Hidden Exceptional Symmetry in the Pure Spinor Superstring

PHYSICAL REVIEW D 101, 026006 (2020)

Hidden exceptional symmetry in the pure spinor superstring

† R. Eager,1,* G. Lockhart,2, and E. Sharpe 3 1Kishine Koen, Yokohama 222-0034, Japan 2Institute for Theoretical Physics, University of Amsterdam, 1098 XH Amsterdam, The Netherlands 3Department of Physics, Virginia Tech, Blacksburg, Virginia 24061, USA

(Received 12 November 2019; published 6 January 2020)

The pure spinor formulation of superstring theory includes an interacting sector of central charge cλ ¼ 22, which can be realized as a curved βγ system on the cone over the orthogonal Grassmannian þ OG ð5; 10Þ. We find that the spectrum of the βγ system organizes into representations of the g ¼ e6 affine algebra at level −3, whose soð10Þ−3 ⊕ uð1Þ−4 subalgebra encodes the rotational and ghost symmetries of the system. As a consequence, the pure spinor partition function decomposes as a sum of affine e6 characters. We interpret this as an instance of a more general pattern of enhancements in curved βγ systems, which also includes the cases g ¼ soð8Þ and e7, corresponding to target spaces that are cones over the complex Grassmannian Grð2; 4Þ and the complex Cayley plane OP2. We identify these curved βγ systems with the chiral algebras of certain two-dimensional (2D) (0,2) conformal field theories arising from twisted compactification of 4D N ¼ 2 superconformal field theories on S2.

DOI: 10.1103/PhysRevD.101.026006

I. INTRODUCTION Nevertheless, we find that the ghost sector partition function [2] can be expressed as a linear combination of The pure spinor formalism [1] is a reformulation of ðeˆ6Þ affine characters, superstring theory which has the virtue that it can be −3 ðˆ Þ ðˆ Þ quantized while preserving manifest covariance with Z ¼ χˆ e6 −3 − χˆ e6 −3 : ð2Þ respect to ten-dimensional super-Poincar´e symmetry. It 0 −3ω1 therefore provides a powerful approach to quantizing the To motivate our results, we will start by briefly recalling superstring in curved backgrounds with Ramond-Ramond different known realizations of the ghost system. A flux and computing multiloop scattering amplitudes. convenient realization is as a curved βγ system on the Focusing on the left movers, the defining feature of this space P of ten-dimensional (10D) pure spinors. A first hint formalism is the presence of a ghost system that is realized λα of the enlarged symmetry follows from the work of in terms of a set of bosonic fields, , transforming in the Levasseur et al. [3] who found that the space of differential 16 of soð10Þ, satisfying the “pure spinor” constraint operators on P enjoys an action of e6; see also [4] and α μ β especially [5]. In the physics literature, the existence of an λ γαβλ ¼ 0; μ ¼ 0; …; 9 ð1Þ e6 finite-dimensional Lie algebra action on the zero modes and contributing cλ ¼ 22 to the left central charge. In this of the pure spinor ghost sector was first observed in [6,7]. paper, we will argue that the pure spinor ghost sector We will find it enlightening to consider a more general ˆ −3 ˆ possesses a hidden affine e6 symmetry at level , albeit family of βγ systems whose target spaces, Xg, have with a choice of stress tensor different from the Sugawara enlarged symmetry g ¼ d4ð¼ soð8ÞÞ; e6, and e7. These one. With this choice of stress tensor, only the currents for varieties can be described as cones over the complex c the soð10Þ−3 ⊕ uˆ ð1Þ−4 subalgebra corresponding to rota- Grassmannian Grð2; 4Þ, the complex orthogonal tional and ghost symmetries have conformal dimension 1. Grassmannian OGþð5; 10Þ, and the complex Cayley plane OP2, respectively. An insightful way to realize the target * ˆ Present address: School of Physics, Korea Institute for spaces Xg is as Lagrangian submanifolds of the moduli Advanced Study, Seoul 02455, Korea. f † spaces M 1 of a single centered g-instanton. These moduli Present address: Theory Department, CERN, CH-1211 g; Geneva 23, Switzerland. spaces are in turn the Higgs branches of a family of 4D N ¼ 2 superconformal field theories (SCFTs) Tg, whose Published by the American Physical Society under the terms of ˆ chiral algebra in the sense of [8] is the vacuum module of gk the Creative Commons Attribution 4.0 International license. affine algebra, where k ¼ −2; −3; −4, respectively, for Further distribution of this work must maintain attribution to ¼ T the author(s) and the published article’s title, journal citation, g d4; e6, and e7. Applying a topological twist to g and DOI. Funded by SCOAP3. and reducing on S2 [9], we will obtain [10] a set of 2D (0,2)

2470-0010=2020=101(2)=026006(9) 026006-1 Published by the American Physical Society R. EAGER, G. LOCKHART, and E. SHARPE PHYS. REV. D 101, 026006 (2020)

