Heterotic sigma models on T 8 and the Borcherds automorphic form Φ12

Journal of High Energy Physics, Oct 2017

We consider the spectrum of BPS states of the heterotic sigma model with (0, 8) supersymmetry and T 8 target, as well as its second-quantized counterpart. We show that the counting function for such states is intimately related to Borcherds’ automorphic form Φ12, a modular form which exhibits automorphy for O(2, 26; ℤ). We comment on possible implications for Umbral moonshine and theories of AdS3 gravity.

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Heterotic sigma models on T 8 and the Borcherds automorphic form Φ12

JHE Heterotic sigma models on T 8 and the Borcherds Sarah M. Harrison 0 1 3 6 7 Shamit Kachru 0 1 3 4 7 Natalie M. Paquette 0 1 3 4 7 Roberto Volpato 0 1 2 3 4 5 7 Max Zimet 0 1 3 4 7 0 Via Marzolo 8 , 35131 Padova , Italy 1 382 Via Puebla Mall , Stanford, CA 94305 , U.S.A 2 Department of Particle Physics and Astrophysics , SLAC 3 17 Oxford St. , Cambridge, MA 02138 , U.S.A 4 Department of Physics, Stanford University 5 Department of Physics and Astronomy, University of Padua 6 Department of Physics, Harvard University 7 2575 Sand Hill Road, Menlo Park, CA 94309 , U.S.A We consider the spectrum of BPS states of the heterotic sigma model with (0; 8) supersymmetry and T 8 target, as well as its second-quantized counterpart. We show that the counting function for such states is intimately related to Borcherds' automorphic Superstring Vacua; Superstrings and Heterotic Strings - HJEP10(27) form 12, a modular form which exhibits automorphy for O(2; 26; Z). We comment on possible implications for Umbral moonshine and theories of AdS3 gravity. 1 Introduction 2 The Borcherds modular form 12 3 T 8 sigma models 3.1 Basic connection 3.2 Second quantization 4 Discussion z ! e iTr ( ) 10( ) = Q Q Q P ! Q P P P = z { 1 { where ( 1)Q P +1D(Q; P ) = parametrizes the Siegel upper half-space of degree 2, and 1 Introduction II string theory on K3 duality group Studies of the entropy of supersymmetric black holes in string theory have led to the discovery of beautiful and unexpected relations between basic objects in the theory of automorphic forms, algebraic geometry, and indices of supersymmetric sigma models. A basic example is the formula of Dijkgraaf, Verlinde and Verlinde [ 1 ], capturing the degeneracy of 1/4-BPS dyons in the N = 4; d = 4 string theory obtained from compactifying type T 2 or, equivalently, the heterotic string on T 6. This theory has SL(2; Z) SO(6; 22; Z); where the electric and magnetic charges are given by 28-dimensional vectors Q; P 2 and the SL(2; Z) factor is the electric-magnetic S-duality group of the theory. The degeneracy of 1/4-BPS dyons, D(Q; P ), with charges (Q; P ) is then 6;22, encapsulates the three T-duality invariants of the black hole charges. The beautiful function in the denominator of the integrand is the Igusa cusp form 10( ), a weight 10 Siegel automorphic form for the modular group SP (2; Z). For large charges this formula for the degeneracies has asymptotic growth D(Q; P ) e pQ2P 2 (Q P )2 ; S = pQ2P 2 (Q P )2: reproducing the expected result for the Bekenstein-Hawking entropy of these black holes, The automorphic function 10 also has a connection to elliptic genera of symmetric powers of the K3 surface, derived in [ 2 ] by considering the D1-D5 system on K3 S1, which of 1= 10;1 upon compacti cation and re-derive the result of [ 1 ]. The purpose of this note is to report analogous formulae governing BPS state counts in the heterotic sigma model with T 8 target. This is the model which would naturally arise on the worldsheet of heterotic string compacti cations preserving half-maximal supersymmetry in two space-time dimensions. We will focus in this note on the physics of the 2d eld theory and its supersymmetry-preserving excitations, and mostly limit any discussion of possible space-time interpretations to the concluding section. This work was largely motivated by trying to develop an understanding of BPS counts at Niemeier points in the moduli space of compacti cations to 3 and 2 dimensions, and their possible interpretation in light of Mathieu and Umbral moonshine [6{8], in the picture advocated in [9]. 