 D0-D4 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 self-dual strings with charges qi coming from open M2-branes, with P units of momenta on them. We also de ne H R0(E viqi), which is the (dimensionless) energy on the self-dual 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 2-planes 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 D0-branes are much heavier than the stretched fundamental strings. Zk is computed as the nonperturbative Witten index of the D0-D4 system with xed k. Zpert comes from the zero modes at P = 0, the perturbative open string modes on the D4-branes. 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 Yang-Mills 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 S-duality from the M-string 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 S-duality of Z, which we comment on in section 2.2. (2.14) has a more natural interpretation as the Witten index of charged W-bosons in the Coulomb phase [7]. However, as an abstract partition function, Zpert is more natural as it is manifestly Weyl-invariant. 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 Gopakumar-Vafa invariants [18, 19]. The factor 2 sinh 2112 sinh 22 comes from the center-of-mass 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 S-duality 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 Yang-Mills 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 1-form 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 S-duality, 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 Seiberg-Witten theory. The perturbative prepotential satis es the following pseudo-scaling 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 Kaluz-Klein 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 S-duality for the partition function on R 4 T 2, it will be helpful to study the interesting decompactifying regime from the well-understood region ! i1. Developing the S-duality of the prepotential F is the goal of this section. (In section 2.2, we also comment on the S-duality of the full partition function.) 2.1 S-duality and its anomaly Following [10], we review the basic set up for studying the S-duality 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 S-dual 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 S-duality exists if FD4d and F 4d are same function, FD4d( ; a; M ) = F 4d( ; a; M ). This S-duality has been tested in detail in [10]. More precisely, it was found that F 4d( ; a; M ) = FS4-ddual( ; a; M ) + Fa4ndom( ; M ) ; where FS4-ddual satis es FS4-ddual( D; aD; M ) = FS4-ddual( ; a; M ) a and Fa4ndom is an anomalous part of S-duality, 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 FS4-ddual are identical in the Seiberg-Witten theory. This establishes the S-duality of the 4d N = 2 theory in the Coulomb branch e ective action. Let us rephrase the 4d S-duality 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 FS4-ddual( D; aD; M ) = 2FS4-ddual( 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, FS4-ddual 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 FS4-ddual( D; aD; M ) = 2FS4-ddual( 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 semi-classical result aD = a. In this limit, we shall discuss what is the natural S-dual variable using the Abelian 6d (2; 0) theory. In 4d, aD = a is a 2F~S4-ddual( D; aD= ; M= ) = F~S4-ddual( ; a; M ) a instead of (2.41). To summarize, by trivially rede ning FS4-ddual and Fa4ndom by the last term of (2.46), one can reformulate the standard S-duality (2.41) as (2.47). Only (2.47) will naturally generalize to the S-duality on R 4 T 2. Now we seek for the S-duality 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 S-duality, 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 S-duality 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 S-duality transformation exchanges R $ R0. So the dual complex structure is D = i R0 = R More precisely, S-duality 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 S-duality, 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~S4-ddual satis es F~S4-ddual( ; a; M ) = F 4d S dual( ; a; M ) jGj 2 r aD = a + natural aspect of S-duality 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 self-dual tensor multiplet, which is noncompact. On the other hand, the imaginary part of a comes from the holonomy of the 2-form 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 S-duality 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 = FS-dual( ; v; m) + Fanom( ; m) where v = Ra, m = RM , and Fanom is independent of the Coulomb VEV. FS-dual satis es 2FS-dual = FS-dual( ; v; m) v We have some freedom to choose Fanom, by adding/subtracting v independent S-dual expressions to Fanom, FS-dual. 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 S-duality 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 Seiberg-Witten 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 S-dual 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 FS-dual. Firstly, replacing F by Fcl, one can check that S-duality 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 Seiberg-Witten theory, f is ambiguous by addition of v independent functions, possibly depending on and m. However, the S-duality requirement (2.58) is sensitive to the value of f , including the v independent part. So when one tries to establish the S-duality 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 S-duality (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 M-string expansion [8]. As studied in the 4d limit [10], there is a natural way of achieving the S-duality requirement (2.58). This is to require that f is expanded in quasi-modular 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 quasi-modular form of weight 2n 2, which means the following. Quasi-modular forms are polynomials of the rst three Eisenstein series E2, E4, E6, where each series has weight 2; 4; 6 respectively under S-duality 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 quasi-modular 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 quasi-modularity is related to the S-duality (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 fS-dual 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 warm-up, 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 S-duality relation (2.58). Therefore, S-duality requirement (2.58) is satis ed if f satis es the quasi-modular 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 fS-dual in a series of m2 which satis es both (2.63) and (2.66). Then we show that f fS-dual is independent of v. We expand fS-dual like (2.59), fS-dual = 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 fS-dual 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 fS-dual is to show that the Coulomb VEV dependent part of f is S-dual. Therefore, while xing the integration constants and f1 in fS-dual 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 fS-dual 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 D0-branes bound to D4-branes but unbound to W-bosons 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 S-duality 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 W-bosons, 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 S-duality of fS-dual. 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 fS-dual at m4 order. We nd that (f N fU(1)) fS-dual 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 S-duality 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 fS-dual 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)) fS-dual 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 simply-laced 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 = fS-dualjm4q1 = 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 D0-branes and 2N D4-branes (in the covering space). The index is a complicated residue sum. One should further subtract the contributions from D0-branes unbound to D4-O4, which was explained in [9]. Following this procedure, we checked that fS-dual (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) = fS-dual( ; 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 S-duality 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) R-symmetry. 