The topologically twisted index of \( \mathcal{N} \) = 4 super-Yang-Mills on T 2 × S2 and the elliptic genus

Journal of High Energy Physics, Jul 2018

Abstract We examine the topologically twisted index of \( \mathcal{N} \) = 4 super-Yang-Mills with gauge group SU(N ) on T 2×S2, and demonstrate that it receives contributions from multiple sectors corresponding to the freely acting orbifolds T2/ℤm × ℤn where N = mn. After summing over these sectors, the index can be expressed as the elliptic genus of a twodimensional \( \mathcal{N} \) = (0, 2) theory resulting from Kaluza-Klein reduction on S2. This provides an alternate path to the ‘high-temperature’ limit of the index, and confirms the connection to the right-moving central charge of the \( \mathcal{N} \) = (0, 2) theory.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

https://link.springer.com/content/pdf/10.1007%2FJHEP07%282018%29018.pdf

The topologically twisted index of \( \mathcal{N} \) = 4 super-Yang-Mills on T 2 × S2 and the elliptic genus

HJE The topologically twisted index of super-Yang-Mills on Junho Hong 0 1 2 James T. Liu 0 1 2 0 Ann Arbor , MI 48109-1040 , U.S.A 1 The University of Michigan , USA 2 Leinweber Center for Theoretical Physics, Randall Laboratory of Physics We examine the topologically twisted index of N = 4 super-Yang-Mills with gauge group SU(N ) on T 2 S2, and demonstrate that it receives contributions from multiple sectors corresponding to the freely acting orbifolds T 2=Zm summing over these sectors, the index can be expressed as the elliptic genus of a twodimensional N = (0; 2) theory resulting from Kaluza-Klein reduction on S2. This provides an alternate path to the `high-temperature' limit of the index, and con rms the connection to the right-moving central charge of the N = (0; 2) theory. Supersymmetric Gauge Theory; AdS-CFT Correspondence; Conformal Field - T 2 S2 and the elliptic genus Theory 1 Introduction 2 3 2.1 2.2 3.1 3.2 Multiple solutions to the BAEs The topologically twisted index The index as an elliptic genus The topologically twisted index in the `high-temperature' limit 4.1 4.2 Expanding Zfm0;n0;r0g in the `high-temperature' limit Examination of the determinant factor 4.3 The full index in the `high-temperature' limit 4.2.1 4.2.2 4.2.3 4.3.1 4.3.2 O(1) order determinant First subexponential order determinant B Invariance of det B under T and S transformations C Proof that the map (3.34) is bijective 1 Introduction Recent advances in supersymmetric eld theories have led to a new era of precision holography through AdS/CFT. It has been driven on the eld theory side by the key developments of rigid supersymmetry [1] and supersymmetric localization [2]. The former allows for a systematic construction of supersymmetric eld theories on a curved background with topological twists. The latter yields exact eld theory results reliable even at strong coupling limit. Combining these two developments, now we can compute exact eld theory results of various topologically twisted SCFTs on curved backgrounds, which can be explored in { 1 { Sd 1 which computes the supersymmetric index as well as Wilson loop observables in various representations of the gauge group. In particular, a three-dimensional topologically twisted index was introduced as the supersymmetric index on S1 S2 with a topological twist on S2 [3]. When applied to the ABJM theory [ 4 ], it has an interesting feature. In the large-N limit, the topologically twisted index of the ABJM theory on S1 S2 matches the entropy of the dual asymptotically AdS4 magnetic black hole, when it is extremized with respect to the chemical potentials [5]. This is regarded as the rst counting of the microstates of a supersymmetric asymptotically AdS4 black hole. Similarly, the four-dimensional topologically twisted index can be introduced as the supersymmetric index on T 2 S2 with a topological twist on S2. In particular, we can apply this to N = 4 SU(N ) super-Yang-Mills (SYM) with a similar goal in mind, namely counting the microstates of the dual asymptotically AdS5 magnetic black string. This is still an open problem, however, and here we review some of the unsolved issues in both the eld theory and supergravity sides of the duality. Field theory side. The topologically twisted N = 4 SU(N ) SYM on T 2 S2 can be constructed by equipping S2 with background gauge elds that couple to the SO(6) Rsymmetry current, satisfying the conditions categorized in [6]. The explicit computation of the topologically twisted index in the large N limit, however, has not yet been performed unlike in the ABJM theory case. Instead, it has been investigated in the `high-temperature' ! 0+, where the modular parameter of the torus is given by studied in [6, 8]. To be speci c, based on the well known duality between N = 4 SU(N ) SYM on R1;3 and Type IIB supergravity in AdS5 S5 background, we may expect that the same eld theory on T 2 S2 with topological twists is holographically dual to Type IIB supergravity in an asymptotically AdS5 magnetic black string background with conformal boundary T 2 S2. The AdS3 S2 near-horizon solution for the string is known and numerical evidence suggests that it can be extended into a full solution [6]. However, a full analytic supergravity solution with such asymptotic conditions has yet to be constructed.1 At this stage, we focus on the eld theory side by taking a closer look at the topologically twisted index of N = 4 SU(N ) SYM on T 2 S2. As demonstrated in [7], the high-temperature limit of the index, when extremized over the chemical potentials a, matches the right-moving central charge of the N = (0; 2) SCFT associated with the AdS5 magnetic black string < log Z = i 2 ; a; na = cr(na); !0+ 6 2 (1.1) 1Asymptotically AdS5 black hole solutions with conformal boundary R S3 have been constructed in [9, 10]. Even in this case, however, matching its entropy with microstate counting in the large-N limit of the dual eld theory has not yet been done due to various issues. { 2 { HJEP07(218) where fnag are integer magnetic charges satisfying P3 negative [6]. In a way, this is not surprising, as (1.1) is just the expected behavior in the Cardy limit of the SCFT. Away from this limit, however, the index must transform as a weak Jacobi form. This can be seen by Kaluza-Klein reducing on S2, whereupon the supersymmetric index on T 2 becomes the elliptic genus [11]. In this paper, we clarify the connection between the topologically twisted index of N = 4, SU(N ) SYM on T 2 S2 and the elliptic genus. As constructed in [7], the index can be computed using Je rey-Kirwan residues. The result