Central charges from the N=1 superconformal index
aa r X i v : . [ h e p - t h ] M a r Central charges from the N = 1 superconformal index Arash Arabi Ardehali and James T. Liu
Michigan Center for Theoretical Physics, Randall Laboratory of Physics,The University of Michigan, Ann Arbor, MI 48109–1040, USA
Phillip Szepietowski
Institute for Theoretical Physics & Spinoza Institute,Utrecht University, 3508 TD Utrecht, The Netherlands
We present prescriptions for obtaining the central charges, a and c , of a four dimensional su-perconformal quantum field theory from the superconformal index. At infinite N , for holographictheories dual to Sasaki-Einstein 5-manifolds the prescriptions give the O (1) parts of the centralcharges. This allows us, among other things, to show the exact AdS/CFT matching of a and c for arbitrary toric quiver CFTs without adjoint matter that are dual to smooth Sasaki-Einstein5-manifolds. In addition, we include evidence from non-holographic theories for the applicability ofthese results outside of a holographic setting and away from the large- N limit. Keywords: Superconformal index, anomalies, AdS/CFT
INTRODUCTION
Given a possibly strongly interacting quantum fieldtheory, one of the basic questions that can be asked iswhat are its degrees of freedom. In general, this appearsto be a difficult problem. However, with the addition ofconformal symmetry, there is a growing body of evidencethat universal information on the spectrum of operatorsis contained in the central charges. In two dimensions,this is evident from the Cardy formula [1], which relatesthe asymptotic density of states to the central charge c , as well as the Zamolodchikov c -theorem [2] governingflows between fixed points. In four dimensions, the cen-tral charges a and c control the entanglement entropy [3],while the a -theorem [4] suggests that a is a proxy for thenumber of degrees of freedom at conformal fixed points.Additional support for a four-dimensional connectionbetween central charges and the spectrum comes fromthe recent observation [5, 6] that the difference c − a can be obtained from the four-dimensional N = 1 su-perconformal index [7, 8]. This index counts the numberof shortened states in the spectrum, and for the right-handed index is given by I R ( t, y ; a i ) = Tr( − F e − βδ t − E + j ) / y j Y a s i i , (1)where δ = E − r − j , and { E, j , j , r } are the quan-tum numbers of the superconformal group SU(2,2 | β regulates the infinite sum but otherwise dropsout of the index since only states with δ = 0 contribute.The final factor above encodes global flavor symmetrieswith quantum numbers { s i } and corresponding fugacities { a i } . The left-handed index I L is similarly defined withthe replacement r → − r and j ↔ j . The insertion of( − F is what ensures that only the shortened spectrumcontributes to the index, and the result of [5, 6] is con-sistent with the central charges a and c being among theunrenormalized (protected) information in the theory [9]. There have been other attempts in the literature to re-late the central charges to the index. In [10], the centralcharge c was noticed to play a role in the modular prop-erties of the N = 2 index, while in [11] a relation wasobtained for 2 a − c of a CFT with N = 2 supersymmetry(see also [12]). Moreover, in [13] the central charges wererelated to the so-called single-letter index, and in [14] itwas observed that the central charges dictate a specificrelation between the supersymmetric partition functionon Hopf surfaces and the index. These results suggestthat it ought to be possible to obtain both of the centralcharges a and c independently from the index.In this letter we demonstrate that the superconformalindex indeed provides information about a and c sepa-rately. This follows from the recent work by Beccariaand Tseytlin [15] that demonstrated that the one-loopcorrections to a and c in the holographic dual only re-ceive contributions from the shortened spectrum. Sinceit is precisely this information that is captured by theindex, it is then possible to extract the corrections to a and c from the index. Following a similar approach as in[6], we find that the central charges are encoded in the t, y → a = 132 ( t∂ t + 1)( − t∂ t ( t∂ t + 2) + ( y∂ y ) −
3) ˆ I ( t, y ) , ˆ c = 132 ( t∂ t + 1)( − t∂ t ( t∂ t + 2) − ( y∂ y ) −
