aa r X i v : . [ h e p - t h ] J a n N = 4 SYM on K3 and the AdS /CFT Correspondence
Kazumi OkuyamaDepartment of Physics, Shinshu UniversityMatsumoto 390-8621, Japan [email protected]
We study the Fareytail expansion of the topological partition function of N = 4 SU ( N ) super Yang-Mills theory on K3. We argue that this expansion corresponds to asum over geometries in asymptotically AdS spacetime, which is holographically dual to alarge number of coincident fundamental heterotic strings.January 2008 . Introduction The AdS/CFT correspondence is a powerful way to study the quantum gravity witha negative cosmological constant. In particular, the AdS /CFT correspondence is in-teresting from the viewpoint of quantum gravity since three dimensional gravity has nopropagating degrees of freedom at the classical level, hence the bulk theory might be sim-pler than the higher dimensional cousins. Recently, Witten proposed a boundary CFTwhich is dual to the pure gravity on AdS [1] (see also [2–8]). It is found that the par-tition function of boundary CFT has a nice interpretation as the sum over geometries inthe bulk. However, there are some left-right asymmetric contributions in the partitionfunctions which are difficult to interpret semi-classically. Moreover, the very existence ofthe pure gravity on AdS as a quantum theory has not been established yet. Therefore,it is desirable to study AdS gravity in the string theory setup. The obvious problem isthat the dual CFT is not known in general. Even if the dual CFT is known, the partitionfunction is usually hard to compute.There are a few cases that we can study the AdS /CFT correspondence quantita-tively. In [9], Type IIB theory on AdS × S × K AdS is avoided sincethe elliptic genus depends only on the left movers.In this paper, we study the partition function of N = 4 SU ( N ) super Yang-Millstheory on K3. Via the string dualities, this is equal to the partition function of BPSstates of N fundamental heterotic strings. Using the technique in [9,10], we show that thispartition function has an expansion as a sum over asymptotically AdS geometries andargue that they are dual to a large number of heterotic strings. In section 2, we review thepartition function of N = 4 SYM on K3 and its relation to the heterotic string. In section3, we write down the Fareytail expansion of the partition function of N = 4 SYM on K3.In section 4, we discuss some questions. N = 4 SYM on K3 and Heterotic Strings: Review
We first review the Vafa-Witten theory of topological N = 4 SYM [11] and its relationto the BPS index of heterotic strings. 1 .1. N = 4 SYM on K3
In [11], it is shown that the topologically twisted SU ( N ) N = 4 SYM on K3 computesthe generating function of the Euler number of moduli space of k instantons Z N ( τ ) = ∞ X k =0 q k − N χ (cid:16) M N,k ( K (cid:17) (2 . q = e πiτ . The N = 4 SU ( N ) SYM with k instantons is realized by the followingbrane configuration in Type IIA theory: N D4 on K × R t ⊕ k D0 , (2 . R t denotes the time direction. In this brane picture, the shift k → k − N of instantonnumber in (2.1) is understood as the contribution of D0-brane charge from the curvatureof K3.The partition function (2.1) is evaluated as follows. Let us first consider the caseof U (1) gauge theory. This is easily obtained by noting that the moduli space of U (1)instantons is equal to the Hilbert scheme of points on K3 M ,k ( K
3) = Hilb k ( K . (2 . α A − n ( A = 1 · · ·
24) at level L = k . Note that α A − corresponds to the generator of H ( K ⊕ H ( K ⊕ H ( K
3) and the higher modes α A − n ( n >
1) correspond to the twistedsector of orbifold ( K k /S k . From this representation, one finds that the partition functionof U (1) theory is given by the partition function of 24 free bosons G ( τ ) = 1 η ( τ ) . (2 . SU ( N ) theory, the partition function is given by an almost Hecke transformof the U (1) partition function G ( τ ) [12,13] Z N ( τ ) = 1 N X ad = N,b ∈ Z d d G (cid:18) aτ + bd (cid:19) . (2 . N = p is prime, this expression simplifies to Z p ( τ ) = 1 p G ( pτ ) + 1 p p − X b =0 G (cid:18) τ + bp (cid:19) . (2 . a factors of N = 1 SU ( d ) pure Yang-Mills. The summation over b ∈ Z d comes from the d vacua of N = 1 SU ( d ) theory.Note that Z N ( τ ) itself is not a modular form, although G ( τ ) is a weight −