ð0 2Þ conformal field theories T ; , whose chiral algebras we formalism [14], where physical states are identified with g ¯ ¼ will identify with the corresponding βγ system on Xˆ . the cohomology of the modified BRST operator Q g þ ∂¯ ∂¯ P We will present a detailed analysis of these theories in a Q , where is a Dolbeault operator acting on . companion paper [10]. As we will see, the symmetry of the quantum mechanics on P ð10Þ ⊕ The global symmetry of the βγ systems is a certain is enlarged from so u1 to e6. In the next section, we will consider similar spaces with quantum mechanical maximal subalgebra g0 of g. However, we will see that from the perspective of geometric representation theory, it is symmetry enhancement considered in [3]. For special target P natural to expect the entire algebra g to act on the states of spaces including , we will argue that the quantum T ð0;2Þ mechanical enhancement will extend to enhancement in the theory. This is indeed the case for the theory soð8Þ, the βγ system. whose chiral algebra was found by Dedushenko and Gukov ˆ [11] to realize the soð8Þ−2 affine algebra. III. βγ SYSTEMS ON COMPLEX CONES We next study how the enlarged symmetry manifests Curved βγ systems [12,15–17] are two-dimensional itself in the partition function for g ¼ soð8Þ and g ¼ e6.In γ∶Σ → ˆ both cases, we will find that the partition function can be sigma models of holomorphic maps X with action ˆ Z expressed as a linear combination of two gk characters. 1 ð0 2Þ ¼ β ∂¯γi ð Þ T ; S i ; 4 These results suggest that the chiral algebra of g 2π Σ receives two types of contributions: one from states arising γ ¼ 1 … ˆ from the reduction of the 4D N ¼ 2 chiral algebra and one where, in a given patch, i, i ; ; dim X serve as local β capturing contributions from a surface defect of the 4D coordinates, i are (1,0)-forms, and T 2 SCFT g that is wrapped along S . dw γiðzÞβ ðwÞ ∼ δi : ð5Þ We will also find an elegant closed form for the partition j j z − w function of the pure spinor ghost system, written in terms ˆ ˆ of e6 Weyl invariant Jacobi forms, which agrees exactly We consider the case where X ¼ Xg is one of the varieties with the first six energy levels computed in [2]. An amusing constructed by Levasseur et al. [3], which is associated to a consequence is that the fields, ghosts, antifields, and anti- Lie algebra g. Constructing these varieties involves a choice ghosts of ten-dimensional supersymmetric Yang-Mills of a minuscule root of g. The minuscule root defines a theory organize into the 27 and 27 of e6. decomposition of g of the form

g ¼ g−1 ⊕ g0 ⊕ g1; ð6Þ II. THE PURE SPINOR GHOST SYSTEM where g0 ¼ u1 ⊕ s, and g1 is a minuscule representation In the pure spinor formalism, the superstring is described ω by a sigma model consisting of maps Σ → M from the Vω associated to the highest weight of the semisimple Lie string worldsheet Σ into ten-dimensional super-Minkowski algebra s [18]. Let Gs be the simply connected complex Lie space M coupled to a ghost system of central charge group corresponding to s, and Pω be the parabolic sub- group corresponding to ω. Then, one defines Xˆ to be the cλ ¼ 22. The matter fields on the worldsheet include a set g μ Pð ˆ Þ¼ of bosonic fields x in the 10 of soð10Þ and, focusing on the complex cone over the base Xg Gs=Pω. The spaces θα 16 ˆ left-movers, a set of periodic fermions in the of Xg have the following homogeneous coordinate ring: soð10Þ, along with their conjugate momenta pα, so that the C½ ˆ ≅ ⨁ ð Þ total left-moving central charge cL ¼ cx þ cθ þ cλ ¼ 0. Xg Vlω: 7 In the original minimal formalism, the ghost sector is l≥0 captured by a sigma model describing maps Σ → P into the We focus on the following cases, where g belongs to the space of ten-dimensional Cartan pure spinors P, which is Deligne-Cvitanović exceptional series, listed in Table I. parametrized by the bosonic fields λα satisfying the con- ⊕ ¼ For e6, g0 is the Lie algebra d5 u1, Vω Vω4 is the straint in Eq. (1). These fields are accompanied by their ˆ spinor representation 16 of soð10Þ, and Xg coincides with conjugate momenta wα. The physical spectrum is given by P, the space of ten-dimensional pure spinors. the cohomology of the nilpotent Becchi-Rouet-Stora- ˆ ˆ The βγ system on X has central charge c ¼ 2 dimC X Tyutin (BRST) operator g g and manifest global symmetry g0 ¼ u1 ⊕ s, where the Z 1 Abelian factor acts by rescaling the cone and s acts on the ¼ λα þðγμθ Þ∂ μ − ðγμθÞ ðθγ ∂θÞ ð Þ base. We consider the following partition function: Q pα α x 2 α μ 3 −1 s F 2πiτH J 2au1 g Zgðt; m ; τÞ¼TrHð−1Þ e t expð2πim Þ; ð8Þ acting on a suitable Hilbert space H. The Hilbert space can be defined using the curved βγ system [2,12,13].A where q ¼ e2πiτ, t ¼ e2πiσ, F is the fermion number, convenient way to do this is to pass to the nonminimal H is the (left) Hamiltonian, J is the u1 generator, and

