USTC-ICTS/PCFT-21-18

Non-Negativity of BMN Two-Point Functions

With Three String Modes

Bao-ning<EMAIL_ADDRESS>Min-xin<EMAIL_ADDRESS>

Interdisciplinary Center for Theoretical Study,

University of Science and Technology of China, Hefei, Anhui 230026, China

Peng Huanwu Center for Fundamental Theory,

Hefei, Anhui 230026, China

Recently, we proposed a novel entry of the pp-wave holographic dictionary,
which equated the Berenstein-Maldacena-Nastase (BMN) two-point functions in
free $\mathcal{N}=4$ super-Yang-Mills theory with the norm squares of the
quantum unitary transition amplitudes between the corresponding tensionless
strings in the infinite curvature limit, for the cases with no more than three
string modes in different transverse directions. A seemingly highly non-
trivial conjectural consequence, particularly in the case of three string
modes, is the non-negativity of the BMN two-point functions at any higher
genus for any mode numbers. In this paper, we further perform the detailed
calculations of the BMN two-point functions with three string modes at genus
two, and explicitly verify that they are always non-negative through mostly
extensive numerical tests.

###### Contents

  1. 1 Introduction
  2. 2 More on The Physical Motivation
  3. 3 Genus One
  4. 4 Genus Two
    1. 4.1 Some Useful Symmetries
    2. 4.2 The Result
  5. 5 Conclusion
  6. A Some Standard Integrals

##  1 Introduction

We continue the studies of free BMN correlators in our recent paper [1]. The
general motivations and the physical settings are explained in the previous
paper. We shall provide a recapitulation with some new comments more relevant
to the current context.

The AdS/CFT correspondence [2, 3, 4] is a conceptual breakthrough in our
understandings of quantum gravity, in particular provides a non-perturbative
definition of string theory in AdS background in terms of $\mathcal{N}=4$
$SU(N)$ super-Yang-Mills theory. We consider the Penrose limit [5], which
gives rise to another maximally supersymmetric background [6], known as the
pp-wave or plane wave geometry

$\displaystyle
ds^{2}=-4dx^{+}dx^{-}-\mu^{2}(\vec{r}^{~{}2}+\vec{y}^{~{}2})(dx^{+})^{2}+d\vec{r}^{~{}2}+d\vec{y}^{~{}2},$
(1.1)

where $x^{+},x^{-}$ are light cone coordinates, $\vec{r},\vec{y}$ are
4-vectors, and the parameter $\mu$ measures the spacetime curvature as well as
the Ramond-Ramond flux $F_{+1234}=F_{+5678}\sim\mu$. This appears to be a
promising ground for quantitative explorations of the holographic duality in
stringy regimes, as the dual theories on both sides can be either free or
weakly coupled.

In the groundbreaking paper [7], Berenstein, Maldacena and Nastase (BMN)
proposed a type of near-BPS operators, which correspond to the type IIB closed
strings on the pp-wave background. The free string spectrum is correctly
reproduced by gauge interactions as the planar conformal dimensions of BMN
operators. The BMN scaling limit with large R-charge $J\sim\sqrt{N}\sim\infty$
appears to be the right Goldilocks limit in this situation, since a smaller
R-charge would not provide finite string interactions in the strict
$N\sim\infty$ limit, while a larger R-charge may blow up the strings into
D-branes, known as giant gravitons, studied in early papers e.g. [8, 9, 10,
11, 12, 13]. Some recent studies relating to the large R-charge limit or
Penrose limit, as well as applications in more general theories can be found
in e.g. [14, 15, 16, 17, 18, 19, 20].

As in our previous papers [21, 22, 23, 24, 25, 1], we focus on free gauge
theory and study the BMN correlation functions. This corresponds to the pp-
wave background with infinite curvature and infinite Ramond-Ramond flux as
$\mu\sim\infty$ in the geometry (1.1), where the strings are tensionless with
completely degenerate spectrum. There are still interesting string
interactions as we identify the finite genus-counting parameter
$g:=\frac{J^{2}}{N}$ as the effective string coupling constant in this case.
Some non-planar BMN correlators are first computed in [26, 27].

The celebrated standard AdS holographic dictionary [4] seems not directly
applicable for string interactions in the pp-wave background, as the
geometries are quite different. So in some cases, certain guessworks may be
required to identify the correct entries of the “pp-wave holographic
dictionary”. In this paper we focus on a probability interpretation of BMN
two-point functions [24, 1]. There are other interesting entries of the pp-
wave holographic dictionary, namely the comparisons of free planar BMN three-
point functions with Green-Schwarz light-cone string field cubic vertices [28,
29, 30, 21, 31], the factorization formulas [22, 23], which are most recently
explored in the recent paper [1] in the context of many string modes. Higher
point correlators, including the planar three-point functions, actually always
vanish in the strict BMN limit, and are now perceived by us as a kind of
virtual processes.

Let us introduce some notations. The BMN vacuum operator is simply
proportional $\textrm{Tr}(Z^{J})$ where $Z$ is a complex scalar in the
$\mathcal{N}=4$ $SU(N)$ super-Yang-Mills theory. One can insert the four
remaining real scalars into the trace with phases, corresponding to string
modes in four of the eight transverse dimensions. The BMN operators are then
denoted as $O^{J}_{(m_{1},m_{2},\cdots,m_{k})}$, where the positive and
negative integer modes represent the left and right moving stringy excited
modes, while the zero modes are supergravity modes representing discretized
momenta in the corresponding traverse direction. We will consider the case of
string modes in different transverse directions, otherwise the BMN operators
are no longer near-BPS and there may be some potential issues as discussed in
[1]. Due to the closed string level matching condition $\sum_{i}m_{i}=0$, the
excited stringy states have at least two string modes. The BMN operators are
properly normalized to be orthonormal at planar level, and the genus $h$ two-
point functions are proportional to $g^{2h}$ as

$\displaystyle\langle\bar{O}^{J}_{(m_{1},m_{2},\cdots,m_{k})}O^{J}_{(n_{1},n_{2},\cdots,n_{k})}\rangle_{0}=\delta_{m_{1},n_{1}}\cdots\delta_{m_{k},n_{k}},$
(1.2)
$\displaystyle\langle\bar{O}^{J}_{(m_{1},m_{2},\cdots,m_{k})}O^{J}_{(n_{1},n_{2},\cdots,n_{k})}\rangle_{h}\sim
g^{2h}.$

As discussed in [1], the BMN two-point functions are real and symmetric, and
there is a nice normalization relation summing over one set of mode numbers

$\sum_{\sum_{i=1}^{k}n_{k}=0}\langle\bar{O}^{J}_{(m_{1},m_{2},\cdots,m_{k})}O^{J}_{(n_{1},n_{2},\cdots,n_{k})}\rangle_{h}=\frac{g^{2h}}{2^{2h}(2h+1)!}.$
(1.3)

We may define a matrix element, summing up all genus contributions with a
proper normalization by the all-genera formula of vacuum correlator

$p_{(m_{1},m_{2},\cdots,m_{k}),(n_{1},n_{2},\cdots,n_{k})}=\frac{g}{2\sinh(\frac{g}{2})}\sum_{h=0}^{\infty}\langle\bar{O}^{J}_{(m_{1},m_{2},\cdots,m_{k})}O^{J}_{(n_{1},n_{2},\cdots,n_{k})}\rangle_{h},$
(1.4)

so that it looks like a probability distribution

$\sum_{\sum_{i=1}^{k}n_{k}=0}p_{(m_{1},m_{2},\cdots,m_{k}),(n_{1},n_{2},\cdots,n_{k})}=1.$
(1.5)

To interpret the matrix elements as a probability distribution, they need to
be non-negative. For the case of two string modes, the non-negativity at any
genus can be easily proven since the two string modes are opposite numbers
[24], while for the case of four string modes, it turns out that the genus one
two-point functions can be negative [1]. There seems to be a rule forbidding
the “crowdedness” of string modes, that we can not holographically use up all
four remaining scalars to fully occupy the four transverse dimensions with
$SO(4)$ rotational symmetry unbroken by the Ramond-Ramond flux in the pp-wave
background. In this paper, we focus on the case of three string modes. Unlike
the case of two string modes, we are not aware of a simple analytic proof of
the non-negativity. Instead, we perform the detailed calculations and
explicitly verify the non-negativity up to genus two.

The normalization relations and non-negativity with two and three string modes
suggest a novel entry of the pp-wave holographic dictionary

$p_{(m_{1},\cdots,m_{k}),(n_{1},\cdots,n_{k})}=|\langle
m_{1},\cdots,m_{k}|\hat{U}(g)|n_{1},\cdots,n_{k}\rangle|^{2},~{}~{}~{}k=2,3,$
(1.6)

where the operator $\hat{U}(g)$ describes the quantum unitary transition
between the degenerate tensionless strings. The BMN single string states form
a complete orthonormal basis of the Hilbert space under such finite string
interactions

$\sum_{\sum_{i=1}^{k}n_{k}=0}|n_{1},n_{2},\cdots,n_{k}\rangle\langle
n_{1},n_{2},\cdots,n_{k}|=I.$ (1.7)

Of course, the probability interpretation only requires the matrix element
(1.4) is non-negative. In flat space it is well known that the string coupling
constant is related to the vacuum expectation value of a complex axion-dilaton
field. In our context of holographic duality, we focus on the correspondence
with gauge theory where the effective string coupling $g=\frac{J^{2}}{N}$ is
an arbitrary real non-negative constant. It would seem rather contrived to
have negative two-point functions at some higher genus but the total
contribution still manages to always remain non-negative. Expecting the same
phenomenon as in the case of two string modes, we make the stronger conjecture
that the BMN two-point functions with three string modes are also always non-
negative separately at each genus.

If our proposal of the entry of pp-wave holographic dictionary (1.6) is
correct, to our knowledge, it would not only provide first examples of
systematic calculations of (the norms of) the higher genus critical
superstring amplitudes, but may also in principle gives exact complete results
for any string coupling, due to the convergence of genus expansion, as
mentioned in [25]. Thus it is important to go through the laborious
calculations with three string modes at genus two, in order to ensure the
previously observed non-negativity at genus one is not just a lucky
coincidence, but more likely a manifestation of the deep mathematical
structures of the underlying holographic duality.

