SU(2) channels the cancellation of K3 BPS states

The conformal field theoretic elliptic genus, an invariant for N = (2, 2) superconformal field theories, counts the BPS states in any such theory with signs, according to their bosonic or fermionic nature. For K3 theories, this invariant is the source of the Mathieu Moonshine phenomenon. There, the net number of 14\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \frac{1}{4} $$\end{document}- BPS states is positive for any conformal dimension above the massless threshold, but it may arise after cancellation of the contributions of an equal number of bosonic and fermionic BPS states present in non-generic theories, as is the case for the class of ℤ2-orbifolds of toroidal SCFTs. Never-theless, the space ℋ̂\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \hat{\mathrm{\mathscr{H}}} $$\end{document} of all BPS states that are generic to such orbifold theories provides a convenient framework to construct a particular generic space of states of K3 theories. We find a natural action of the group SU(2) on a subspace of ℋ̂\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \hat{\mathrm{\mathscr{H}}} $$\end{document} which is compatible with the cancellations of contributions from the corresponding non-generic states. In fact, we propose that this action channels those cancellations. As a by-product, we find a new subspace of the generic space of states in ℋ̂\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \hat{\mathrm{\mathscr{H}}} $$\end{document}.


Introduction
The BPS spectrum of quantum field theories with extended supersymmetry has long been recognised to yield crucial and in fact useful information. Indeed, it is the key to the construction of invariants which both allow extrapolations from weak to strong coupling and relate abstractly defined theories to geometry. K3 theories provide a rich family of examples. They are superconformal field theories with N = (2, 2) worldsheet supersymmetry at central charges (c, c) = (6, 6), with spacetime supersymmetry and integer eigenvalues of the operators J 0 and J 0 -the zero modes of the two u(1) currents generating an affine subalgebra of the left and right N = 2 superconformal algebras -and without holomorphic BPS states at weight 1 2 . These requirements imply that every K3 theory has N = (4, 4) worldsheet supersymmetry, and that its conformal field theoretic elliptic genus E(τ, z) equals the complex elliptic genus of K3 surfaces [1]. The moduli space M K3 of K3 theories has dimension 80 and possesses at least one connected component whose structure is well understood [2,3] and which contains all Z 2 -orbifold conformal field theories that JHEP04(2020)184 state T diag built on a twisted ground state |α diag whose orthogonal complement in the 16-dimensional space of twisted ground states is a 15-dimensional subspace A. We denote by H ⊥ the space of all massive ground states in the Fock space over A. H ⊥ is graded and we write H ⊥ = ∞ n=1 H ⊥ n where H ⊥ n is the space of level n states in H ⊥ , n ∈ N. At level 1, it so happens that the net number of massive 1 4 -BPS states counted by the elliptic genus corresponds to states solely built on twisted ground states in A. Moreover, all the massive 1 4 -BPS states built on |α diag cancel in the counting against all massive 1 4 -BPS states from the untwisted sector. This property helped us pinning down the octad group action on the remaining massive 1 4 -BPS states at level 1, using Margolin's construction of a 45-dimensional representation of M 24 as a guide [17], but the situation is not typical at higher level. Yet the octad group continues to act on H ⊥ , as elegantly demonstrated by Gaberdiel, Keller and Paul [18]. Moreover, they argue that G also acts on the space H rest consisting of some 1 4 -BPS states built on the twisted ground state |α diag and some 1 4 -BPS states from the untwisted sector such that the graded dimension of H rest ⊕ H ⊥ agrees precisely with the massive ground state contributions to E(τ, z). Since the action of G arises from symmetry surfing, this beautifully supports the symmetry surfing programme.
Let us describe the 1 4 -BPS states in a given K3 theory in more detail. Here and in the following, by this abbreviation we mean those BPS states which saturate the BPS bound for half of the antiholomorphic N = 4 worldsheet supersymmetries and which are massive with respect to the holomorphic N = 4 superconformal algebra. By H n , n ∈ N \ {0}, we denote the space of all 1 4 -BPS ground states at level n in the given K3 theory. The findings of [18] imply that H n ∼ = H ⊥ n ⊕ H rest n ⊕ H + n , where as above, the dimension of H ⊥ n ⊕ H rest n agrees with the massive ground state contributions at level n to E(τ, z). Accordingly, H + n accounts for "excess" states whose contributions to the conformal field theoretic elliptic genus cancel out. While H ⊥ n is well under control by the results of [18], the spaces H rest n ⊕ H + n have not yet released all their secrets. In particular, for n > 1 neither H rest n nor H + n has been constructed explicitly so far in any K3 theory. In a very recent article [19], Keller and Zadeh have deformed Z 2 -orbifold CFTs on K3 away from the orbifold point using second order conformal perturbation methods. They have shown that under a deformation by T diag all the 1 4 -BPS states in H ⊥ 1 remain 1 4 -BPS, while those in H + 1 cease to satisfy the bound once the initial theory is deformed away from the orbifold. Deforming away from the orbifold into a different direction lifts the states in a different space H + remains at the BPS bound. This fits very well in the overall picture of the elliptic genus providing information that remains unchanged regardless of the regime (perturbative or not) one is interested in, and regardless of the point in M K3 one considers. It also fits nicely with the symmetry surfing predictions: depending on the direction of deformation, different symmetry groups remain unbroken. Hence different subspaces of H = ∞ n=1 H n remain stable under different deformations, but each of them is isomorphic, as a representation of the Virasoro-and u(1)-current zero modes L 0 , L 0 , J 0 , J 0 , to the massive ground state contributions to the generic space of states H 0 introduced in [12]. This in particular shows that the inclusion H ⊥ n ⊕H rest n → H of representations is not uniquely determined in general.