2 The Borcherds modular form 12 The hero of our story will be the Borcherds modular form 12 [10]. A nice description of the relevant aspects of this form can be found in the work of Gritsenko [11], from which we borrow heavily. modular form with this property. 12. Consider the split Let 2;26 denote the (unique) even unimodular lattice of signature (2; 26). The Borcherds 12 is of weight 12 with respect to O+( 2;26). It is the unique cusp form Of great interest for us will be the following explicit multiplicative lift formulae for 2;26 = 1;1 1;1 number of the root system R associated with N , where we set h(R) = 1 when N is the Leech lattice. To each possible choice of N , we can associate the re ned lattice theta series N ( ; ) = X e i ( ; )+2 i( ; ) : 2N Here, (v; w) denotes the lattice inner product of v; w 2 N , and 12( ; ; ) = qArB~ pC (1 qnr pm)f(mn; ) Y n;m2Z (n; 2;mN)>0 p e2 i ; r = e2 i( ; ): In the above, we have de ned A n > 0, or m = n = 0 and < 0. Furthermore, > 0 (or < 0) means that 48 1 P 2N f (0; )( ; ),2 and used the notation (n; ; m) > 0 to mean m > 0, or m = 0 and positive (respectively, negative) scalar product with a reference vector x 2 N 2 N has R. The 24 2N f (0; ); B~ 2 vector x must be chosen so that (x; ) 6= 0 for all related to one each other by automorphisms in O( 2;26). 2 N , and di erent choices of x are More precisely, the Niemeier and Leech points de ne cusps in the domain of de nition of 12. These formulae should be thought of as expansions of the modular form around the cusps. To make contact with the earlier work [9], it is useful to specialize the chemical potentials as follows. Choose a xed lattice vector 2 N . Then we can de ne N; ( ; z) = X q ( 2; ) y( ; ) 2N 2In the interpretation of 12 as a denominator for the fake Monster Lie algebra, one views (A; B~ ; C) as a Weyl vector; see [10, 11] for details and section 4 for further comments on potential applications of the algebraic structure to physics. { 3 { with y = e2 iz. One now obtains a Jacobi form of weight 0 and index ( ; )=2: F N; ( ; z) = N; ( ; z) 24( ) = X n;l2Z f (n; l)qnyl f (n; l) = f (n; ) : X 2N; ( ; )=l where as well: These Jacobi forms, for suitable choices of , are the BPS counting functions discussed in [9]. That is, they control the coe cients in the expansion of a certain \F 4" term in the low-energy e ective action of heterotic string compacti cation to three dimensions, when the moduli are deformed a slight distance away from a point with Niemeier symmetry (the enhanced symmetry point itself having singular couplings). A specialized form of 12 can be obtained as a lift of these BPS counting functions HJEP10(27) 1N2; ( ; z; ) = qAyBpC (1 qnylpm)f (mn;l) Y n;m;l2Z (n;l;m)>0 with the prefactors A and C as above and B automorphic form on the Siegel upper half-space 2 >0 f (0; )( ; ). This object is an f( ; z; ) 2 C3 j =( )=( ) =(z)2 > 0; =( ) > 0g for the group SO+(L ), where L is the lattice of signature (2,3) and with quadratic form B0 1 1 0 ( ; ) 0 0C 0 0 0 0 T 8 sigma models Basic connection The Narain moduli space [12] of compacti cations of heterotic strings on T 8 is the double 8;24(M). The worldsheet bosons propagating on the relevant lattice. M = O(8; 24; Z)nO(8; 24)=O(8) O(24) : This structure also arises in the non-perturbative description of heterotic strings on T 7 [13]. One can think of M as parametrizing even unimodular lattices of signature (8; 24), eld theory at a given point in moduli space consists of 2d Let us consider this theory on a toroidal worldsheet. At any point in moduli space, there are 24 abelian currents of conformal dimension (1; 0). One can consider coupling { 4 { these to background chemical potentials (\Wilson lines"). In this setup, the parameter of section 2 can be considered as the modular parameter of the torus, while the 24 complex chemical potentials should be thought of as these Wilson line degrees of freedom. The parameter p will emerge upon second-quantization. Now, let us make a precise connection to the discussion of section 2. Consider a point in M where where N is again one of the 24 even unimodular positive-de nite lattices, and 8;0 the E8 lattice. We will call this CFT VN . One obtains