is thus given in terms of a sum over solutions to a set of algebraic equations, commonly referred to as the `Bethe ansatz equations' (BAEs). In contrast to the S2 S1 index, where there is only a single solution to the BAEs (up to permutations) [3, 5], here we nd multiple solutions, where the `eigenvalues' are uniformly distributed over the T 2. Furthermore, the existence of these multiple solutions is fundamental in order for the index to be an elliptic genus. Once the index is understood as an elliptic genus, we revisit the high-temperature ! i0+, by performing the modular transformation ! our results are left at the conjecture level, we reproduce the Cardy limit (1.1), where 1= . Although some of cr(na) = 3(N 2 1) 1 n1n2n3 (n1n2 + n2n3 + n3n1) ; (1.2) in agreement with [7]. Since this expression is valid for arbitrary N , it also holds in the large-N case with holographic dual. More generally, however, it would be interesting to explore the large-N limit at arbitrary values of the modular parameter . Unfortunately, this still appears to be a rather challenging problem, as the only expression we have for the index at arbitrary N is given as a sum over sectors, each corresponding to a di erent solution to the BAEs. The outline of the paper is as follows. In section 2, we rst review the topologically twisted index of N = 4 SYM on T 2 S2, then demonstrate that the BAEs admit multiple solutions. In section 3, we connect the index to the elliptic genus and in particular demonstrate that it transforms as a weak Jacobi form. Given this understanding of its modular properties, we then revisit the high-temperature limit in section 4. Finally, we conclude with some comments on the large-N limit in section 5. 2 The topologically twisted index of N = 4 SYM on T 2 S2 The topologically twisted index of N = 4 SYM with gauge group SU(N ) was de ned in [3, 7] as the supersymmetric index of the theory on T 2 S2 with a topological twist on S2. The index depends on the modular parameter q = e2 i as well as avor chemical potentials a and magnetic uxes na, and may be written as [7] Z( ; a; na) = A X 1 I2BAEs det B N Y 3 Y j6=k a=1 1(uj 1(uj uk; ) uk + a; ) 1 na ; where the prefactor A is given by (2.1) (2.2) A = iN 1 ( )3(N 1) Y 1( a; ) (N 1)(1 na): 3 a=1 { 3 { The sum in (2.1) is over all solutions, I = fu0; u1; : : : ; uN 1; vg, of the `Bethe ansatz the ui's sum to zero. constrained to satisfy While this Jacobian is explicitly constructed from N 1 of the N eigenvalues ui, it is easily seen that it does not depend on which one is omitted because of the SU(N ) condition that According to [7], the avor chemical potentials a and the magnetic uxes na are eiBj = 1 a; ) a; ) : B ; BN ) ; uN 1; v) : 3 X a=1 Note that the uj 's are also constrained to satisfy the SU(N ) condition PjN=01 uj = 0. In up to sign and we will x these degrees of freedom later according to our purpose. { 4 { (2.3) (2.4) (2.5) (2.6) (2.7) (2.8) =N . Here we exclude a = 2 Z in order to avoid issues with the vanishing of 1(0; ) but we do not necessarily assume invariant under a 2 R or 0 < < a < 2 . Instead, the twisted index (2.1) is 2.1 Multiple solutions to the BAEs A solution to the BAEs, (2.3), was obtained in the `high-temperature' limit, under the condition P3 a=1 a = 2 . It can be written as ! i0+, in [7] uj = u 2 ~j; v = (N + 1) ; where u is a constant chosen to enforce the SU(N ) condition Pj uj = 0, and ~ = While this solution was obtained in the high-temperature limit, it actually satis es the BAEs for any in the upper half plane and for arbitrary a's satisfying the constraint P3 a=1 Furthermore, we show below that (2.8) is in fact a special case of a larger set of BAE solutions. The key observation is that the uj variables are doubly periodic, as they are de ned on T 2, with periods uj ! uj + 2 and uj ! uj + 2 . Based on this periodicity, the solution (2.8) then corresponds to the uj 's being evenly distributed along the thermal circle. This de nes the torus T 2=ZN with modular parameter ~ = =N . Then modular invariance suggests that having the uj 's evenly distributed along the other cycle of the T 2 ought to yield another solution, this time with modular parameter ~ = N . Taking this one step further, we expect that any uj 's evenly distributed over the torus T 2 satisfy the BAEs, (2.3). In this case, the set of uj 's de nes a freely acting orbifold T 2=Zm Zn where fmg is the set of all positive divisors of N with N = mn. The corresponding uj 's can be written explicitly as Note that we have introduced a double index notation uj 1 is a constant that, along with m and n, speci es the orbifold. In order to prove that (2.9) indeed satis es the BAEs, we substitute it into (2.4), so that the BAEs reduce to the claim that (2.10) (2.11) (2.12) ! : (2.13) eiv =! Y 3 m 1 n 1 1 Y Y We now use the double periodicity of 1, (A.5), to shift the product over ^j and k^ as an appropriately chosen v. determine v by choosing ^j0 = m 1 and k^0 = n the identity a + 2 ^j+mk^~ ; Inserting this into the r.h.s. of (2.11) and using the constraint P3 a=1 a = 2 Z, then gives eiv =! Y 3 m 1 n 1 1 Y Y In particular, the r.h.s. is now manifestly independent of ^j0 and k^0, thus demonstrating that the full set of BAEs reduce to a single equation that can be consistently satis ed for While this is su cient to demonstrate that (2.9) satis es the BAEs, we can explicitly { 5 { 1 in (2.12) with the upper sign to obtain = ei[(N+1) +(n 1)m a]: (2.14) Inserting this into (2.13), taking the product over a and reducing the exponent then gives v = (N + 1) , which is also compatible with the solution (2.8) of [7]. As a result, we have found multiple solutions to the BAEs, (2.9), labeled by three integers m, n, and r such that N = mn and r = 0; : : : ; n 1. While we have not proven that these are the complete set of solutions to the BAEs (up to permutations), we argue below in section 3 that they are in fact complete based on modular covariance of the index. We now compute the topologically twisted index for a particular sector labeled by fm; n; rg by inserting the solution (2.9) into (2.1). Making this substitution gives where the primes indicate that ^j1 = k^1 = 0 is to be omitted from the double product. The product over ^j1 and k^1 can be shifted using (2.12) as follows: m ^j2 1 n k^2 1 Y0 Y0 ^j1= ^j2 k^1= k^2 1 As a result, we have 1 2 ^j1+k^1 ~ ; m a + 2 ^j1+k^1 ~ ; m = e imk^2 a Y0 Y0 The product of the theta functions can be simpli ed by using the product form of 1(u; ) given in (A.2). We nd mY10 nY10 1 u + 2 ^j+mk^~ ; Substituting these expressions and (2.2) into (2.17) then gives Zfm;n;rg = i 3 " Y det Bfm;n;rg a=1 it is convenient to maintain the original single index notation for the uj 's. Noting that (2.5) singles out u0 as the constrained variable, the entries of the matrix are uj ; a; ) g(u u ; a; ) + g(u u0; a; ); (2.21a) uj ; a; ) g(u0 u ; a; ) + g(0; a; ); where ; 2 f1; 2; : : : ; N 1g. Here we have de ned Since g(u; a; ) is an even function of u, we can derive the identities g(u; a; ) log 1( a + u; ) 1( a u; ) : and Bj;0 = N: X j=0 det B = N det ; BN 1) ; uN 1) B ; B0; B ;0 B0;0 (2.22) (2.23) (2.24) (2.25) # ; (2.26) (2.27) (2.28) HJEP07(218) Consequently, we have Therefore it is enough to study the determinant of the (N (N 1)-square matrix whose entries are given by (2.21a). At this stage, we return to index pair notation given in (2.9) by u ; a; ) ! then de ne the G-function as ; N j=0 3 i X a=1 log 1 a + 2 Accordingly, the sum in (2.21a) can be written in terms of index pair notation as where = n^j + k^ . Now changing the summation over ^j and k^ into a product within the log and inserting (2.12) then gives so that the sum in (2.21a) is in fact independent of which entry is being considered. Simplifying the product of theta functions within the log using (2.18), we get where the prime denotes di erentiation with respect to the rst argument of 1. Finally, inserting (2.30) into (2.21a), we can rewrite (2.21a) as [Bfm;n;rg] ; = ! h IN 1 + B~fm;n;rg ; i (2.31) where B~fm;n;rg is an (N 1) 1) square matrix with entries [B~fm;n;rg] ; = Gfm;n;rg(^j ; k^ ; a; ) Then (2.24) leads to det Bfm;n;rg = N det(1 + B~fm;n;rg): (2.33) Finally, the contribution to the topologically twisted index from the sector labeled by fm; n; rg is given by combining (2.20) with (2.33), Zfm;n;rg( ; a; na) = 3 " Y a=1 " 3 a=1 #N 1 : (2.34) 3 The index as an elliptic genus As we have seen above, there are multiple solutions to the BAEs, each labeled by a set of integers fm; n; rg, corresponding to the modding out of the original T 2 by a freely acting Zm Zn action. The sum over these multiple solutions I 2 BAEs in (2.1) is non-trivial, and explicitly takes the form Gfm;n;rg(^j; k^; a; ) = 2i X @ a log 1(m a; ~) (2.30) X ZfN=n;n;rg( ; a; na); { 8 { (3.1) where Zfm;n;rg is given in (2.34). In this section, we study this expression further for arbitrary and N . In particular, we show explicitly that the index is an elliptic genus, which can be seen based on reduction over the S2 [11]. Here the sum in (3.1) is crucial to ensure proper modular behavior of the index, since modular transformations permute the individual sectors labeled by fm; n; rg. For example, consider the case N = 6, where the index (3.1) is a sum over the twelve fm; n; rg = f1; 6; 0g; f1; 6; 1g; f1; 6; 2g; f1; 6; 3g; f1; 6; 4g; f1; 6; 5g; with corresponding modular parameters ~ = 6 ; + 1 6 ; + 2 6 ; + 3 6 ; + 1 6 ; + 2 6 ; + 3 6 ; + 4 6 ; + 4 6 ; + 5 6 Y " ( )3 more involved. Consider, for example, the action of S on the f1; 6; 2g sector, with ~ = ( + 1)=6 , and then perform a SL(2; Z) transformation 2)=6. We rst take S: ~ ! ~0 = (2 ~0 ! (2~0 1)=(3~0 1) to bring this into the form (2 + 2)=3, corresponding to the f2; 3; 2g sector. Of course, the detailed modular properties of the topologically twisted index depends on how precisely the various building blocks of Zfm;n;rg transform. Before considering the general case, we gain additional insight from the example of N = 2. In this case, there are only three sectors, denoted by f1; 2; 0g, f1; 2; 1g and f2; 1; 0g. The topologically twisted index is then given by the sum where i = 2; 3; 4 correspond to the f2; 1; 0g, f1; 2; 1g and f1; 2; 0g sectors, respectively. Then the modular properties of the index can be derived from those of the elliptic theta functions, i. Turning to the general case, for the index to be an elliptic genus, it must transform as a weak Jacobi form of weight zero. Here it is worth recalling that, for a single chemical potential, a Jacobi form of weight k and index m transforms according to a^ + 2 for a single a^. Since 1 picks up a minus sign for every 2 shift, the numerator of (2.20) picks up a sign ( 1)(1 mN)(1 na^), while the denominator is unchanged since the logarithmic derivatives of 1 are not sensitive to the sign. As a result, we nd Zfm;n;rg ! ( 1)(1 mN)(1 na^)Zfm;n;rg = ( 1)2ma^ ( 1)N(N m)(1 na^)Zfm;n;rg; where we substituted in the index ma^ from (3.8). Writing N = mn then gives N (N m2n(n 1), which is an even integer. Thus the second factor above is simply +1, and we are left with Zfm;n;rg ! ( 1)2ma^ Zfm;n;rg, in agreement with (3.7a). Note that this result , we rst consider the numerator factors in (2.34) usis valid even if we only shift a single a^ . For the shift a^ ! a^ + 2 ing (A.5). For 1( a^; ), we nd simply where ya^ = ei a^ . For 1(m a^; ~), we rst write and use the relation q~n = e2 in~ = e2 irqm to obtain 1( a^ + 2 ; ) = q 1=2ya^ 1 1( a^; ); 1(m( a^ + 2 ); ~) = 1(m a^ + 2 (n~ r); ~) = ( 1)r+nq~ n2=2ya^ N 1(m a^; ~); (3.11) and indices under the constraint P a + 2 and T : ! a ! + 1 and S : a + 2 ! both N and na are even, and an integer otherwise. It is straightforward to generalize this to the case of three chemical potentials, and we verify below that the index (3.1) indeed transforms as a weak Jacobi form of weight zero ma = N 2 2 1 (1 na); a a = 0. To do so, we rst consider the periodic shifts for (3.7a), and next consider the modular transformations 1= for (3.7b). Note that the index ma is a half-integer when (3.8) a ! (3.9) m) = (3.10) (3.12) (3.13) (3.14) (3.15) HJEP07(218) 1(m( a^ + 2 ); ~) = ( 1)n+r(n+1)q N=2ya^ N 1(m a^; ~): This demonstrates that the numerator picks up an overall factor h ( 1)1 N(n+r(n+1))q(N2 1)=2yN2 1i1 na^ ; a^ under a shift of . As above, the sign factor can be rewritten as 1 N (n + r(n + 1)) = = (N 2 (N 2 1) + N (n(m 1) + n2m(m 1) 1) r(n + 1)) rmn(n + 1): Since the last two terms in the nal expression are even, they do not contribute to the overall sign, and we are left with Zfnmum;ne;rrg ! ( 1)2ma^ q ma^ ya^ 2ma^ Zfnmum;ne;rrg; which is the expected result for a Jacobi form of index ma^ given by (3.8). Since the numerator by itself transforms properly under the shift of that the denominator must be inert under this shift. This is not entirely obvious, though, as the logarithmic derivatives of 1 transform as i; a^; ~) iN; (3.16) as can be seen directly from (3.10) and (3.12). The sum of logarithmic derivatives, however, is invariant so long as we simultaneously shift another chemical potential, say ^b, by 2 , since then these additional factors will cancel. Therefore the denominator is invariant under this combined shift, and hence (3.13) extends