[Paper Review] Big Numbers in String Theory

A.N. Schellekens|arXiv (Cornell University)|Jan 11, 2016

Particle physics theoretical and experimental studies44 references3 citations

TL;DR

This paper reflects on the origin of the large number $10^{1500}$ in early string theory work, analyzing its derivation in the context of even self-dual lattices via the free fermion construction. It demonstrates that free fermionic models constitute an exponentially small fraction—less than $10^{-370}$—of all such lattices in dimension 88, implying most even self-dual lattices are unreachable by this method, highlighting the vastness of the string theory landscape.

ABSTRACT

This paper contains some personal reflections on several computational contributions to what is now known as the "String Theory Landscape". It consists of two parts. The first part concerns the origin of big numbers, and especially the number $10^{1500}$ that appeared in work on the covariant lattice construction (with W. Lerche and D. Luest). This part contains some new results. I correct a huge but inconsequential error, discuss some more accurate estimates, and compare with the counting for free fermion constructions. In particular I prove that the latter only provide an exponentially small fraction of all even self-dual lattices for large lattice dimensions. The second part of the paper concerns dealing with big numbers, and contains some lessons learned from various vacuum scanning projects.

Motivation & Objective

• To re-express and correct the derivation of the large number $10^{1500}$ from the 1986 free fermion construction paper.
• To estimate the fraction of even self-dual lattices realizable via free fermionic models, especially in high dimensions.
• To demonstrate that free fermionic constructions represent an exponentially small subset of all even self-dual lattices for large dimensions.
• To reflect on computational challenges posed by the string theory landscape and the role of big numbers in vacuum scanning.
• To argue that the vast majority of even self-dual lattices cannot be captured by free fermion techniques, implying a fundamental limitation of this method.

Proposed method

• Derives an upper bound on the number of even self-dual lattices (ESDLs) using combinatorial counting of basis vectors with even norms and integer inner products.
• Applies statistical estimates to the probability of random vectors having even norm ($\sim 1/8$) and integer inner products ($\sim 1/4$), refining these with empirical data from $D_1$ conjugacy classes.
• Computes suppression factors for norm and inner product conditions: $8^{-N}4^{-\frac{1}{2}N(N-1)}$ for the first $N$ vectors and $2^{-M}2^{-NM}$ for $M$ order-2 vectors.
• Combines these suppression factors into a total bound: $2^{8k(2k-1)}$ for $N=4k$, leading to an upper bound $F_{8k} < 4k \cdot 2^{8k(2k-1)}$.
• Estimates overcounting due to permutations, automorphisms, and basis redundancies, including $(4k)!^2 2^{4k}$ for $N=4k$, which reaches $10^{122}$ at $k=11$.
• Uses empirical overcounting factors ($1024$, $10^{15}$, $10^{36}$ for $k=1,2,3$) to infer that overcounting grows factorially, suggesting a reduction factor of $\sim 10^{200}$ at $k=11$.

Experimental results

Research questions

• RQ1 What is the correct derivation and magnitude of the number $10^{1500}$ originally derived in the 1986 free fermion paper?
• RQ2 What fraction of all even self-dual lattices in dimension 88 can be constructed using the free fermion method?
• RQ3 How do statistical estimates of norm and inner product distributions affect the counting of valid lattice basis vectors?
• RQ4 To what extent do overcounting symmetries (permutations, automorphisms) reduce the effective number of distinct free fermionic models?
• RQ5 Why do the special cases $k=1$ and $k=2$ give a misleading impression of the completeness of free fermionic constructions?

Key findings

• The number $10^{1500}$ arises from a combinatorial estimate of even self-dual lattices via free fermionic constructions, but the derivation contains a large but inconsequential error that is corrected in this work.
• For dimension 88 ($k=11$), the upper bound on the number of free fermionic models is approximately $10^{558}$, significantly smaller than $10^{1500}$.
• After accounting for overcounting symmetries, the effective number of distinct free fermionic models is reduced by a factor of at least $10^{200}$ at $k=11$, suggesting a total overcount of $\sim 10^{200}$.
• The fraction of even self-dual lattices realizable via free fermionic constructions is less than $10^{-370}$ in dimension 88, indicating they represent an exponentially small subset.
• Even with corrections, the free fermionic method fails to capture the vast majority of even self-dual lattices, especially in high dimensions, due to combinatorial suppression and overcounting.
• Empirical overcounting factors grow rapidly with $k$, suggesting that the $k=1$ and $k=2$ cases are unrepresentative of the full landscape.