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While the structure of H ⊥ has been understood by the results of [16,18], we now proceed to uncover more structure on the spaces H rest n ⊕ H + n , with the hope of finding how Z 2 -orbifold CFTs earmark excess BPS states. The results of [12] show that independently of the deformation, if deforming to a generic theory, then the bosonic excess states in H n are precisely those states in H n that transform in the vacuum representation of the antiholomorphic N = 4 superconformal algebra. Under deformation, the N = 4 representations built on these states combine with representations with fermionic 1 4 -BPS ground states to form long representations away from the BPS bound. By the above, these excess fermionic ground states depend on the choice of deformation. To describe them for the deformation by T diag , we construct a "geometric" action of the group SU(2), henceforth denoted SU (2) geom , under which the four free Dirac fermions of our theories and their superpartners transform as doublets and whose action commutes with that of the N = 4 superconformal algebra. All 1 4 -BPS states in the graded space under SU(2) geom . We argue that for deformations in the diagonal direction T diag , to be lifted to a long representation, both the fermionic and bosonic excess 1 4 -BPS states in H + must transform in isomorphic representations of SU(2) geom .
Before proceeding to the heart of our matter, we provide the context in which M 24 emerged in K3 theories. This has the double aim of providing a quick overview for the reader who is not familiar with the subject, and of introducing some of the definitions and notations that will be used later in the paper.
Let τ, z ∈ C, with τ in the upper complex halfplane. 2 The genus one partition function Z N = (4,4) of an N = (4, 4) superconformal field theory at central charge c = c = 6 is a modular covariant function where the traces are taken over the subspaces H NS and H R of H, the (Z 2 × Z 2 )-graded complex vector space of all states in the superconformal field theory. The gradings split H into the Neveu-Schwarz and Ramond sectors, 3 each containing (worldsheet) bosons and fermions, that is 2 By τ , z ∈ C we denote the complex conjugates of τ, z ∈ C, which we occasionally include as arguments of a non-holomorphic function of τ, z. This is convenient whenever such a function allows a power series expansion in q := exp(2πiτ ), q := exp(−2πiτ ), y := exp(2πiz), y := exp(−2πiz), where q, q, y, y may also be viewed as independent formal variables. 3 Referring to both left and right movers, as in a K3 theory, there exist no NSR or RNS states, by definition. Note that our description of K3 theories involves the internal CFT only.

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Here, as representations of the N = (4, 4) superconformal algebras, the spaces H NS and H R are isomorphically mapped to each other under spectral flow. The operators J 0 and J 0 are the zero modes of the u(1) currents which, together with the currents J ± (z) and J ± (z), form two copies of the affine su(2) subalgebra of the (small) N = (4, 4) superconformal algebra. The partition function (1.1) may be expressed in terms of sesquilinear combinations of N = 4 unitary irreducible characters at central charges c = 6 and c = 6. These characters are generating functions for short and long representations and were coined 'massless' and 'massive' respectively in [21] to signify that the corresponding representations have non-zero or zero Witten index respectively. We recall the expressions for these characters in appendix A.3, where we label the two massless characters in the Ramond sector as χ R 0 (τ, z) and χ R with a, b running over massless and massive N = 4 characters and with the term containing the vacuum of the theory having n 00 = 1. The conformal field theoretic elliptic genus E(τ, z) of a K3 theory is defined as the specialisation of Z R where the antiholomorphic R characters are projected to their Witten index value by setting z = 0. We thus have As a consequence of the theory enjoying N = (2, 2) worldsheet supersymmetry, E is a holomorphic function of τ and z, and it counts (with opposite signs) the RR fermionic and bosonic states whose antiholomorphic signature is the Witten index of the massless representation they belong to. The first explicit calculation of this topological invariant was carried out within the framework of Gepner models and Z 2 -orbifold CFTs in [1, (3.8)], where a spectral-flowed version of the conformal field theoretic elliptic genus (1.5) was used, namely Φ(τ, z) := q 1/4 y E(τ, z + τ +1 2 ). This was in order to make a direct parallel with the work of Witten [22]. The elliptic genus presented in [1, (5.10)-(5.12)] is the z = 0 specialisation of With the help of (A.16a)-(A.16c) and (A.17), Φ(τ, z) may be expressed as an infinite sum of irreducible N = 4 Neveu-Schwarz characters in the following way, where A(τ ) := 2 − 8q A n q n = 2 · (45q + 231q 2 + +770q 3 + 2277q 4 + · · · ), (1.8) , the weakly holomorphic mock modular form on SL(2, Z) presented in [23, (7.16)].
A list of the first 8 coefficients A n , n ∈ {1, . . . , 8}, counting 1 4 -BPS ground states, was given in [24], but the significance of these coefficients has only been realised since 2010, 4 after the observation in [4] that they coincide with dimensions of representations of the sporadic group M 24 . The existence of an infinite-dimensional M 24 module underlying A(τ ) was proven in [9]. Yet the role of M 24 in the context of strings compactified on K3 surfaces remains a mystery, and this phenomenon has been named Mathieu Moonshine.
We have structured the remainder of this work as follows.
In section 2, we recall the ingredients from Z 2 -orbifold CFTs relevant to our analysis. Moreover, we present an explicit construction of excess BPS states pertaining to H + n , n ∈ {1, 2}, including their decomposition into SU(2) geom multiplets. The information at level n = 1 was already provided in our work [16], but the full significance of SU(2) geom was not recognised then. The information at level n = 2 is new and requires a careful and detailed analysis of the data encoded in the partition function of Z 2 -orbifold CFTs on K3. The states in H rest 2 are listed in appendix B. Section 3 takes stock of the group theoretic information gleaned in the previous section, gives our rationale behind our construction of the SU(2) geom × SU(2) geom action, and provides analytic expressions for untwisted and twisted partition functions that encode the SU(2) geom action.
A discussion and outlook is given in section 4. Appendix A gathers helpful Jacobi theta function identities, as well as expressions for the N = 4 characters at central charge c = 6 involving Appell functions, whose definitions are also presented. This appendix also offers explanations for the analytic expressions appearing in section 3.