to Zfm;n;rg itself. Note that this simultaneous shift is in fact required to maintain the condition that the a^'s sum to 2 Z. 3.2 Modular transformations We now turn to the properties of the topologically twisted index under modular transformations. Since a general transformation can be generated by a combination of T and S, it is su cient for us to demonstrate the following properties: These follow from the de nition (3.7b) for a Jacobi form of weight zero and indices ma for the chemical potentials T transformation We begin with the T transformation. As indicated in (3.17a), we expect the partition function to be invariant under T . Nevertheless, the individual sectors labeled by fm; n; rg will get permuted, as in the N = 6 example shown in (3.4). We thus work one sector at a time, and in particular consider the T transformation of Zfm;n;rg. To proceed, we consider the expression (2.20), and observe that the numerator is built from the combination (3.18) (3.19) (3.20) which transforms as a weak Jacobi form of weight 1 and index 1=2, as can be seen from (A.4). For ( a; ), we have simply However, the transformation is not as direct for (m a; ~), since T : ~ ! ~ + m=n, which is not a SL(2; Z) transformation on ~. In this case, it is more useful to note that where r0 = r + m (mod n). Since is invariant under integer shifts of the modular parameter, we end up with T : (m a; ~) ! fm0; n0; r0g = fm; n; r + m (mod n)g: n n g ; Gfm;n;rg(^j; k^; a; + 1) = Gfm0;n0;r0g(^j0; k^0; a; ); ^j0 = ^j + k^ r + m n (mod m); k^0 = k^: S : ( a; ) ! 1 i 2a e 4 Then since the above (^j; k^) ! (^j0; k^0) is a bijective map from Zm and hence the denominator transforms in the expected manner as well. As a result, T permutes the sectors without any additional factors, T : Zfm;n;rg ! Zfm0;n0;r0g. Finally, since fm; n; rg ! fm0; n0; r0g is bijective, it is clear that the full partition function (3.1) is indeed invariant under T transformations, (3.17a). 3.2.2 S transformation We now turn to the S transformation, which takes a ! a= along with ! 1= . Once again, we start with the numerator. Since (u; ) de ned in (3.18) is a weak Jacobi form of weight 1 and index 1=2, we immediately have where with The combination of (3.19) and (3.21) then demonstrates the simple transformation T : Zfnmum;ne;rrg ! Zfnmum0;ner0;r0g; as anticipated in (3.4). To be complete, we must also investigate the T transformation on the denominator of (2.20), which comes from the determinant of Bfm;n;rg. Here we use the double periodicity (A.5) and the modular property (A.3b), to obtain the map For (m a; ~), it is important to realize that S does not simply take ~ to 1=~. Instead, we want to map ~ into a new ~0, at least up to a SL(2; Z) transformation. In particular, we demand where ~0 = (m0 + r0)=n0. The resulting SL(2; Z) transformation is given by a = r g ; c = ad bc = 1; g gcd(n; r); (3.21) (3.22) (3.23) Inserting this expression along with (3.27) into (2.20) then gives S : Zfnmum;ne;rrg ! N 1e 2 i Pa ma 2a Znumer fm0;n0;r0g; with ma given in (3.8). The extra factor of N 1 is canceled by a similar factor arising from det B in the denominator. For this determinant, we use the double periodicity (A.5) and the modular property (A.3b), along with the requirement P a a = 0 to obtain the map with Gfm;n;rg(^j; k^; a= ; 1= ) = Gfm0;n0;r0g(^j0; k^0; a; ) ^j0 = k^0 = g (k^ + dk^0) n g n ^j + r g ^ k mod mod g; N g : In appendix C, we show that the above (^j; k^) ! (^j0; k^0) is a bijective map from Zm Zn to Zm0 Zn0 . Therefore, we get (see appendix B) S : det Bfm;n;rg ! N 1 det Bfm0;n0;r0g; which cancels the extra factor of N 1 in the numerator. As a result, S permutes the sectors with a common factor, S : Zfm;n;rg ! e 2 i P a ma 2a Zfm0;n0;r0g. Then since fm; n; rg ! fm0; n0; r0g is self-inverse and therefore bijective, the full partition function (3.1) transforms under S transformation as (3.17b). Finally, we wish to explain why the chemical potentials must sum to zero in order for the index to be a proper modular form, in particular under the S-transformation: since S takes a to a= , we must demand the simultaneous conditions and ~0 takes the form Here b and d are uniquely determined as the solution to (3.29) under the constraint for r0, r0 < n0. Also note that we can make use of the simple relation c~0 + d = m0 =m, which can be derived without explicit knowledge of b and d. Given (3.28), we then nd S : (m a; ~) ! m a r ; n m = ; m0 a a~0 + b c~0 + d c~0 + d = m m0 e iN4 2a (m0 a; ~0): to satisfy the rst constraint given in (2.6) for both Z( ; a; na) and Z( 1= ; a= ; na), which only makes sense when P3 of the degrees of freedom introduced in (2.7), is not a serious restriction on the index. a=1 a = 0. Of course, we can always use the second type a ! a + 2 Z, to set P3 a=1 a = 0, so this (3.30) (3.31) (3.32) (3.33) (3.34a) (3.34b) (3.35) where where Given the construction of the index as a sum over sectors, (3.1), we now revisit the `hightemperature' limit, ! 0+ with = i =2 , rst investigated in [7] for the single sector Zf1;N;0g. Note that, in what follows, we restrict to purely imaginary , corresponding to a square torus, and real chemical potentials a. In order to explore this limit, it is natural to perform an S transformation (3.17b) assuming P modular parameter has large imaginary part. In particular, we write a a = 0 so that the transformed The partition function Z( 0; 0a; na) receives contributions from individual sectors Zfm0;n0;r0g as we have seen in (3.1), and we generically expect only one or a handful of sectors to dominate. To see this, we rst work on the expression for a xed sector, and then look for the dominant contribution to the sum over sectors. 