The paper is organized as the followings. In Sec. 2 we provide some more
discussions on the physical motivation. In Sec. 3, we review some calculations
at genus one. We provide a complete proof of the non-negativity including some
cases with mode number degeneracies which require a little extra attentions.
In Sec. 4, we perform the detailed calculations at genus two, utilizing some
symmetries. Although the calculations are quite complicated, we report the
result in a relatively compact form in terms of standard integrals, defined in
the Appendix A. We then numerically verify the non-negativity for small modes
numbers and give an argument for large mode numbers. We conclude in Sec. 5
with some potential future directions.

##  2 More on The Physical Motivation

Since the paper consists of rather technical calculations, it is useful to
explain further about the physical motivation. A main goal of the theories of
quantum gravity is to understand the physics in the regime beyond the reach of
classical gravity, e.g. in the highly curved spacetime region near the black
hole singularity. For $AdS_{d}$ space with $d>2$, the scalar curvature is
negative $R=-\frac{d(d-1)}{r^{2}}$ where $r$ is the radius of the AdS space,
so the spacetime is highly curved if the radius $r$ is very small (compared to
the string or Planck length). For the pp-wave background (1.1), the scalar
curvature actually vanishes, while the non-vanishing component of the Ricci
curvature is proportional to $\mu^{2}$ [6]. As we mentioned, we will focus on
the $\mu\sim\infty$ or infinite curvature limit. Certainly the effective
action of classical gravity completely breaks down in this case, but this is
not necessarily a bad situation as we can instead probe the fundamental nature
of spacetime. As the background remains maximally supersymmetric, the physics
is much more amenable to studies than e.g. those near black hole or big bang
singularities. Thanks to the helps from holographic duality, the physical
string amplitudes can still be subjected to quantitative studies with some
reasonable conjectures and indirect constrains such as unitarity, which is a
fundamental principle that we expect to remain valid in the infinite curvature
limit.

In our context, the main goal of the current calculations is to better
understand the conjectured entry (1.6) of pp-wave holographic dictionary. This
seemingly simple equation has eluded research on the topic for many years, as
the physical picture of string dynamics in the infinitely curved pp-wave
background turns out to be drastically different from those familiar in flat
spacetime or AdS space with large radius. In particular, there is no finite
physical process of multiple particles or strings scattering to and from
asymptotic region of spacetime, as the higher point functions always vanish in
the strict $N\sim\infty$ limit. Instead, the tensionless string directly jumps
from one excited state to another through a quantum unitary transition, much
as in a S-matrix where the incoming and outgoing states have the same energy.
Of course, the BMN higher point functions are still very useful since
infinitely many of them may combine to make a finite contribution. The cubic
string vertices are in fact the fundamental building blocks of the string
(loop) diagrams in the factorization formulas studied in [22, 23], but are not
needed in the current paper. Since the only finite BMN two-point functions are
always real and symmetric, unitarity of string interactions rules out the
naive possibility that they are directly identified with quantum transition
amplitudes on the string theory side [24]. So we arrive at the otherwise
seemingly most natural conjecture (1.6).

There is no technical obstruction for our calculations on the free gauge
theory side at any genus, however the available tools on the string theory
side are very limited. Much progress for the calculations of higher genus
critical superstring amplitudes focused on using the RNS (Ramond-Neveu-
Schwarz) formalism in flat space, and is already quite difficult at genus two,
see e.g. an early review [32]. Some obstructions to higher genus calculations
were discussed more recently in [33]. Our conjecture (1.6) gives the norms of
certain critical superstring amplitudes including all genus contributions. Of
course, the string amplitudes are complex, and consist of the norms and phase
angles. As discussed in [24], unitarity can in principle determine a large
part of the phase angles, but not completely. Our studies thus provide a long
term motivation for future research to develop techniques that can deal with
string theory on highly curved background, with flux, and including highly
excited stringy states, for the purposes of a direct verification of the
conjecture (1.6) as well as the complete determination of string amplitudes
including the phase angles. Some recent studies concerning highly excited
strings are [34, 35]. For the moment, our verifications of non-negativity on
the gauge theory side provide indirect non-trivial evidence of the conjecture
(1.6).

We should note that the BMN two-point functions are exactly zero in the cases
of the mode numbers $m_{i}\neq 0,n_{i}=0$ for an index $i$, as can be easily
seen from the integral formula at any genus. This can be understood as a
consequence of momentum conservation since a zero mode represents a
discretized unit of momentum in the corresponding transverse direction. If we
extend the mode numbers to be real, generically there is no indication that
these would all be extremal points of the two-point function formula, which
may no longer even be real. So the non-negativity seems to be an intrinsically
stringy phenomenon, valid only for integer mode numbers satisfying the level
matching condition, but can not be extended to real mode numbers. The usual
technique of extremization in dealing with functions of complex or real
variables is probably not much helpful in our context.

##  3 Genus One

The calculations of the torus BMN two-point functions with many string modes
are explained e.g. in our recent paper [1]. The formula is

$\displaystyle\langle\bar{O}^{J}_{(m_{1},m_{2},\cdots,m_{k})}O^{J}_{(n_{1},n_{2},\cdots,n_{k})}\rangle_{\textrm{torus}}$
(3.1)
$\displaystyle=g^{2}\int_{0}^{1}dx_{1}dx_{2}dx_{3}dx_{4}\delta(x_{1}+x_{2}+x_{3}+x_{4}-1)\int_{0}^{x_{1}}dy_{k}e^{2\pi
i(n_{k}-m_{k})y_{k}}\times$
$\displaystyle\prod_{i=1}^{k-1}(\int_{0}^{x_{1}}+e^{2\pi
in_{i}(x_{3}+x_{4})}\int_{x_{1}}^{x_{1}+x_{2}}+e^{2\pi
in_{i}(x_{4}-x_{2})}\int_{x_{1}+x_{2}}^{1-x_{4}}+e^{-2\pi
in_{i}(x_{2}+x_{3})}\int_{1-x_{4}}^{1})dy_{i}e^{2\pi i(n_{i}-m_{i})y_{i}}$

The result for three string modes were calculated in [23], and can be written
in terms of the standard integrals in the Appendix A. For convenience we
factor out the coupling constant and denote
$\langle\bar{O}^{J}_{(m_{1},m_{2},m_{3})}O^{J}_{(n_{1},n_{2},n_{3})}\rangle_{\textrm{torus}}\equiv
F_{1}g^{2}$, with

$\displaystyle F_{1}$ $\displaystyle=\sum_{i\neq
j}[I_{(5,1,1)}(0,m_{i}-n_{i},-m_{j}+n_{j})+I_{(2,2,2,1)}(0,-m_{j},-m_{j}+n_{j},m_{i}-n_{i})$
(3.2)
$\displaystyle+I_{(2,2,2,1)}(0,n_{j},-m_{j}+n_{j},m_{i}-n_{i})+I_{(2,2,1,1,1)}(0,-m_{j}+n_{j},-m_{j},n_{j},m_{i}-n_{i})$
$\displaystyle+I_{(1,1,1,1,1,1,1)}(0,m_{i},n_{i},-m_{j},-n_{j},m_{i}-n_{j},-m_{j}+n_{i})].$

Since this is a 7-dimensional integral, the indices in the standard integrals
always sum to 7. One can explicitly compute the standard integrals using the
formulas (A.3). In the generic case where none of $m_{i},n_{i},m_{i}\pm
n_{j}$’s is zero, there is no further degeneracy in the arguments of the
standard integrals in (3.2), the result is

$F_{1}=\frac{\sum_{i=1}^{3}(m_{i}-n_{i})^{2}}{32\pi^{4}\prod_{i=1}^{3}(m_{i}-n_{i})^{2}},$
(3.3)

which is of course manifestly positive.

We check also the numerous degenerate cases where some of
$m_{i},n_{i},m_{i}\pm n_{j}$’s vanishes. Most cases also have manifestly non-
negative results. However, there are several degenerate cases where the
results are somewhat complicated to check the non-negativity, namely the cases
3,7,8 in [23]. In the followings we perform a more careful analysis to show
that they are always positive.

  1. 1.

$n_{3}=m_{3}$ and everything else generic. Using the level matching conditions
there are three independent modes numbers. We express the result in terms of
$m_{1},n_{1},m_{3}$ as

$\displaystyle F_{1}$
$\displaystyle=\frac{1}{48\pi^{2}(m_{1}-n_{1})^{2}}+\frac{1}{16\pi^{4}(m_{1}-n_{1})^{4}}$
(3.4)
$\displaystyle-\frac{m_{3}^{4}+(m_{1}+n_{1})m_{3}^{3}+m_{1}n_{1}m_{3}^{2}-m_{1}n_{1}(m_{1}+n_{1})m_{3}-m_{1}^{2}n_{1}^{2}}{16\pi^{4}(m_{1}-n_{1})^{2}m_{1}n_{1}m_{3}^{2}(m_{3}+m_{1})(m_{3}+n_{1})}$
$\displaystyle+\frac{1}{16\pi^{4}(m_{1}-n_{1})^{2}m_{1}^{2}n_{1}^{2}m_{3}^{2}(m_{3}+m_{1})^{2}(m_{3}+n_{1})^{2}}\\{m_{3}^{6}(m_{1}^{2}+n_{1}^{2})$
$\displaystyle+2m_{3}^{5}(m_{1}^{2}n_{1}+m_{1}n_{1}^{2}+m_{1}^{3}+n_{1}^{3})+m_{3}^{4}(4m_{1}^{3}n_{1}+2m_{1}^{2}n_{1}^{2}+4m_{1}n_{1}^{3}+m_{1}^{4}+n_{1}^{4})$
$\displaystyle+2m_{3}^{3}m_{1}n_{1}(m_{1}^{3}+n_{1}^{3})-8m_{3}^{2}m_{1}^{3}n_{1}^{3}-4m_{3}m_{1}^{3}n_{1}^{3}(m_{1}+n_{1})-2m_{1}^{4}n_{1}^{4}\\}.$

The expression looks more complicated even than the formula for the generic
case (3.3). However after some manipulations we can write it as

$\displaystyle F_{1}$
$\displaystyle=\frac{1}{16\pi^{4}(m_{1}-n_{1})^{2}}\\{(\frac{\pi^{2}}{3}-\frac{1}{m_{3}^{2}})+\frac{1}{(m_{1}-n_{1})^{2}}+\frac{m_{1}^{2}-m_{1}n_{1}+n_{1}^{2}}{m_{1}^{2}n_{1}^{2}}$
(3.5)
$\displaystyle+\frac{1}{2(m_{3}+m_{1})^{2}}+\frac{1}{2(m_{3}+n_{1})^{2}}\\}+\frac{1}{32\pi^{4}(m_{3}+m_{1})^{2}(m_{3}+n_{1})^{2}}.$

Now in this form it is clear that each term is manifestly positive.

