Recounting dyons in N = 4 string theory

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Citation Strominger, Andrew, David Shih, and Xi Yin. 2006. “Recounting Dyons in N = 4 String Theory.” Journal of High Energy Physics 2006 (10): 087–087. https://doi.org/10.1088/1126-6708/2006/10/087.

Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:41417362

Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA JHEP10(2006)087 hep102006087.pdf March 31, 2006 October 30, 2006 is used to derive September 15, 2006 September 27, 2006 , Received: Published: Revised: ients of the unique weight Accepted: rlinde. rsity, Princeton, NJ 08544, USA. University, Cambridge, MA 02138, USA. ). This result agrees exactly with a Z , =4stringtheoryfromthewell-known5D ∗ string theory N http://jhep.sissa.it/archive/papers/jhep102006087 /j [email protected] , =4 and Xi Yin N † Published by Institute of Physics Publishing for SISSA David Shih, ∗ Black Holes in String Theory, D-branes. Arecentlydiscoveredrelationbetween4Dand5Dblackholes [email protected] Permanent address: Department of Physics, Princeton Unive Permanent address: Jefferson Physical Laboratory, Harvard † ∗ Center of Mathematical Sciences,Hangzhou Zhejiang 310027 University China E-mail: [email protected] SISSA 2006 c ⃝ AndrewStrominger, Abstract: weighted BPS black holedegeneracies. degeneracies They are for found10 4D to be automorphic given form by the of Fourier the coec ffi modular group Sp(2 Keywords: conjecture made some years ago by Dijkgraaf, Verlinde and Ve Recounting dyons in JHEP10(2006)087 1 S (4) (1) (2) 3-wrapped D4 -wrapped NS5 D1 branes and 1 1 K : (K3 cycle) and 4 S compactification, the configuration Q 23 ompactification on is × 2 × hat the BPS states q ingle D6 brane with =0) (3) 2 kgraaf, Verlinde and T 3 23 i T q q 0 cation has an M-theory K × q and angular momentum 3 ack hole degeneracies. For rap = lity which acts on electric etry. This observation was ll known. In this paper we K A =0; q m q 23 Φofthemodulargroup cform =4stringtheory,therelevant D2-branes. T-dualizing the · q confirms their old conjecture. ies, which turn out to be Fourier ; e A A N q . q ) ; , 0 Z ,q ) q +1 (5) j 23 and q ). 2 e q 22; ; 1 q 27 transforming in the 28 of the second -wrapped D2 charge , =( Z 2 23 1 2 S = e q 3; T ; 2 m = ,... .ThisisthenU-dualtoa K A q 3-wrapped D2 charge, SO(6 ( 5 2 m q ,q 1 2 2 ; q K –1– Q × 0 1 2 1 H ,and q ) , A Q Z =0) =4blackholes,andthiswasshowntopassseveral q 3-wrapped M2 charge B =0 ,Λ i =( q 22 is ≡ q K Λ m e A N ; q q q , N 23 SL(2; ,... 2 m q AB q 1 2 C F-strings winding =1 1 2 =0; 2 m with q = ,A A 27 are momentum and winding modes of 1 1 2 q 2 e A S q q 1 2 × ,... compactification, corresponding to momentum/winding modes. 