Soft Particles and Infinite-Dimensional Geometry Daniel Kapec Center of Mathematical Sciences and Applications Harvard University Cambridge Massachusetts

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Soft Particles and Infinite-Dimensional Geometry

Daniel Kapec

Center of Mathematical Sciences and Applications, Harvard University, Cambridge, Massachusetts

02138, USA

Center for the Fundamental Laws of Nature, Harvard University, Cambridge, Massachusetts

02138, USA

danielkapec@fas.harvard.edu

Abstract

In the sigma model, soft insertions of moduli scalars enact parallel transport of S-matrix elements

about the ﬁnite-dimensional moduli space of vacua, and the antisymmetric double-soft theorem

calculates the curvature of the vacuum manifold. We explore the analogs of these statements

in gauge theory and gravity in asymptotically ﬂat spacetimes, where the relevant moduli spaces

are inﬁnite-dimensional. These models have spaces of vacua parameterized by (trivial) ﬂat

connections on the celestial sphere, and soft insertions of photons, gluons, and gravitons parallel

transport S-matrix elements about these inﬁnite-dimensional manifolds. We argue that the

antisymmetric double-soft gluon theorem in d`2bulk dimensions computes the curvature of a

connection on the inﬁnite-dimensional space MappSd, Gq{G, where Gis the global part of the

gauge group. The analogous metrics in abelian gauge theory and gravity are ﬂat, as indicated

by the vanishing of the antisymmetric double-soft theorems in those models. In other words,

Feynman diagram calculations not only know about the vacuum manifold of Yang-Mills theory,

they can also be used to compute its curvature. The results have interesting implications for

ﬂat space holography.

arXiv:2210.00606v2 [hep-th] 6 Jul 2023

Contents

1. Introduction 2

2. Kinematics and celestial CFTd5

3. Sigma model 6

4. Abelian gauge theory 7

5. Gravity 11

6. Non-abelian gauge theory 14

7. Discussion and implications for CCFT 17

A. Antisymmetric double-soft limit 20

B. Coordinate systems on the space of vacua 21

1 Introduction

This paper is about an apparent ambiguity in the deﬁnition of the Yang-Mills S-matrix. In the

Coulomb phase, these S-matrix elements can be computed in perturbation theory and are unam-

biguous for ﬁnite, nonzero momenta. However, the multi-soft limits of the S-matrix, in which the

wavelengths of two or more gluons approach inﬁnity, depend on the order of limits and are there-

fore ambiguous. This ambiguity is quantiﬁed by the commutator of consecutive soft limits, also

known as the antisymmetric double-soft theorem. These antisymmetric soft limits are universal (at

tree-level) but non-vanishing for any non-abelian gauge group G:

„lim

qIÑ0,lim

qJÑ0ȷAK1¨¨¨Kn;IJ,ab

n`2‰0.(1.1)

There is some confusion regarding the interpretation of this fact. Strictly speaking, the soft limit

requires one to consider the boundary of the domain upon which the S-matrix is deﬁned, and one

could phrase (1.1) as an ambiguity in the extension of the S-matrix to the boundary of kinematic

space. Although one is tempted to look for the “right deﬁnition” of the multi-soft limits, compactiﬁ-

cations of this sort arising in moduli problems are often non-canonical1and any prescription is bound

to appear somewhat arbitrary. Although this is really just a question about the four-dimensional

S-matrix, the issue takes on added importance in the celestial CFT (CCFT) approach to ﬂat space

holography, where soft gluons are related to non-abelian conserved currents in a lower-dimensional

boundary dual. Due to (1.1), the correlation functions of these currents appear ambiguous.

1We thank Andrew Strominger for emphasizing this point.

In this paper we will advocate for an (inﬁnite-dimensional) geometric interpretation of (1.1)

which combines observations on the vacuum structure of gauge theory [1–4] with recent geometric

results on soft limits in the sigma model [5,6]. In a word, (1.1) is true because the amplitude

AK1¨¨¨Kn;IJ,ab

n`2is actually a section of a bundle over an inﬁnite-dimensional space of vacua Mlabeled

by ﬂat (trivial) G´connections Capxqon the celestial sphere. Insertions of soft gluons in the S-

matrix enact parallel transport about this space, and (1.1) simply computes the holonomy around an

inﬁnitesimal closed curve, also known as the Riemann curvature RpX, Y qZ“ r∇X,∇YsZ´∇rX,Y sZ

„lim

qIpxqÑ0ω, lim

qJpyqÑ0ω1ȷAK1¨¨¨Kn;IJ,ab

n`2“R˜δ

δĂ

apxq

,δ

δĂ

bpyq¸AK1¨¨¨Kn

n.(1.2)

In this formula, the (shadow-transformed) ﬂat connections r

Capxqare local coordinates on M,δ

δĂ

apxq

is an associated vector ﬁeld, and the fact that the antisymmetric double-soft limit does not vanish

simply means that the vacuum manifold for Yang-Mills theory in asymptotically ﬂat space is curved.