  2. 2.

$n_{3}=m_{3},n_{2}=m_{2}$ and everything else generic. The level matching
condition also requires $m_{1}=n_{1}$. The result is

$\displaystyle
F_{1}=\frac{1}{120}+\frac{5}{16\pi^{4}}(\frac{1}{m_{1}^{4}}+\frac{1}{m_{2}^{4}}+\frac{1}{m_{3}^{4}})-\frac{(m_{1}^{2}+m_{2}^{2}+m_{3}^{2})^{2}}{192\pi^{2}m_{1}^{2}m_{2}^{2}m_{3}^{2}}.$
(3.6)

There are two independent mode numbers as $m_{1}+m_{2}+m_{3}=0$. Without loss
of generality we can assume $m_{1},m_{2}>0$. The negative last term can be
estimated

$\frac{(m_{1}^{2}+m_{2}^{2}+m_{3}^{2})^{2}}{192\pi^{2}m_{1}^{2}m_{2}^{2}m_{3}^{2}}=\frac{1}{48\pi^{2}}[\frac{m_{1}^{2}+m_{2}^{2}+m_{1}m_{2}}{m_{1}m_{2}(m_{1}+m_{2})}]^{2}<\frac{3}{64\pi^{2}}.$
(3.7)

This is clearly less than the first term, so the result must be overall
positive.

  3. 3.

$n_{3}=m_{3},n_{2}=m_{1}$ and everything else generic. The level matching
condition also requires $n_{1}=m_{2}$. Correcting a typo in the expression in
[23], the result is

$\displaystyle F_{1}$
$\displaystyle=\frac{1}{48\pi^{4}m_{1}^{2}n_{1}^{2}(m_{1}-n_{1})^{4}(m_{1}+n_{1})^{2}}[9(m_{1}^{6}+n_{1}^{6})-12m_{1}n_{1}(m_{1}^{4}+n_{1}^{4})$
(3.8)
$\displaystyle-9m_{1}^{2}n_{1}^{2}(m_{1}^{2}+n_{1}^{2})+36m_{1}^{3}n_{1}^{3}+\pi^{2}m_{1}^{2}n_{1}^{2}(m_{1}^{2}-n_{1}^{2})^{2}].$

It is convenient to denote the positive integers $a=|m_{1}|,b=|n_{1}|$. Since
$m_{1}\neq\pm n_{1}$, we can estimate
$(m_{1}^{2}-n_{1}^{2})^{2}=(m_{1}+n_{1})^{2}(m_{1}-n_{1})^{2}\geq(a+b)^{2}$.
So the numerator is no less than

$9(m_{1}^{6}+n_{1}^{6})-12m_{1}n_{1}(m_{1}^{4}+n_{1}^{4})+36m_{1}^{3}n_{1}^{3}.$
(3.9)

We can further discuss two cases

    1. (a)

$m_{1},n_{1}$ have the sam sign. Then the expression (3.9) is

$9(a^{6}+b^{6})-12ab(a^{4}+b^{4})+36a^{3}b^{3}$ (3.10)

Using the inequalities

$2a^{6}+a^{3}b^{3}\geq 3a^{5}b,~{}~{}~{}~{}2b^{6}+a^{3}b^{3}\geq 3ab^{5},$
(3.11)

it is easy to see the result is positive.

    2. (b)

$m_{1},n_{1}$ have different signs. Then the expression (3.9) is

$9(a^{6}+b^{6})+12ab(a^{4}+b^{4})-36a^{3}b^{3}\geq 6a^{3}b^{3}>0.$ (3.12)

Thus we have provided a complete proof of non-negativity at genus one for any
mode numbers.

##  4 Genus Two

In this section, we will calculate the genus two BMN two-point functions. At
genus $h$, we need to divide the trace in the BMN vacuum operator
$\textrm{Tr}(Z^{J})$ into $4h$ parts, and there are $\frac{(4h-1)!!}{2h+1}$
cyclically inequivalent permutations of $(12\cdots(4h))$ representing the Wick
contractions of Feynman diagrams [36]. The analysis is quite complicated at
genus two. Fortunately, we find some symmetries that will be useful to
simplify the calculations. First we will give some explanations about the
symmetries, then we use them to write the results in terms of standard
integrals.

###  4.1 Some Useful Symmetries

Figure 1: The genus 2 diagrams

We draw the 21 cyclically inequivalent permutations
$\sigma_{i},i=1,2,\cdots,21$ in the Fig. 1. For convenience we denote the
total genus two contrition in terms of individual permutations as

$\langle\bar{O}^{J}_{(m_{1},m_{2},m_{3})}O^{J}_{(n_{1},n_{2},n_{3})}\rangle_{2}=g^{4}\sum_{i=1}^{21}F_{\sigma_{i}}(\vec{m},\vec{n}),$
(4.1)

where $\vec{m}=(m_{1},m_{2},m_{3}),\vec{n}=(n_{1},n_{2},n_{3})$. Here we
always fix $\sigma(1)=1$ by cyclic symmetry. We can then insert the three
string modes and sum over all positions with phases. We also choose to put the
third string mode in the first segment using again cyclic symmetry. The
contribution of a particular diagram can be explicitly written as

$\displaystyle F_{\sigma}(\vec{m},\vec{n})$
$\displaystyle=\int_{0}^{1}(dx_{1}\cdots
dx_{8})\delta(x_{1}+\cdots+x_{8}-1)\int_{0}^{x_{1}}dy_{3}e^{2\pi
i(n_{3}-m_{3})y_{3}}\times$ (4.2)
$\displaystyle\prod\limits_{k=1}^{2}(\int_{0}^{x_{1}}+\sum_{j=2}^{8}e^{2\pi
in_{k}(\sum\limits_{i=2}^{8}s_{\sigma}(i,j)x_{i})}\int_{x_{1}+\cdots+x_{j-1}}^{x_{1}+\cdots+x_{j}})dy_{k}e^{2\pi
i(n_{k}-m_{k})y_{k}},$

where $s_{\sigma}(i,j)$ is a sign function encoding the phase shifts of the
string modes

$s_{\sigma}(i,j)=\begin{cases}0,&i=j,~{}~{}\textrm{or}~{}~{}i<j,\sigma(i)<\sigma(j),~{}~{}\textrm{or}~{}~{}i>j,\sigma(i)>\sigma(j),\\\
1,&i>j,\sigma(i)<\sigma(j),\\\ -1,&i<j,\sigma(i)>\sigma(j).\end{cases}$ (4.3)

For example, the contribution of the first diagram $\sigma=(14863275)$ in Fig.
1 is

$\displaystyle F_{\sigma}(\vec{m},\vec{n})$
$\displaystyle=\int_{0}^{1}(dx_{1}\cdots
dx_{8})\delta(x_{1}+\cdots+x_{8}-1)\int_{0}^{x_{1}}dy_{3}e^{2\pi
i(n_{3}-m_{3})y_{3}}\times$ (4.4)
$\displaystyle\prod\limits_{i=1}^{2}(\int_{0}^{x_{1}}+e^{2\pi
in_{i}(x_{5}+x_{6})}\int_{x_{1}}^{x_{1}+x_{2}}+e^{-2\pi
in_{i}(x_{1}+x_{2}+x_{3})}\int_{x_{1}+x_{2}}^{x_{1}+x_{2}+x_{3}}+$
$\displaystyle e^{2\pi
in_{i}(x_{5}+x_{6}+x_{8}-x_{3})}\int_{\sum\limits_{j=1}^{3}x_{j}}^{\sum\limits_{j=1}^{4}x_{j}}+e^{2\pi
in_{i}(x_{6}-\sum\limits_{j=2}^{4}x_{j})}\int_{\sum\limits_{j=1}^{4}x_{j}}^{\sum\limits_{j=1}^{5}x_{j}}+e^{-2\pi
in_{i}\sum\limits_{j=2}^{5}x_{j}}\int_{\sum\limits_{j=1}^{5}x_{j}}^{\sum\limits_{j=1}^{6}x_{j}}$
$\displaystyle+e^{2\pi
in_{i}(x_{8}-x_{3})}\int_{\sum\limits_{j=1}^{6}x_{j}}^{1-x_{8}}+e^{-2\pi
in_{i}(x_{3}+x_{4}+x_{7})}\int_{1-x_{8}}^{1})dy_{i}e^{2\pi
i(n_{i}-m_{i})y_{i}}.$

Note this formalism also works for the simpler genus one case, where the
formula (LABEL:freetorus1) comes from the only permutation $(1432)$ in this
case.

A convenient way to check the normalization of (4.1) is to compare with the
summation formula (1.3). Summing over the $\vec{m}$ string modes can be done
by the Poisson summation formula $\sum_{m=-\infty}^{\infty}e^{-2\pi
imy}=\sum_{p=-\infty}^{\infty}\delta(y-p)$. The delta functions constrain the
non-vanishing contributions to the cases where all string modes are inserted
in the same segment. Using the special formula (A.5) it is straightforward to
check that for general genus $h$ the sum is indeed
$\frac{(4h-1)!!}{2h+1}\frac{1}{(4h)!}=\frac{1}{2^{2h}(2h+1)!}$.