4.1 Expanding Zfm0;n0;r0g in the `high-temperature' limit In order to expand Zfm0;n0;r0g, we rewrite (2.34) as Zfm0;n0;r0g( 0; 0a; na) = Q a ( 0a; 0) (m0 0a; ~0) N 1 na n0 det 1 + B~fm0;n0;r0g h The numerator can be easily treated using the asymptotic expression for , (A.9), as ( 0a; 0) = i( 1)Da exp 4 i 2a + i (m0 0a; ~0) = i( 1)Xa ei nr00 Xa(Xa+1) exp da(1 da) 1 + O(e 22 min(da;1 da)) ; (4.4a) iN 4 2 a + i m0 n0 xa(1 xa) 1 + O(e 22 mn00 min(xa;1 xa)) ; da xa a 2 n0 a 2 Note that these expressions break down if da = 0 or xa = 0, so from now on we assume da's are not integer multiples of 1=n0 where this does not occur. For the denominator, we rst examine the logarithmic derivative term in (4.3). So long as we avoid the special cases xa = 0, the asymptotic expression (4.4b) is di erentiable with respect to its rst argument, and we obtain (4.6) Since P a a = 0 and we avoid special cases, we see that Xa must generically sum to either 1 or 2. Therefore (4.6) is in fact just The remaining term, namely det(1 + B~fm0;n0;r0g), is more di cult to analyze. So for the moment we leave it implicit. In this case, combining the numerator terms (4.4) with (4.6) and taking into account the prefactor in (4.1) gives where ' is a phase independent of , and the transformed quantities fm0; n0; r0g are given Examination of the determinant factor The asymptotic expression for the index, (4.7), is now complete up to the expansion of the determinant. Unfortunately, its structure is rather intricate, and we have been unable to nd a simple universal formula describing its asymptotics. The main issue is the observation that the high temperature limit of log det(1 + B~fm0;n0;r0g) can be of either O(1) or O(1= ). This term is relatively unimportant in the former case, but will contribute to the leading order behavior in (4.7) in the latter case. However, which case the determinant factor is in depends in a non-obvious manner on the chemical potentials a and is not easily obtained. We now proceed with a closer look at the matrix B~fm0;n0;r0g de ned in (2.32). To avoid unnecessary notation, we will omit the universal arguments ( 0a; 0) = ( a= ; 1= ) and occasionally the sector labels fm0; n0; r0g, in what follows. In this case, the B~ matrix entries can be simply written as [B~fm0;n0;r0g] ; = Gfm0;n0;r0g(^j0 ; k^0 ) ; ^j0=0 Pm0 1 Pn0 1 k^0=0 Gfm0;n0;r0g(^j0; k^0) (4.8) where we have the index pair associations ! (^j0 ; k^0 ) and ! (^j0 ; k^0 ). At this stage it is convenient to note that while this is originally an (N 1) (N 1) square matrix, it can be extended to an N N square matrix by including the = 0 and = 0 entries. This is equivalent to allowing ^j0 and k^0 to independently run over 0 : : : m0 1 and 0 : : : n0 1 without removing the (0; 0) pair. Since the rst column of B~ with entries [B~] ;0 vanishes identically, however, the determinant det(1+ B~) can be viewed either as an (N 1) (N 1) or an N N determinant. Taking the logarithmic derivative of (u; ) and using the asymptotic expansion (A.9) gives the high-temperature expansion of G temperature limit of the matrix B~fm0;n0;r0g with entries (4.8). We keep the O(1) and the , which is necessary to study the highrst subexponential term, G(^j0; k^0) = G0(^j0; k^0) + Gexp(^j0; k^0) + , where 1 + Da(k^0=n0) + Da( k^0=n0) ; G0(^j0; k^0) = G exp(^j0; k^0) = 3 X X a=1 = and we have de ned where the determinant on the left is that of an N N matrix, while that on the right is of an n0 n0 matrix. since B~0f1;1;0g = 0, so that At this point, we are still left with the n0 n0 determinant to evaluate. However, there is an important special case corresponding to fm0; n0; r0g = fN; 1; 0g. This case is trivial det(1 + B~0fN;1;0g) = 1: The situation is more complicated when n0 6= 1. While we do not have a proof, numerical evidence indicates that the O(1) order determinant only takes on two possibilities, depending on the chemical potentials: da(x) 2 a + x (mod 1); Da(x) 2 a + x : one will dominate, depending on the relative magnitudes of da( k^0=n0). Note that, while Gexp is a sum of twelve exponentially small terms, generically only a single Given the asymptotic form of G(^j0; k^0), the B~ matrix can be expanded into the sum of an O(1) matrix and a subexponential one, B~ = B~0 + B~exp. If det(1 + B~0) 6= 0, then we are essentially done, as it will not contribute at the O(1= ) order in the high-temperature limit. However, if this vanishes, the subexponential contribution becomes important. We thus consider the O(1) order determinant rst, before turning to the subexponential one. 4.2.1 O(1) order determinant For the B~0 matrix, we note that its entries are built from Gfm0;n0;r0g(^j0; k^0), where here we 0 have restored the sector labels fm0; n0; r0g. However, examination of (4.9a) demonstrates that it is actually independent of m0 and r0 as well as the index ^j0. As a result, we can write the matrix expression where U is the m0 m0 square matrix whose entries are all unity. Since U has only one non-vanishing eigenvalue equal to m0, we then see that det(1 + B~0fm0;n0;r0g) = det(1 + B~0f1;n0;0g); ~0 Bfm0;n0;r0g = 1 m0 U ~0 det(1 + B~0fm0;n0;r0g) = 0 or n02: (4.9a) (4.9b) (4.10) (4.11) (4.12) (4.13) (4.14) HJEP07(218) ~0 = 0, while the gray regions correspond to non-trivial 1+B~0, but still with vanishing determinant. B The determinant evaluates to n02 in the unshaded regions. The yellow triange corresponds to the region 0 < d1 d2 In order to investigate where the determinant vanishes, we take da = a=2 (mod 1) and assume none of them are integer multiples of 1=n0 as in (4.5a). Furthermore, without loss of generality, we let Pa da = 1, which follows from the requirement P a a = 0. (The other a da = 2, can be mapped to this one by taking into account the invariance For small values of n0, the regions in (d1; d2) parameter space where the determinant vanishes are shown in gure 1. Here the (n0 1) (n0 identically in the black regions. For prime n0, this appears to be the only places where the determinant vanishes, while for composite n0 there are additional regions with vanishing determinant but with non-trivial 1 + B~0, represented by the gray regions. 1) matrix 1 + B~0f1;n0;0g vanishes For general n0, consider that the B~0 matrix is obtained from G0(^j0; k^0) = <>>0; 8 > > > > >>1; >:>2; 1; da2 > min(k^0) and (da1 da1 < min(k^0) and (da2 da2 < min(k^0) < da3 < max(k^0); da3 < min(k^0); min(k^0))(da3 min(k^0))(da3 max(k^0)) < 0; max(k^0)) < 0; (4.15) which is a direct consequence of (4.9a). Here da's are ordered as 0 < da1 1 and we have de ned min(k^0) and max(k^0) as the min and max of fk^0=n0; 1 da2 da3 < k^0=n0g, respectively. In particular, note that G0(^j0; k^0) = 0;k^0 where l0 n0 l0 + 1 n0 ; < da2 < da1 + da2 < with l0 = 0; 1; ; (4.16) n0 1 2 ; which corresponds to the black regions in gure 1. Inserting G0(^j0; k^0) = then explains why 1 + B~0 vanishes identically in these regions. However, the resulting B~0 0;k^0 into (4.8) matrix outside of the black regions is rather di cult to work with. Nevertheless, for prime n0, we conjecture based on numerical evidence that det(1 + B~0) = n02 everywhere outside of the black regions speci ed by (4.16). The case for composite n0 is clearly more complicated, as can be seen from the gure. First subexponential order determinant ponentially suppressed contributions to B~. For prime n0, we can derive Whenever the O(1) order determinant vanishes, it becomes necessary to examine the ex1 + B~f1;n0;r0g = exp 2 i f1;n0;r0g(da) r0 n0 f1;n0;r0g(da) C( f1;n0;r0g(da)); (4.17) at leading order