In other words, there is no “right deﬁnition” for multi-soft limits in non-abelian gauge theory, just

as there is no unique parallel transport between two points in a curved space. The result is path

dependent, and (1.1) is just an inﬁnite-dimensional manifestation of that geometric fact. Note that

in this language, the vanishing of the antisymmetric double-soft limit in gravity is a statement

about ﬂatness of the space of supertranslation vacua, and a similar statement holds for abelian

gauge theory.

The claim that there exists an inﬁnite-dimensional space of vacua for gauge theory in asymp-

totically ﬂat space is not new. In fact, this observation [2], and its analog in gravity [1], was the

genesis of the “celestial CFT” program which seeks to construct a boundary dual for asymptotically

ﬂat gravity. The existence of these vacua is a subtle boundary eﬀect which only holds for gauge

theories in the Coulomb phase on non-compact spaces, but the interpretation of (1.1) advanced in

this paper makes it clear that the moduli space of vacua is detectable using standard perturbative

calculations: the formula (1.1) can be derived using standard Feynman rules with no reference to

asymptotic symmetries, boundary conditions, or gauge transformations with non-compact support.

Perturbative S-matrix calculations in Yang-Mills theory (which are by deﬁnition performed in the

Coulomb phase, with long-range gauge ﬁelds) know about the inﬁnite-dimensional space of vacua.

In fact they know more: they can calculate its curvature.

To put these statements in context, note that the antisymmetric double-soft limit has historically

been a discovery mechanism for hidden symmetries of the S-matrix (see for instance [7] for the case

of E7p7qin N“8supergravity in four dimensions). The gauge theory examples discussed in this

work are really just inﬁnite-dimensional examples of this phenomena. Indeed, these statements

are all simpler and more familiar when the moduli space of vacua Mis ﬁnite-dimensional. In this

case, the vacuum is determined by boundary conditions (vacuum expectation values, or vevs) at

spatial inﬁnity (i0) for gapless local ﬁelds xΦIyi0“vI, and the dynamics of the long-wavelength

ﬂuctuations about these vevs is described by a sigma model with target space M. The single soft

insertion of a moduli scalar deﬁnes an operator whose matrix element is a derivative on the moduli

space [5,6]

xSIpxqO1¨ ¨ ¨ Onyv“∇IxO1¨ ¨ ¨ Onyv.(1.3)

Here, SIpxqis a soft moduli scalar, the Oirepresent hard particles, and the subscript x¨yvindicates

that the S-matrix element is calculated with the asymptotic boundary condition xΦIyi0“vI. The

physical interpretation of this formula is simple: the zero mode of the moduli scalar is just the vev, so

exciting this mode inﬁnitesimally shifts the boundary condition. In the celestial CFT formalism, the

existence of this operator is equivalent to the existence of a marginal deformation in CCFT [6]: SIpxq

indicates a continuous, ﬁnite-dimensional family of celestial CFTs labeled by boundary conditions

and related by marginal perturbations (i.e., constant background ﬁelds for the marginal operator).

Multi-soft limits in the sigma model do not commute, and the antisymmetric double-soft theorem

measures the inﬁnitesimal holonomy around a closed loop in the space of vacua [5]

„lim

qIÑ0,lim

qJÑ0ȷAK1¨¨¨KnIJ

n`2““∇I,∇J‰AK1¨¨¨Kn

n“

i“1

RIJKi

KAK1¨¨¨K¨¨¨Kn

n.(1.4)

