String theory (mathematical physics)

String theory replaces point particles with vibrating strings and is, before anything else, a proposal for a quantum theory of gravity. Many of its concrete questions come down to a single well-defined quantity, an integral or an enumeration over integers. Where earlier work on such a quantity relied on approximations, special cases or conjectures, an exact computation can settle the question, and each of the three papers here carries out one such computation.

Project pages: The Grassmannian string integral  ·  Flux vacua  ·  Modular graph functions

Exact questions in string theory

String theory began with an exact formula: in 1968 Veneziano proposed, for the strong interactions, the scattering amplitude

where $s$ and $t$ encode the collision energy and scattering angle, $\alpha(s)$ is a linear function of $s$, and $\Gamma$ is Euler's Gamma function. The integral runs over the position $x$ of one point on a line; it was later understood as the amplitude for four open strings to scatter, its poles giving the masses of the string's excited states. String amplitudes in general are integrals of this kind, over the positions of several points on a line or a sphere and, at the first quantum correction, over the shape of the torus a closed string sweeps out. Their low-energy expansions have particular constants as coefficients, among them the values $\zeta(n)=\sum_{k\ge1}k^{-n}$ of Riemann's zeta function and their nested generalizations, the multiple zeta values.

String theory also corrects Einstein's equations by an infinite series of terms whose coefficients it fixes completely, and at the first quantum correction each coefficient comes from integrating special functions called modular graph functions over the shape of the torus. For a long thin torus these functions reduce to Laurent polynomials (polynomials allowing negative powers) whose coefficients are built from zeta values and multiple zeta values, conjectured to belong to a restricted class of such numbers.

The theory's vacuum states are labeled by integers: the extra dimensions must be curled into a compact space, and fluxes, generalized magnetic fields quantized in whole units, hold its shape fixed. A consistency condition, the tadpole boundFor M-theory on a Calabi–Yau fourfold the flux charge plus the number of space-filling membranes must equal one twenty-fourth of the Euler number of the compact space, so once anti-membranes are excluded the flux charge cannot exceed that number., caps the total flux charge at a budget set by the topology, and questions about the whole set of vacua, the landscape, become questions about integers: whether any flux within the budget fixes every shape parameter, or how small a vacuum's charge can be.

In all three settings the quantity in question has a definite value, a closed form or a whole number, yet much of what was known came from numerical searches, special families and conjectures. A floating-point search cannot prove that the smallest flux charge it finds is the minimum, and a family of functions containing only odd zeta values cannot test a conjecture about which other numbers appear. An exact computation replaces the search by an enumeration proved complete, the floating-point value by a closed form or an interval guaranteed to contain the answer, and the conjecture, within a finite family, by a direct check.

The three projects

The Grassmannian string integral. The first paper asks which properties of string amplitudes survive in $K(3,6)$, a generalization of the Veneziano integral to six points in the projective plane, written down in 2019 by Arkani-Hamed, He and Lam, that no known string theory produces. It proves that $K(3,6)$ still factorizes at every pole as a string amplitude does, even though cluster decompositionThe principle that experiments far apart do not influence each other; in field theory, together with unitarity, it forces the residue at a pole to split into a product of two smaller amplitudes. cannot be formulated for it, finds numerically that the Kawai–Lewellen–Tye double copyRelations that express closed-string amplitudes as sums of products of pairs of open-string amplitudes. also holds, and proves, granted results of Brown and Panzer, that the low-energy coefficients are multiple zeta values to all orders.

Exact methods for flux vacua. The second paper starts from the question of whether any flux within the budget, 24 units for the compact space K3×K3, can fix every shape parameter; the numerical searches of Bena, Blåbäck, Graña and Lüst had found none. It proves, under stated hypotheses, that no flux supported on a lattice of rank at most threeRoughly, a flux built from at most three independent directions in the lattice of allowed fluxes on each K3 factor; the paper calls this sublattice the support of the flux. does, constructs one of higher rank with charge 22 that does, and, for the benchmark Kachru–Kallosh–Linde–Trivedi vacuum, proves that the defining critical point exists and computes its superpotentialA function $W$ of the shape parameters, determined by the fluxes; the vacua are its critical points, and $W_0$ denotes its value at the vacuum. with a proven error bound.

Modular graph functions and depth-three zeta values. Zerbini conjectured that the thin-torus coefficients of modular graph functions belong to a restricted class, Brown's single-valued multiple zeta values, and they are expected to map into one another under a symmetry called the cosmic Galois group; the complete families computed before contain only odd zeta values, for which both properties are automatic. The third paper computes in closed form the thin-torus Laurent coefficients of a complete family of forty-nine such functions and finds multiple zeta values of depth three (triple nested sums), proved confined to one coefficient per function, exactly as many independent ones as the Broadhurst–Kreimer conjecture allows, mapped by the Galois group back into the family, and single-valued in every case.

The papers

• Factorization without clustering and the six-point Grassmannian string integral (PDF) — Matthew D. Schwartz; currently marked preliminary. Computes $K(3,6)$, proves that it factorizes onto its boundaries, tests the double copy numerically and identifies the zeta values in its low-energy coefficients.
• Exact methods for string flux vacua (PDF) — Matthew D. Schwartz; currently marked preliminary. Proves, under stated hypotheses, the K3×K3 bound for fluxes whose support has rank at most three, locates over a stated range the first vacuum in the symmetric sector of a Hulek–Verrill fourfold, and computes $|W_0|$ for the benchmark vacuum of Demirtas, Kim, McAllister and Moritz with a proven error bound.
• Irreducible multiple zeta values of depth three in four-edge modular graph functions (PDF) — Matthew D. Schwartz; currently marked preliminary. Gives in closed form the Laurent polynomials of all 49 two-vertex four-edge functions of weights seven through twelve and analyzes their depth-three zeta values.

Supplementary material

• Supplementary material for the first paper (PDF) — the full account of the numerical computations that its Appendix B summarizes: the kinematic points used, the conventions and values of each numerical check, and the inputs and outputs of the integer-relation searches (Sections S1–S8).
• Supplementary material for the second paper (PDF) — Sections S1–S7: the proofs of the existence and isolation criteria, the enumeration of vacua in ten flux sectors, notes on the KKLT constructions with the numerical data for the benchmark computation, the tests of the implementation, and the lattice, charge-22 and Hulek–Verrill data in full.

Code and data for each project are linked from its own page: the Grassmannian string integral, flux vacua and modular graph functions.

References

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