Z -orbifold CFTs
Since the elliptic genus E is an invariant on the moduli space M K3 of K3 theories, it encodes properties that all K3 theories share. In particular, apart from states in massless representations with respect to the holomorphic and the antiholomorphic N = 4 superconformal algebra, which will not be our concern here, E counts a net number of 1 4 -BPS states at each integer conformal weight strictly above threshold. By the results of [9], the corresponding contributions to E agree with the graded character of a space

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where H N =4 n is an irreducible massive N = 4 representation at conformal weight n for each n ∈ N \ {0}, and H n is a finite dimensional representation of M 24 and J 0 . H BPS is the subspace of massive states in the generic space of states H 0 introduced in [12]. Each H n is an invariant of K3 theories, although the dimensions of the H n in the space of all massive 1 4 -BPS states may vary from one K3 theory to another. In generic K3 theories, we have H n ∼ = H n for all n [11,12], while in non-generic theories dim H n ≥ dim H n for all n. When one deforms away from a non-generic theory, excess states, whose contributions to the elliptic genus cancel, are lifted into non-BPS representations off threshold. Although the results of [11,12] imply that H BPS has a geometric description in terms of the chiral de Rham cohomology of K3, this space remains difficult to access. For any deformation of a non-generic theory, it is therefore valuable to gain insight on the subspace of excess states in H BPS whose contributions to E cancel, and to identify the driver of such cancellations in non-generic yet accessible K3 theories.

Free fermions and bosons as building blocks
Our prototype of non-generic K3 theories is the class of Z 2 -orbifold superconformal field theories, which we denote by C = T /Z 2 with T a toroidal SCFT at central charges c = c = 6. (2. 3) The construction of C is induced by the standard Kummer construction which minimally resolves the singularities of the Z 2 -quotient of a complex 2-torus T Λ := C 2 /Λ, with Λ ⊂ C 2 a rank 4 lattice over Z. Unlike generic K3 theories, these provide a framework to test the symmetry surfing idea explicitly and garner further clues for the construction of the putative VOA(s) associated with the M 24 Moonshine module. To this effect, we restrict our attention to the symmetry groups G i induced by geometric symmetries -including those stemming from shifts by half lattice vectors -of the underlying toroidal conformal field theories T . As was detailed in [14][15][16], this is meaningful after the choice of a geometric interpretation for the theory T on some torus T Λ . The symmetry surfing programme also requires a choice of generators for the lattice Λ ⊂ C 2 ∼ = R 4 . All in all, these choices induce an identification 1 2 Λ/Λ ∼ = F 4 2 , such that every geometric symmetry group G i acts on the twisted ground states T a , a ∈ F 4 2 , as permutation group by means of affine linear maps on the space of labels F 4 2 . The underlying toroidal CFT T possesses two holomorphic free Dirac fermions χ a + (z) and their complex conjugates χ a − (z) with standard OPEs, Their bosonic superpartners j a ± (z) are built out of a set of four real holomorphic U(1)currents j I (z), I ∈ {1, 2, 3, 4}, whose zero modes generate infinitesimal translations on the torus T Λ . Here, j I (z) is the Noether current for the translation along the I th coordinate JHEP04(2020)184 axis in the standard coordinate system that T Λ inherits from C 2 ∼ = R 4 around each point. One has Under the action of the Z 2 -orbifold group, the fields χ a ± (z) and j a ± (z) flip sign, while the N = 4 SCA is invariant under this orbifold action, as follows from the following free field representation: The currents J ± and J 3 generate the su(2) affine subalgebra of the N = 4 superconformal algebra, under which the Dirac fermions χ a ± have charges ± 1 2 , while their bosonic superpartners are uncharged, as is immediate from the form of the Cartan subalgebra current J 3 in (2.7). In contrast, the symmetry groups G i also act linearly as subgroups of SU(2) on χ a ± and j a ± . More precisely, χ a + and j a + transform as doublets 2 under this SU(2), which will be referred to as 'geometric' SU(2) geom while χ a − and j a − transform as complex conjugate doublets 2. In other words, if χ a + and j a + transform with then χ a − and j a − transform with M . The action of SU(2) geom commutes with the N = 4 action, as can be inferred from the SU(2) geom invariance of the fields in (2.7).
In the antiholomorphic sector of the theory, the two Dirac fermions χ a + (z) and their superpartners j a + (z) transform as doublets under a right-moving group SU(2) geom whose action commutes with that of the antiholomorphic N = 4 superconformal algebra, while they are singlets under SU(2) geom . Their complex conjugates χ a − (z) and j a − (z) also transform as doublets under SU(2) geom and as singlets under SU(2) geom .

The Neveu-Schwarz partition function
With an eye to prepare the ground for future work on the VOA(s) expected to underlie the Mathieu Moonshine module, we choose to work in the Neveu-Schwarz sector. The Z 2 -orbifold partition function in this sector is given by contributions from the two complex NS fermions and their bosonic superpartners (2.6), both untwisted and twisted by the Z 2 action as in [1],