2) ˆ I ( t, y ) , (2)where ˆ I = (1 − yt − )(1 − y − t − ) I + s.t. is the single-traceindex with descendants removed and I + s.t. ≡ ( I Rs.t. + I Ls.t. ). (The single-trace index is obtained from Eq. (1)by restricting the sum to the single-trace spectrum and isnatural from a holographic point of view.) The fugacitiesare taken to one after acting with the differential operatoron ˆ I and the central charges are extracted as a = lim t → ˆ a ( t, y = 1) , c = lim t → ˆ c ( t, y = 1) . (3)Note also that the difference of these equations repro-duces the c − a prescription of [6].Since Eq. (2) was derived from a one-loop computationin the holographic dual, it only computes the subleading O (1) parts of a and c in holographic theories. Curiously,however, it is possible to recover the full values of a and c from these expressions for some classes of large- N non-holographic theories. In any case, the result obtainedfrom these equations may be divergent when working inthe large- N limit, in which case the appropriate prescrip-tion is to take the finite term in the Laurent expansionsof ˆ a and ˆ c about t = 1.In order to highlight the potential divergences in a and c , we consider a series expansion of I + s.t. , first around y = 1 and then around t = 1. Generically, the expansiontakes the form (see Sec. IV of [6]) I s.t. = (cid:18) a t − a + a ( t −
1) + · · · (cid:19) +( y − (cid:18) b ( t − + b ( t − + b t − · · · (cid:19) + · · · . (4)We have dropped the + superscript of I s.t. assuming thatwe are dealing with CP invariant theories; this will be therunning assumption in the rest of this letter. ApplyingEq. (2) to this expression givesˆ a | y =1 = 9( a − b )32( t − − a + 12 a ) − b − b + b )32+ · · · , ˆ c | y =1 = − a − b )32( t − − a + 12 a ) + 3( b − b + b )32+ · · · . (5)Provided the single-trace index has the structure of (4),this demonstrates that ˆ a and ˆ c have at most a doublepole and no single pole. The prescription for removingthe divergence then amounts to dropping the double pole. LARGE- N THEORIES WITH HOLOGRAPHICDUAL
We first examine the holographic case, since that isthe framework in which the expressions for a and c werederived. More specifically, we focus on four-dimensionalSCFTs dual to IIB theory on AdS × SE (and leavethe study of other holographic settings to future work).For these examples Eq. (2) gives only an O (1) sublead-ing correction to the central charges, so in this sectionwe expect to only reproduce this subleading contributionwhich we denote by δa and δc. Of course, for such theo-ries the computation of the O (1) part of a and c from thelarge- N single-trace index of the SCFT (or equivalently,the single-particle index of the gravity side) follows di-rectly from the work of Beccaria and Tseytlin [15] on the one-loop contributions of bulk one-particle states to theboundary central charges. The use of the superconformalindex in Eq. (2) is at one level simply a rewriting of thesum of the contributions over all bulk states. However,the index does provide an alternative method for regu-larizing the divergent sum over the KK towers in termsof keeping the finite term in an expansion about t = 1.In principle, the application of Eq. (2) to a holographicSCFT can also be viewed as a one-loop test of AdS/CFT.In this sense, the result of [15] can be interpreted as a testfor the N = 4 theory, confirming and refining the earlierresults of [16, 17]. This can be easily generalized to thecase of arbitrary toric quiver CFTs without adjoint mat-ter that are dual to smooth Sasaki-Einstein 5-manifolds.The index of such a toric theory is [18, 19] I s.t. = X i t r i / − , (6)where r i are the R -charges of extremal BPS mesons. Ap-plying (2) to (6) givesˆ a = − t − n v X i =1 r i − n v X i =1 r i + · · · (7)in an expansion about t = 1. Keeping the finite piece andnoting that P r i = 6( δa = −
316 ( . (8)This matches the expected result for the O (1) part of a based on the decoupling of a U(1) at each node inthe quiver; since there are no adjoint matter fields inthe quiver, there are no additional O (1) contributionsto a in the field theoretical computation through a = (9Tr R − R ). The successful matching for the O (1)part of c can now be deduced either from a similar appli-cation of Eq. (2) to (6) or from the successful matchingof c − a reported in [6].We have also checked that Eq. (2) successfully repro-duces the O (1) part of the central charges of all the otherholographic theories discussed in [6]. These include the N = 4 theory which has adjoint matter, the singular Z orbifold, and the non-toric SPP and del Pezzo theories.(Of course, a test of AdS/CFT only occurs if the indexwas computed on the gravity side. Since this was not thecase for the SPP and del Pezzo theories, in these casesthe matching is only a confirmation of Eq. (2), and nota true test of AdS/CFT.)It is of course possible to perform a one-loop test bydirectly performing the KK sum, and not going throughthe index as a regulator. In particular, one could proceedalong the lines of [20–22] by introducing a z p regulatorwhere p is the KK level, and then taking the limit z → N = 4theory in [15] in terms of the ten-dimensional spectral ζ -function, and we have verified that it continues to providea successful O (1) matching of both a and c for all the N = 1 cases discussed in [20–22]. The second order pole in a and c As seen in (5), the coefficients of the second order polein ˆ a and ˆ c , and hence ˆ c − ˆ a are all determined by thecombination a − b . A relation was given in [6] for thepole in ˆ c − ˆ a in terms of curvature invariants of the dualgeometry. Therefore similar relations may be obtainedfor the coefficients of the pole terms that Eq. (2) givesfor ˆ a and ˆ c of a holographic SCFT. The relation proposedin [6] implies a negative coefficient for the pole in ˆ c − ˆ a ,and hence a positive one for ˆ a and a negative one for ˆ c .Because of the universal behavior of the second orderpole, for all SCFTs dual to IIB theory on AdS × SE the combination3ˆ c + ˆ a = −