whenever we are in the black regions speci ed by (4.16). Here C( ) is an (n0 1) (n0 1)-square matrix de ned by [C( )] ; = 2 ; + ; + ;n0 ; ;n0 ; ; (n0 ) ; (4.18) and f1;n0;r0g(da) and f1;n0;r0g(da) are given by f1;n0;r0g(da) = <da3 f1;n0;r0g(da) = 8 (n0 l0 + 1; n0n0 1 ; n l0; :min da2 nl00 ; da3 l0 = 0; we have det C( ) = n02 for any = 1; ; n0 1. Combining this result with the conjecture for the O(1) behavior made above, we nd for prime n0 (excluding the special case) det(1 + B~f1;n0;r0g) = n02 exp 2 i(n0 1) f1;n0;r0g(da) r0 n0 f1;n0;r0g(da) ; (4.22) Note that here we are excluding the special case da2 nl00 = da3 For prime n0, we can prove det C( ) = n02 for any = 1; : : : ; n0 1. In particular, we can rst show det C(1) = n02 by mathematical induction. Then since with a permutation ( ) = (mod n0) and the corresponding permutation matrix C( ) = ~ 1C(1)~ ; [~ ] ; = ( ); ; (4.19a) (4.19b) (4.20) (4.21) at leading order, where we set regions speci ed by (4.16). f1;n0;r0g(da) to vanish outside of the black We now have all the components needed to work out the high temperature expansion of the index in the fm; n; rg sector provided the corresponding m0 is unity and n0 is prime. For such a sector, substituting (4.22) into (4.7) yields da) and cf1;n0;r0g(da) > 0 is a positive function away from special values of the chemical potentials da. We expect that this expression continues to hold for arbitrary values of fm; n; rg, although we have been unable to obtain a general expression for the determinant factor fm0;n0;r0g(da) apart from the above case. 4.2.3 The N = 2 and 3 cases We now give a couple of examples supporting the results (4.22) and (4.23). For notational convenience, here we set 0 < d1 d2 d3 < 1 without loss of generality and therefore the domain in (d1; d2) parameter space shrinks down to the yellow triangle in gure 1. For the N = 2 case, we have a total of three sectors, labeled by fm0; n0; r0g = f2; 1; 0g, f1; 2; 0g and f1; 2; 1g. The determinant in the f2; 1; 0g sector is trivial as seen in (4.13), so we focus on the f1; 2; r0g case. From (4.8) and (4.9a), we have det(1 + B~f1;2;r0g) = 1 + 2 P3 a=1 Da(1=2) up to higher order terms. Due to the constraint P restricted as exp Gf1;2;r0g(0; 1) and therefore (4.24) leads to (\ " denotes the non-vanishing leading order) det(1 + B~f1;2;r0g) When d3 > 1=2, we use the expansion (4.9b) to obtain exp Gf1;2;r0g(0; 1) 2 exp 2 i d3 1=2 where we used the fact that minfda(1=2); 1 da(1=2)g = d3(1=2) = d3 (d3 > 1=2); 1 2 ; 8 <4; : r0 2 (4.25) (4.26) (4.27) (4.28) which is valid for d3 > 1=2. Consequently, we have which is consistent with (4.22). As a result, the N = 2 index is given by (4.23) with (4.29) (4.30) (4.31) (4.32) (4.33) (4.34) (4.35) (4.36) (4.37) and therefore we have det(1 + B~f1;3;r0g) (9; d2 < 1=3 < d3 < 2=3; 2Gexp(0; 1)2 + 5Gexp(0; 1)Gexp(0; 2) + 2Gexp(0; 2)2; otherwise: result is independent of k^0, and is given by We can pull out the leading order behavior of Gexp(0; k^0) with k^0 = 1; 2 from (4.9b). The exp Gf1;3;r0g(0; k^0) 8 d2 < 13 ; d3 > 23 ; ; d2 > 13 ; d3 < 23 ; where we made use of minfda(1=3); 1 da(1=3); da( 1=3); 1 da( 1=3)g G exp(0; 1) $ G exp(0; 2) ; We now turn to the N = 3 index. Here there are four sectors, given by fm0; n0; r0g = f3; 1; 0g and f1; 3; r0g with r0 = 0; 1; 2. Since the determinant in the f3; 1; 0g sector is trivial as seen in (4.13), we focus on the f1; 3; r0g case. From (4.8) and (4.9a), we have up to higher order terms, where we have suppressed the f1; 3; r0g subscript from G(^j0; k^0). Due to the constraint P 1, the sum Pa(Da(1=3) + Da(2=3)) is restricted as 3 a=1 X(Da(1=3) + Da(2=3)) = ( 1; d2 < 1=3 < d3 < 2=3; 0; otherwise; det(1 + B~f1;3;r0g) 9 2 3 ; 1 3 ; = ( d3(1=3) = d3 d2( 1=3) = d2 d2 < 1=3; d3 > 2=3; ; d2 > 1=3; d3 < 2=3: 8>1; > <exp h4 i r30 >>:exp h4 i 2r0 3 d3 2=3 i d2 1=3 i d2 < 1=3; d3 < 2=3; d2 < 1=3; d3 > 2=3; ; d2 > 1=3; d3 < 2=3; which is consistent with (4.22). As a result, the N = 3 index is given by (4.23) with f3;1;0g = 0; f1;3;r0g = max(0; d2 1=3; d3 2=3): 4.3 After the above examination of the individual fm; n; rg sectors, we now return to the full index, (3.1), in the high temperature limit. From (4.23), we expect the leading behavior of each individual sector Zfm;n;rg to scale exponentially in 1= . Thus, the sectors with the largest positive coe cient of 1= will dominate the full index, and the other sectors will be exponentially suppressed. As a result, we are left with identifying the dominant sectors and their contribution to the index. Note that the degeneracy, if any, of the dominant sectors does not contribute to the leading order expansion of the full index. The high temperature limit was investigated in [7], where the BAE were solved in the 0; constraint P3 and the result is As discussed in [7], this is to be extremized with respect to the potentials, da under the a=1 da = 1. This can be performed by the method of Lagrange multipliers, log Zf1;N;0g da = cr(na) + ; where the extremum values, da, are given by da = na(na 2 1) ; 1 (n1n2 + n2n3 + n3n1); and cr(na) is the right-moving central charge of the 2d N = (0; 2) SCFT arising from the KK compacti cation of the topologically twisted N = 4 SYM over S2 [7]. Here we have assumed that two of na's are negative so cr(na) given in (1.2) is positive. Note that the left-hand side of (4.40) only corresponds to a single sector of the full index. Nevertheless, this connection to the right-moving central charge suggests that the f1; N; 0g sector is a dominant one, so that 6 2 log Z da = cr(na) + ; where Z is the full index, and is indeed the only physically relevant quantity to connect to the central charge. Note that, if log Z is truly dominated by log Zf1;N;0g, then da can be considered not just an extremum of log Zf1;N;0g, but the full index as well. Hence the identi cation of the central charge with the extremized index in the `high-temperature' limit [7] remains valid in the presence of multiple BAE solutions. Of course, it is still necessary to demonstrate that the f1; N; 0g sector is a dominant one. To do so, we must show that (4.39) (4.40) (4.41) (4.42) at leading order in 1= for any fm; n; rg. This inequality can be written explicitly by inserting (4.40) into (4.23): 3 X(1 a=1 na) xa(1 xa) n02 da(1 da) 2(n0 1) N 2 fm0;n0;r0g(da); (4.43) which originate from the determinant of 1 + B~. for any fm0; n0; r0g. The di culty in proving this inequality lies in the fm0;n0;r0g(da) factors fm0;n0;r0g(da) factors depend in a complicated manner on the extremized potentials da. However, they are always non-negative and in fact vanish in the white regions of gure 1. In