Title: Stringy canonical forms

Abstract

Canonical forms of positive geometries play an important role in revealing hidden structures of scattering amplitudes, from amplituhedra to associahedra. In this paper, we introduce “stringy canonical forms”, which provide a natural definition and extension of canonical forms for general polytopes, deformed by a parameter α'. They are defined by real or complex integrals regulated with polynomials with exponents, and are meromorphic functions of the exponents, sharing various properties of string amplitudes. As α'→ 0, they reduce to the usual canonical form of a polytope given by the Minkowski sum of the Newton polytopes of the regulating polynomials, or equivalently the volume of the dual of this polytope, naturally determined by tropical functions. At finite α', they have simple poles corresponding to the facets of the polytope, with the residue on the pole given by the stringy canonical form of the facet. There is the remarkable connection between the α'→ 0 limit of tree-level string amplitudes, and scattering equations that appear when studying the α'→ ∞ limit. We show that there is a simple conceptual understanding of this phenomenon for any stringy canonical form: the saddle-point equations provide a diffeomorphism from the integration domain to the interior of the polytope,more » and thus the canonical form can be obtained as a pushforward via summing over saddle points. When the stringy canonical form is applied to the ABHY associahedron in kinematic space, it produces the usual Koba-Nielsen string integral, giving a direct path from particle to string amplitudes without an a priori reference to the string worldsheet. We also discuss a number of other examples, including stringy canonical forms for finite-type cluster algebras (with type A corresponding to usual string amplitudes), and other natural integrals over the positive Grassmannian.« less

Authors:

Arkani-Hamed, Nima ^[1]; He, Song ^[2]; Lam, Thomas ^[3]

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+ Show Author Affiliations

1. Institute for Advanced Study, Princeton, NJ (United States); Harvard University, Cambridge, MA (United States)
2. Chinese Academy of Sciences (CAS), Beijing (China); Hangzhou Institute for Advanced Study, UCAS (China); ICTP-AP International Centre for Theoretical Physics Asia-Pacific, Beijing/Hangzhou (China); University of Chinese Academy of Sciences, Beijing (China)
3. University of Michigan, Ann Arbor, MI (United States); Massachusetts Institute of Technology (MIT), Cambridge, MA (United States)

Publication Date:

Tue Feb 09 00:00:00 EST 2021

Research Org.:

Institute for Advanced Study, Princeton, NJ (United States)

Sponsoring Org.:

USDOE Office of Science (SC)

OSTI Identifier:

1839609

Grant/Contract Number:  

SC0009988

Resource Type:

Accepted Manuscript

Journal Name:

Journal of High Energy Physics (Online)

Additional Journal Information:

Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2021; Journal Issue: 2; Journal ID: ISSN 1029-8479

Publisher:

Springer Nature

Country of Publication:

United States

Language:

English

Subject:

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Differential and Algebraic Geometry; Scattering Amplitudes