932 ( t∂ t + 1)(2 t∂ t ( t∂ t + 2) + 1) ˆ I ( t, y ) , (9)is finite at t = 1. In fact, assuming the expansion (4) forthe index, this combination is always finite. The finite-ness can be traced to the absence of the y -dependent operator in (9). This particular combination of a and c has been shown to be proportional to a supersymmetricCasimir energy in [13] and further discussed and arguedto be regularization scheme independent in [14, 23]. Herewe find explicit evidence for these statements of schemeindependence. In particular, we see that this quantity re-ceives no contributions from states with arbitrarily largedimension in the large- N limit which would give rise tothe second order pole in ˆ a and ˆ c individually. It would beinteresting to understand this behavior more completely. NON-HOLOGRAPHIC SCFTS
Although the expression (2) was derived from a holo-graphic computation of the O (1) contributions to a and c , we can nevertheless ask whether it can apply to non-holographic SCFTs as well. Since the single-trace indexis inherently a large- N construct, we start the discussionwith large- N SCFTs.Our primary example are the A k theories, the simplestof which has k = 1; this is SQCD without adjoint matter.The single-trace index can be obtained in the Venezianolimit [6, 24], and application of Eq. (2) then givesˆ a = 9(2 k − k + 1)128 k ( t − + − k − k + N c (6 + 3 k + 15 k ) − N c /N f k + 1) + · · · , ˆ c = − k − k + 1)128 k ( t − + − k − k + N c (7 + 5 k + 16 k ) − N c /N f k + 1) + · · · , (10)with the finite terms giving the full values of a and c , andnot just their O (1) components [25].This example demonstrates that the divergence at t = 1 remains, presumably as a large- N effect, regard-less of holography. However here the finite term recoversthe full O ( N ) values of both a and c in contrast with theholographic examples where the result only gave the O (1)contributions. The difference presumably lies in the typeof large- N limit taken. For the A k theories we have takenthe Veneziano limit, i.e. N c ≫ N c /N f fixed. Toemphasize one reason why this is different from a holo-graphic ’t Hooft limit, a distinction should be made inthe number of types (or flavors) of single-trace operators.In the holographic setting this corresponds to the num-ber of Kaluza-Klein towers that exist in the reduction sowe will refer to each flavor as an individual tower. In theVeneziano limit there are an infinite number of towers ofsingle-trace operators, as opposed to a finite number oftowers arising in the holographic examples. This featureis presumably what allows the index to capture the fullexpressions for a and c , including the N terms. Moving away from large- N Since the index is well-defined even away from thelarge- N limit, Eq. (2) ought to be applicable to finite- N theories as well. However, in this case the single-traceindex is not well defined, and a natural choice is to re-place it by the plethystic log [26] of the full index.Obtaining tractable analytic expressions for the super-conformal index of interacting theories at finite N is gen-erally difficult, so instead we first comment on the generalstructure. Following [6], we assume that the result of theplethystic log gives a reduced index ˆ I that is a regularfunction with a first order zero at t = 1 when y = 1. Inthis case, one can Taylor expand around t = y = 1ˆ I ( t, y ) = f ( t −
1) + f ( t − + f ( t − + · · · +( y − (cid:0) g + g ( t −
1) + · · · (cid:1) + · · · , (11)where we have kept only the terms relevant for the calcu-lation of ˆ a and ˆ c in (5). Comparison with (4) then yields a = b = f . Examination of (5) then demonstratesthat the expressions for ˆ a and ˆ c in (2) remain finite.Consistency with the result of Di Pietro and Komar-godski [5] then gives a = b = 32( c − a ), along with afurther condition g + g = 0 that was obtained in [6].Combining this information with (5) we findˆ a = −
332 ( a + 9 a ) + O ( t − , ˆ c = −
132 (2 a + 27 a ) + O ( t − . (12)Remarkably, only the a and a coefficients enter theexpressions for a and c . This means, in particular, thatthe central charges can be obtained from I finite- Ns.t. ( t, y is set to unity. Since y = 1 corresponds to acomputation of the supersymmetric partition function onthe round S [27–29], we see that no squashing is neededto have a and c separately encoded. One can also turnEq. (12) around and write it as an expansion of the finite- N single-trace index. The result is I finite- Ns.t. ( t,