BPS states in the heterotic sigma model by putting right-movers in their ground state and performing a trace counting left-moving excitations. We restrict the counting to BPS states carrying no right-moving momentum, i.e. carrying zero charge with respect to the right-moving E8 lattice: TrVN ( 1)F qL0 qL0 y a Ja (with Ja the 24 left-moving abelian currents). A subtlety arises because of the extra right-moving zero modes of this string background; without inserting extra factors of the right-moving fermion number operator FR in the trace, the computation will formally vanish due to the existence of fermion zero modes. One can circumvent this vanishing by inserting powers of F in the trace to soak up zero modes, as in [14]; however, this will alter the modularity properties and spoil the connection we intend to make to 12. Instead, we consider, following [15], a `twisted' computation where we trace over left-moving excitations but consider the ground states of the Z2 orbifold of the right-moving sector | that is, we consider the right-movers to live in the Ramond sector of the Conway module constructed by Duncan [16].3 The orbifold action kills the extra zero modes, but leaves the left-moving sector untouched. The ground-state degeneracy of the right-movers gives an overall factor of 24, as in [15]. Including this degeneracy, the result (for a given N ) is BPS counting function(V~N ) TrV~N ( 1)F qL0 qL0 y a Ja = 24 (where V~N is the twisted Hilbert space described above). Specializing chemical potentials as before by choosing a 2 N , we see that the BPS counting function is F N; ( ; z). We note here that the need to consider the twisted index computation to nd a non-zero answer plays well with the structure of 12. While the heterotic compacti cation without the twist has a moduli space obtained by choosing left and right moving sub-lattices of 8;24, after twisting the E8 lattice is rigidi ed. Therefore, compactifying on an additional T 2 to compute the index, one expects a moduli space based on 2;26. This makes it natural to expect a connection to an automorphic form for the Narain modular group O(2; 26; Z), such as 12. 3This is related by modular invariance to computations with a Z2 insertion in the trace, which give character valued indices of the original model | hence, we view it as a trick to extract certain BPS degeneracies of the original theory. { 5 { Now, let us consider the second-quantized version of this counting function. To do this, consider the heterotic sigma model based on the conformal eld theory Symn(V~N ). (V~N appears because again, one can count BPS states without additional insertions to absorb fermion zero modes by quotienting by the natural Z2 action on the E8 lattice, before taking the symmetric product and putting right movers into their ground state). We still restrict ourselves to BPS states carrying zero charge with respect to the right-moving U(1) currents. Following the logic of Dijkgraaf, Moore, Verlinde and Verlinde [ 2 ], we see that 1 n=0 log X pnF Symn(V~N )( ; z) = 24 log ! N; ( ; z; ) 1N2; ( ; z; ) ! : Here, F Symn(V~N ) is the specialized BPS counting function for the CFT based on the nth symmetric product of the (twisted) heterotic sigma model, and N; = ph(R) 24( ) Y 2R+ 1( ; z( ; )) ( ) where R+ is the set of positive roots of the lattice N , and the sign depends on the choice of the set of positive roots. The factor of 24 again arises from the right-moving ground state degeneracy, as in [15]. correction of) 1 10 We see that there is a precise analogy to the ndings in [ 1, 2 ]: just as an (automorphic governs the BPS states of the sigma models on the Hilbert scheme of K3 surfaces, (an automorphic correction of) 1 sigma models Symn(V~N ). The role of the factor 12 governs the BPS states of the heterotic N; is super cially similar to that of the automorphic correction 10;1 in the former story. As mentioned earlier, this factor achieves an interpretation in the 4d/5d lift [4, 5], and it would be interesting to give a precise similar interpretation to N; . 