3 with 2 =(1; 2 1 K T T m =24 3-wrapped D2 charge S q × K × ,i 3 i , 3 and magnetic charges q 0 is the intersection matrix on K q K Λ D3-branes carrying momentum is D0-charge, e 2. We now use one of the compactification circles to interpret .Thedualitygroupisconjecturedtobe AB q -wrapped M2 charge / 2 0 A 2 =4stringtheoryinfourdimensionscanbeobtainedfromIIAc 0 q C q T q T Now consider a black hole corresponding to a bound state of a s By lifting this to M-theory on Taub-NUT, it was argued in [1] t Almost a decade ago an inspired conjecture was made [5] by Dij N AgeneralD0-D2-D4-D6blackholeina4DIIAstringcompactifi × D5 branes on = 3 5 L with D0 charge types of F-string The duality invariant charge combinations are Q of this system are the same as those of a 5D black hole in a J as IIA on yields where lift to a 5D blackused hole in configuration [1] to in derive a athe multi-Taub-NUT simple geom case relation between of 5D and 4D BPS bl K expansion coefficents of a well-studiedof weight a 10 automorphi genus 2 Riemman surface [4, 5]. Verlinde forconsistency the checks. 4D We will degeneracies see that our analysis of precisely 5D black holes weretranslate found this in into [2, an 3] exact expression and for the the degeneracies 4D degenerac are we branes. The magnetic objects are 24 types of D-branes which w charges The first factor may be described as an electromagnetic S-dua charge, and where factor. For the electric objects, we may take JHEP10(2006)087 3 (6) (7) (8) (9) (10) (11) (12) 3) in . K . ( $ Z N m 1 2 q · e q 1 2 , 3asgivenin[7]. in the definition of K , ) +1 . Ts on Hilb L 2 2 e J over U-duality orbits. " m 2 2 q sbulkmodesobtained arge, and the center-of- 0 − ) 2 racies needed in (9), for 1 2 q 1) ν ite , kl ; L − (4 2 J m A masasinglefivebraneand c the Fourier expansion of the q ) ing to [1] the 4D degeneracy has a product representation: ,q +2 1 2 ) ) nfromtheellipticgenusofa ht abuse of language, we mean ρ 23 0 # mν q L 5 ( + ! d 1 πi 5 lσ ρ,,σ ν m 2 of the D1-D5 system are well known d ( q + e · ′ [3], and is hence takes values in 0 ) 5 e q kρ q Φ d L L ( m 1) . F 1) q πi = 2 2 m − · − e q e 2. ,N,J 1 2 q − 0 1 2 πiNσ L = ≤− )=( –2– 2 (1 ( e = 0 L Z ′ 5 n : ) 1 ∈ d L L ;0;0)=( S L J ,m A ρ,ν 0 ,J q ( 0 ,N,J % & ; 0 L ,l> N 0 0 L q )=0for χ | ( is half the R-charge ≥ n 0 5 k ( )= L ;0 c ≥ d J % L N 23 cientsforasingle ffi )theellipticgenuscoe J comes from the extra insertion of ( q ρ,ν )= 2 ( 0 3): ,m 1) q N 1 χ − K 1) ( (1; 0; k, ρ,,σ ν 4 ( − ( N ′ ′ 5 d 1 d )=( (0) = 20, and Φ c m q denotes fixed-charge degeneracies and does not involve a sum )= · 2 n e d m ,q 1) = 2, 2 e − − is given by ( ′ c k ,q 2 m (4 q c ( Equation (11) is the generating function for BPS states of CF Of course these microscopic BPS degeneracies 4 One should keep in mindNote that that Note D1 branes.) Second, it misses the contribution from massles 1 2 3 d single fivebrane. (By U-duality,N we are free to view the syste the D5 worldvolume.two However it reasons. does not First, quite give it the leaves degene out the decoupled contributio the 5D index. Since the degeneracies are U-dual we may also wr It is shown in [6] that the weighted sum of the elliptic genera [2, 3]. Their mainelliptic contribution genus comes of from Hilb the coefficients in and left-moving momentum along the Hence, with the aboveof relations states between with parameters, charges accord (3) and 5D degeneracies are related by angular momentum The extra factor of ( with where Φ the number of bosonsmass minus multiplet the is number factored of out. fermions of a given ch Here and elsewhere in this paper by “degeneracies,” in a slig JHEP10(2006)087 . and 20 (15) (14) (13) (18) (16) (17) − ourier