1) = 32( c − a ) t − a − c − a )( t − · · · . (13)Note that the Hofman-Maldacena bound 3 c ≥ a [30]guarantees that the coefficient of t − N gauge group and N χ neutral chiral multiplets(along with their conjugates) having R -charges R i . Thisclass includes the magnetic dual description of SQCDwith N f = N c + 1. The index is I finite- Ns.t. = N (cid:18) − − t − (1 − t − y )(1 − t − y − ) (cid:19) + N χ X i =1 t − R i − t R i − (1 − t − y )(1 − t − y − ) . (14)Expanding I finite- Ns.t. ( t,
1) around t = 1 yields I finite- Ns.t. = − N + P ( R i − t − − X ( R i − − (cid:16)X ( R i − − X ( R i − (cid:17) ( t − · · · . (15)Comparing with (4), it is now easy to see that a = − R = 32( c − a ) and a = − (Tr R − Tr R ) = − (3 c − a ), thus confirming (13).Taking the expression (13) one step further, we nowconsider the plethystic exponential of the finite- N single-trace index near t = 1 . For this it is convenient to define t = e β and expand near β = 0 . From (13) we find I finite- N ( e β , ∞ X n =1 n I finite- Ns.t. ( e nβ , ! = exp ∞ X n =1 c − a ) n β + a ′ n −
827 (3 c + a ) β + · · · ! = exp (cid:18) π ( c − a )3 β + 427 (3 c + a ) β + · · · (cid:19) , (16)where in the final equality we have replaced the infinitesums on n with their ζ -function regularized values andthrown away the divergent harmonic series, i.e. ∞ X n =1 n = π , ∞ X n =1 n → , ∞ X n =1 → − . (17)Note that with this regularization we are neglecting po-tential O ( β ) and O (log β ) terms in the exponent of theindex (see Eq. (4.9) in [5] which demonstrates the exis-tence of such terms in the index of a free vector multi-plet). Nonetheless the O (1 /β ) and O ( β ) terms in (16)appear to be unambiguous. In particular, the leadingbehavior of the result (16) is consistent with the genericresults of [5] on supersymmetric partition functions. Fur-thermore, the O ( β ) term in the exponential of (16) isprecisely the supersymmetric Casimir energy (9) whichwas originally obtained in [13, 14] from the single-letterindex.The above discussion provides evidence that the ex-pression (2), when applied to finite- N theories, yields a and c directly, without any needed subtraction. With theassumption in (11) on the form of the single-trace index(whose validity is worth exploring), we can turn our con-jecture into one for the coefficient of the linear term inthe expansion of the single-trace index around t = 1; thisis shown as the last term in Eq. (13). The fact that thisterm reproduces the precise behavior in [13, 14] providesa non-trivial test of this statement.At large- N , the expression for ˆ a and ˆ c formally di-verges at t = 1. This divergence is related to the infinitesum encountered when computing the single-trace index,which at finite- N would terminate at O ( N ) due to traceidentities. In the holographic examples, Eq. (2) computesthe O (1) contribution to a and c , while in the A k theo-ries in the Veneziano limit, it yields the complete O ( N )behavior. The distinction between these two cases ap-pears to be related to the number of types (or flavors)of single-trace operators present in the theory, with theholographic cases having a finite O (1) number and the A k theories having an infinite O ( N ) number.It would be interesting to explore the pole structureand validity of our prescriptions in (2) and (3) for the in-dices of strongly coupled theories with a six dimensionalorigin [31]. These theories have O ( N ) degrees of free-dom and also admit a dual holographic description inthe large- N limit [32]. Results on the indices are alreadyavailable [33, 34], although their behavior near t = 1 re-mains to be explored.Finally, for the holographic examples, the leading O ( N ) contributions to a and c are well understood fromthe gravity dual in terms of the geometry of the inter-nal manifold [35, 36]. In the field theory they appear inthe behavior of the Hilbert series for mesonic operatorsin the CFT [37]. Therefore, the leading order centralcharges are encoded in the spectrum as well; our resultssuggest that they are not, however, encoded in the large- N superconformal index. A proper understanding of thisdistinction may shed light on the manifestation of a holo-graphic dual directly within field theory.JTL wishes to thank Cyril Closset for useful discus-sions. PS is grateful to A. Gadde and A. Tseytlin forinsightful email correspondence. We would also like tothank L. Di Pietro and Z. Komargodski for useful com-ments. This work is part of the D-ITP consortium, aprogram of the Netherlands Organisation for ScientificResearch (NWO) that is funded by the Dutch Ministryof Education, Culture and Science (OCW), and is alsosupported in part by the US Department of Energy un-der grant DE-SC0007859. [1] J. L. Cardy, Operator Content of Two-Dimensional Con-formally Invariant Theories,
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