12 modularform. This is related to the fact that the Montonen-Olive S-duality maps the SU ( N )theory to a theory with different gauge group SU ( N ) / Z N . Therefore, Z N does not comeback to itself under the action of S-duality.However, we can regard Z N as a member of more general class of partition functions Z ( v ) N with ’t Hooft flux v ∈ H ( K , Z N ) turned on , and identify Z N = Z ( v =0) N . Thepartition function with ’t Hooft flux v is given by [14] Z ( v ) N ( τ ) = 1 N X ad = N,b ∈ Z d d G (cid:18) aτ + bd (cid:19) δ dv, e − πi bv · vaN , (2 . v · v ′ = R K v ∧ v ′ is the intersection number. One can show that Z ( v ) N transform asa vector-valued modular form of weight −
12 [14] Z ( v ) N ( γ ( τ )) = ( cτ + d ) − X v ′ ∈ H ( K , Z N ) M vv ′ ( γ ) Z ( v ′ ) N ( τ ) . (2 . γ ∈ SL (2 , Z ) and its action on τγ = (cid:18) a bc d (cid:19) , γ ( τ ) = aτ + bcτ + d . (2 . M ( γ ) for S = (cid:18) − (cid:19) and T = (cid:18) (cid:19) is given by M vv ′ ( S ) = 1 N e πiN v · v ′ , M vv ′ ( T ) = δ v,v ′ e πiN v · v . (2 . q -expansion of partitionfunctions (2.4), (2.5) G = q − + 24 + 324 q + 3200 q + 25650 q + · · · ,Z = 14 q − + 30 + 3200 q + 176337 q + 5930496 q + · · · ,Z = 19 q − + 803 + 25650 q + 5930496 q + 639249408 q + · · · ,Z = 116 q − + 632 + 176256 q + 143184800 q + 42189811200 q + · · · . (2 . One can introduce the theta series for the lattice Γ , by summing over the ’t Hooft fluxes.This corresponds to considering U ( N ) gauge theory instead of SU ( N ) gauge theory [11]. Z N has a ‘gap’ between q − N and q , i.e. , the coefficient of q n vanishes in the range − N + 1 ≤ n ≤ −
1. This is true for general N : Z N = 1 N q − N + 24 X a | N a + O ( q ) . (2 . M N,k ( K
3) = 4 N ( k − N ) + 4 , (2 . k < N . This implies that there is no contribution to Z N from the instantons with the instanton number k < N . By the duality chain, we can dualize the D4-D0 configuration in (2.2) to a configurationin heterotic string theory. To see this, we first lift the IIA configuration (2.2) to the M-theory configuration: N M5 on K × R t × S ⊕ k units of momentum along S . (2 . S denotes the M-theory circle in the eleventh direction. In order to relate thisconfiguration to the topological N = 4 SYM, we perform a Wick rotation of the timedirection R t and compactify it to a thermal circle S β . Then the worldvolume of M5-branebecomes K × T where T = S β × S . More generally, we replace the two-dimensionalpart of M5-brane worldvolume by a torus Σ τ with an arbitrary modular parameter τ R t × S −→ Euclidean torus Σ τ . (2 . N = 4 SYM, the moduli τ of torus Σ τ is identified as the coupling constant of N = 4 SYM τ = θ π + i πg . (2 . Curiously, the q term of Z N is 24 times the integral of matrix model obtained by thedimensional reduction of D = 10 super Yang-Mills to zero dimension [15]. N = 4 SYM on K3 and the heterotic string follows fromthe identification of M5-brane wrapping around K3 and the fundamental heterotic string.Therefore, the M5-brane configuration (2.14) is dual to N heterotic strings on Σ τ ⊕ k units of momentum along S ⊂ Σ τ . (2 . Z N is given by the index of BPS states(Dabholkar-Harvey states) in the N = (0 ,
8) superconformal field theory of N fundamentalheterotic strings. This is computed by setting the right-moving SUSY part to the groundstate and summing over the left-moving bosonic side. For the single string case, thissummation gives η ( τ ) − , as expected from the result of U (1) N = 4 SYM (2.4). For N >
1, the Hecke structure of SU ( N ) SYM partition function (2.5) is interpreted in theheterotic picture as the effect of multiple winding of genus one worldsheet around thetarget space torus Σ τ [12,16].