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s m ∈ hðsÞC are fugacities for s. Our definition for the TABLE I. Relevant varieties. partition function differs from that of [2,19,20] by a factor 1 ω ˆ Pð ˆ Þ ðPð ˆ ÞÞ − a g s dim g1 dimC Xg Xg c1 Xg of t 2 u1 , where d4 ω4 a3 65Grð2; 4Þ 4 ˆ ω þð5 10Þ a ¼ −c1ðPðX ÞÞ ð9Þ e6 1 d5 16 11 OG ; 8 u1 g 2 e7 ω7 e6 27 17 OP 12 is the u1 symmetry anomaly appearing in the operator product expansion (OPE) a partial N ¼ −1 topological twist on the N ¼ 2 SCFT ð Þ au1 J z along the lines of [9], and reduce the theory on a two-sphere JðyÞTðzÞ ∼ þ : ð10Þ ðy − zÞ3 ðy − zÞ2 S2, leading to a two-dimensional theory preserving (0,2) supersymmetry. The u1 level, which appears in the OPE, Four-dimensional N ¼ 2 SCFTs have both a Higgs branch and an associated vertex operator algebra (VOA) k ð Þ ð Þ¼ u1 ð Þ [8]. These invariants are intricately related to each other J y J z 2 11 ðy − zÞ [26]. The Higgs branch HiggsðT gÞ is the minimal (non- zero) nilpotent orbit O of g, which is also the centered is given in this class of models by moduli space of one G-instanton [27,28], and has complex dimension 2h∨ − 2. Algebraically, the minimal nilpotent 1 k ¼ a : ð12Þ orbit is the associated variety of the Joseph ideal J 0 [29]. u1 2 u1 ˆ The spaces Xg are Lagrangian submanifolds of HiggsðT gÞ. ¼ The u1 symmetry anomaly and level can be extracted from To see this, we first fix a triangular decomposition g − ⊕ ⊕ þ the unrefined Hilbert series of Xˆ [19]. n h n of g. The irreducible components of the g intersection of O with nþ are called minimal orbital The partition function displays the field-antifield sym- varieties. They are isotropic subspaces with respect to metry 1 the Killing form, of dimension 2 dim O, hence Lagrangian ˆ subvarieties of O by theorem 3.3.6 of [30], and play an Z ðt; ms; τÞ¼ð−1ÞdimC Xg Z ð1=t; −ms; τÞð13Þ g g important role in geometric representation theory [3, ⋆ 31–33]. Smooth orbital varieties of the minimal nilpotent and -conjugation symmetry ˆ orbit are minuscule varieties [34]. The spaces Xg are 1 −1 1 ðPð ˆ ÞÞ Z ðt; ms; τÞ¼−ðq2t Þ2c1 Xg Z ðq=t; −ms; τÞð14Þ minuscule varieties for Gs. g g ˆ The associated VOA, VkðgÞ, is the affine algebra gk at ∨ ∨ of the βγ system [2,13]. The ground states contribute level k ¼ −h =6 − 1, where h is the dual Coxeter number of g [35]. The holomorphic twist of the (0,2) theory is a c 1 ˆ −24 2c1ðXgÞ chiral theory whose spectrum organizes into representa- q t HSg ð15Þ tions of the VOA. In particular, from the growth of states, P ∞ l s s one can argue that the spectrum must include a represen- to the partition function, where HS ¼ l¼0 t χ ðm Þ is g Vlω ¼ 1 ð4 4D − 5 4DÞ 0 ˆ tation of dimension hmin 2 a c < [9], where the refined Hilbert series of Xg. 4D 4D ða ;c Þ are the anomaly coefficients of Tg listed in IV. βγ SYSTEM FROM 4D/2D SCFT Table II. Thus, the central charge of the (0,2) theory is shifted from the Sugawara value SCFT provides an additional vantage point from which βγ βγ the curved systems can be studied. Indeed, the curved ¼ dim gk ð Þ ˆ cSug ∨ 16 system with target Xg can be identified with the holomor- h þ k phic twist of a two-dimensional (0,2) sigma model on Xˆ , g to the effective central charge which also implies equality between the partition function of the former and the elliptic genus of the latter [21–23]. ˆ T To realize the (0,2) sigma models with the targets Xg TABLE II. Properties of the g theories. listed in Table I, we begin with a triplet of four-dimensional T ∨ 4D 4D g h cSug a c hmin ceff SCFTs T g, where g denotes the Lie algebra of the flavor T T N ¼ 2 T −14 23 7 −1 symmetry group of g. The theory d4 is the super d4 6 24 6 10 ð2Þ T −26 41 13 −2 Yang-Mills theory with gauge group SU and four e6 12 24 6 22 T T 59 19 flavors, while e6 and e7 are the rank-one e6 and e7 T −38 −3 e7 18 24 6 34 Minahan-Nemeschansky theories [24,25]. We next perform