this case the claim (4.43) reduces to da(1 da) 0; (4.44) for any integers na with the constraints Pa na = 2 and two of them being negative. Note that the latter is necessary for the 2d SCFT arising from the KK compacti cation to have a positive right-moving central charge [6]. Here we prove this reduced claim under the same constraints, but without the integer condition. To begin with, note that the map (4.40) is in fact invertible between fna : X na = 2; two of them are negativeg $ fda : X da = 1; (1=2 d1 d2)2 > d1d2g; a for any da within the domain given in (4.45). Now we de ne f (d1; d2) as the l.h.s. of the above inequality. Then within the subdomain of xed bn0dac, where @daf is well de ned, we can consider the extremum of f which satis es ) (1 2 da)2 xa(1 xa) n02 da(1 da) + 1 1 2da 1 2xa n0 (1 2da) = k; (4.49) with the inverse map Hence, using the above claim can be rewritten equivalently as na = 2da(2da 1) 1 4(d1d2 + d2d3 + d3d1) : (1 na)(1 2da) = Q3a=1(1 na) > 0; 3 X a=1 1 1 2da xa(1 xa) n02 da(1 da) 0; (4.45) (4.46) (4.47) (4.48) where k is some constant independent of a. At this extremum, the determinant of the Hessian is given by 16k2 (1 2d1)(1 2d2)(1 2d3) < 0; (4.50) so it is in fact a saddle point. Note that we have used da1 da2 < 1=2 < da3 , ordered as before, which is valid in the domain given in (4.45). This implies the minimum of f within the subdomain of xed bn0dac must stay on its boundary. If one investigates the values of f on this boundary, it is straightforward (though tedious) to check that f is minimized where xa1 ! 0+; xa2 ! 0+; xa3 ! 1 xa1 ! 0+; xa2 ! 1 ; xa3 ! 1 for for 3 X a=1 3 X a=1 bn0dac = n0 bn0dac = n0 1; 2: For both cases, we have f ! 3 X a=1 da(1 1 2da da) na 1 na 0; which proves (4.48) and thereby the claim (4.43) in the white regions of gure 1 where fm0;n0;r0g(da) vanishes. 4.3.2 The N = 2 and 3 cases Of course, we are left to deal with the regions where fm0;n0;r0g(da) is strictly positive. In this case, the inequality (4.43) is stronger than the reduced claim (4.44), and the above proof no longer applies. In the absence of a general expression for verify (4.43) for N = 2 and 3, and leave the general case for N we set 0 < d1 d2 d3 < 1 without loss of generality as in 4.2.3. fm0;n0;r0g(da), we only 4 as a conjecture. Here For N = 2, it su ces to prove the inequality (4.43) for fm0; n0; r0g = f1; 2; r0g. Inserting fx1; x2; x3g = f2d1; 2d2; 2d3 1g and f1;2;r0g(da) = d3 1=2 into (4.43) then reduces the claim to (n3 1)(n1n2 1) 0; which is true since n2 n1 1 and n3 4. Hence the claim is proven for N = 2. For N = 3, it su ces to prove the inequality (4.43) for fm0; n0; r0g = f1; 3; r0g. Inserting fx1; x2; x3g = < f1;3;r0g(da) = 8>>f3d1; 3d2; 3da f3d1; 3d2; 3da >>:f3d1; 3d2 ( d3 d2 into (4.43) and examining the resulting expression then proves the claim for N = 3. 1 ; g 2 ; g (d2 < 1=3; d3 < 2=3); (d2 < 1=3; d3 > 2=3); g 1; 3da 1 ; (d2 > 1=3; d3 < 2=3); 2 3 1 3 ; (d2 < 1=3; d3 > 2=3); ; (d2 > 1=3; d3 < 2=3); (4.51) (4.52) (4.53) HJEP07(218) (4.54) (4.55a) (4.55b) Our main observation is that the BAEs for the topologically twisted index for N = 4, SU(N ) SYM on T 2 S2 have multiple solutions labeled by three integers m, n, and r such that N = mn and r = 0; : : : ; n 1. Modular covariance of the index is only achieved after summing over a complete set of these solutions. Taking this into account, we veri ed that the index gives the elliptic genus of the (0; 2) theory [11, 12], which transforms as a weak Jacobi form of weight zero. Based on this observation, we expect that the BAEs for general supersymmetric indices where there is a T 2 factor will similarly admit multiple solutions. This is equivalent to having multiple saddle points in the matrix integrals that arise from localization of the path integral. Multiple solutions of the BAEs, however, make it rather di cult to compute the index explicitly. This is because we have to sum over all possible contributions to get the full index (3.1). We conjecture that the contribution from a single sector, namely Zf1;N;0g, will dominate in the `high-temperature' limit when extremized with respect to the avor chemical potentials a, giving the result (4.41), which connects the index to the central charge of the (0,2) theory. However, we have been unable to demonstrate this in full generality because of the di culty in computing det Bfm0;n0;r0g in this limit. This connection between the high-temperature limit of the topologically twisted index and the right-moving central charge, (1.2), was derived with the assumption that cr(na) > 0. On the holographic side, positivity of the central charge is necessary for a good AdS3 supergravity solution to exist. However, it may be interesting to explore the case when S2 a single magnetic charge na is negative, corresponding to cr(na) < 0 after extremization. While the holographic dual is not obviously well-de ned, the eld theory may still be interesting on its own. In this situation, the f1; N; 0g sector may no longer dominate, and additional sectors will have to be considered as well. We were initially drawn to the topologically twisted T 2 S2 index because of our interest in its large-N limit. This limit, however, is somewhat delicate, as the sum over sectors involves the modular parameter ~ = (m + r)=n with N = mn. The di erent sectors then have =~ ranging from (= )=N ! 0 to N = ! 1 for xed limit. Similar to the high-temperature limit, we may expect O(N 2) contributions to arise from the =~ ! 0 sectors, and in particular the f1; N; 0g sector. However, for nite modular parameter , the nal result ought to remain a weak Jacobi form of weight zero, as the in the large-N Cardy limit would not yet have been taken. Assuming progress can be made with the large-N limit, this would allow us to investigate the partition function for microstate counting of the dual magnetic black string, in analogy with the AdS4 black hole story of [5]. However, an analytic supergravity solution has not yet been constructed. (See [13] for a singular magnetic string and [6] for a numerical solution.) So in order to complete the picture, it would be worth obtaining such a solution that interpolates from an AdS3 S2 near-horizon geometry [6] to asymptotic AdS5 with conformal boundary T 2 S2. If such an analytic solution can be found, an interesting follow up would be to compare the log N term in the index with the corresponding oneloop supergravity result. (See [14{16] for recent work on the topologically twisted index for ABJM theory.) This would, however, require a more careful computation of log det B than what we considered above, and