There are two types of involution actions on a permutation $\sigma$, denoted
as $\sigma^{-1}$ and $\sigma^{\prime}$. The $\sigma^{-1}$ is the usual inverse
permutation, i.e. $\sigma(\sigma^{-1}(i))=i$, where $i=1,2,\cdots,8$, while
$\sigma^{\prime}$ is a conjugate action discussed in [1], which maps the
permutation $\sigma=(1,a_{1},a_{2},\cdots,a_{7})$ to
$\sigma^{\prime}=(1,10-a_{7},10-a_{6},\cdots,10-a_{1})$. The two types of
actions can be visualized as flipping the diagrams vertically for
$\sigma^{-1}$ and horizontally for $\sigma^{\prime}$ in Fig. 1. Some pairs are
related by both actions at the same time, and such relations are not
completely explicitly captured in the Fig. 1. Namely we have
$\sigma_{6}=\sigma_{3}^{-1}=\sigma_{3}^{\prime},\sigma_{11}=\sigma_{10}^{-1}=\sigma_{10}^{\prime},\sigma_{18}=\sigma_{17}^{-1}=\sigma_{17}^{\prime}$.
We also have some permutations invariant under the involution actions
$\sigma_{7}=\sigma_{7}^{-1},\sigma_{8}=\sigma_{8}^{-1},\sigma_{9}=\sigma_{9}^{\prime},\sigma_{12}=\sigma_{12}^{\prime},\sigma_{19}=\sigma_{19}^{-1}=\sigma_{19}^{\prime},\sigma_{20}=\sigma_{20}^{-1}=\sigma_{20}^{\prime},\sigma_{21}=\sigma_{21}^{-1}=\sigma_{21}^{\prime}$.

The contributions of the permutations related by the involution actions are
related by a simple transformation of the string mode numbers. Specifically,
we have

$F_{\sigma}(\vec{m},\vec{n})=F_{\sigma^{-1}}(-\vec{n},-\vec{m})=F_{\sigma^{\prime}}(-\vec{m},-\vec{n}).$
(4.5)

To illustrate we consider the example of the first diagram
$\sigma=(14863275)$, which has
$\sigma^{-1}=(16528473),\sigma^{\prime}=(15387426)$. Using the general
formula, we can explicitly write

$\displaystyle F_{\sigma^{-1}}(-\vec{n},-\vec{m})$ (4.6)
$\displaystyle=\int_{0}^{1}(dx_{1}\cdots
dx_{8})\delta(x_{1}+\cdots+x_{8}-1)\int_{0}^{x_{1}}dy_{3}e^{2\pi
i(n_{3}-m_{3})y_{3}}\times$
$\displaystyle\prod\limits_{i=1}^{2}(\int_{0}^{x_{1}}+e^{-2\pi
im_{i}(x_{3}+x_{4}+x_{6}+x_{8})}\int_{x_{1}}^{x_{1}+x_{2}}+e^{-2\pi
im_{i}(x_{4}+x_{6}+x_{8}-x_{2})}\int_{x_{1}+x_{2}}^{x_{1}+x_{2}+x_{3}}+$
$\displaystyle e^{2\pi
im_{i}(x_{2}+x_{3})}\int_{\sum\limits_{j=1}^{3}x_{j}}^{\sum\limits_{j=1}^{4}x_{j}}+e^{-2\pi
im_{i}\sum\limits_{j=6}^{8}x_{j}}\int_{\sum\limits_{j=1}^{4}x_{j}}^{\sum\limits_{j=1}^{5}x_{j}}+e^{-2\pi
im_{i}(x_{8}-x_{2}-x_{3}-x_{5})}\int_{\sum\limits_{j=1}^{5}x_{j}}^{\sum\limits_{j=1}^{6}x_{j}}$
$\displaystyle+e^{-2\pi
im_{i}(x_{8}-x_{5})}\int_{\sum\limits_{j=1}^{6}x_{j}}^{1-x_{8}}+e^{2\pi
im_{i}(x_{1}+x_{4}+x_{8})}\int_{1-x_{8}}^{1})dy_{i}e^{2\pi
i(n_{i}-m_{i})y_{i}}.$

We can change the phase factors in (4.4) from $\vec{n}$ to $\vec{m}$ by a
shift of the $y_{i}$ integration variable. For example we have

$\displaystyle e^{2\pi
in_{i}(x_{5}+x_{6})}\int_{x_{1}}^{x_{1}+x_{2}}dy_{i}e^{2\pi
i(n_{i}-m_{i})y_{i}}$ (4.7) $\displaystyle={}$ $\displaystyle e^{2\pi
im_{i}(x_{5}+x_{6})}\int_{x_{1}+x_{5}+x_{6}}^{x_{1}+x_{2}+x_{5}+x_{6}}dy_{i}e^{2\pi
i(n_{i}-m_{i})y_{i}}.$

We then rename the integration variables in (4.4) according to the permutation
as $x_{i}\rightarrow x_{\sigma(i)}$, or more explicitly $x_{1}\rightarrow
x_{1},x_{2}\rightarrow x_{4},x_{3}\rightarrow x_{8},x_{4}\rightarrow
x_{6},x_{5}\rightarrow x_{3},x_{6}\rightarrow x_{2},x_{7}\rightarrow
x_{7},x_{8}\rightarrow x_{5}$. It is easy to check that the expressions (4.4)
and (LABEL:ex1a) are indeed equal.

The case of $\sigma^{\prime}$ was discussed in [1], and we give some more
details here. For the example we can write

$\displaystyle F_{\sigma^{\prime}}(-\vec{m},-\vec{n})$ (4.8)
$\displaystyle=\int_{0}^{1}(dx_{1}\cdots
dx_{8})\delta(x_{1}+\cdots+x_{8}-1)\int_{0}^{x_{1}}dy_{3}e^{-2\pi
i(n_{3}-m_{3})y_{3}}\times$
$\displaystyle\prod\limits_{i=1}^{2}(\int_{0}^{x_{1}}+e^{-2\pi
in_{i}(x_{3}+x_{6}+x_{7})}\int_{x_{1}}^{x_{1}+x_{2}}+e^{-2\pi
in_{i}(x_{7}-x_{2})}\int_{x_{1}+x_{2}}^{x_{1}+x_{2}+x_{3}}+$ $\displaystyle
e^{-2\pi
in_{i}(\sum\limits_{i=5}^{8}x_{i})}\int_{\sum\limits_{j=1}^{3}x_{j}}^{\sum\limits_{j=1}^{4}x_{j}}+e^{-2\pi
in_{i}(x_{6}+x_{7}+x_{8}-x_{4})}\int_{\sum\limits_{j=1}^{4}x_{j}}^{\sum\limits_{j=1}^{5}x_{j}}+e^{-2\pi
in_{i}(x_{7}-x_{2}-x_{4}-x_{5})}\int_{\sum\limits_{j=1}^{5}x_{j}}^{\sum\limits_{j=1}^{6}x_{j}}$
$\displaystyle+e^{2\pi
in_{i}(\sum\limits_{j=2}^{6}x_{j})}\int_{\sum\limits_{j=1}^{6}x_{j}}^{1-x_{8}}+e^{2\pi
in_{i}(x_{4}+x_{5})}\int_{1-x_{8}}^{1})dy_{i}e^{-2\pi i(n_{i}-m_{i})y_{i}}.$

We change the integration variables in (4.4) by $y_{i}\rightarrow
1+x_{1}-y_{i},~{}i=1,2,3$. This gives an overall phase which is trivial by
closed string level matching conditions $\exp[2\pi
i\sum\limits_{i=1}^{3}(n_{i}-m_{i})x_{1}]=1$. The first integration range
$[0,x_{1}]$ is not changed since the $y_{i}$ can be always shifted by an
integer. The other 7 integrals are reversed in order as visualized by
horizontally flipping the diagram. We further change the integration variables
$x_{i}\rightarrow x_{10-i},~{}i=2,3,\cdots 8$ with $x_{1}$ unchanged. It is
not difficult to check that the integrals (4.4) and (LABEL:ex1b) are also
equal.

It is clear that the $\vec{m}\rightarrow-\vec{m},\vec{n}\rightarrow-\vec{n}$
transformation is the complex conjugate action. Using together with the
symmetry of inverse permutation (4.5), since the set of 21 diagrams in Fig. 1
is closed under both involution actions, we have explicitly verified the two-
point functions are real and symmetric at genus two. Similarly this works at
any higher genus $h$ since flipping the diagrams vertically or horizontally
does not change the genus

$\displaystyle\langle\bar{O}^{J}_{(m_{1},m_{2},m_{3})}O^{J}_{(n_{1},n_{2},n_{3})}\rangle_{h}^{*}=\langle\bar{O}^{J}_{(m_{1},m_{2},m_{3})}O^{J}_{(n_{1},n_{2},n_{3})}\rangle_{h}$
(4.9)
$\displaystyle\langle\bar{O}^{J}_{(m_{1},m_{2},m_{3})}O^{J}_{(n_{1},n_{2},n_{3})}\rangle_{h}=\langle\bar{O}^{J}_{(n_{1},n_{2},n_{3})}O^{J}_{(m_{1},m_{2},m_{3})}\rangle_{h}.$

Figure 2: The symmetry relations with three string insertions

In fact, there are some more refined symmetries with specified positions for
string modes. First we introduce a notation
$F_{\sigma}(\vec{m},\vec{n})(i,j,k)$, denoting the contribution of a
permutation $\sigma$ with the three string modes inserted in the $i,j,k$’th
segments. The formula (4.2) is then simply
$F_{\sigma}(\vec{m},\vec{n})=\sum_{i,j=1}^{8}F_{\sigma}(\vec{m},\vec{n})(i,j,1)$.
Suppose $P$ is one of 6 permutations of $(123)$, then we note that the
contribution is obviously invariant under the permutation of the three
positions $(i,j,k)$ and mode numbers simultaneously, namely
$F_{\sigma}(P(\vec{m}),P(\vec{n}))(P(i,j,k))=F_{\sigma}(\vec{m},\vec{n})(i,j,k)$.