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¼ − 24 ¼ 2 ˆ ð Þ ceff cSug hmin dim Xg; 17 which coincides with the central charge of the βγ system on ˆ Xg. The shift in central charge can be traced back to the fact that the stress-energy tensor differs from the Sugawara stress tensor by a correction term [11,13,19], The relation between the βγ system and the twisted S2 T T ¼ T þ ∂J; ð18Þ compactification of the g theory provides a further reason Sug ˆ to expect the appearance of the affine gk algebra. Indeed, as which gives rise to the u1 anomaly of Eq. (10). Since the we have seen in the previous section, the chiral algebra of ð Þ ð Þ J y TSug z OPE has no anomaly, the u1 symmetry the resulting (0,2) model provides a representation of the anomaly and level are proportional, with relation given VOA VkðgÞ. by Eq. (12). Similarly, after the modification of the stress In the remainder of this section, we discuss in detail how tensor, the currents in g0 retain conformal dimension 1, the symmetry enhancement manifests itself in the partition c while the currents in g1 and g−1 acquire the new conformal function in the soð8Þ−2 and ðeˆ6Þ−3 cases. Zhu’s theorem dimensions 2 and 0, respectively. [37] relates the classification of simple highest weight Altogether, these considerations suggest that the (0,2) VkðgÞ-modules to Joseph’s classification [33] of simple theories we constructed flow in the IR to a sigma model on highest weight UðgÞ=J 0-modules [36]. We use this clas- ˆ ðT Þ the Lagrangian submanifold Xg of Higgs g and that their sification in the following examples. elliptic genus coincides with the partition function of the corresponding βγ system. Indeed, for g ¼ d4, the twisted ˆ A. Enhancement to (d ) − in the Gr(2;4) cone βγ system compactification of T is the Dedushenko-Gukov (0,2) 4 2 d4 βγ model, which has been argued to flow to a (0,2) sigma We start with the system on the complex cone over Grð2; 4Þ. There exist at least two convenient UV descrip- model on Xˆ [11]. We conjecture that an analogous result d4 tions of the corresponding (0,2) sigma model, for which the holds for g ¼ e6 and e7 as well. ˆ appearance of an affine ðd4Þ−2 algebra was found in [11]: the first is as a two-dimensional (0,2) SUð2Þ gauge theory V. SYMMETRY ENHANCEMENT with four fundamental chiral multiplets, which arises ˆ The βγ system with target X has a manifest affine gˆ0 ⊂ gˆ directly from the twisted compactification of the 4D g N ¼ 2 T symmetry. In this section, we argue that in fact the βγ theory soð8Þ; the second is as a (0,2) Landau- system enjoys affine gˆ symmetry. First, let us review how Ginzburg model consisting of a single Fermi superfield, Ψ Φ ∧2 4 ¼ 6 enhancement to g ¼ g−1 ⊕ g0 ⊕ g1 occurs in the quantum and a set of chiral superfields in the repre- ð4Þ mechanics on Xˆ , a fact which has been studied in [6,7]. sentation of SU , coupled via a J-type superpotential g ¼ Ψ ðΦÞ Dð ˆ Þ ˆ interaction term J Pf . The differential operators, Xg ,onXg are equivalent to Classically, the vacuum moduli space of this model is a ð Þ J ð Þ 6 U g = 0, where U g is the universal enveloping algebra quadric in C , specifically the affine cone over Gð2; 4Þ, of g and J 0 is the Joseph ideal [3]. Infinitesimal rotation ˆ ˆ which is the closure of Xsoð8Þ. Quantum corrections will and dilatation symmetries of Xg are generated by differ- modify this picture in the interior. As we will discuss in ential operators transforming in g0. Differential operators [10], in analogy with the pure spinor case, we conjecture realizing g−1 and g1 also have a simple description: those in [38] that quantum corrections move the singular vertex g−1 correspond to coordinate multiplication, while those in of the affine cone infinitely far away, realizing a two- ˆ g1 act like generalized special conformal transformations. dimensional (0,2) sigma model on X ð8Þ. Dð ˆ Þ so Next let us discuss how the identification between Xg For this theory, g0 ¼ u1 ⊕ a3 is the manifest global ð Þ J βγ and U g = 0 suggests a relationship between the symmetry, while g−1 ¼ g1 is the Vω ¼ 6 representation ˆ ð Þ 2 system on Xg and the VOA Vk g . On the one hand, the of a3. The partition function can be computed straight- ˆ operators realizing the affine gk symmetry in the βγ system, forwardly, either as the elliptic genus of the (0,2) SQCD whose explicit construction we defer to future work [10], theory [39] following [40–42], or directly as the partition ˆ function of the curved βγ system on Grð2; 4Þ [13].Itis are expected to reduce to differential operators on Xg in the limit of quantum mechanics. On the other hand, the given by chiralization of the algebra UðgÞ=J W ≅ C × UðgÞ=J 0 5 iηðτÞ θ1ð2σ; τÞ a3 is the VOA V ðgÞ, where the ideal J W of UðgÞ is defined Z ðt; m ; τÞ¼Q : ð19Þ k d4 θ ðσ þðma3 Þ τÞ w∈6 1 ;w ; in [36]. This means that the Zhu algebra of VkðgÞ is C × UðgÞ=J 0. This suggests that we can view the VOA The product in the denominator is over the weights in 6 of ˆ VkðgÞ as an algebraic analog of the curved βγ system on Xg. a3. We now proceed to express the partition function in ˆ The various relations are summarized in the following terms of ðd4Þ−2 characters. The embedding of u1 ⊕ a3 into diagram: d4 implies the following mapping of parameters:

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2 ˆ ˆ d4 a3 t ðd4Þ ðd4Þ m ¼ m for i ¼ 1; 2; 3; χˆ −2 ðτ − σ; ma3 ; τÞ¼ξˆ −2 ðσ; ma3 ; τÞ; ð27Þ i i q 0 0 ma3 ma3 md4 ¼ σ − 1 − ma3 − 3 4 2 2 2 : which guarantees that the partition function obeys the ⋆-conjugation symmetry of Eq. (14). ˆ The algebra ðd4Þ−2 is known to possess four irreducible (ˆ ) βγ highest weight representations, corresponding to the fol- B. Enhancement to e6 − 3 in the pure spinor system lowing choices of highest weight [36,43]: We now turn to the pure spinor βγ system and discuss how the affine ðeˆ6Þ symmetry manifests itself at the level 0 −2ω −2ω −2ω ð Þ −3 ; 1; 3; 4: 20 of the partition function. The partition function has been computed up to energy level five by fixed point techniques The three nonvacuum representations are related by triality. in [13], and an all-order expression in the md5 → 0 limit While each of these highest weights is not dominant, it is still the case that there exists a unique dominant weight Λ in was found in [46] using local algebra [47]. In what follows, the shifted Weyl orbit of the highest weight. As a we will be able to give a very simple closed form for the me6 consequence [44,45], the corresponding affine characters partition function for arbitrary values of fugacities. ⊕ are determined in terms of Kazhdan-Lusztig polynomials For this theory, g0 is the u1 d5 ghost and rotational [44,45]. By an explicit computation, we find that the symmetry of the pure spinor ghost system. The components partition function is given by a sum of two characters, g−1 and g1 correspond to the 16 and 16 representations of d5, respectively. We begin by discussing the e6 → u1 ⊕ d5 ˆ ˆ ðd4Þ ðd4Þ P Z ðt; ma3 ; τÞ¼χˆ −2 ðmd4 ; τÞ − χˆ −2 ðmd4 ; τÞ: ð21Þ branching rules. The realization of the space of pure d4 0 −2ω4 ˆ spinors as the variety Xe6 implies the following mapping of The vacuum character has the following q expansion: parameters between e6 and u1 ⊕ d5: 0 1 0 1 0 1 ˆ u1 ðd4Þ 14 e6 1 3 3 5 −2 24 d4 d4 d4 2 m χˆ0 ¼ q ð1 þ χ28q þðχ300 þ χ28 þ 1Þq þ …Þ: ð22Þ m1 −3 − 2 −1 − 2 − 4 − 4 B C B C B d5 C B e6 C B C B m1 C B m2 C B 000010C B C B C B C B C The nonvacuum character has conformal dimension e6 d5 B C B 000001C B m2 C h ¼ −1, consistent with Table II. As noted in [26],it B m3 C B C B C B C ¼ B C · B C has the property that an infinite number of states appear at B me6 C B 000100C B md5 C B 4 C B C B 3 C each energy level. In particular, its lowest energy level B e6 C B 001000C B d5 C @ m5 A @ A @ m A component, expressed in u1 ⊕ a3 fugacities, has the 4 e6 010000 d5 following series expansion: m6 m5 X∞ 14 −1 2 l a3 −q24 t t χ ðma3 Þ; ð23Þ (see Appendix A for our conventions). At the level of Vlω2 l¼0 representations, the 27 and adjoint of e6 decompose as follows, where the subscript denotes u1 charge, which encodes the ground states of the partition function of 27 → 1 þ 16 þ 10 the βγ system, Eq. (15). At the next energy level, one finds −4 −1 2; the following contributions: 78 → 16−3 þ 10 þ 450 þ 163:

14 2 a3 2 a3 3 a3 a3 Furthermore, to match the pure spinor formalism conven- q24ð2 − t ðχ þ 1Þ − t ðχ þ 1Þ − t ðχ þ 2χ Þþ…Þ: 15 15 64 6 tions, in Eq. (8), we must set ð24Þ t ¼ e2πi·ð−3mu1 Þ: ð28Þ Interestingly, it appears natural to define the following new combination of characters: The algebra ðeˆ6Þ−3 possesses a finite number of irreduc-

ˆ ˆ ˆ ible modules [36] corresponding to the highest weights ðd4Þ ðd4Þ ðd4Þ ξˆ −2 ðmd4 τÞ¼−χˆ −2 ðmd4 τÞþ2χˆ −2 ðmd4 τÞ ð Þ −2ω4 ; −2ω4 ; 0 ; ; 25 0; −3ω1; −3ω6; ω1 −2ω3; ω6 −2ω5; −2ω2;−ω4: in terms of which ð29Þ ˆ ˆ ðd4Þ ðd4Þ ð ma3 τÞ¼−χˆ −2 ðmd4 τÞþξˆ −2 ðmd4 τÞ ð Þ Zd4 t; ; 0 ; −2ω4 ; : 26 The nonvacuum representations have conformal dimension −2 which equals the value of h given in Table II. We find ˆ min The ðd4Þ−2 characters, once expressed in terms of u1 ⊕ a3 that the pure spinor partition function is given by the fugacities, satisfy the following simple relation: following combination of ðeˆ6Þ−3 characters:

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ðeˆ6Þ ðeˆ6Þ e6 ð ma3 τÞ¼χˆ −3 ðme6 τÞ − χˆ −3 ðme6 τÞ ð Þ m˜ 1 ¼ −3m1 − 3m2 − 5m3 − 6m4 − 4m5 − 2m6: ð36Þ Ze6 t; ; 0 ; −3ω1 ; : 30 → ⊕ The lowest energy component of the nonvacuum term is the Under e6 d5 u1, this shift corresponds to setting u1 → −3 u1 md5 Hilbert series of the Wallach representation of the e6 finite- m m , while keeping invariant. dimensional Lie algebra [5] corresponding to highest Using the modular transformation properties of the 26−2 4 −3ω − 24 numerator [and taking into account the shift (36)], one weight 1, up to an overall factor of q t . τ → −1 τ finds that under = Ze6 transforms as follows: Expressed in terms of u1 ⊕ d5 fugacities, it is given by me6 1 πi X∞ e6 e6 e6 22 − ¼ −3 ðm m Þ ðm τÞ − 4 l d5 Ze6 ; exp ; Ze6 ; ; −q 24t t χ ðmd5 Þ; ð31Þ τ τ τ Vlω5 l¼0 where the phase factor is consistent with the occurrence of in agreement with the Hilbert series of the cone over the the ðeˆ6Þ−3 algebra [48]. þ orthogonal Grassmannian OG ð5; 10Þ, which is the space It is now straightforward to express the partition function of pure spinors in ten dimensions. (35) in terms of the pure spinor fugacities t and md5 via Again, it appears natural to define the following combi- Eq. (B6); after doing so, we find an exact match with [2], nation of characters: where Ze6 was computed up to the fifth energy level by fixed point techniques. ðeˆ6Þ ðeˆ6Þ ðeˆ6Þ ξˆ −3 ðme6 τÞ¼−χˆ −3 ðme6 τÞþ2χˆ −3 ðme6 τÞ ð Þ Rewriting the partition function as −3ω1 ; −3ω1 ; 0 ; ; 32

