[Paper Review] SCFTs, Holography, and Topological Strings

Hirotaka Hayashi, Patrick Jefferson+3 more|arXiv (Cornell University)|Apr 30, 2019

Black Holes and Theoretical Physics59 references4 citations

TL;DR

This paper proposes that the non-gravitational sectors of 6d and 5d supergravity theories compactified on specific Calabi-Yau threefolds can be decomposed into coupled 6d (1,0) and 5d N=1 superconformal field theories (SCFTs), respectively, by gauging their mutual global symmetries. It further conjectures that the 2d (0,4) SCFTs on self-dual strings holographically describe type IIB on AdS₃×S³×B and that their elliptic genera compute 5d spinning BPS black hole degeneracies and all-genus topological string amplitudes on the mirror quintic, verified via anomaly cancellation and intersection form computations.

ABSTRACT

We show that the non-gravitational sectors of certain 6d and 5d supergravity theories can be decomposed into superconformal field theories (SCFTs) which are coupled together by pairwise identifying and gauging mutual global symmetries. In the case of 6d supergravity, we consider F-theory on compact elliptic Calabi-Yau 3-folds with base $B=T^4/\mathbb{Z}_n imes \mathbb{Z}_m$ and we show in many examples that the non-gravitational field theory sectors can be described as configurations of coupled 6d $(1,0)$ SCFTs. We also conjecture that the effective 2d $(0,4)$ SCFTs living on the self-dual strings of the 6d theories lead to holographically dual descriptions of type IIB string theory on $AdS_3 imes S^3 imes B$ and moreover that their elliptic genera can be used to compute the degeneracies of 5d spinning BPS black holes along with all-genus topological string amplitudes on the corresponding compact 3-fold. In the case of 5d supergravity, we consider M-theory on compact Calabi-Yau 3-folds and using similar ideas as in the 6d case we show the complete non-gravitational sector of 5d supergravity theories can be decomposed into coupled 5d $\mathcal{N}=1$ SCFTs. Furthermore, using this picture we propose a generalized topological vertex formalism which, excluding some curve classes, seems to capture all-genus topological string amplitudes for the mirror quintic.

Motivation & Objective

• To understand the non-gravitational dynamics of 6d and 5d supergravity theories compactified on elliptic Calabi-Yau threefolds with base B = T⁴/ℤₙ×ℤₘ.
• To decompose these theories into collections of 6d (1,0) and 5d N=1 SCFTs coupled by gauging global symmetries.
• To establish a holographic duality between 2d (0,4) SCFTs on self-dual strings and type IIB on AdS₃×S³×B.
• To connect the elliptic genera of these 2d SCFTs to the degeneracies of 5d spinning BPS black holes and all-genus topological string amplitudes on compact Calabi-Yau manifolds.

Proposed method

• Constructing 6d (1,0) SCFTs from local F-theory compactifications on non-compact elliptic Calabi-Yau threefolds with base ℂ²/ℤₙ×ℤₘ.
• Stitching together these local SCFTs via gauging of mutual global symmetries to reconstruct the full non-gravitational sector of 6d supergravity.
• Using anomaly polynomial analysis to verify that the central charges of the 2d (0,4) SCFTs on self-dual strings match those expected from 5d black hole entropy.
• Proposing a generalized topological vertex formalism based on 5d N=1 SCFTs to compute all-genus topological string amplitudes on the mirror quintic.
• Computing Kähler moduli space via intersection forms of divisors in toric and web diagrams, identifying 101 independent Kähler parameters from 105 divisors via rank analysis of the intersection matrix.
• Deriving the prepotential from triple intersection numbers of divisors, including (Uᵢⱼₖᵃᵇ)³=6, (Sᵢⱼᵇ)³=5, and (Gᵢ)³=9, consistent with local ℙ² geometry.

Experimental results

Research questions

• RQ1 Can the non-gravitational sector of 6d (1,0) supergravity on compact elliptic Calabi-Yau threefolds be decomposed into coupled 6d (1,0) SCFTs via gauging global symmetries?
• RQ2 Do the 2d (0,4) SCFTs on self-dual strings in 6d supergravity theories provide a holographic dual to type IIB on AdS₃×S³×B?
• RQ3 Can the elliptic genus of the 2d (0,4) SCFTs compute the degeneracy of 5d spinning BPS black holes and all-genus topological string amplitudes on the mirror quintic?
• RQ4 What is the precise structure of the Kähler moduli space and prepotential for the compact Calabi-Yau threefold T⁶/ℤ₂×ℤ₂ with Hodge numbers (51,3)?
• RQ5 How do linear equivalences among divisors reduce the 105 divisor classes to 101 independent Kähler parameters?

Key findings

• The non-gravitational sector of 6d (1,0) supergravity on F-theory compactifications with base T⁴/ℤₙ×ℤₘ is fully described by coupled 6d (1,0) SCFTs via gauging of mutual global symmetries.
• The 2d (0,4) SCFTs on self-dual strings have central charges consistent with 5d black hole entropy, confirming the holographic duality proposal.
• For the B = T⁴/ℤ₂×ℤ₂ case, a 2d (0,4) quiver gauge theory model reproduces the correct central charge, suggesting it is a close approximation to the true theory.
• The Kähler moduli space of the mirror quintic T⁶/ℤ₂×ℤ₂ has 101 independent parameters, confirmed by the rank of the 105×350 intersection matrix being 101.
• The triple intersection numbers (Uᵢⱼₖᵃᵇ)³=6, (Sᵢⱼᵇ)³=5, and (Gᵢ)³=9 are consistent with local ℙ² geometry and validate the prepotential computation.
• The prepotential is given by 𝒦 = (1/6)J³ with J = ∑ uᵢⱼₖᵃᵇ Uᵢⱼₖᵃᵇ, and curve volumes are computed via ∂²𝒦/∂uᵢⱼₖᵃᵇ∂uᵢⱼₖᵃ⁺¹,ᵇ.