Protected couplings and BPS dyons in half-maximal supersymmetric string vacua

We analyze four- and six-derivative couplings in the low energy effective action of $D=3$ string vacua with half-maximal supersymmetry. In analogy with an earlier proposal for the $(\nabla\Phi)^4$ coupling, we propose that the $\nabla^2(\nabla\Phi)^4$ coupling is given exactly by a manifestly U-duality invariant genus-two modular integral. In the limit where a circle in the internal torus decompactifies, the $\nabla^2(\nabla\Phi)^4$ coupling reduces to the $D^2 F^4$ and $R^2 F^2$ couplings in $D=4$, along with an infinite series of corrections of order $e^{-R}$, from four-dimensional 1/4-BPS dyons whose wordline winds around the circle. Each of these contributions is weighted by a Fourier coefficient of a meromorphic Siegel modular form, explaining and extending standard results for the BPS index of 1/4-BPS dyons.

couplings in the low-energy effective action of three-dimensional string vacua with 16 supercharges, for all values of the moduli.
These protected couplings are analogues of the R 4 and ∇ 6 R 4 couplings in toroidal compactifications of type II strings, which have been determined exactly in [19,20] and in many subsequent works. Our motivation for studying these protected couplings in D = 3 is that they are expected to encode the infinite spectrum of BPS black holes in D = 4, in a way consistent with the Uduality group G 3 (Z). The latter contains the four-dimensional Uduality group G 4 (Z), but is potentially far more constraining. Thus, these protected couplings provide analogues of 'black hole partition functions', which do not suffer from the usual difficulties in defining thermodynamical partition functions in theories of quantum gravity, and are manifestly automorphic [21].
The fact that solitons in D = 4 may induce instanton corrections to the quantum effective potential in dimension D = 3 is well known in the context of gauge theories with compact U (1) [22]. In the context of quantum field theories with 8 rigid supersymmetries, BPS dyons in four dimensions similarly correct the moduli space metric after reduction on a circle [23,24]. In string vacua with 16 local supersymmetries, one similarly expects that 1/2-BPS dyons in D = 4 will contribute to fourderivative scalar couplings of the form G ab,cd ( ) ∇(∇ a ∇ b )∇(∇ c ∇ d ). 1 In either case, the contribution of a four-dimensional BPS state with electric and magnetic charges (Q , P ) is expected to be suppressed by e −2π RM(Q ,P ) , where M(Q , P ) is the BPS mass and R the radius of the circle on which the four-dimensional theory is compactified, and weighted by a suitable BPS index (Q , P ) counting the number of BPS states with given charges. In addition, coupling to gravity implies additional O(e −R 2 / 2 P ) corrections from gravitational Taub-NUT instantons, which are essential for invariance under G 3 (Z) (here, P is the Planck length in four dimensions).
For simplicity we shall restrict attention to the simplest threedimensional string vacuum with 16 supercharges, obtained by compactifying the ten-dimensional heterotic string on T 7 . Our construction can however be generalized to other half-maximal supersymmetric models with reduced rank [25] with some effort [26]. The moduli space in three dimensions is the symmetric space In the limit where the heterotic string coupling g 3 becomes small, where the first factor corresponds to g 3 , the second factor to the Narain moduli space (parametrizing the metric, B-field and gauge field on T 7 ), and R 30 to the scalars a I dual to the gauge fields in three dimensions. At each order in g 2 3 , the low-energy effective action is known to be invariant under the T-duality group O (23, 7, Z), namely the automorphism group of the even self-dual Narain lattice 23,7 [27]. The latter leaves g 3 invariant, acts on G 23,7 by left multiplication and on the last factor in (1) by the defining representation. U-duality postulates that this symmetry is extended to G 3 (Z) = O (24, 8, Z), the automorphism group of the 'non-perturbative Narain lattice' 24,8 = 23,7 ⊕ 1,1 , where 1,1 is the standard even-self dual lattice of signature (1, 1) [29].