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The dependence on the moduli of the underlying toroidal theory becomes apparent in Z NS untwisted , which depends on Γ(Λ, B) ⊂ R 4 ⊕ R 4 , the signature (4, 4) Narain lattice associated with the lattice Λ and the B-field of the underlying toroidal theory. Indeed, we have 5 Different tori T Λ lead to different Narain lattices, but the 1 4 -BPS states emerging from non-zero momentum or winding are non-generic in the class of Z 2 -orbifold CFTs (see, for example, [12, (3.7)-(3.8)] for the precise argument). The remaining spectrum of states is generic to Z 2 -orbifold CFTs on K3, however, and is the object of our present analysis.
Restricting the graded trace that usually yields the Neveu-Schwarz partition function to the states with vanishing winding and momentum and the twisted sector thus yields The BPS states we are interested in are Neveu-Schwarz states which under the antiholomorphic N = 4 superconformal algebra transform like elements of the chiral ring. In other words, we need to project to ker 2L 0 − J 0 . These states are thus encoded in the conformal field theoretic elliptic genus of K3 spectral-flowed from the R sector to the NS sector, that is, in It may also be obtained from the generic Neveu-Schwarz partition function by inserting Note that our conventions ensure that the vacuum contributes to E NS (τ, z) with a positive sign, which we find natural, since it is bosonic. 6 We now rewrite E NS (τ, z) in terms of N = 4 characters and Appell functions. To do so, we use the following notations introduced in [18, (B.2)-(B.4)]: by U = 1 2 (z) we denote the 5 The dependence on τ will often be understood but not explicitly referred to for easy reading of formulas. 6 There is a global sign difference between equation (2.14) and [18, (2.5)], or equivalently, (1.6); we view this as a different choice of conventions, due to the fact that in [1,18], the elliptic genus is expressed in the NS R sector, while we work in the NSNS sector.

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generating function for BPS states that transform as the vacuum under the antiholomorphic N = 4 superconformal algebra. To be invariant under the Z 2 -orbifold action, there must be an even number of modes acting on the vacuum: Similarly, U =0 (z) is the generating function for untwisted BPS states that transform as massless matter ground states under the antiholomorphic N = 4 superconformal algebra. These states are created from the vacuum by the action of a single mode of weight 1 2 of an antiholomorphic Dirac fermion. Hence for such BPS states to be invariant under the Z 2 action, there must be an odd number of holomorphic modes acting on the ground state: In analogy with our notation (1.8), we introduce f n q n and g inv (τ ) := ∞ n=1 g inv n q n , and using (A.16c) and (A.17) we obtain the following analytic expressions for these functions, (2.17) Here, the function h 2 is a specialisation of a level one Appell function (see appendix A.2). By (2.14), the contributions to E NS (τ, z) from the untwisted sector thus may be written as Proceeding in a similar fashion in the twisted sector, we use the function T =0 (z) introduced in [ We introduce the function whose Fourier modes, alongside those of f (τ ) and g inv (τ ), provide crucial data for our analysis. By (2.14) the contributions to E NS (τ, z) from the twisted sector thus read where the factor 16 accounts for the number of linearly independent ground states in the twisted sector of the theory. Indeed, the twisted ground states are localised at the 16 singular points of the quotient T Λ /Z 2 . Altogether, (2.18) and (2.21) yield a decomposition of the conformal field theoretic elliptic genus according to and thus, by comparison with (1.7), With the decomposition (2.22) of the NS-elliptic genus E NS in hand, we will from now on continue to work in the Neveu-Schwarz sector, focussing on massive primary states that contribute to E NS . Note that in this sector, the level n accounted for by A n agrees with the conformal weight.
2.3 Decomposition of the space H BPS of massive 1 4 -BPS states As already mentioned in the introduction to this section, in non-generic K3 theories, the space of massive 1 4 -BPS states H BPS is larger at every level than the space H BPS of generic massive 1 4 -BPS states, as the conformal field theoretic elliptic genus counts BPS states with signs. Ultimately, we wish to know to what extent one can identify the very states in the class of Z 2 -orbifold CFTs, whose contributions cancel in the net count of E under selected JHEP04(2020)184 deformations. The better we understand them, in particular the type of group action they may enjoy, the more we can hope to uncover the VOA structure(s) on the generic space of states that does contribute to the net count. Our guiding principle in this quest is symmetry surfing [14,15], a programme we have developed over a period of years and that has passed a number of non-trivial tests, either through explicit calculations within Z 2orbifold CFTs [16] or through a process of deformations away from the Z 2 -orbifold point in two very interesting papers [18,19].
In [16], symmetry surfing identifies a special one-dimensional subspace of the 16dimensional space of twisted ground states in Z 2 -orbifold conformal field theories on K3. Indeed in such theories, corresponding to the 16 fixed points of the standard Z 2 action on T Λ , there are 16 pairwise orthogonal twisted ground states |α β , labelled by β ∈ F 4 2 . By construction, the "diagonal" state is invariant under all symmetries induced by geometric symmetries of the torus T Λ , including shifts by elements of 1 2 Λ. It is thus invariant under the full overarching affine group which contains all the groups G i of finite symplectic automorphisms on Kummer surfaces. Conveniently, in [16] we found that at massive level one, the orthogonal complement of the Fock space built on |α diag echoes the construction of a 45-dimensional representation of the group M 24 by Margolin [17]. Inspired by his construction, we thus define H ⊥ ⊂ H BPS as the Fock space built on the 15-dimensional orthogonal complement of |α diag in the space of twisted ground states. This prompts the following ansatz, 7 introduced similarly in [18], as representations of J 0 and the octad group G, for all n ∈ N, n > 0, (2.26) and where H n was defined in (2.1). Ultimately, (2.26) should extend to an isomorphism of representations of J 0 and M 24 . As was pointed out in [18], the ansatz (2.25) can be interpreted as identifying the massive contributions to the generic space of states as the subspace H ⊥ ⊕ H rest of H BPS which remains at the BPS bound under deformations of our CFT by the exactly marginal deformation T diag built on |α diag . Table 1 summarises data for the first four conformal weights above threshold in terms of the Fourier coefficients of the generating functions f (τ ), g inv (τ ) and g tw (τ ) for massive  dim H rest n = g tw n + g inv n − 2f n for all n ∈ N, n > 0. In other words, the excess states in H + n , which are lifted from the BPS bound under a deformation by T diag , belong to the twisted sector generated by |α diag and the untwisted sector. While the results of [11,12] imply that the holomorphic untwisted states accounted for by 2f n all belong to H + n , it is not possible at this stage to identify whether the remaining states in H + n come from the diagonal twisted sector, the untwisted sector, or both. We will return to this point in section 3.

