Four-derivative couplings and BPS dyons in heterotic CHL orbifolds

Abstract

Three-dimensional string models with half-maximal supersymmetry are believed to be invariant under a large U-duality group which unifies the S and T dualities in four dimensions. We propose an exact, U-duality invariant formula for four-derivative scalar couplings of the form F (Φ) (∇Φ) 4 in a class of string vacua known as CHL Z N heterotic orbifolds with N prime, generalizing our previous work which dealt with the case of heterotic string on T 6 . We derive the Ward identities that F (Φ) must satisfy, and check that our formula obeys them. We analyze the weak coupling expansion of F (Φ), and show that it reproduces the correct treelevel and one-loop contributions, plus an infinite series of non-perturbative contributions. Similarly, the large radius expansion reproduces the exact F 4 coupling in four dimensions, including both supersymmetric invariants, plus infinite series of instanton corrections from half-BPS dyons winding around the large circle, and from Taub-NUT instantons. The summation measure for dyonic instantons agrees with the helicity supertrace for half-BPS dyons in 4 dimensions in all charge sectors. In the process we clarify several subtleties about CHL models in D = 4 and D = 3, in particular we obtain the exact helicity supertraces for 1/2-BPS dyonic states in all duality orbits.

Heterotic-string amplitudes at one loop: modular graph forms and relations to open strings

“…The structure of higher-derivative invariants in super Yang-Mills theory was studied for example in [80][81][82] and the three operators above are of 1/2-, 1/4-and non-BPS type, respectively. General references on the effective action of heterotic string theories include [24,26,27,[83][84][85][86][87][88][89].…”

Exact effective interactions and 1/4-BPS dyons in heterotic CHL orbifolds

“…We went on to analyze the limit R → ∞ and demonstrated that the O(e −R ) corrections to F (24,8) abcd and to G (24,8) ab,cd were proportional to the known helicity supertraces of 1/2-and 1/4-BPS four-dimensional black holes, respectively. In particular, the DVV formula for the index of 1/4-BPS states [5], with the correct contour prescription [9], emerges in a transparent fashion after unfolding the integral over the fundamental domain Sp(4, Z)\H 2 onto the full Siegel upper-half plane H 2 .In [22], we extended the study of the 1/2-BPS saturated coupling F (24,8) abcd to the case of CHL heterotic orbifolds of prime order N = 2, 3, 5, 7. All these models have 16 supercharges, and their moduli space in D = 3 or 4 is of the form (1.2), (1.3) with k = 24/(N + 1).…”

“…Furthermore, for simplicity we restrict to Z N orbifolds with N ∈ {1, 2, 3, 5, 7} prime, in which case the gauge symmetry in D = 4 is reduced from U (1) 28 in the original 'maximal rank' model (namely, heterotic string compactified on T 6 ) to U (1) 2k+4 with k = 24/(N + 1). The lattice of electromagnetic charges in D = 4 is a direct sum Λ em = Λ e ⊕ Λ m where Λ m is an even, lattice of signature (2k − 2, 6) and Λ e ≡ Λ * m its dual (see Table 1 in [22]). While Λ m is not self-dual for N > 1, it is N -modular in the sense that Λ * m = Λ m [1/N ] [27], in particular we have the chainwhere SL(2, R)/SO(2) is parametrized by the heterotic axiodilaton S and the Grassmannian G 2k−2,6 is parametrized by the scalars ϕ in the vector multiplets.…”

Some comments on symmetric orbifolds of K3

“…These identities might be relevant in trying to understand the relationship between the K3 twining genera and the weak Jacobi forms appearing in Mathieu [12,[48][49][50][51][52] and Umbral moonshine [13][14][15]53]. The Siegel modular forms appear in certain low energy effective couplings in compactifications of type II superstrings on CHL models [54][55][56]. Such effective couplings might be important in trying to relate the moonshine conjectures to low dimensional string models whose symmetry groups are exactly the umbral groups.…”