The analog of (1.3) in gauge theory is the soft gluon theorem

xSI

apxqO1¨ ¨ ¨ OnyC“0“igYM

k“1

pk¨εapxq

pk¨ˆqpxqTI

kxO1¨ ¨ ¨ OnyC“0,(1.5)

where SI

apxqis a soft gluon with transverse polarization index a, color index I, and momentum in

the direction ˆqpxq, and the subscript x¨yCindicates that the S-matrix element is calculated with the

asymptotic boundary condition xAay “ Capxq. We would like to interpret this formula by analogy

with the the sigma model: the zero mode of the gluon shifts the boundary condition, so (1.5) is

really a functional derivative on the space of ﬂat G´connections on the celestial sphere. It is by

now understood that SI

apxqis related to the non-abelian current in the CCFT dual [8–10], and in

this language the preceding statement seems very similar to the deﬁnition of the current correlator

in the presence of a background source (in this case, a ﬂat connection)

xJI

apxqO1¨ ¨ ¨ OnyC“0“δ

δCa

IpxqxO1¨ ¨ ¨ OnyCˇˇC“0.(1.6)

This formula obviously resembles (1.3). Proceeding with the analogy, we will be led to interpret

the antisymmetric double-soft gluon theorem as an inﬁnite-dimensional version of (1.4). The fact

that (1.3) and (1.4) have no momentum dependence (position dependence in CCFT) while (1.1)

and (1.5) do is a signal that the moduli space in gauge theory is inﬁnite-dimensional, i.e., a space

of functions. This is related to the position dependence of the boundary conditions xAay “ Capxq

which is absent in the sigma model.

The organization of this paper is as follows. In section 2we review the celestial CFT formalism.

Section 3reviews the ﬁnite-dimensional case of the sigma model and its relation to marginal de-

formations of CCFT. Sections 4and 5treat the inﬁnite-dimensional ﬂat examples of abelian gauge

theory and gravity. Section 6treats the non-abelian case, and section 7concludes with a discussion

of the implications for celestial CFT.

2 Kinematics and celestial CFTd

The results of this paper are logically independent of the celestial CFT formalism, but they have

interesting implications for CCFT and are most easily phrased in that language.

We will work on Rd`1,1. The Lorentz group SOpd`1,1qis isomorphic to the d-dimensional

Euclidean conformal group. If we parameterize a generic massless on shell momentum as

qµpω, xq “ ωˆqµpxq,ˆqµpxq “ 1

2`1`x2,2xa,1´x2˘,(2.1)

then Lorentz transformations act as global conformal transformations on the xacoordinates (the

transverse cuts of the null momentum cone, or equivalently the transverse cuts of null inﬁnity). The

dindependent transverse polarization vectors are

εµ

apxq”Baˆqµpxq“pxa, δb

a,´xaq,(2.2)

and they satisfy

n¨εapxq “ 0,ˆqpxiq ¨ εapxjq “ pxij qa, εapxiq ¨ εbpxjq “ δab ,´2ˆqpxq ¨ ˆqpyq“px´yq2,(2.3)

with n“1

2p1,0a,´1q. Similarly, the transverse traceless polarization tensor for gravitons is

εµν

ab pxq ” 1

2rεµ

apxqεν

bpxq ` εν

apxqεµ

bpxqs ´ 1

dδabΠµν pxq,(2.4)

where

Πµν pxq ” εµ

apxqεa,ν pxq “ ηµν `2nµˆqνpxq ` 2nνˆqµpxq.(2.5)

The operators that create single particle states live on the null cone

Oipωi, xiq ” aout

ipqpωi, xiqqθpωiq ` ¯ain:

ip´qpωi, xiqqθp´ωiq,(2.6)

and the S-matrix amplitudes can be written in the suggestive form

An“ xO1pω1, x1q ¨ ¨ ¨ Onpωn, xnqy .(2.7)

The amplitude (2.7) does not transform like a correlator of conformal primaries in CFTdsince

momentum eigenstates are not boost eigenstates. For massless particles, this is ﬁxed by performing

a Mellin transform

Op∆, xq “ żC

dω ω∆´1Opω, xq(2.8)

for some contour Cand scaling dimension ∆. We will be interested in the “residue operators” arising

from Mellin transforms with integer dimensions and compact contours surrounding the origin

Opn, xq “ ¿dω

2πi ωn´1Opω, xq.(2.9)

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SoftParticlesandInfinite-DimensionalGeometryDanielKapecCenterofMathematicalSciencesandApplications,HarvardUniversity,Cambridge,Massachusetts02138,USACenterfortheFundamentalLawsofNature,HarvardUniversity,Cambridge,Massachusetts02138,USAdanielkapec@fas.harvard.eduAbstractInthesigmamodel,softinsertions...

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