1 4 -BPS states at level one and two
To investigate the elusive properties of H + n in general, we begin by studying the spaces H n = H ⊥ n ⊕ H rest n ⊕ H + n at levels n = 1 and n = 2 more closely. We also introduce a consistent action of SU(2) geom × SU(2) geom on H rest n ⊕ H + n at these levels. The rationale behind our construction will be explained in section 3 -based on the data collected at levels n = 1 and n = 2. Here and in the following, we denote the modes of the four free fermions and of their superpartners as (χ k ± ) and (a k ± ) m with and m either integers or half-integers in accordance with the boundary conditions imposed by the Z 2orbifold construction. Following a wide-spread tradition, and m account for the negative contributions to the energy. Level 1. At conformal weight n = 1, since A 1 = 90 = 15g tw 1 and g inv 1 = 0, our ansatz is compatible with the claim that the contributions to E NS from the six-dimensional (2f 1 = 6) space of untwisted massive 1 4 -BPS states cancels those from the six-dimensional (g tw 1 = 6) space of massive 1 4 -BPS states in the diagonal twisted sector. This was already discussed in [16], where the geometric action of the group SU(2) geom was mentioned. We reproduce our results here in the Neveu-Schwarz sector, not the least because some interesting lessons can be drawn from this case.
To work in the Neveu-Schwarz sector of the Z 2 -orbifold CFTs, we spectral flow from the Ramond sector and choose chiral-chiral ground states, i.e. states in the kernel of (2L 0 − J 0 ) and (2L 0 − J 0 ).

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In the subspace of the untwisted sector accounted for by the partition function U = 1 2 (z) (see (2.15)), this amounts to building states from the bosonic highest weight states Ω with conformal weights (h, h) = (0, 0) and q Ω = (χ 1 (2.7)). The six-dimensional space of states contributing to 2f 1 = 6 is generated by the following N = 4 primaries: The vacuum Ω is by definition invariant under SU(2) geom × SU(2) geom and so is q Ω, since χ a + with a ∈ {1, 2} is a doublet under SU(2) geom and a singlet under SU(2) geom . Hence each row in (2.27) generates a (3, 1) representation of the group SU(2) geom × SU(2) geom . In the diagonal twisted sector, the six dimensional space of states accounted for by g tw 1 = 6 is built on the SU(2) geom × SU(2) geom invariant twisted ground state |α diag and also transforms as the sum of two (3, 1) representations of the SU(2) geom × SU(2) geom action, generated by (2.28) So at level 1, the "excess" 1 4 -BPS states in H + belong to a pair of isomorphic representations of SU(2) geom × SU(2) geom of opposite fermion number. A detailed analysis of the fate of this 12-dimensional space of states when the K3 theories are deformed away from the Z 2orbifolds has recently been carried out in [19]. The conclusion is that under the diagonal deformation T diag , these states combine into non-BPS representations and thus cease to be accounted for by E NS ; in other words, deformations of a non-generic theory 'lift' a number of massive 1 4 -BPS states, including all those contributing to U = 1 2 (z). That this must happen under deformations to generic theories follows already from the analysis of [12].
Level 2. At conformal weight n = 2, in H + 2 there is only a two-dimensional space of untwisted holomorphic excess states counted with one sign (2f 2 = 2). A priori we must find a two-dimensional subspace of the space H rest 2 ⊕H + 2 of dimension g inv 2 +g tw 2 = 44, contributing to E NS with the opposite sign, in order to identify the subspace of H + 2 matching the untwisted holomorphic excess states. The latter are singlets of SU(2) geom ×SU(2) geom given by (2.29)

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Remarkably, there are exactly two massive 1 4 -BPS singlets with respect to the SU(2) geom × SU(2) geom action in the diagonal twisted sector to match |s (2) and |q s (2) . These are also the only singlets under the action of the diagonal SU(2) in SU(2) geom × SU(2) geom . There are none in the untwisted sector accounted for by U =0 , a fact which serves as a justification in section 3 when we generalise our arguments. With the notation 30) the two SU(2) geom × SU(2) geom singlets in the diagonal twisted sector are given by The remaining space of massive level two 1 4 -BPS states in the diagonal twisted sector is 26dimensional (g tw 2 = 28) and is presented in appendix B.1, while the 16-dimensional space of massive 1 4 -BPS states in the untwisted sector accounted for by U =0 (z) is presented in appendix B.2. This detailed analysis supports the conjecture that the matching of excess 1 4 -BPS states respects the SU(2) geom and the SU(2) geom actions on these states, both in the untwisted and twisted sectors. We thus expect that the four states |s (2) , |q s (2) , |s (2) , | q s (2) are lifted from the BPS bound under a deformation by T diag . It would be interesting to confirm this prediction by conformal perturbation methods along the lines of [19]. In the next section we present general results corroborating our conjecture at low levels.