3. Fareytail Expansion of N = 4 SYM on K3
As discussed in [17,18], a large number of coincident fundamental heterotic strings hasa near horizon geometry of the form
AdS × M , hence it is expected to have a holographicdual two-dimensional CFT. In the previous section, we saw that the partition function Z N of N = 4 SYM on K3 captures the BPS spectrum of N fundamental heterotic strings.Therefore, it seems natural to identify Z N as the BPS index of string theory on the AdS dual of heterotic strings. Since we have Wick-rotated the time direction, the dual AdS should be understood as the Euclidean AdS and the torus Σ τ is interpreted as theboundary of AdS . The modular parameter τ should be fixed as a boundary condition forthe bulk metric. Note that the large N limit with τ fixed is different from the ’t Hooftlimit of N = 4 SYM, which in turn implies that the AdS dual in question is not AdS but AdS .The Euclidean AdS is topologically a solid torus. There are many ways to fill insidethe torus Σ τ to make a solid torus. The bulk geometry is distinguished by the homologycycle of Σ τ which becomes contractible. For instance, the spatial circle is contractible forthe thermal AdS and the temporal circle is contractible for the BTZ black hole.To see the relation of the partition function Z N to the bulk AdS geometry , it isuseful to rewrite Z N as a Poincar´e series. A general procedure is developed in [9,10] and The relation between the partition function G ( τ ) of U (1) theory and the black holes in N = 4string theories is studied in [19]. The gravity dual of a single heterotic string is studied in [20]. M ( γ ) in(2.8) and the coefficient c v ( n ) of the polar part of Z ( v ) N = P n c v ( n ) q n . Applying the generalformula in [10] to our case, the Fareytail expansion of Z N reads Z N ( τ ) = 12 X a | N a + 12 X γ ∈ Γ ∞ \ Γ ( cτ + d ) X v ∈ H ( K , Z N ) M − ( γ ) v × X n< c v ( n ) exp (cid:18) πin aτ + bcτ + d (cid:19) R (cid:18) πi | n | c ( cτ + d ) (cid:19) , (3 . ∞ = (cid:26)(cid:18) t (cid:19) , t ∈ Z (cid:27) is the parabolic subgroup of Γ = SL (2 , Z ), and R ( x ) isdefined by R ( x ) = 1(12)! Z x dt t e − t . (3 . N limit, we expect that the expansion (3.1) can be interpreted as a sumover semi-classical geometries. One can see that in the large N limit the summation over’t Hooft flux is dominated by the v = 0 term, since the leading term of Z ( v =0) N is q n with n = O ( N ), while Z ( v =0) N starts with the term N q − N . Therefore, in the the large N limitwe can approximate Z N as Z N ∼ N X γ ∈ Γ ∞ \ Γ ( cτ + d ) M − ( γ ) exp (cid:18) − πiN aτ + bcτ + d (cid:19) R (cid:18) πiNc ( cτ + d ) (cid:19) . (3 . SL (2 , Z ) family of BTZ black holes [21] S = − πN Im (cid:18) aτ + bcτ + d (cid:19) . (3 . Z N of N = 4 SYM on K3 admits a semi-classical expansionof sum over geometries in the AdS background, which is holographically dual to heteroticstrings. As we move τ on the upper half plane, the dominant term in the sum (3.3) changes.Since a large factor of N is multiplied in the classical action (3.4), this change of dominant The sum over the coset Γ ∞ \ Γ should be defined as a limit [10] X Γ ∞ \ Γ ≡ lim K →∞ X (Γ ∞ \ Γ) K = lim K →∞ X | c |≤ K X | d |≤ K, ( c,d )=1 . N limit. This is interpreted asthe Hawking-Page transition [22] in the bulk gravity side. The phase diagram is the sameas that of the pure gravity on AdS (see Fig.3b in [7]).