Similarly as the symmetry formulas (4.5), we can derive the more refined
formulas

$\displaystyle F_{\sigma}(\vec{m},\vec{n})(i,j,k)$
$\displaystyle=F_{\sigma^{{}^{\prime}}}(-\vec{m},-\vec{n})(10-i,10-j,10-k)$
(4.10)
$\displaystyle=F_{\sigma^{-1}}(-\vec{n},-\vec{m})(\sigma(i),\sigma(j),\sigma(k)),$

where the three positions are understood to be equivalent mod 8. As an
example, in Figure (2), $\sigma=(14863275)$, three string modes
$(\phi_{1},\phi_{2},\phi_{3})$ is inserted into the segments $(2,3,1)$, and we
have the symmetry relation

$\displaystyle
F_{\sigma}(\vec{m},\vec{n})(2,3,1)=F_{\sigma^{{}^{\prime}}}(-\vec{m},-\vec{n})(8,7,1)=F_{\sigma^{-1}}(-\vec{n},-\vec{m})(4,8,1).$
(4.11)

So from one contribution, we can infer the results of the other two. In this
case their formulas in terms of standard integrals are

$\displaystyle F_{\sigma}(\vec{m},\vec{n})(2,3,1)$
$\displaystyle=I_{(3,2,2,2,2)}(0,n_{1},-n_{2},-m_{2},n_{1}+m_{3}),$ (4.12)
$\displaystyle F_{\sigma^{{}^{\prime}}}(\vec{m},\vec{n})(8,7,1)$
$\displaystyle=I_{(3,2,2,2,2)}(0,-n_{1},n_{2},m_{2},-n_{1}-m_{3}),$
$\displaystyle F_{\sigma^{-1}}(\vec{m},\vec{n})(4,8,1)$
$\displaystyle=I_{(3,2,2,2,2)}(0,-m_{1},m_{2},n_{2},-n_{3}-m_{1}).$

Figure 3: The cyclical symmetry

Finally, there is a useful residual cyclical symmetry. When writing two-point
function, we have fixed the third string mode into the first segment using the
cyclicity, which breaks the symmetry between different modes. But due to the
permutation symmetry of string modes, there is still some useful relations for
calculations. For a permutation $\sigma$, if we insert
$(\phi_{1},\phi_{2},\phi_{3})$ into the positions $(i,j,1)$, and rotate the 8
segments such that the $\sigma(i)$ or $\sigma(j)$ is put into the first
segment, the integral contribution is still the same, so we have

$\displaystyle F_{\sigma}(\vec{m},\vec{n})(i,j,1)$
$\displaystyle=F_{\sigma-\sigma(i)+1}(\vec{m},\vec{n})(1,j-i+1,2-i)$ (4.13)
$\displaystyle=F_{\sigma-\sigma(j)+1}(\vec{m},\vec{n})(i+1-j,1,2-j)$

where $\sigma-i$ means that we subtract $i$ from each number of the
permutation $\sigma$, and using the cyclicality to put 1 into the first place.
Again it is implicit that the numbers are equivalent mod 8.

As an example, consider the three permutations in Fig. 3, where
$\sigma_{1}=(14863275)$ and $\sigma_{2}=\sigma_{1}-3,\sigma_{3}=\sigma_{1}+1$.
We have the following relations also with results in terms of a standard
integral

$\displaystyle F_{\sigma_{1}}(\vec{m},\vec{n})(2,3,1)=$ $\displaystyle
I_{(3,2,2,2,2)}(0,n_{1},-n_{2},-m_{2},n_{1}+m_{3})$ (4.14) $\displaystyle=$
$\displaystyle F_{\sigma_{2}}(\vec{m},\vec{n})(1,2,8)$ $\displaystyle=$
$\displaystyle F_{\sigma_{3}}(\vec{m},\vec{n})(8,1,7).$

We see this symmetry relates the different diagrams in the group with the same
string diagram multiplicity in the classification in [1]. In our calculations
we always put $\phi_{3}$ into the first segment in every diagram, we can use
the above formulas to nicely combine the contributions into sums over the
permutations of the three string modes. For the current example we have

$\displaystyle
F_{\sigma_{1}}(\vec{m},\vec{n})(2,3,1)+F_{\sigma_{1}}(\vec{m},\vec{n})(3,2,1)+F_{\sigma_{2}}(\vec{m},\vec{n})(2,8,1)$
(4.15) $\displaystyle+$ $\displaystyle
F_{\sigma_{2}}(\vec{m},\vec{n})(8,2,1)+F_{\sigma_{3}}(\vec{m},\vec{n})(8,7,1)+F_{\sigma_{3}}(\vec{m},\vec{n})(7,8,1)$
$\displaystyle=$
$\displaystyle\sum\limits_{(i,j,k)}I_{(3,2,2,2,2)}(0,n_{i},-n_{j},-m_{j},n_{i}+m_{k}),$

where the sum $(i,j,k)$ is over the 6 permutations of (123).

###  4.2 The Result

We can classify the 21 diagrams in Fig. (1) into four groups according to [1],
then we can use the two involution symmetries to further separate the diagrams
into small groups as

$\displaystyle
8:\\{\\{(1),(2),(4),(5)\\},~{}~{}\\{(3),(6)\\},~{}~{}\\{(7),(8)\\}\\}$ (4.16)
$\displaystyle
8:\\{\\{(9),(12)\\},~{}~{}\\{(10),(11)\\},~{}~{}\\{(13),(14),(15),(16)\\}\\}$
$\displaystyle 4:\\{\\{(17),(18)\\},~{}~{}\\{(19)\\},~{}~{}\\{(20)\\}\\}$
$\displaystyle 1:\\{(21)\\}.$

To calculate the total contribution (4.1), we only need to choose one
permutation in each small group, e.g.

$(1),(3),(7),(9),(10),(13),(17),(19),(20),(21).$

The contributions of other diagrams can then be obtained by symmetry. Each
permutation consists of $\frac{8\cdot 9}{2}=36$ different standard integrals,
so the computation is still very complicated.

After some very lengthy calculations, we finally obtain the result in terms of
standard integrals, which is organized into four parts as

$\displaystyle~{}~{}I_{1}(\vec{m},\vec{n})$ (4.17)
$\displaystyle=\sum\limits_{(i,j,k)}21I_{(9,1,1)}(0,n_{i}-m_{i},-n_{j}+m_{j})+I_{(5,5,1)}(0,n_{i}-m_{i},-n_{j}+m_{j})$
$\displaystyle+I_{(5,2,2,1,1)}(0,n_{i}-m_{i},-n_{j}+m_{j},n_{i}+m_{j},-n_{j}-m_{i})$
$\displaystyle+I_{(5,1,1,1,1,1,1)}(0,n_{i}-m_{k},-n_{k}+m_{i},m_{i},n_{i},-m_{k},-n_{k})$
$\displaystyle+6I_{(3,3,2,2,1)}(0,n_{i}-m_{i},n_{i},-m_{i},-n_{j}+m_{j})$
$\displaystyle+2I_{(3,2,2,1,1,1,1)}(0,n_{j}-m_{j},-n_{k}+m_{k},n_{j},-n_{k},-m_{j},m_{k})$
$\displaystyle+I_{(2,2,1,1,1,1,1,1,1)}(0,-n_{k}+m_{k},-n_{k},m_{k},n_{i},-m_{j},-n_{k}-m_{j},n_{i}+m_{k},n_{i}-m_{j}),$
$\displaystyle~{}~{}I_{2}(\vec{m},\vec{n})$ (4.18)
$\displaystyle=\sum\limits_{(i,j,k)}5I_{(6,2,2,1)}(0,-m_{i},-n_{i},n_{k}+m_{j})+5I_{(6,2,1,1,1)}(0,-n_{i}+m_{i},-n_{i},m_{i},n_{k}-m_{k})$
$\displaystyle+I_{(5,2,2,2)}(0,n_{i}+m_{j},n_{i}-m_{i},-n_{j}+m_{j})+I_{(5,2,2,1,1)}(0,m_{i},n_{i},n_{i}+m_{i},-n_{j}-m_{k})$
$\displaystyle+I_{(5,1,1,1,1,1,1)}(0,-n_{k},m_{j},-n_{k}+m_{j},n_{i}+m_{j},n_{i}-m_{i},-n_{k}-m_{i})$
$\displaystyle+2I_{(4,3,3,1)}(0,-m_{i},-n_{i},n_{k}+m_{j})$
$\displaystyle+3I_{(4,2,2,2,1)}(0,m_{i},-n_{i},-n_{i}+m_{i},n_{k}-m_{k})$
$\displaystyle+I_{(4,1,1,1,1,1,1,1)}(0,-m_{k},n_{k},-n_{j}-m_{k},n_{k}+m_{i},m_{i},-n_{j},n_{k}-m_{k})$
$\displaystyle+4I_{(3,3,3,1,1)}(0,n_{i}-m_{i},-m_{i},n_{i},-n_{j}+m_{j})$
$\displaystyle+2I_{(3,2,2,2,2)}(0,-n_{k},-m_{j},n_{j}-m_{j},-n_{k}+m_{k})$
$\displaystyle+2I_{(3,2,2,2,1,1)}(0,n_{k},m_{k},-n_{i}-m_{j},-n_{i},-m_{j})$
$\displaystyle+I_{(3,2,2,1,1,1,1)}(0,n_{i},-m_{j},-n_{k}-m_{j},n_{i}-m_{j},n_{i}+m_{k},-n_{k}+m_{k})$
$\displaystyle+I_{(3,2,2,1,1,1,1)}(0,n_{j},-m_{i},-n_{i},m_{j},n_{j}-m_{i},-n_{i}+m_{j})$
$\displaystyle+I_{(3,2,2,1,1,1,1)}(0,n_{j},m_{j},-n_{i},-m_{i},n_{j}-m_{i},-n_{i}+m_{j})$
$\displaystyle+I_{(2,2,2,2,2,1)}(n_{j}+m_{i},m_{i},-n_{k},n_{j},-m_{k},0)$
$\displaystyle+I_{(2,2,2,2,1,1,1)}(0,n_{j}-m_{j},-n_{k}+m_{k},m_{k},-n_{k},-m_{j},n_{j}+m_{k})$
$\displaystyle+I_{(2,2,2,2,1,1,1)}(0,-n_{k},-m_{j},-n_{k}-m_{j},n_{i},n_{i}+m_{k},-n_{k}+m_{k})$
$\displaystyle+2I_{(2,2,2,1,1,1,1,1)}(0,n_{j}-m_{j},-n_{k}+m_{k},-n_{k},n_{j},-m_{j},-n_{k}-m_{j},n_{j}+m_{k})$
$\displaystyle+I_{(2,2,2,1,1,1,1,1)}(0,-n_{k},-n_{k}+m_{k},n_{i},-m_{j},-n_{k}-m_{j},n_{i}-m_{j},n_{i}+m_{k})$
$\displaystyle+2I_{(2,2,2,1,1,1,1,1)}(0,-n_{k},-n_{k}+m_{k},n_{i},-n_{k}-m_{j},-m_{j},m_{k},n_{i}+m_{k}),$