L ) J 4 ) ρ πin 2 e ρ πin 2 − ,orbytheU- e ) 4 2 (1 − 2 m . NUT − ) =0,theproductis − (1 . ω ) 2 ) l kl ν − . (4 ) lculated. (See [9] for a − c = 1 ) 24 ν " nρ ρ,,σ ν k ) )followsbyconstruction,and − ( using candfermioniccollective − la proposed in [5] for the ) Z modes consist of 24 scalars, πi Φ( ) nρ mν 2 ( e .Theydonotcarry + function 22; = ρ πin πi , lσ 2 2 ion (keep in mind that it carries − ) σ πi e e + ν 2 ρ,,σ ν e L (1 econtributionsintheoriginalversionof kρ − − in Taub-NUT space [8]. These two J ( he duality group (1), and the extra factor 2 Φ( − πi (1 (1 +2 ) 2 1 2 σ ) e = =4stringtheory—uptoanoverall ν NUT ≥ − 1) & under SO(6 ) ) n + ω − − ) ) and in the case N ν 1 nρ m N ( q ρ πi + · ! Z 2 e ν,ρ +( πi 0 q nρ ( ρ − 2 ρ,,σ ν ( ∈ > 0 0 e e ( 1) ) –3– ′ L πi Z ( − 2 m − Φ = & e πi 2 k,l,m 0, (1 24 ( e − → 1 ) − ) ≥ ) ν ≥ ) L =0andtherearenoD1branesatall. (1 & Taub-NUT. This reduction relies on the existence of a n ρ 1 + ( σ × ≥ N η & 3 k, l + n ρ πi 1 ρ 2 2 ,N,J ( K ρ,,σ ν − 0 − ( ) πi )= ′ e ) L 2 ρ 2 ρ )istheuniqueautomorphicformofweight10ofthemodular ( Φ ( e ( − 5 ν πi ) d − L bulk bulk e )= ν πi Z Z ρ,,σ ν − )followsfromtheS-dualitytransformationsofthecharges ) − e ,N,J Z % 5 0 . 0meansthat − L ν πi ν,ρ ρ,,σ ν m ( e q 0. Φ( · 5 > )hasaproductrepresentation e Φ( ν πi to D )andwasstudiedin[4].The5DBPSdegeneraciesarethentheF q ) e ′ Z Z 1) )=( , does not spoil this. Invariance of ( m< − =( ρ,,σ ν m momentum and transverse angular momentum) q )= ρ,ν · k, l, m ( e 1 q 5 S ν,ρ D 1) Now let us briefly sketch how these partition functions are ca ( We thank the refereeNote for that pointing the formula of [5] out was manifestly an invariant under error t regarding thes Z − 0 4 5 Z This shifts Φ contribute a partition function dual description in terms of a fundamental heterotic string Combining these two contributions, we obtain the partition only over where the infinite productcoordinates comes of from the the D5which brane. 4 could Meanwhile, transverse be the bosoni either left seen moving bulk by reducing the IIB massless fields invariance under SL(2; group Sp(2 contributions remain even when more careful derivation.) The singleboth D5 has partition funct coefficients in factor of ( Inserting the 4D-5Dmicroscopic relation degeneracy (9), of (18) BPS agrees black with holes the of formu the paper. of ( from the reduction of IIB on unique, normalizable, self-dual, harmonic 2-form where ( where Φ( JHEP10(2006)087 , , Nucl. 120 , Phys. , ]. string theory Invent. Math. , 54. We are grateful to =4 (1997) 197 N espondence. 185 Elliptic genera of symmetric ]. e, surface and generalized Kac-Moody hep-th/9708086 D-branes and spinning black holes 3 K ]. and infinite products ) R Counting dyons in ]. ( (1997) 283 [ 2 , 1 Unwinding strings and T-duality of Kaluza-Klein hep-th/9512046 +2 ]. s –4– ]. O New connections between 4D and 5D black holes ]. Commun. Math. Phys. , Product representation of dyon partition function in hep-th/0602254 (1996) 59 [ hep-th/9602065 Microscopic origin of the Bekenstein-Hawking entropy hep-th/9607026 B372 (2006) 064 [ hep-th/9601029 hep-th/0503217 06 (1997) 93 [ Adv. Theor. Math. 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