4. Discussion
In this paper, we studied the Fareytail expansion of the partition function of N = 4SYM on K3 and interpreted it as a sum over geometries dual to fundamental heteroticstrings. It is observed in [23] that the contribution of BTZ black hole is reproduced bytaking the saddle point of instanton sum (2.1). To see this, recall that when the instantonnumber becomes large the Euler number of instanton moduli space scales as χ (cid:16) M N,k ( K (cid:17) ∼ e π √ N ( k − N ) ( k − N ≫ . (4 . c = 24 N CFT. Then thepartition function (2.1) is approximated as Z N ∼ X k e π √ N ( k − N ) q k − N . (4 . k = k of the above sum is given by k − N = − Nτ , (4 . e π √ N ( k − N ) q k − N = e πi Nτ . (4 . N q − N corresponds to thethermal AdS . It would be interesting to understand what happens when adding k unitsof momentum to the fundamental heterotic string and see what triggers the phase transitionin the heterotic string picture. It would also be interesting to study the zeros of Z N ( τ )and see if the Hawking-Page transition is associated with a condensation of Lee-Yangzeros [7]. Finally, it would be interesting to identify the ( c L , c R ) = (24 N, N ) CFT of N fundamental heterotic strings. Acknowledgment
This work is supported in part by MEXT Grant-in-Aid for Scientific Research The phase diagram of N = 4 SYM on K3 was studied in [23]. However, the motivation of[23] seems to be different from ours. eferences [1] E. Witten, “Three-Dimensional Gravity Revisited,” arXiv:0706.3359 [hep-th].[2] D. Gaiotto and X. Yin, “Genus Two Partition Functions of Extremal Conformal FieldTheories,” JHEP , 029 (2007) [arXiv:0707.3437 [hep-th]].[3] M. R. Gaberdiel, “Constraints on extremal self-dual CFTs,” JHEP , 087 (2007)[arXiv:0707.4073 [hep-th]].[4] J. Manschot, “AdS Partition Functions Reconstructed,” JHEP , 103 (2007)[arXiv:0707.1159 [hep-th]].[5] X. Yin, “Partition Functions of Three-Dimensional Pure Gravity,” arXiv:0710.2129[hep-th].[6] X. Yin, “On Non-handlebody Instantons in 3D Gravity,” arXiv:0711.2803 [hep-th].[7] A. Maloney and E. Witten, “Quantum Gravity Partition Functions in Three Dimen-sions,” arXiv:0712.0155 [hep-th].[8] D. Gaiotto, “Monster symmetry and Extremal CFTs,” arXiv:0801.0988 [hep-th].[9] R. Dijkgraaf, J. M. Maldacena, G. W. Moore and E. P. Verlinde, “A black hole fareytail,” arXiv:hep-th/0005003.[10] J. Manschot and G. W. Moore, “A Modern Farey Tail,” arXiv:0712.0573 [hep-th].[11] C. Vafa and E. Witten, “A Strong coupling test of S duality,” Nucl. Phys. B , 3(1994) [arXiv:hep-th/9408074].[12] J. A. Minahan, D. Nemeschansky, C. Vafa and N. P. Warner, “E-strings and N= 4 topological Yang-Mills theories,” Nucl. Phys. B , 581 (1998) [arXiv:hep-th/9802168].[13] J. M. F. Labastida and C. Lozano, “The Vafa-Witten theory for gauge group SU(N),”Adv. Theor. Math. Phys. , 1201 (1999) [arXiv:hep-th/9903172].[14] T. Sasaki, “Hecke operator and S-duality of N = 4 ADE gauge theory on K3,” JHEP , 024 (2003) [arXiv:hep-th/0303121].[15] G. W. Moore, N. Nekrasov and S. Shatashvili, “D-particle bound states and general-ized instantons,” Commun. Math. Phys. , 77 (2000) [arXiv:hep-th/9803265].[16] R. Dijkgraaf, G. W. Moore, E. P. Verlinde and H. L. Verlinde, “Elliptic genera ofsymmetric products and second quantized strings,” Commun. Math. Phys. , 197(1997) [arXiv:hep-th/9608096].[17] J. M. Lapan, A. Simons and A. Strominger, “Nearing the Horizon of a HeteroticString,” arXiv:0708.0016 [hep-th].[18] P. Kraus, F. Larsen and A. Shah, “Fundamental Strings, Holography, and NonlinearSuperconformal Algebras,” JHEP , 028 (2007) [arXiv:0708.1001 [hep-th]].[19] A. Dabholkar, “Exact counting of black hole microstates,” Phys. Rev. Lett. , 241301(2005) [arXiv:hep-th/0409148]. 820] A. Sen, “How does a fundamental string stretch its horizon?,” JHEP , 059 (2005)[arXiv:hep-th/0411255].[21] J. M. Maldacena and A. Strominger, “AdS(3) black holes and a stringy exclusionprinciple,” JHEP , 005 (1998) [arXiv:hep-th/9804085].[22] S. W. Hawking and D. N. Page, “Thermodynamics Of Black Holes In Anti-De SitterSpace,” Commun. Math. Phys.87