4 Discussion In this note, we've described how the (inverse of the) Borcherds modular form 12 serves as a generating function for the BPS state degeneracies of heterotic sigma models on T 8 (after suitably twisting to kill right-moving zero modes), and their symmetric products. We conclude with several comments and possible avenues for further development. In light of AdS3/CFT2 duality, it is natural to conjecture that these heterotic conformal eld theories, Symn(VN ), (at least at large central charge, i.e. large n) are dual to AdS3 gravity theories with discrete symmetry groups corresponding to those of the associated Niemeier lattice | i.e. the Umbral groups and Conway's largest sporadic group. A criterion was developed in [17], using BPS degeneracies of 2d SCFTs to test for a possible large radius gravity dual. By modifying and checking this criterion for the case at hand, it has been found that these theories will have (at best) `stringy' gravity duals | they will not achieve a (parametric) separation between the inverse AdS radius and Mstring as the central charge c ! 1 [18]. { 6 { The 2d heterotic compacti cations governed by the CFTs VN are the dimensional reductions of the Niemeier points studied in [9] in relation to moonshine. It is tempting to try and connect Borcherds' modular form 12 to Umbral moonshine. Making these thoughts precise is di cult because the notion of counting BPS states in gravity theories in d 3 at dimensions is fraught with subtlety; charged or gravitating particles have strong infra-red e ects in low dimensions. Possibly, nding an interpretation of the present formulae in the setting of the supersymmetry-protected amplitudes studied in [19] is a route forward. While the models VN depend on the choice of a Niemeier lattice N , the function counting the second quantized BPS states is the same in all cases. Indeed, these counting functions are just Fourier expansions at di erent cusps of the same automorphic function 12. This is non-trivial: the points where the lattice 8;24 splits into an E8 lattice plus a Niemeier lattice are isolated points in the Narain moduli space of perturbative heterotic strings on T 8. The basic reason behind this phenomenon is that these isolated points are the di erent decompacti cation limits in the Narain moduli space of heterotic strings on T 9, with the condition that the Narain lattice 9;25 splits as an orthogonal sum of E8 and 1;25. This suggests that there might be an interpretation of 12 in terms of compacti cation of the heterotic string to one dimension, possibly along the lines suggested in [15]. It is known that 1= 10 is related to the square of the denominator of a generalized Kac-Moody (GKM) algebra. There is a beautiful story relating this algebra to wallcrossing of 1/4-BPS dyons in N = 4; d = 4 string theory [20]. As we mentioned earlier, 1= 12 is also the denominator of a GKM algebra-the fake monster Lie algebra. It would be interesting to explore a similar story relating this algebra to the BPS states in the theories we discuss. The Gromov-Witten theory of K3 T 2 has recently been conjectured to be governed by the Igusa cusp form [21]. This follows from the role of 10 in black hole entropy counts, and string duality. Given that the heterotic string on T 8 is dual to type IIA on K3 T 4, it is tempting to think that the appearance of 12 in BPS counts in the present setting presages a similar role for 12 in ( avored) enumerative geometry, perhaps of K3 Acknowledgments We are grateful to N. Benjamin and A. Tripathy for discussions. S.M.H. is supported by a Harvard University Golub Fellowship in the Physical Sciences and DOE grant DESC0007870. The research of S.K. was supported in part by the National Science Foundation under grant NSF-PHY-1316699. N.M.P. is supported by a National Science Foundation Graduate Research Fellowship under grant DGE-114747. R.V. is supported by a grant from `Programma per giovani ricercatori Rita Levi Montalcini' 2013. M.Z. is supported by the Mellam Family Fellowship at the Stanford Institute for Theoretical Physics. { 7 { Open Access. 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Sarah M. Harrison, Shamit Kachru, Natalie M. Paquette, Roberto Volpato, Max Zimet. Heterotic sigma models on T 8 and the Borcherds automorphic form Φ12, Journal of High Energy Physics, 2017, 121, DOI: 10.1007/JHEP10(2017)121