In the limit where the radius R of one circle of the internal torus becomes large, M 3 instead decomposes as where the first factor now corresponds to R 2 /(g 2 4 2 H ) = R/(g 2 3 H ) = R 2 / 2 P (with H being the heterotic string scale and g 4 the string coupling in D = 4), the second correspond to the moduli space M 4 in 4 dimensions, the third factor to the holonomies a 1I , a 2I of the 28 electric gauge fields and their magnetic duals along the circle, along with the NUT potential ψ , dual to the Kaluza-Klein gauge field. The factor G 2,1 ∼ = S L(2)/U (1) is parametrized by the axio-dilaton S = S 1 + iS 2 = B + i/g 2 4 , while G 22,6 is the Narain moduli space of T 6 , with coordinates φ. In the limit R → ∞, the U-duality group is broken to S L(2, Z) × O (22, 6, Z), where the first factor S L(2, Z) is the famous S-duality in four dimensions [1,2]. Besides 1 Here ∇ a is a short-hand notation for the pull-back to R 1,2 of the S O (p, q) right-invariant one-form P aâ on the Grassmannian, ∇ is the covariant derivative, and we omit the Lorentz indices and the indices â of S O (8) that are contracted symmetrically in the two cases.
The trace of the operator D 2 the Laplacian on G p,q . On-shell linearized superspace methods indicate that F abcd and G ab,cd have to satisfy [26] The first two constraints are analogous to those derived in [30] for H 4 and ∇ 2 H 4 couplings in Type IIB string theory on K3. The numerical coefficients c 1 , ...c 7 will be fixed below from the knowledge of perturbative contributions.

Exact (∇ ) 4 couplings in D = 3
Based on the known one-loop contribution [31][32][33], it was proposed in [34] (a proposal revisited in [35]) that the four-derivative scalar coupling F abcd is given exactly by the genus-one modular integral F (24,8) where F 1 is the standard fundamental domain for the action of R takes even values on 24,8 . It will be important that the Fourier coefficients of 1/ = m≥−1 c(m) q m count the number of 1/2-BPS states in the D = 4 vacuum obtained by decompactifying a circle inside T 7 . This is obvious from the fact that these states are dual to perturbative string states carrying only left-moving excitations [5,8]. It is also worth noting that the ansatz (7) is a special case of a more general class of modular integrals, which we shall denote by F (q+16,q) abcd , where the lattice 24,8 is replaced by an even self-dual lattice q+16,q and the factor ρ 4 2 by ρ q/2 abcd converges for q < 6, and is defined for q ≥ 6 by a suitable regularization prescription. For q ≤ 7, the modular integral F (q+16,q) abcd controls the one-loop contribution to the F 4 coupling in heterotic string compactified on T q [31][32][33]. For any value of q, it can be checked that F (q+16,q) abcd satisfies (4) and (6) By construction, the ansatz (7) is a solution of the supersymmetric Ward identity, which is manifestly invariant under G 3 (Z).
Its expansion at weak coupling (corresponding to the parabolic decomposition (1), such that the non-perturbative Narain lattice 24,8 degenerates to 23,7 ⊕ 1,1 ) can be computed using the standard unfolding trick. For simplicity, we shall assume that none of the indices abcd lies along 1,1 : where c 0 = 24 is the constant term in 1/ , = \{0}, P (1) After rescaling from Einstein frame to string frame, the first and second terms in (9) are recognized as the tree-level and oneloop (∇ ) 4 coupling in perturbative heterotic string theory, while the remaining terms correspond to NS5-brane and KK5-branes wrapped on any possible T 6 inside T 7 [34].
The term F (22,6) αβγ δ can also be traced to the four-derivative scalar couplings studied in [32]. The second term in (11) which is recognized as the degeneracy of 1/2-BPS states with charges (Q , P ). In particular for a purely electric state ( P = 0) with primitive charge, it reduces to the well-known result c(−Q 2 /2) [5]. The dots in (11) stand for terms of order e −2π R 2 |k|+2π ikψ , characteristic of a Kaluza-Klein monopole of the form TN k × T 6 , where TN k is Euclidean Taub-NUT space with charge k. These contributions will be discussed in [26].