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 D0-branes become light. The key technique of computation will be the anomalous S-duality that we developed in section 2. Our convention is that the strong coupling theory of our interest is the `S-dualized' theory. So we take D ! i0+, and = 1D ! i1. Recall fS-dual satis es (2.133) (2.134) (2.135) fS-dual( D; vD; mD) = fS-dual( ; v; m) + (3.1) 1 4 i where D = 1 , vD = v + 2 1i @@fv , mD = m . We replaced fS-dual by f when it appears with v derivatives, since fanom is independent of v. Inserting fS-dual = 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 S-dual 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 5-brane web realization of the 5d N = 1 system. More precisely, we realize the `S-dualized' setting at ! i1, using weakly coupled type IIB string theory. The brane web rst consists of N D5-branes and 1 NS5-brane, all (3.17) (3.18) (3.19) HJEP12(07) extended on 01234 directions, transverse to 789, and forming a web on the x5-x6 plane. One makes a twisted compacti cation (x5; x6) (x5 + 2 RIIB; x6 + M `s2). The D5-branes wrap x5 direction, and form a web with the NS5-brane 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 D5-branes across the web have mass 2M . D1-branes wrapping x5 ending on NS5-brane are identi ed as Yang-Mills instantons. Unit instanton's mass is given by D1 2 RIIB = 22 R0IgIBs , which should be identi ed with R10 in our M5-brane setting. So one obtains 2 RIIB = 2 R00gs . On the other hand, = gis , which should be identi ed in our M5-brane setting as i RR0 . So one nds gs = RR0 . These provide the relations between the parameters RIIB, gs and the M5-brane parameters is given at zero axion by R; R0. The slope of the (N; 1) 5-brane 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) 5-brane shrinks at M = 0. Here, the perturbative hypermultiplet particle becomes massless, corresponding to the fundamental strings connecting D5-branes across the NS5-brane. This causes the so-called op phase transition. The singular term proportional to jxj3 in (3.19) is caused by fpert in the S-dual 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 bottom-right 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 D1-brane segment extended along the shrinking segment being massless. As one continues to change M , transitions due to non-perturbative 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 D1-branes. 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 5-brane 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 D5-brane 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 non-perturbative 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 multiple-wrapping of M5-branes on S1, analogous to the x + 2N . It will be interesting to see if the mutiple-winding 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 D0-branes bound to N D4-branes 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 S-dualization 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 multiple-wrapping of M5-branes, 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 S-duality based approach. In the previous S-duality based approach, we rst took 1; 2 ! 0 limit of the partition function, to focus on the prepotential only. Then using the S-duality, 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 S-duality-based 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 Omega-deformed R 4 T 2 In this subsection, we shall discuss the connection between the S-duality 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 S-duality 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 S-duality 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 S-dual, then its exact S-duality 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 S-duality 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 self-dual tensor multiplet theory. This corresponds to the system of N M5-branes including the decoupled center-of-mass multiplet. The anomaly polynomial 8-form 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 M5-brane 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 matrix-valued 2-form, 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 re-compute this term using chiral anomaly only. This will provide another strong test of our ndings from the D0-D4 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 inverse-temperature 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 gravi-photon 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 non-local 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 Chern-Simons 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 Wick-rotate 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 non-invariant 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 Chern-Simons 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 Chern-Simons terms. SC(2S) is part of the gauge non-invariant 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 Chern-Simons 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 non-invariant 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 gauge-invariant 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 Chern-Simons 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 Chern-Simons term is allowed precisely because we allow gauge non-invariant Chern-Simons 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 D0-D4 results to constrain the 5d e ective action. Namely, we know from our D0-D4 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 gravi-photon Chern-Simons 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 D0-D4 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 S-duality of the prepotential of the 6d (2; 0) theories compacti ed on T 2, on the Coulomb branch. We found evidences of S-duality 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) R-symmetry. 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 M5-branes [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 inverse-radius 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 D0-D4 calculus. Our studies based on D0-D4 system also shows that the light D0-brane particles are responsible for the UV enhancement of degrees of freedom. Since D0-branes are the key objects which construct M-theory 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 D0-D4 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 M5-brane theory compacti ed on a lightlike circle can be studied using the D0-D4 quantum mechanics [41, 42]. Its index at DLCQ momentum k can be computed by integrating the D0-D4 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 multi-particles 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 `self-dual string expansion,' similar to the M-string 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 self-dual strings in the tensor branch. Those elliptic genera have been studied for various 6d (1; 0) theories [43{48]. The S-duality 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 S-duality 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. S-dualities of other defect operators should also be interesting. 3 Finally, one may ask if a suitable M2-brane 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 D0-branes (instanton solitons). Acknowledgments We thank Prarit Agarwal, Joonho Kim, Kimyeong Lee, Jaemo Park, Jaewon Song, Shuichi Yokoyama for helpful discussions, and especially Hee-Cheol 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 Mong-Koo Foundation (JN). Open Access. 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Seok Kim, June Nahmgoong. Asymptotic M5-brane entropy from S-duality, Journal of High Energy Physics, 2017, 120, DOI: 10.1007/JHEP12(2017)120