$\displaystyle~{}~{}I_{3}(\vec{m},\vec{n})$ (4.19)
$\displaystyle=\sum\limits_{(i,j,k)}I_{(3,3,1,1,1,1,1)}(0,n_{i},-n_{k},-m_{j},-n_{k}-m_{j},n_{i}+m_{k},-n_{k}+m_{k})$
$\displaystyle+2I_{(3,2,2,2,2)}(0,n_{i},-n_{j},n_{i}-m_{i},-n_{j}+m_{j})$
$\displaystyle+I_{(3,2,2,1,1,1,1)}(0,m_{j},-m_{i},n_{j},-n_{i},-m_{i}+n_{j},m_{j}-n_{i})$
$\displaystyle+I_{(2,2,1,1,1,1,1,1,1)}(0,m_{k},n_{j},n_{j}-m_{j},-m_{j},-n_{i}-m_{j},-n_{i}+m_{k},n_{j}+m_{k},-n_{i}),$

$\displaystyle~{}~{}I_{4}(\vec{m},\vec{n})$ (4.20)
$\displaystyle=\sum\limits_{(i,j,k)}5I_{(6,2,2,1)}(0,-m_{i},n_{i}-m_{i},-n_{j}+m_{j})+4I_{(5,3,2,1)}(0,m_{i},-n_{i}+m_{i},n_{k}-m_{k})$
$\displaystyle+4I_{(5,3,2,1)}(0,n_{i},m_{i},-n_{j}-m_{k})+4I_{(5,2,2,1,1)}(0,n_{i},n_{i}-m_{i},-m_{i},-n_{j}+m_{j})$
$\displaystyle+I_{(5,1,1,1,1,1,1)}(0,-n_{k},n_{i},m_{j},n_{i}+m_{j},n_{i}-m_{i},-n_{k}-m_{i})$
$\displaystyle+3I_{(4,4,2,1)}(0,n_{i},n_{i}-m_{i},-n_{j}+m_{j})+3I_{(4,3,2,1,1)}(0,n_{i},n_{i}-m_{i},-m_{i},-n_{j}+m_{j})$
$\displaystyle+2I_{(4,3,2,1,1)}(0,-m_{i},-n_{i},-n_{i}-m_{i},n_{k}+m_{j})$
$\displaystyle+I_{(4,2,1,1,1,1,1)}(0,m_{i},-m_{k},-n_{j},-m_{k}-n_{j},m_{i}+n_{k},n_{k}-m_{k})$
$\displaystyle+I_{(4,2,1,1,1,1,1)}(0,n_{j},m_{k},-n_{i},n_{j}-m_{j},-n_{i}-m_{j},n_{j}+m_{k})$
$\displaystyle+I_{(4,2,1,1,1,1,1)}(0,-m_{j},-n_{j},m_{k},n_{k},n_{k}-m_{j},-n_{j}+m_{k})$
$\displaystyle+I_{(4,2,1,1,1,1,1)}(0,-n_{k},-m_{i},m_{j},n_{i}+m_{j},n_{i}-m_{i},-n_{k}-m_{i})$
$\displaystyle+I_{(4,2,1,1,1,1,1)}(0,m_{k},-n_{i},n_{j}-m_{j},-n_{i}-m_{j},-n_{i}+m_{k},n_{j}+m_{k})$
$\displaystyle+I_{(3,3,1,1,1,1,1)}(0,-n_{k},n_{i},-m_{j},-n_{k}-m_{j},n_{i}+m_{k},-n_{k}+m_{k})$
$\displaystyle+I_{(3,3,1,1,1,1,1)}(0,n_{i},-m_{j},-n_{k}-m_{j},n_{i}-m_{j},n_{i}+m_{k},-n_{k}+m_{k})$
$\displaystyle+2I_{(3,2,2,2,2)}(0,n_{i},-m_{j},-n_{j},n_{i}+m_{k})$
$\displaystyle+2I_{(3,2,2,2,1,1)}(0,n_{j}-m_{j},-n_{k}+m_{k},m_{k},n_{j},-m_{j})$
$\displaystyle+I_{(3,2,2,1,1,1,1)}(0,n_{i},-n_{k},n_{i}+m_{j},n_{i}-m_{i},-n_{k}-m_{i},m_{j})$
$\displaystyle+I_{(3,2,2,1,1,1,1)}(0,m_{i},n_{j},-m_{k},n_{j}+m_{i},-n_{i}+m_{i},-n_{i}-m_{k})$
$\displaystyle+I_{(3,2,2,1,1,1,1)}(0,n_{i},-m_{j},-n_{k}-m_{j},n_{i}+m_{k},m_{k},-n_{k}+m_{k})$
$\displaystyle+I_{(3,2,1,1,1,1,1,1)}(0,-n_{k},n_{i}+m_{j},n_{i},n_{i}-m_{i},-m_{i},-n_{k}-m_{i},m_{j})$
$\displaystyle+I_{(3,2,1,1,1,1,1,1)}(0,m_{j},-n_{j}+m_{j},-n_{j},n_{k},-n_{j}-m_{i},n_{k}+m_{j},-m_{i})$
$\displaystyle+I_{(2,2,2,2,2,1)}(0,-n_{k},n_{j}-m_{j},-n_{k}-m_{j},-n_{k}+m_{k},n_{j})$
$\displaystyle+I_{(2,2,2,2,2,1)}(0,n_{j},-n_{k},n_{j}-m_{j},-n_{k}+m_{k},-m_{j})$
$\displaystyle+2I_{(2,2,2,2,1,1,1)}(0,n_{j}-m_{j},-m_{j},-n_{k}+m_{k},-n_{k},-n_{k}-m_{j},m_{k})$
$\displaystyle+I_{(2,2,2,1,1,1,1,1)}(0,-n_{k}-m_{j},-m_{j},-n_{k},n_{i},n_{i}-m_{j},n_{i}+m_{k},-n_{k}+m_{k})$
$\displaystyle+I_{(2,2,2,1,1,1,1,1)}(0,-m_{j},m_{k},-n_{j},n_{k},n_{k}-m_{j},-n_{j}+m_{k},m_{k}+n_{k})$
$\displaystyle+I_{(2,2,2,1,1,1,1,1)}(0,n_{j},-m_{i},m_{j},-n_{i},-n_{i}-m_{i},n_{j}-m_{i},-n_{i}+m_{j}),$

where $(i,j,k)$ sums over the 6 permutations of (123), so each term is
apparently invariant under permutations of the three string modes. There are
7, 20, 4, 28 elements in $I_{1},I_{2},I_{3},I_{4}$ with a total of 59
elements. The total contribution to the two-point function includes the
elements in $I_{1},I_{2},I_{3},I_{4}$ and their transformations as

$\displaystyle~{}~{}~{}~{}g^{-4}\langle\bar{O}^{J}_{(m_{1},m_{2},m_{3})}O^{J}_{(n_{1},n_{2},n_{3})}\rangle_{2}$
(4.21)
$\displaystyle=I_{1}(\vec{m},\vec{n})+I_{2}(\vec{m},\vec{n})+I_{2}(-\vec{m},-\vec{n})+I_{3}(\vec{m},\vec{n})+I_{3}(-\vec{n},-\vec{m})$
$\displaystyle+I_{4}(\vec{m},\vec{n})+I_{4}(-\vec{m},-\vec{n})+I_{4}(-\vec{n},-\vec{m})+I_{4}(\vec{n},\vec{m})$
$\displaystyle=I_{1}(\vec{m},\vec{n})+I_{3}(\vec{m},\vec{n})+I_{3}(-\vec{n},-\vec{m})+2\Re[I_{2}(\vec{m},\vec{n})+I_{4}(\vec{m},\vec{n})+I_{4}(\vec{n},\vec{m})].$

So together with their transformations, each element in
$I_{1},I_{2},I_{3},I_{4}$ contributes 1,2,2,4 times. In the second equality we
have used the fact that the transformation
$(\vec{m},\vec{n})\rightarrow-(\vec{m},\vec{n})$ is the complex conjugation.

We can analyze the reality property. Each element in
$I_{1}(\vec{m},\vec{n}),I_{3}(\vec{m},\vec{n}),I_{3}(-\vec{n},-\vec{m})$
summing over the $(i,j,k)$ permutations can be easily shown to be real using
some obvious symmetries of the standard integrals. On the other hand, each
element in
$I_{2}(\vec{m},\vec{n}),I_{4}(\vec{m},\vec{n}),I_{4}(\vec{n},\vec{m})$ summing
over the $(i,j,k)$ permutations is not always real with the exception of one
element in $I_{2}$, namely we can explicitly check that the reality of the
following sum

$\sum\limits_{(i,j,k)}I_{(2,2,2,1,1,1,1,1)}(0,n_{j}-m_{j},-n_{k}+m_{k},-n_{k},n_{j},-m_{j},-n_{k}-m_{j},n_{j}+m_{k}),$
(4.22)

though this is not easily derived by obvious symmetries of the standard
integrals.

For the generic case where none of $m_{i},n_{i},m_{i}\pm n_{j}$ is zero, we
note that the element with indices at most 2, i.e. the standard integral
$I_{(2,\cdots)}$ appearing here, always has vanishing real part. This is not
obvious from symmetries of the standard integrals but can be checked
explicitly. This property is no longer always true if there are some mode
number degeneracies.

We can perform some tests on the rather complicated final result
(LABEL:totalresult). For example, when two string modes are zero
$m_{3}=n_{3}=0$, the result correctly reduces to the case of two string modes,
previously first obtained in [27]. For the case $m_{3}=0,n_{3}\neq 0$, the
result correctly vanishes. We can also test numerically the summation formula
(1.3). We fix one set of string modes and perform the sum over the other set
of modes numerically. As an example, we check that the two-point functions
with $\vec{m}=(1,1,-2)$ and all $\vec{n}$ modes in the range
$\max(|n_{i}|)\leq 1424$ are always non-negative, and their numerical sum is
0.999001 of the exact value $\frac{1}{1920}$.