Geometric SU(2) as a guiding principle
Let us now give a general definition of the SU(2) geom × SU(2) geom action on the space H rest ⊕H + , along the lines indicated in section 2.1. We have defined this space as a space of massive ground states common to all Z 2 -orbifold conformal field theories on K3, containing both twisted and untwisted contributions, mindful however that its decomposition into H rest ⊕ H + has not been carried out so far. All states in H rest ⊕ H + are elements of the Fock space representations obtained from the vacuum Ω and the diagonal twisted ground state |α diag by the action of the modes of the free fermionic fields χ a ± , a ∈ {1, 2}, and their superpartners j a ± , along with their antiholomorphic analogues. It thus suffices to state the action of SU(2) geom × SU(2) geom on these fields, on the vacuum Ω and on |α diag , yielding an action on the entire Fock space built on these states. For each Z 2 -orbifold conformal field theory on K3 we do this by using the left-and the right-moving action of the group SU(2) which contains the linear part of the geometric symmetry group of our theory. In other words, we use the standard action of the group SU(2) on C 2 , which is a subgroup of JHEP04(2020)184 Table 2. Charges of free bosons and fermions under SU(2) geom and under the affine su(2) subalgebra of the N = 4 superconformal algebra.
Both Ω and |α diag are invariant under every geometric symmetry group for any Z 2 -orbifold conformal field theory on K3. We therefore choose both these states to be invariant under SU(2) geom × SU(2) geom , while the fields χ a ± and j a ± , a ∈ {1, 2}, transform as doublets under the action of SU(2) geom and trivially under SU(2) geom . Note that these fields also carry a U(1) charge associated with the affine subalgebra su(2) of the N = 4 superconformal algebra. Our conventions for the two sets of charges, which are summarised in table 2, inform on how to refine the holomorphic partition functions for the untwisted and twisted sectors so they encode the action of the group SU(2) geom on 1 4 -BPS states. The charges of χ a ± and j a ± under SU(2) geom and su(2) are analogous. From our derivation of the SU(2) geom × SU(2) geom action on H rest ⊕ H + , we do not expect any meaningful extension to H ⊥ . Indeed, although a well-defined action of SU(2) exists on H ⊥ , which is trivial on every twisted ground state, this action does not extend the action of the linear parts of our geometric symmetry groups to SU(2), since these groups act non-trivially on H ⊥ .
We introduce the complex variable w := e 2πiν , ν ∈ C, to track the SU(2) geom charges of 1 4 -BPS states by refining the partition functions U = 1 2 , U =0 , T =0 introduced in (2.15), (2.16) and (2.19) to Given the action of SU(2) geom introduced above, the only states accounted for by the above partition functions which carry a non-trivial action of this group are those built on the Z 2 -orbifold odd ground states Ω 1 := (χ 1 + ) − 1 2 Ω and Ω 2 := (χ 2 + ) − 1 2 Ω. U =0 is the graded character for the space of Z 2 -orbifold invariant states in the Fock space built on these two states, which transform as a doublet under SU(2) geom . To encode this action as well, we JHEP04(2020)184 therefore multiply U =0 (z, ν) by 1 2 w + w −1 . We treat w as a formal variable separately from w, to keep the actions of SU(2) geom and SU(2) geom apart, in the spirit of separating the action of the left-and the right-moving N = 4 superconformal algebras.
The action of the diagonal SU(2) in SU(2) geom × SU(2) geom is then captured by identifying w as the complex conjugate of w. Indeed, by what was said above, for any given K3 theory the action of SU(2) geom and SU(2) geom induces the action of the linear part of the geometric symmetry group G i mentioned in section 2.3 on holomorphic and antiholomorphic fields, respectively. If G i acts on the holomorphic fields by the representation , then it acts by the complex conjugate representation on the antiholomorphic partner fields. This is used, for example, in the construction of the corresponding partition functions, where the contributions from the antiholomorphic fields are simply obtained as the complex conjugates of the contributions coming from their holomorphic partners (see, for example, [1, (5.2)]). We remark that the partition functions [18, (C.5), (C.8), (C.10)] at z = 0 and at three specific values of w, with complex conjugates w, which were interpreted as eigenvalues of the elements g of the linear parts of the geometric symmetry groups G i . In that situation, w = w −1 , and the expressions in [18, (C.5), (C.8), (C.10)] are invariant under w ↔ w. Indeed, by construction, at these special values our partition functions are the characters of g. In [18], the latter play a crucial role in providing evidence for the symmetry surfing programme.
Let us first restrict our attention to the action of SU(2) geom . Since the SU(2) geom action commutes with that of N = 4 and using (2.15), (2.16), (2.19), the refined partition functions enjoy a decomposition in N = 4 superconformal characters of the form As before, and by abuse of notation, f (ν) is our shorthand notation for f (τ, ν), and f (τ ) = f (τ, ν = 0), etc. By construction, SU(2) geom maps the spaces of massive ground states that contribute to U = 1 2 , U =0 , T =0 , respectively, to themselves, such that these three spaces decompose into direct sums of irreducible representations of SU(2) geom . Therefore, with the SU(2) character of the representation with isospin p ∈ 1 2 N given by

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and f n,p , g inv n,p and g tw n,p ∈ N the multiplicity of the (2p + 1)-dimensional representation at level n in the two untwisted and the twisted diagonal sectors, respectively.
In fact, as explained in appendix A.5, using identities amongst Jacobi theta functions and Appell functions, one may rewrite (3.3a)-(3.3c) as The three functions f (τ, ν), g inv (τ, ν) and g tw (τ, ν) are the graded characters of certain subspaces of the space of massive 1 4 -BPS ground states, whose decomposition into H rest ⊕ H + is at the heart of our investigation. Recall that we have defined the space H + as to contain pairs of ground states at opposite fermion numbers whose contributions to the elliptic genus E NS cancel. Upon deformation of our K3 theory by T diag , each such pair is lifted to a common long N = 4 representation off the BPS bound. On the other hand, for every Z 2 -orbifold CFT on K3, SU(2) geom × SU(2) geom acts on H rest ⊕ H + as to restrict to the action of the linear part of the geometric symmetry group of the theory, which remains unbroken under deformations by T diag . This, together with the symmetry surfing proposal, prompts us to postulate that the states in H + are paired up according to their transformation properties under SU(2) geom × SU(2) geom . In other words, we postulate that H + decomposes into pairs of isomorphic representations under SU(2) geom and SU(2) geom with opposite fermion numbers.
To determine which subspaces of the spaces accounted for by f (ν), g inv (ν) and g tw (ν) may contribute to H + , we recall from the above that by the results of [12], all untwisted holomorphic states accounted for by f (ν) are non-generic and thus belong to H + . All of them are bosonic, while g inv (ν) and g tw (ν) account for fermionic states only. In other words, we must find SU(2) geom × SU(2) geom representations matching those accounted for by f (ν) within the spaces whose graded characters are g inv (ν) and g tw (ν). Observe that by (A.11) and (3.5a)-(3.5c), f (ν) and g tw (ν) are invariant under a shift of the variable ν by 1 2 , while g inv (ν + 1 2 ) = −g inv (ν). This shows that g inv (ν) only accounts for representations of SU(2) geom with half-integer spin, while f (ν) and g tw (ν) only account for representations with integer spin. Moreover, all states accounted for by f (ν) and g tw (ν) transform trivially under SU(2) geom , in contrast to those accounted for by g inv (ν). This, together with the evidence provided by the explicit calculations at levels n = 1, 2, prompts us to postulate that, at any level n ∈ N, n ≥ 1, only 1 4 -BPS states belonging to the diagonal twisted sector can pair up with those occurring in the untwisted sector and accounted for by f (ν). In light of the decompositions (3.3a)-(3.3c) into characters of SU(2) geom , this implies the claim that ∀p, n ∈ N : g tw n,p − 2f n,p ≥ 0. (3.6) We have expanded the functions f (ν) and g tw (ν) as q-power series and verified (3.6) up to O(q 101 ), supporting our postulates. We present the data up to O(q 16 ) in table 3 for reference. We hope to provide an analytic proof of (3.6) in the near future.