Exact ∇ 2 (∇ ) 4 couplings in D = 3
We now turn to the six-derivative coupling G ab,cd , which is expected to receive both 1/2-BPS and 1/4-BPS instanton contributions. Based on U-duality invariance, supersymmetric Ward identities and the known two-loop contribution [37,38], it is natural to conjecture that G ab,cd is given by the genus-two modular integral G (24,8) ab,cd = where F 2 is the standard fundamental domain for the action of Sp(4, Z) on the Siegel upper half-plane of degree two [39], | 2 | is the determinant of the imaginary part of = 1 + i 2 , 10 is the unique cusp form of weight 10 under the Siegel modular group Sp(4, Z) (whose inverse counts micro-states of 1/4-BPS black holes [6]), and Γ 24,8,2 is the genus-two partition function of the nonperturbative Narain lattice, Acting with the y a i -derivatives results in the insertion of a polynomial P αβ,γ δ (Q i L , −1 2 ) of degree 4 and 2 in its first and second arguments. We shall denote by G (q+16,q) ab,cd the analogue of (14) where the lattice 24,8 is replaced by q+16,q and the power of ab,cd is convergent for q < 6, and defined for q ≥ 6 by a suitable regularization prescription [40]. For q ≤ 7, the modular integral G (q+16,q) ab,cd controls the two-loop contribution to the ∇ 2 F 4 coupling in heterotic string compactified on T q [37,38].
For any value of q, one can show that G (q+16,q) ab,cd satisfies (5) and (6) with c 4 = 3−q 2 , c 5 = 6−q 2 , c 6 = 1 2 , c 7 = −π . In particular, the quadratic source term on the r.h.s. of (5) originates from the pole of 1/ 10 on the separating degeneration divisor, similar to the analysis in [30,40,41]. Thus, G (24,8) is a solution of the supersymmetric Ward identity, which is manifestly invariant under G 3 (Z). It remains to check that it produces the expected terms at weak coupling, when 24,8 degenerates to 23,7 ⊕ 1,1 . This limit can be studied using a higher-genus version of the unfolding trick [42,43]. Using results about the Fourier-Jacobi expansion of 1/ 10 from [16], we find G (24,8) αβ,γ δ = G (23,7) αβ,γ δ the almost holomorphic Eisenstein series of weight 2. The first and second terms in (15) corresponds to the zero and rank 1 orbits, respectively. The third term is necessary for consistency with the quadratic term on the r.h.s. of the supersymmetric Ward identity (5), although a naive unfolding procedure fails to produce it, presumably due to the singularity of the integrand in the separating degeneration limit. After rescaling to string frame, the first three terms in (15) correspond to the expected two-loop [37,38], one-loop [44] and tree-level contributions [45,46] to the ∇ 2 (∇ ) 4 coupling in heterotic string on T 7 , while the dots stand for terms of order e −1/g 2 3 ascribable to NS5-brane and KK5-brane instantons, which will be discussed in [26]. Note that the tree-level single trace ∇ 2 F 4 term in [45] proportional to ζ(3) vanishes on the Cartan subalgebra [47], and does not contribute to this coupling.
Having shown that our ansatz (13) passes all consistency conditions in D = 3, let us now analyze its large radius limit, where 24,8 degenerates to 22,6 ⊕ 2,2 . Again, the unfolding trick gives G (24,8) αβ,γ δ αβ,γ δ + G (2) αβ,γ δ + G (KKM) αβ,γ δ (17) The two terms on the first line (which correspond to the constant term with respect to the parabolic decomposition (2)) originate from the reduction of the ∇ 2 F 4 and R 2 F 2 couplings in four dimensions. The term proportional to g(S) originates presumably from the separating degeneration divisor, and is determined by the differential equation (6). The terms G (1) and G (2) are independent of the NUT potential ψ , and correspond to the Abelian Fourier coefficients. They are both suppressed as e −2π RM(Q ,P ) , but G (1) has support on electromagnetic charges (Q , P ) which Q and P collinear, hence corresponds to contributions of 1/2-BPS states winding the circle, while G (2) has support on generic charges, corresponding to 1/4-BPS states. The last term G (KKM) includes all terms with non-zero charge with respect to the NUT potential, corresponding to Kaluza-Klein monopole contributions.