hence may remain an open challenge. Acknowledgments The motivation to explore the large-N limit of the topologically twisted index on T 2 S2 arose out of conversations with L. Pando Zayas. We wish to thank S. M. Hosseini, F. Larsen, L. Pando Zayas and V. Rathee for enlightening discussions and N. Bobev for interesting comments. This work was supported in part by the US Department of Energy under Grant No. DE-SC0007859. A Elliptic functions Let q = e2 i and x = eiu. Then the Dedekind eta function is given by 1 1 1 n= 1 1 n= 1 1 x 21 ) Y(1 n=1 1(x; q) = 1(u; ) = iq 8 (x 2 x 1qn) = i X ( 1)nxn+ 12 q 12 (n+ 12 )2: The Jacobi theta function 1 is given by (A.1) (A.2) (A.3a) (A.3b) (A.4a) (A.4b) (A.5) (A.6) These elliptic functions satisfy the following modular properties ( + 1) = ei =12 ( ); 1(u; + 1) = ei =4 1(u; ); ( 1= ) = p i ( ); 1(u= ; 1= ) = i p i eiu2=4 1(u; ): These modular properties, (A.3), can be extended to general SL(2; Z) transformations u 1 ; = c + d ( ); = 3pc + d e 4 i(ccu2+d) 1(u; ); where is a 24-th root of unity. In addition, 1 is quasi-doubly periodic with (p; q 2 Z) 1(u + 2 (p + q ); ) = ( 1)p+qe ique i q2 1(u; ): In the text, we have introduced the weak Jacobi form of weight 1 and index 1=2, (u; ) (This is the square-root of the unique weak Jacobi form of weight 2 and integer index 1, sometimes denoted ' 2;1.) This can be expanded for = 1 (ie jqj 1), with the result (u; ) Note that this expansion breaks down for integer `. We can also rewrite this expansion as (u; ) 0 = (mod 1) = 1 ; `0 = = 1: u2 2 2 B Invariance of det B under T and S transformations Here we demonstrate that det B transforms according to (3.26) and (3.35) under T and S transformations, respectively. We rst note that the eigenvalues for the BAE solution denoted by fm; n; rg are canonically ordered according to (2.25). The key step here is then to order the eigenvalues for the BAE solution denoted by fm0; n0; r0g di erently, according to fm; n; rg sector : un^j+k^ fm0; n0; r0g sector : un^j+k^ ! ! Note that (^j; k^) ! (^j0; k^0) is a bijective map from Zm S transformation cases so the above ordering for fm0; n0; r0g sector is valid. Furthermore, it does not a ect the determinant of the B matrix as the determinant does not depend on eigenvalue ordering. Now we prove, with respect to the above ordering, Bfm;n;rg( a; + 1) = Bfm0;n0;r0g( a; ); Bfm;n;rg( a= ; 1= ) = N 1 Bfm0;n0;r0g( a; ); (A.7) (A.8) (A.9) (A.10) (B.1a) (B.1b) (B.2a) (B.2b) (B.3a) (B.3b) as unity for l 2 f1; su ces to show which automatically yields (3.26) and (3.35) respectively. Note that fm0; n0; r0g are di erent for T and S cases. From (2.21), the (l; N ) entries of the l.h.s. and the r.h.s. are the same ; N g. In order to prove that the remaining entries also match, it k^0; a; + 1) = Gfm0;n0;r0g(^j0 k^0; a= ; 1= ) = for any ^j; ^j0 2 Zm and k^; k^0 2 Zn. Note that these are not trivial from (3.24) or (3.33) but can be proved based on those relations and the following properties of the G-function: Gfm;n;rg(^j + m; k^; a; ) = Gfm;n;rg(^j; k^; a; ); l:h:s: = Gfm;n;rg ^j0 + r ; fk^ k^0; ng; a ; Proof of (B.3a). l:h:s: = Gfm;n;rg = Gfm0;n0;r0g = Gfm0;n0;r0g = r:h:s: ( ^j + k^ ^j0 + r ^j0 + r $ ^ k $ ^ k m + r k^0 % ; m k^0 % ; m Here fA; Bg denotes A mod B (0 the 3rd lines. The 2nd line comes from (3.24). Proof of (B.3b). ; fk^ k^0; ng; a; + 1 m + r ; m ; fk^ k^0; ng; a; ; m ; k^ k^0; a; A < B). Note that (B.4) has been used in the 1st and ! ) 1 ! N g ; g ; (B.4b) ! (B.5) Gfm0;n0;r0g Gfm0;n0;r0g n (^j g n n k^ + d n k^0 + d r (k^ n g ^j + g ^j + r g ^ N g n (^j N g r g ^ N g r (k^ ! ^j0) + ; a; ; g n g ^j0 + r g k^0; g N n g ^j0 + r g k^0; g N ; g ; ; a; ! = r:h:s: (B.6) Note that (B.4) has been used in the 1st and the 4th lines. The 2nd line comes from (3.24) followed by the identity M fA; Bg = fM A; M Bg. C Proof that the map (3.34) is bijective First we prove that (3.34) is one-to-one, i.e. ^j10 = ^j20 & k^10 = k^20 ) ^j1 = ^j2 & k^1 = k^2: (C.1) To begin with, note that (3.34a) implies which means k^1 = k^2 in fact. Combined with this fact, (3.34b) implies Zn0 (m0 = g; n0 = N=g). To begin with, recall that we have Next we prove that (3.34) is onto, i.e. there exists (^j; k^) 2 Zm Zn satisfying (3.34) ) ) g n ( b) + (d) = 1: Then for any given (^j0; k^0) 2 Zm0 Zn0 , we have This can be rewritten as g bk^0 + r ^j0 + g = k^0: n ^j0 r dk^0 k^0 = n ( bk^0 + r ^j0 + r $ dk^0 ng ^j0 % ; m n ^j0; n ; N ) : As in appendix B, fA; Bg denotes A mod B (0 check that A < B). Now it is straightforward to ^j = k^ = r ^j0 + r $ dk^0 ng ^j0 % ) ; m 2 Zm; n ^j0; n 2 Zn; (C.2) (C.3) (C.4) (C.5) (C.6) (C.7a) (C.7b) truly satisfy (3.34), so (3.34) is onto. Open Access. Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. [1] G. Festuccia and N. Seiberg, Rigid supersymmetric theories in curved superspace, JHEP 06 (2011) 114 [arXiv:1105.0689] [INSPIRE]. [2] V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE]. [3] F. Benini and A. Za aroni, A topologically twisted index for three-dimensional supersymmetric theories, JHEP 07 (2015) 127 [arXiv:1504.03698] [INSPIRE]. [arXiv:0806.1218] [INSPIRE]. localization, JHEP 05 (2016) 054 [arXiv:1511.04085] [INSPIRE]. JHEP 06 (2013) 005 [arXiv:1302.4451] [INSPIRE]. and black strings in AdS5, JHEP 04 (2017) 014 [arXiv:1611.09374] [INSPIRE]. and a no go theorem, Int. J. Mod. Phys. A 16 (2001) 822 [hep-th/0007018] [INSPIRE]. [hep-th/0401042] [INSPIRE]. 048 [hep-th/0401129] [INSPIRE]. arXiv:1504.04355 [INSPIRE]. S2 and elliptic genus, S2 and supersymmetric localization, JHEP 03 (2014) 040 [arXiv:1311.2430] [INSPIRE]. large N in localization and the dual one-loop quantum supergravity, JHEP 01 (2018) 026 [arXiv:1707.04197] [INSPIRE]. [4] O. Aharony , O. Bergman , D.L. Ja eris and J. Maldacena , N = 6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals , JHEP 10 ( 2008 ) 091 [5] F. Benini , K. Hristov and A. Za aroni, Black hole microstates in AdS4 from supersymmetric [6] F. Benini and N. Bobev , Two-dimensional SCFTs from wrapped branes and c-extremization , [9] J.B. Gutowski and H.S. Reall , Supersymmetric AdS5 black holes , JHEP 02 ( 2004 ) 006 [10] J.B. Gutowski and H.S. Reall , General supersymmetric AdS5 black holes , JHEP 04 ( 2004 ) [11] M. Honda and Y. Yoshida , Supersymmetric index on T 2 [12] C. Closset and I. Shamir , The N = 1 chiral multiplet on T 2


This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2FJHEP07%282018%29018.pdf

Junho Hong, James T. Liu. The topologically twisted index of \( \mathcal{N} \) = 4 super-Yang-Mills on T 2 × S2 and the elliptic genus, Journal of High Energy Physics, 2018, 18, DOI: 10.1007/JHEP07(2018)018