It seems far from obvious the expression (LABEL:totalresult) is always non-
negative. We check numerically for all cases with mode numbers
$\max(|m_{i}|,|n_{i}|)\leq 30$. Furthermore, we also test some random mode
numbers in larger range. For example, we take random integers
$m_{i},n_{i},i=1,2$ from a Gaussian distribution with expectation value 0 and
standard deviation 100, while $m_{3}=-m_{1}-m_{2},n_{3}=-n_{1}-n_{2}$ are then
determined. We compute the result (LABEL:totalresult) for more than a million
random sets of mode numbers in this way. In all tests we have not found any
negative result.

For the generic case, an analytic expression can be obtained, but it is too
long to write down completely for analysis. We find that the result
(LABEL:totalresult) can be written as

$f_{-4}(\vec{m},\vec{n})+f_{-6}(\vec{m},\vec{n})+f_{-8}(\vec{m},\vec{n}),$
(4.23)

where $f_{k}(\vec{m},\vec{n})$ is a rational function of the string mode
numbers with homogeneous degree $k$, which is also the degree of $\pi$, the
only irrational number appearing in the final result. The expressions
$f_{-6},f_{-8}$ are quite complicated, and can be either positive or negative.
However, we find that the $f_{-4}$ expression is quite simple and is actually
related to the genus one result (3.3) up to a numerical constant

$f_{-4}(\vec{m},\vec{n})=\frac{7\sum_{i=1}^{3}(m_{i}-n_{i})^{2}}{3840\pi^{4}\prod_{i=1}^{3}(m_{i}-n_{i})^{2}}.$
(4.24)

We also note that $f_{-4}$ comes from the element with most degenerate
indices, namely the first element $I_{(9,1,1)}(0,n_{i}-m_{i},-n_{j}+m_{j})$ in
$I_{1}$ in (LABEL:I1), which in turns comes from the contributions with all
three string modes in the same of the 8 segments. For string mode numbers with
large absolute values, the manifestly positive term $f_{-4}(\vec{m},\vec{n})$
should be dominant, so it is indeed plausible that the total result should
remain positive in this case.

We expect this argument works also at general genus $h$. For generic mode
numbers with large absolute values, the dominant contributions should come
from the cases in the $\frac{(4h-1)!!}{2h+1}$ permutations where all string
modes are in the same segment. For generic case we can first perform the $dy$
integrals in e.g. (4.2) for the genus two case. Using the formulas in Appendix
A, it is not difficult to calculate the leading term

$\displaystyle~{}~{}~{}\langle\bar{O}^{J}_{(m_{1},m_{2},m_{3})}O^{J}_{(n_{1},n_{2},n_{3})}\rangle_{h}$
(4.25) $\displaystyle=g^{2h}\frac{(4h-1)!!}{2h+1}\int_{0}^{1}(dx_{1}\cdots
dx_{4h})\delta(x_{1}+\cdots+x_{4h}-1)\prod_{i=1}^{3}\frac{e^{2\pi
i(n_{i}-m_{i})x_{1}}-1}{2\pi i(n_{i}-m_{i})}+\cdots$
$\displaystyle=\frac{g^{2h}}{(2\pi
i)^{3}}\frac{(4h-1)!!}{2h+1}(\prod_{i=1}^{3}\frac{1}{n_{i}-m_{i}})\sum_{i=1}^{3}[I_{(4h-1,1)}(0,n_{i}-m_{i})-I_{(4h-1,1)}(0,m_{i}-n_{i})]+\cdots$
$\displaystyle=g^{2h}\frac{h(4h-1)}{2^{2h+2}(2h+1)!}\frac{\sum_{i=1}^{3}(m_{i}-n_{i})^{2}}{\pi^{4}\prod_{i=1}^{3}(m_{i}-n_{i})^{2}}+\cdots,$

which is also proportional to the genus one formula (3.3) and manifestly
positive. Of course, this argument does not apply to the cases with small
absolute values of string modes, where the non-negativity seems highly non-
trivial.

The similar phenomenon actually also appears in the calculations in the case
of two and four string modes in previous papers. The dominant contribution for
generic mode numbers with large absolute values also comes from the situations
where all string modes are inserted in the same segment. In these cases of an
even number $k$ of string modes, the dominant contribution has the same sign
as $(2\pi i)^{k}\prod_{i=1}^{k}(n_{i}-m_{i})$, which is always positive for
$k=2$, but can be either positive or negative for $k\geq 4$.

For the case of two string modes, the non-negativity is valid separately for
groups of diagrams related by cyclic symmetry, since the contributions in a
group can be combined into an integral of the form
$\int_{0}^{1}\cdots\prod_{i=1}^{4h}dx_{i}\delta(\sum_{i=1}^{4h}x_{i}-1)$,
where the $\cdots$ denotes a product of two complex conjugate factors. One may
wonder whether this is also true for the case of three string modes. At genus
two, the 21 diagrams are divided into four groups (LABEL:biggroups). In this
case, each group is closed under the two involution actions
$\sigma\rightarrow\sigma^{-1},\sigma^{\prime}$, so their individual
contributions with three string modes are still real and symmetric 111This
would be no longer true for higher genus $h\geq 3$. At genus three, there are
1485 diagrams, i.e. permutations of $(1,2,\cdots,12)$ with 1 fixed at the
first position, divided into 131 groups according to cyclic symmetry. Most
groups, i.e. 118 of them, have 12 diagrams, while the remaining groups may
have 6,4,1 diagrams. There is also a group with a single diagram
$(1,8,3,10,5,12,7,2,9,4,11,6)$, which is invariant under both
$\sigma\rightarrow\sigma^{-1},\sigma^{\prime}$ actions. Under the action
$\sigma\rightarrow\sigma^{-1}$, 27 groups are closed while the others form 52
pairs. Under the action $\sigma\rightarrow\sigma^{\prime}$, 33 groups are
closed while the others form 49 pairs. . We further test the non-negativity
for the four groups (LABEL:biggroups) with random mode numbers. In our tests
we have not found any negative result for the first three groups. However, for
the last group with a single diagram, there are some negative results, though
such occurrences are quite rare, e.g. with random mode numbers
$m_{i},n_{i}\sim[-30,30]$, about 1/500 of the test results are negative. Of
course, the total contributions of the four groups are always non-negative in
all our tests.

##  5 Conclusion

The main result of the paper is the explicit form (LABEL:totalresult) of BMN
two-point functions at genus two. We are able to verify numerically and give
an analytic argument in the cases of generic mode numbers with large absolute
values that they are always non-negative. Now that we are more confident in
the validity of the conjectured non-negativity for three string modes, it
would be certainly better to search for a universal analytic proof, at genus
two and further at any higher genus. Although the integrals are completely
elementary, it seems a complete proof may require some advanced mathematical
techniques, probably relating to the moduli space of higher genus Riemann
surfaces. Such a proof may elucidate the surprising entry (1.6) of the pp-wave
holographic dictionary.

It is well known that string theory has a Hagedorn temperature inverse
proportional to the string length, which may be interpreted as a maximal
temperature or a temperature of phase transition. In our physical setting, the
tensionless strings have effectively infinite length, so the Hagedorn
temperature is zero. This is not necessarily a problem, since for example an
extremal black hole emits no Hawing radiation, has zero temperature, but can
still have finite event horizon and entropy. In [25], we consider the entropy
of these tensionless BMN strings, which can be interpreted as the Von-Neumann
entropy of a mixed BMN state due to decoherence. It would be interesting to
see whether we can further study some interesting thermodynamics under such
extreme conditions, e.g. along the line of the classic paper [37].

It would be very interesting to make connections with the influential
Swampland Conjectures [38], which has been studied extensively in the
literature, e.g. [39, 40]. In particular, the Swampland Distance Conjecture
[41] states that at large distance in the moduli space, an infinite tower of
exponentially light states will emerge, invalidate the effective action
description. In our setting, the tensionless strings are an infinite tower of
degenerate states with quantum transitions between them without energy cost.
Such systems do not have local effective action description but can be still
regarded as consistent quantum theories. It would be interesting to better
understand the relative place where our present setting fits in the string
landscape and the swampland.

It is widely believed that strings may not be the right fundamental degrees of
freedom to formulate the still mysterious M-theory, since there are non-
perturbative objects like D-branes and M-branes. On the contrary, if our
conjecture (1.6) is correct, it seems that in our very special setting with
infinite spacetime curvature and Ramond-Ramond flux, the tensionless closed
strings do provide the proper complete degrees of freedom for physics at any
coupling constant. Although we focus on a highly unrealistic special
situation, our studies may provide some insights for the non-perturbative
formulation of string/M-theory on general backgrounds.

Acknowledgments

We thank Jun-Hao Li, Jian-xin Lu, Gao-fu Ren, Pei-xuan Zeng for helpful
discussions. This work was supported in parts by the national Natural Science
Foundation of China (Grants No.11947301 and No.12047502).

##  Appendix A Some Standard Integrals

We will use some standard integrals as in [1], which is defined by

$\displaystyle I(u_{1},u_{2},\cdots,u_{r})\equiv\int_{0}^{1}dx_{1}\cdots
dx_{r}\delta(x_{1}+\cdots+x_{r}-1)e^{2\pi i(u_{1}x_{1}+\cdots u_{r}x_{r})}.$
(A.1)

It is clear that the integral is unchanged if we add an integer to all the
arguments. If some of the $u_{i}$’s are identical, one uses the following
notation

$\displaystyle I_{(a_{1},\cdots,a_{r})}(u_{1},u_{2},\cdots,u_{r})\equiv
I(u_{1},\cdots,u_{1},u_{2},\cdots,u_{2},\cdots,u_{r},\cdots,u_{r}),$ (A.2)

where $a_{i}$’s are integers representing the numbers of the $u_{i}$’s in the
right hand side, and for $a_{i}=0$ we can just eliminate the corresponding
argument. The integral can be calculated by some recursion relations, and one
can obtain the following explicit formulas

$\displaystyle I(u_{1},u_{2},\cdots u_{r})$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{r}e^{2\pi iu_{i}}\prod_{j\neq i}\frac{1}{2\pi
i(u_{i}-u_{j})},$ (A.3) $\displaystyle
I_{(a_{1}+1,\cdots,a_{r}+1)}(u_{1},\cdots,u_{r})$ $\displaystyle=$
$\displaystyle\prod_{i=1}^{r}\frac{(\partial/\partial u_{i})^{a_{i}}}{(2\pi
i)^{a_{i}}a_{i}!}I(u_{1},\cdots,u_{r}),$ (A.4)

where the $u_{i}$’s are different. We note we have used the $i$ symbol for
both the pure imaginary number and the product index, which are easy to
distinguish and should not cause confusion. In our calculations, the arguments
$u_{i}$’s will be linear combinations of the integral string mode numbers, so
the end results are always rational functions of string mode numbers without
the exponential functions.