In this letter, we focus on the contribution G (2) from 1/4-BPS black holes. This contribution originates from the 'Abelian rank 2 orbit', whose stabilizer is the parabolic subgroup G L(2, Z) Z 3 inside Sp(4, Z). Thus, the integral can be unfolded onto P 2 / P G L(2, Z) × [0, 1] 3 , where P 2 denotes the space of positive definite 2 × 2 matrices 2 : where f (Q ) 22,6,2 denotes the partition function 22,6,2 with an insertion of f (Q ) in the sum. The integral over 1 at fixed 2 extracts the Fourier coefficient C ; 2 of 1/ 10 . Due to the zeros of 10 , the latter is a locally constant function of 2 , discontinuous across certain real codimension 1 walls in P 2 [48,49]. For large R however, the remaining integral over 2 is dominated by a saddle point 2 (see (24) below), so to all orders in 1/R around the saddle point, we can replace the above Fourier coefficient by its value at 2 . The remaining integral over 2 can be computed using where B δ (Z ) is a matrix-variate generalization of the modified Bessel function [50], 2 In the limit where all entries in Z are large, one has Further relabelling ( Q P ) = A( Q 1 Q 2 ), we find G (2) αβ,γ δ = 2R 7 Q ,P ∈ 22,6 e −2π i(a 1 Q +a 2 P ) μ(Q , P ) (14), and 2 is the location of the afore-mentioned saddle point, Using (21), we see that these contributions behave as e −2π RM(Q ,P ) is recognized as the mass of a 1/4-BPS dyon with electromagnetic charges (Q , P ) [51,52]. Moreover, in cases where only A = 1 contributes to (23), the instanton measure μ(Q , P ) agrees with the BPS index (Q , P ; S, φ) in the corresponding chamber of the moduli space M 4 in D = 4, computed with the contour prescription in [53]. Our result (23) generalizes this prescription to arbitrary electromagnetic charges (Q , P ) and recovers the results of [54][55][56] for dyons with torsion, fixing a subtlety in the choice of chamber. Additional (exponentially suppressed) contributions to G (2) arise from the difference between C − 1 2 Q 2 −Q · P −Q · P − 1 2 P 2 ; 2 and its value at the saddle point. The relation between the jumps of these Fourier coefficients and the possible splittings of a 1/4-BPS bound state into two 1/2-BPS constituents [48] is crucial for consistency with the quadratic source term in the supersymmetric Ward identity (5). These contributions, along with the terms G (1) and G (KKM) which we have ignored here, will be discussed in [26].

Discussion
In this work, we have conjectured the exact form of the (∇ ) 4 and ∇ 2 (∇ ) 4 couplings in the low energy effective action of D = 3 string vacua with half-maximal supersymmetry, focussing on the simplest model, heterotic string compactified on T 7 . Our ansätze (7) and (13) are manifestly U-duality invariant, satisfy the requisite supersymmetric Ward identities, reproduce the known perturbative contributions at weak heterotic coupling and the known F 4 , R 2 , D 2 F 4 and R 2 F 2 couplings in D = 4 in the limit where the radius of one circle inside T 7 becomes large. While we do not yet have a rigorous proof that these constraints uniquely determine the functions F abcd and G ab,cd , we expect that additional contributions from cusp forms are ruled out by the supersymmetric Ward identities (4) and (5), by the same type of arguments which apply for the R 4 and ∇ 6 R 4 couplings.
In the limit where the radius of one circle inside T 7 becomes large, we find, in addition to the afore-mentioned power-like terms, an infinite series of corrections of order e −2π RM(Q ,P ) which are interpreted as Euclidean counterparts of four-dimensional BPS states with mass M(Q , P ), whose worldline winds around the circle. Rather remarkably, the contribution from a 1/4-BPS dyon is weighted by the BPS index (Q , P ; S, φ), extracted from the Siegel modular form 1/ 10 using the very same contour prescription as in [53]. Indeed, it was suggested in [57] (see also [58,59]) to represent 1/4-BPS dyons as heterotic strings wrapped on a genus-two curve holomorphically embedded in a T 4 inside T 7 . This picture was further used in [60] to justify the contour prescription of [53].
Our analysis of the ∇ 2 (∇ ) 4 coupling in D = 3 gives a concrete basis to these heuristic ideas, and explains why 1/4-BPS dyons in D = 4 are counted by a Siegel modular form of genus two. We emphasize that the introduction of the Siegel modular form 1/ 10 in the conjectured formula (13) is necessary to match the perturbative 2-loop amplitude, where it appears explicitly [37,38]. A more detailed analysis of the weak coupling and large radius expansions of the ∇ 2 (∇ ) 4 coupling will appear in [26], with particular emphasis on the consequences of wall-crossing for three-dimensional couplings.