JHEP04(2020)184 4 Discussion
The VOA(s) underlying Mathieu Moonshine remain elusive to the extent that even a consensus on whether or not to expect a link between Mathieu Moonshine and K3 theories has not been reached. The works of [11,12] indicate that such a link could involve a generic space of states of K3 theories, in accordance with the ideas behind our symmetry surfing programme [13][14][15][16]. The present work is a contribution to the study of generic properties of K3 theories which we find interesting in their own right. This can be viewed as a preparation for a new attempt at the construction of a Mathieu Moonshine VOA on the generic space of states.
In [12], crucially building on the results of [11], it was shown that indeed there exists a generic space of states H 0 for all K3 theories, roughly defined through the property that it embeds into the space of BPS states of every K3 theory as a representation of the holomorphic N = 4 superconformal algebra, extended by the zero modes L 0 , J 0 of the Virasoro field and u(1) current in the antiholomorphic N = 4 superconformal algebra. Although H 0 can be modelled by the chiral de Rham cohomology of a K3 surface [11,12], its detailed structure has not been studied so far. Approaching the space from the perspective of nongeneric yet accessible K3 theories, we have, in the present work, studied more closely the structure of the 1 4 -BPS states of Z 2 -orbifold CFTs of toroidal theories. We propose a strategy to earmark the 1 4 -BPS states that move off the BPS bound under the most symmetric deformation T diag of Z 2 -orbifold CFTs on K3, away from the Z 2 -orbifold limit. Such states must come in pairs of opposite fermion numbers, such that their contributions cancel each other in the conformal field theoretic elliptic genus of K3, in order to be part of the same (long) non-BPS representation after deformation. In [16], where the first concrete study of BPS states contributing to the count at level n = 1 in the conformal field theoretic elliptic genus of K3 was undertaken, we identified a 15-dimensional space of twisted ground states carrying a Fock space representation H ⊥ which turns out to be generic in the above sense, that is, along deformations by T diag . In the twisted sector, H ⊥ is the orthogonal complement of a 'diagonal' subspace from which T diag arises. The findings were guided by the symmetry surfing programme [14,15] and were inspired by Margolin's construction of a 45dimensional representation of M 24 [17]. They highlighted the action of the octad group, a maximal subgroup of M 24 , on the 45-dimensional subspace of H ⊥ at level 1. Evidence of the octad group action at all levels was provided in the work [18], fueling the symmetry surfing programme whose aim is to exhibit an M 24 action on the generic space of states. More recently, Keller and Zadeh [19] deformed the Z 2 -orbifold CFTs away from the Kummer point by a marginal operator and showed that if the deformation is taken in the diagonal direction T diag , then indeed all the BPS states in the twisted sector of the original non-generic K3 theory that are orthogonal to H ⊥ move off the BPS bound under the deformation.
It remains that beyond level n = 1, we do not have total control on which BPS states move off the BPS bound under a given deformation. By the results of [12], we know that under deformations to generic K3 theories this happens for every untwisted massive state accounted for by the partition function U = 1 2 . Since each state that moves off the BPS bound pairs up with a state of opposite fermion number, to become part of the same long JHEP04(2020)184 non-BPS representation, one needs to identify the correct partners in either the untwisted subsector U =0 or the twisted sector. Following the results of [18], we know that under the deformation by T diag , of the twisted states only those in the diagonal twisted sector can move off the BPS bound. We postulate that none of the untwisted states accounted for in U =0 do. At level n = 1 this trivially holds as the level one contribution to U =0 is zero. In fact, all states in the twisted diagonal sector move off the BPS bound under deformation by T diag . This is a very special situation that does not persist at higher levels.
In order to identify potential states to pair up with the states accounted for in U = 1 2 and move off the BPS bound under deformation by T diag , we postulate compatibility with a geometric action of the group SU(2), denoted SU(2) geom in this work. Indeed, as already pointed out in [16], the holomorphic Dirac fermion fields χ a ± , a ∈ {1, 2}, and their superpartners, which are the building blocks of Z 2 -orbifold CFTs, transform as doublets under SU(2) geom . The group acts trivially on the vacuum and on the twisted diagonal ground state, and also on the antiholomorphic partners of the χ a ± , a ∈ {1, 2}, and their superpartners. We have introduced refined partition functions that keep track of that group action on 1 4 -BPS states in the untwisted and twisted diagonal sectors, and we were able to show that at level n ≤ 2 only states stemming from the twisted diagonal sector carry representations of SU(2) geom that match those in the untwisted subsector accounted for by U = 1 2 . The SU(2) geom action on the 1 4 -BPS states thus helps to identify sets of states in the diagonal twisted sector that may lift off the BPS bound under the deformation by T diag . We note that a certain degree of indetermination remains, as the multiplicities of isospin p representations at any fixed level n in the diagonal twisted sector quickly exceed by far twice the multiplicities of isomorphic representations at the same level in the untwisted sector, as evidenced in table 3. Therefore, except for levels n = 1, 2, our postulate is not powerful enough to pin down the exact states that are generic along the deformation by T diag .
As explained above, SU(2) geom in particular leaves the Z 2 -orbifold odd ground states in the untwisted subsector accounted for by U =0 invariant. From the perspective of symmetry surfing, it is perhaps more natural to consider the action of a diagonal SU(2) in SU(2) geom × SU(2) geom , where SU(2) geom is the antiholomorphic analog of SU(2) geom . In particular, under this diagonal action of SU(2), the ground states in the untwisted sector accounted for by U =0 transform non-trivially. We argue that representations from this sector do not pair up with states accounted for by U = 1 2 to form long representations off the BPS bound under any deformation. Indeed, if this were the case, then there should be a deformation in the underlying toroidal theory that would lift these states off the BPS bound. However, such states are generic to all toroidal theories and hence such deformations do not exist.
Altogether we expect the role of SU(2) geom to be helpful in understanding the generic space of states of K3 theories, independently of Mathieu Moonshine. An analysis of deformations beyond level n = 1 along the lines of those followed in [19] would certainly shed more light on the relevance of SU(2) geom . The analysis of [19] already shows that any sufficiently small deformation away from the Z 2 -orbifold conformal field theories on K3 reduces the space of 1 4 -BPS states at massive level one to a generic space. Given the results of [12], the same must hold at arbitrary level. It would be interesting to understand the JHEP04(2020)184 structure of the underlying VOAs and their dependence on the details of the deformation. Indeed, is it possible that the dependence on the choice of deformation drops out entirely? Ultimately, this could answer some of the open questions of Mathieu Moonshine.

