The holographic landscape of symmetric product orbifolds

Abstract

We investigate the growth of coefficients in the elliptic genus of symmetric product orbifolds at large central charge. We find that this landscape decomposes into two regions. In one region, the growth of the low energy states is Hagedorn, which indicates a stringy dual. In the other, the growth is much slower, and compatible with the spectrum of a supergravity theory on AdS 3 . We provide a simple diagnostic which places any symmetric product orbifold in either region. We construct a class of elliptic genera with such supergravity-like growth, indicating the possible existence of new realizations of AdS 3 /CFT 2 where the bulk is a semi-classical supergravity theory. In such cases, we give exact expressions for the BPS degeneracies, which could be matched with the spectrum of perturbative states in a dual supergravity description.

$\mathcal{N}=2$ minimal models: A holographic needle in a symmetric orbifold haystack

“…with t/b ∈ have a basis of unwrapped elliptic genera of N = (2, 2) CFTs. We note that since [23] showed that if b > t, the symmetric product cannot grow slowly, for such cases U t,b = 0. This means that the CFTs in the stronger conjecture have to have c ≤ 6.…”

“…This implies that for a fixed t, there is only a relatively small number of such terms, roughly ∼ t 2 /12. The analysis of (2.12) in [23] also established that b 2 > t implies Hagedorn growth. This implies that if ϕ is the elliptic genus of a bona fide CFT, i.e.…”

“…The authors of [23,24] give a simple criterion to determine which seed theories X yield elliptic genera which exhibit slow growth at large N . Let us briefly state this criterion, which is stated for the unwrapped elliptic genus (2.6).…”

“…3 Moreover, we can write down a generating function for the low-lying states in the large N symmetric product which exhibit this slow growth. The q h y term in the NSsector elliptic genus of the symmetric product can be formed into a generating function we call χ NS ∞ (q, y); see [23]. In Appendix C, we give explicit expressions for χ NS ∞ (q, y) for all minimal models.…”

“…To deal with the first hurdle, we simply concentrate on CFTs with c ≤ 6. To deal with the second hurdle, we use the fact that [22][23][24] found mathematical exceptions to the generic behavior established in [21]: that is, there are specific so-called weak Jacobi forms, which could describe the elliptic genus of a CFT, whose symmetric orbifold exhibits a slow, supergravity-like growth. In this paper we build on these observations to find explicit CFTs that pass both diagnostics: N = 2 minimal models.…”