ðeˆ6Þ1 ðeˆ6Þ1 in terms of which χˆω ðme6 ; τÞ − χˆω ðme6 ; τÞ Z ðme6 ; τÞ¼Q1 6 ; ð37Þ e6 16 ηðτÞ−1θ ððm α∨ Þ τÞ i¼1 1 ; 16;i ; ðeˆ6Þ ðeˆ6Þ Z ðt; md5 ; τÞ¼−χˆ −3 ðme6 ; τÞþξˆ −3 ðme6 ; τÞ: ð33Þ e6 0 −3ω1 ðˆ Þ χˆ e6 1 ¼ Θe6 η6 where ω1;6 ω1;6 = are level 1 e6 affine characters, The ðeˆ6Þ−3 characters, expressed in terms of u1 ⊕ d5 suggests a possible alternative interpretation as a level 1 e6 fugacities, satisfy the following relation: sector coupled to 16 complex bosons; we do not pursue this direction further in this paper. 4 t ðeˆ6Þ ðeˆ6Þ χˆ −3 ðτ − σ; md5 ; τÞ¼ξˆ −3 ðσ; md5 ; τÞ; ð34Þ q2 0 0 VI. CONCLUSIONS We have found that the states in the βγ system with target which guarantees that the partition function obeys the Xˆ organize into a direct sum of irreducible modules of an ⋆-conjugation symmetry described by Eq. (14). The g affine gˆ symmetry algebra. When the target is the space of significance of writing the partition function as in P ðˆ Þ ten-dimensional pure spinors, , the symmetry algebra is ξˆ e6 −3 Eq. (33) is that the −3ω1 character captures the contribution ðeˆ6Þ−3. This knowledge led us to find a compact closed of the globally defined operators, which are identified with form expression for the full partition function of the ghost 0 ¯ ðeˆ6Þ−3 the zeroth cohomology H ð∂Þ, while χˆ0 captures the system of the pure spinor superstring in Eq. (35). We leave contribution from the “missing states” in the Hilbert space it to future work to study possible implications for the of the pure spinor system, which are built out of the b ghost computation of operator product expansions and scattering and correspond to H3ð∂¯Þ. amplitudes in superstring theory. We also find that the pure spinor partition function, While we have given several arguments for the appearance of the gˆ symmetry, it should be possible to explicitly written in terms of e6 fugacities, can be written in the k βγ following very simple closed form: realize its generators in the curved system. We have focused on three different smooth targets for the βγ system.

−1 e6 It would also be natural to extend our analysis to other ð2iÞ α−5 1ðm˜ ; τÞ ðme6 τÞ¼ Q ; ð Þ Ze6 ; : 35 targets, possibly with singularities. ηðτÞ22 16 φ ððme6 α∨ Þ τÞ ¼1 −1;1=2 ; 16 ; ˆ j ;j We have seen that the appearance of gk symmetry has ∨ a natural explanation from the perspective of the four- α 2 The product in the denominator is over the subset 16 of dimensional SCFT, T , dimensionally reduced on S . This ¼ 16 g coroots of e6 that belong to g−1 under the grading in also explains the appearance of the vacuum module of gˆ Eq. (6). On the other hand, the numerator is given in terms in the elliptic genus. It remains however to explain the ½ αe6 ðme6 τÞ of the Weyl e6 -invariant Jacobi form −5;1 ; (see occurrence of a second module. A possible hint is that Appendix B), with the following subtlety: theP argument the unflavored limit of the vacuum character of the VOA me6 m˜ ¼ ˜ e6 ω ð Þ ć is replaced by the shifted e6 fugacities i mi i, V gk , for g belonging to the Deligne-Cvitanovi excep- ˜ e6 ¼ ¼ 2 … 6 where mi mi for i ; ; ,but tional series, satisfies a second order linear modular

026006-6 HIDDEN EXCEPTIONAL SYMMETRY IN THE PURE SPINOR … PHYS. REV. D 101, 026006 (2020) differential equation [26,49]; the other solution has been Bourbaki’s numbering convention for the ωi. We denote the conjectured by Beem and Rastelli to arise from a surface irreducible representations associated to the highest weight operator in the Tg theory. This suggests an interpretation of ω either by Vω or by its dimension, following the the second module in the (0,2) theory as originating from a conventions of [51] (see, e.g., Fig. 1). The character of a surface defect wrapping the S2. We plan to return to this highest weight representation R of g is given by issue in a separate work [10]. X g g χRðm Þ¼ expð2πiðm ;wÞÞ; ðA2Þ ACKNOWLEDGMENTS w∈R P mg ¼ gω ∈ ¼ We would like to thank C. Beem, T. Creutzig, J. Distler, where i mi i hC. The character of a g u1 R. Donagi, I. Melnikov, M. Movshev, B. Pioline, I. Saberi, representation of charge k is just e2πikmu1 . For an affine Lie and J. Song for valuable discussions and correspondence. algebra, we denote a highestP weight representation simply R. E. thanks the Korean Institute for Advanced Study for ω ¼ ω by the finite part i ni i of its highest weight ˆ hospitality. The work of G. L. is supported by ERC starting gk ðω;k;nÞ. We denote by χˆω ðmg; τÞ the corresponding affine Grant No. H2020 ERC StG #640159. The work of E. S. is character. partially supported by NSF Grant No. PHY-1720321.