Acknowledgments
We gratefully acknowledge support from the Simons Center for Geometry and Physics, Stony Brook University, at which a major part of the research for this paper was performed. AT thanks the Mathematics Institute at the University of Freiburg for their warm welcome and the support received from the DFG Research training group GRK 1821 "Cohomological methods in geometry". We also thank Christoph Keller and Ida Zadeh for helpful correspondences. An anonymous referee deserves our thanks for diligent reading and helpful suggestions.

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The Dedekind η function is defined as and is related to the Jacobi theta functions by the identity For future reference, we note (A.6)

A.3 N = 4 characters at c=6
The characters for unitary, irreducible representations of the N = 4 superconformal algebra were first derived in [21]. When the central charge is c = 6, the representations fall into an infinite class of 'long' or 'massive' representations with Neveu-Schwarz characters of the form with h ∈ R, h > 0, the conformal weight of the highest weight state alongside two 'short' or 'massless' representations labelled by the su(2) 'spin' ∈ {0, 1 2 } and the conformal weight h = of their highest weight states. The corresponding Neveu-Schwarz characters are expressible in a variety of ways. Here, we use their expressions in terms of the Appell functions h i (z) defined in (A.8) and (A.9). We have This form of the characters was first presented in [27] and expresses the branching of N = 4 characters in an infinite sum of N = 2 characters at central charge c = 6. Inserting (A.12) and (A.13) into (A.15), one gets The second massless N = 4 character in the Neveu-Schwarz sector is given by

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Twisting by the fermion number operator one obtains Under spectral flow, the Neveu-Schwarz and Ramond characters flow to each other as Given (A.15) and (A.17), all N = 4 characters may be expressed in terms of Appell functions. In particular, A. 4 The functions f (τ ), g inv (τ ) and g tw (τ ) The functions (2.17) and (2.20) can all be obtained by standard manipulations, and A.5 The functions f (τ, ν), g inv (τ, ν) and g tw (τ, ν) One now uses some of the relations (A.12) and (A.13) between Appell functions as well as the theta function identities (A.2) and (A.3) to obtain (3.5a)-(3.5c).
We also note that the functions f (ν), g inv (ν) and g tw (ν) may be written in terms of the massless characters of an N = 4 SCA at central charge c = 6. Taking advantage A. 6 Fourier coefficients of f (τ, ν) and g tw (τ, ν) The data in table 3 is presented in support of our claim that under deformation of our Z 2orbifold CFT on K3 by T diag , only 1 4 -BPS ground states stemming from the diagonal twisted sector pair up with those in the untwisted subsector accounted for by U = 1 2 to form long representations off the BPS bound. For reference, we also present data in table 4 relating to the action of SU(2) geom on 1 4 -BPS ground states from the untwisted subsector accounted for by U =0 . We also note that for the action of the diagonal SU(2) in SU(2) geom × SU(2) geom (see section 3 and appendix B.2), the information in table 5 should be used. However, at level 1 (resp. 2), the two triplets (resp. the two singlets) from the untwisted subsector accounted for by U = 1 2 only match the SU(2) representation contributions of the 1 4 -BPS ground states in the diagonal twisted sector: at level n = 1, there are just no states available in the untwisted subsector U =0 , and there are no singlets in that sector at level n = 2.   15/2 13/2 11/2 9/2 7/2 5/2 3/2 1/2 Table 5. SU(2) geom × SU(2) geom representations and multiplicities at level n, 1 ≤ n ≤ 15. An entry of type b m at level n is understood as multiplicity m = g inv n,p of the representation of dimension b = 2p + 1 of the diagonal SU(2) in SU(2) geom × SU(2) geom . with where q k := 3 − k.
The four quintuplets. On the other hand, we find four quadruplet representations of SU(2) geom at level 2 in this sector. We introduce the notation where each row generates a (4, 2) representation of SU(2) geom ×SU(2) geom . Note that under the action of the diagonal subgroup SU(2) of SU(2) geom × SU(2) geom , we obtain two copies of 4 ⊗ 2 ∼ = 3 ⊕ 5, yielding no singlets altogether.
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