Note added.—We wish to thank B. Pioline and M. Movshev APPENDIX B: MODULAR AND JACOBI FORMS for bringing references [4,6,7] to our attention, where the The Dedekind η function is defined as follows: existence of an action of the e6 finite Lie algebra on the ground states of the pure spinor system was discussed. Y∞ 1=24 1=24 k M. Movshev has informed us of unpublished work [50] in ηðτÞ¼q ðq; qÞ∞ ¼ q ð1 − q Þ: ðB1Þ which the presence of an affine e6 symmetry in the pure k¼1 spinor system has also been studied. The Jacobi theta functions are given by

APPENDIX A: LIE ALGEBRAS X 2 X 2 n n n n n θ1ðζ; τÞ¼i ð−1Þ z q 2 ; θ2ðζ; τÞ¼ z q 2 ; ð Þ Given a Lie algebra g, let h g be its Cartan subalgebra, n∈Zþ1 n∈Zþ1 Δg α ¼ 1 … ð Þ X 2 X 2 its root lattice, i, i ; ; rank g a choice of simple 1 2 1 2 α∨ θ ðζ τÞ¼ n 2n θ ðζ τÞ¼ ð−1Þn n 2n roots, and i the corresponding coroots. The fundamental 3 ; z q ; 4 ; z q ; n∈Z n∈Z weights ωi are defined by 2π ζ ðω α∨Þ¼δ ð Þ where z ¼ e i . Closely related is the weak Jacobi form of i; j ij; A1 1 weight −1 and index 2, where (·; ·) is the invariant bilinear form on hðgÞ , nor- C θ ðζ τÞ ðα; αÞ¼2 1 ; malized such that for the long roots. We adopt φ−1 1 2ðζ; τÞ¼ : ð Þ ; = ηðτÞ3 B2

We also make use of Weyl-invariant weak Jacobi forms [52–55]. Under a modular transformation, a Weyl½g- invariant weak Jacobi form φw;n∶hðgÞ × H → C of weight w and index n transforms as z aτ þ b φ ; w;n cτ þ d cτ þ d πinc ¼ðcτ þ dÞw exp ðz; zÞ φ ðz; τÞ; ðB3Þ cτ þ d w;n

while for s ∈ Weyl½g

φw;nðsz; τÞ¼φw;nðz; τÞ: ðB4Þ

g Let Jw;n be the vector space of Weyl½g-invariant weak Jacobi forms of weight w and index n.Forg ≠ e8, the g ¼ ⨁ g bigraded ring J; w;nJw;n is a polynomial ring over FIG. 1. Dynkin diagrams showing our labeling conventions. the ring of SLð2; ZÞ modular forms, which is known to be

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g ⊕ generated by rkðgÞþ1 independent forms αw;n of specified In terms of d5 u1 fugacities, weight and index. For g ¼ e6, the seven generators 1=6 X4 Y5 e6 e6 e6 e6 e6 e6 e6 e6 q −2πiμ α0 1; α−2 1; α ; α ; α−8 2; α−9 2; α−12 3 Θω ðm; τÞ¼ σ e θ ð3μ − τ; 3τÞ θ ðμ ; τÞ; ; ; −5;1 −6;2 ; ; ; 1 2 k k k j k¼1 j¼1 have been constructed in [53,54]. In this paper, we make 1=6 X4 Y5 e6 q 2π μ use of the unique Weyl½e6-invariant weak Jacobi form of Θω ðm; τÞ¼ σ e i θ ð3μ þ τ; 3τÞ θ ðμ ; τÞ; 6 2 k k k j weight −5 and index 1, k¼1 j¼1

e6 e6 ðB6Þ 2iðΘω ðme6 ; τÞ − Θω ðme6 ; τÞÞ αe6 ðme6 ; τÞ¼ 1 6 ; ðB5Þ −5;1 ηðτÞ16 u1 where −σ1 ¼ σ2 ¼ σ3 ¼ −σ4 ¼ 1, μ ¼ −2m and where the level 1 e6 theta functions X 1 1 μ ¼ d5 þ d5 þ d5 þ d5 þ d5 Θe6 ðme6 τÞ¼ ðπ τð Þþ2π ð me6 ÞÞ 1 m1 m2 m3 m4 m5 ; ω1;6 ; exp i w; w i w; 2 2 ∈Δe6 þω w 1;6 1 1 μ ¼ d5 þ d5 þ d5 þ d5 2 m2 m3 2 m4 2 m5 ; have the following q-series expansions: 1 1 d5 d5 d5 μ3 ¼ m3 þ m4 þ m5 ; Θω1 ðm τÞ 2 2 e6 ; e6 e6 2 e6 e6 3 ¼ χ27 þ qχ351 þ q ðχ1728 þ χ3510 ÞþOðq Þ; 1 1 1 1 11=24ηðτÞ5 d5 d5 d5 d5 q μ4 ¼ m4 þ m5 ; μ5 ¼ − m4 þ m5 ω 2 2 2 2 Θ 6 ðm; τÞ e6 ¼ χe6 þ qχe6 þ q2ðχe6 þ χe6 ÞþOðq3Þ: 11=24ηðτÞ5 27 351 1728 3510 q are d5 fugacities expressed in the orthogonal basis.

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