A useful special case is when all arguments are degenerate at an integer.
Applying the formulas we have

$I_{n+1}(0)=\int_{0}^{1}dx_{1}\cdots
dx_{n+1}\delta(x_{1}+\cdots+x_{n+1}-1)=\frac{1}{n!},$ (A.5)

which is simply the volume of the standard $n$-dimensional simplex

$\Delta_{n}=\\{(x_{1},x_{2},\cdots,x_{n})\in\mathbb{R}^{n}|x_{i}\geq
0,\sum_{i=1}^{n}x_{i}\leq 1\\}.$ (A.6)

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# Impact of Different Infrastructures and Traffic Scenarios on Behavioral and
Physiological Responses of E-scooter Users

Dong Chen1, Arman Hosseini1, Arik Smith1, David Xiang1, Arsalan Heydarian1,
Omid Shoghli2, Bradford Campbell1 1Dong Chen, Arman Hosseini, Arik Smith,
David Xiang, Arsalan Heydarian, and Bradford Campbell are with the University
of Virginia, Charlottesville, VA, 22903, USA. Email: {dqc4vv, ufy5tc, qgn5zs,
mwd4wm, heydarian<EMAIL_ADDRESS>Shoghli is with Civil
Engineering Technology and Construction Management, University of North
Carolina at Charlotte, Charlotte, NC 28223, USA. Email:
<EMAIL_ADDRESS>
###### Abstract

As micromobility devices such as e-scooters gain global popularity, emergency
departments around the world have observed a rising trend in related injuries.
However, the majority of current research on e-scooter safety relies heavily
on surveys, news reports, and data from vendors, with a noticeable scarcity of
naturalistic studies examining the effects of riders’ behaviors and
physiological responses. Therefore, this paper aims to study the responses of
e-scooter users under different infrastructures and scenarios through
naturalistic riding experiments. The findings indicate that different speed
profiles, infrastructural elements, and traffic scenarios significantly
influence riding dynamics. The experimental results also reveal that
e-scooters face amplified safety challenges when navigating through areas with
speed variations and without dedicated riding spaces. The study underscores
the importance of considering infrastructure design and its influence on
e-scooter safety, providing insights that could inform future urban planning
and policy-making to enhance the safety of these increasingly popular
vehicles.

###### Index Terms:

E-scooter, micromobility, safety, physiological responses, eye tracking, gaze

##  I Introduction

The integration of emerging modes of transportation, such as e-scooters,
presents substantial safety challenges that hinder their smooth integration
into our urban ecosystems. Between 2017 and 2021, injuries associated with
micromobility vehicles surged by 127% to 77,200. During this time, e-scooters
witnessed the most significant rise in injuries and fatalities [1].
Specifically, according to the U.S. Consumer Product Safety Commission (CPSC),
e-scooters have seen a continuous year-over-year increase in injuries, with a
22% rise in 2022 compared to 2021 [2]. A comprehensive study by the Centers
for Disease Control and Prevention (CDC) [3] found that 20 out of every
100,000 e-scooter trips resulted in injuries. Half of these injuries are head
injuries, with a significant 15% related to severe traumatic brain injuries.
Even more alarmingly, e-scooter-related injuries among children have risen
noticeably, with hospitalizations increasing from 4.2% in 2011 to 12.9% in
2020 [4, 5, 6]. Given these alarming trends, the urgent need for innovative
research to bridge the safety knowledge gap becomes evident, paving the way
for safer e-scooter integration.

Previous research has primarily used emergency room records and media reports
to gather crash information and conduct safety studies [7, 8]. For instance,
the study by [7] analyzed treatment records from two emergency departments,
revealing that 44% of e-scooter incidents took place on sidewalks. Similarly,
[8] utilized a comprehensive collection of news reports detailing 169
e-scooter-involved crashes across the US between 2017 and 2019, finding that
the majority of these incidents occurred on streets, intersections, and
sidewalks. Additionally, survey and observational studies have been pivotal in
this realm [9, 10, 11, 12, 13]. For example, targeted research in Washington,
DC, USA, involving roadside observations of 2,004 e-scooters, found that
riders more commonly used sidewalks (60%) rather than roads (40%) in areas
devoid of bike lanes. These observations stress the importance of exploring
innovative strategies to comprehend the behavioral patterns and psycho-
physiological responses of e-scooter users in various roadway conditions and
infrastructures to enhance safety measures [13].

Despite these valuable insights, the aforementioned studies have not
adequately captured the critical situational interactions between riders and
urban infrastructure that precipitate accidents. This gap underscores the
critical need for naturalistic studies that observe and record behaviors and
responses in the complex, uncontrolled conditions of real-world settings [14].
Such studies are uniquely positioned to reveal the nuanced dynamics of
e-scooter usage, offering insights into how different infrastructural elements
and traffic scenarios directly impact rider safety. Nevertheless, focused
research on e-scooter safety through naturalistic methods remains relatively
scarce [14]. Notably, [15] and [16] delved into the naturalistic studies of
risks associated with e-scooter vibrations, which can impact riding comfort
due to varying road surfaces and wheel sizes. Moreover, [17] analyzed
e-scooter interactions with urban infrastructure, revealing that vibrations,
speed variations, and environmental constraints elevate safety risks,
emphasizing the necessity for research to enhance infrastructure and policies.
Similarly, [18] investigated the visual attention of pedestrians, cyclists,
and e-scooter users through mobile eye tracking, showing that while fixation
rates per minute were comparable across these groups, the distribution of
their visual attention varied significantly. However, this research did not
cover diverse traffic conditions, and its analysis was confined to fixation
metrics, omitting critical metrics like gaze entropy and variability [19, 20].

This paper aims to investigate the interplay between e-scooter users’ behavior
and physiological responses across various traffic scenarios and
infrastructures to enhance our understanding of how these factors contribute
to overall road safety and user experience. Specifically, we examine the
influence of riding on different road layouts, such as sidewalks, pedestrian
trails, and roads with or without bike lanes. Moreover, we assess human
reactions across different traffic scenarios, ranging from bike lane
transitions to pedestrian interactions, to capture a comprehensive picture of
e-scooter navigation challenges. Furthermore, this paper utilizes diverse
metrics such as gaze density heat maps, the percentage of road center focus,
gaze entropies, gaze variations, and fixation details, offering a nuanced view
of e-scooter users’ visual and cognitive interaction with their surroundings.
By highlighting the necessity for custom-tailored infrastructure and well-
informed policy decisions, this study aims to advance urban safety and
navigability for e-scooter users, fostering a more integrated and safe
coexistence among all road participants.

##  II METHODOLOGY

###  II-A Experiment setup

This paper investigates e-scooter riders’ behavioral and physiological
reactions to varying roadway designs and infrastructural elements. The
naturalistic studies were conducted using a Ninebot MAX KickScooter, equipped
with a 350W motor capable of reaching speeds up to 18.6 mph and a range of 25
miles, supporting riders up to 220 lbs. Additionally, the SpeedTracker app
further enhances data collection, recording vital travel information such as
GPS, speed, elevation, and timestamps at one-second intervals (see Figure 1).

Instrumentation also included Tobi Pro Glasses 3, which captured a wealth of
data on gaze behavior, including gaze patterns, eye movements, saccades, and
visual fixations. This device, equipped with a 3-axis gyroscope and
accelerometer, recorded continuous forward-facing video at a resolution of
$1920\times 1080$ pixels with a 90-degree field of view at 25 Hz.
Additionally, a Samsung Galaxy smartwatch was used to record heart rate
variations. Although this study primarily focuses on the visual data, a
detailed analysis of heart rate variations across different scenarios is
planned for future research.

Figure 1: E-Scooter instrumentation for naturalistic study.

###  II-B Experiment procedure & Data cleaning

All riding experiments were carried out by two male researchers (coauthors of
this paper), each approximately 6 feet tall and experienced in e-scooter
riding. These trials involved navigating diverse routes around the University
of Virginia (UVA) and the downtown area of Charlottesville, VA, USA, to
explore various road structures and traffic scenarios. To guarantee a thorough
evaluation, each experiment covered different routes. All experiments took
place during daylight, lasting about an hour each, under either sunny or
cloudy weather conditions, avoiding any trials in the rain. Prior to each
ride, pre-ride safety inspections and equipment calibrations were meticulously
performed to ensure the reliability of all data collection tools. It is
important to note that this paper does not focus on the impact of human
factors (i.e., riders’ weight and age) on riding risk, which are topics
planned for future research.

In total, we collected data from 16 trips, amounting to approximately 16 hours
of riding data. Each trip provided extensive information, encompassing speed,
GPS, elevation, and gaze data. The collected data was meticulously
synchronized for analysis, aligning with the frame rate of the gaze data. The
data cleaning phase involved several crucial steps: filtering out noise from
the GPS and gaze data, managing any instances of missing data, and rectifying
discrepancies within the datasets to enhance the dataset’s quality for
analysis. Subsequently, various traffic scenarios encountered across these
trips were methodically classified by our research team, alongside the
corresponding trip data, facilitating a structured analysis framework [21].
For detailed information on our methodology and data processing techniques,
readers are referred to our open-source
codes111https://github.com/DongChen06/Escooter_ITSC.

###  II-C Physiological data analysis

Eye gaze patterns have been employed across various studies to deduce human
states. Features like fixation duration, direction-specific gaze variability,
and gaze entropies have demonstrated correlations with states including
workload, stress levels, and emotions [22, 19].

Fixations, which occur when the eyes cease scanning and maintain central
foveal vision on specific points, represent the most prevalent aspect utilized
in eye tracking to deduce cognitive processes or states. The Tobii Pro Lab
software, designed for Tobii glasses, incorporates a built-in algorithm that
considers a single gaze lasting 0.02 seconds as a fixation. When three or more
gazes fall within a 0.5∘ radius of each other, they are classified as a
fixation. The Mean Fixation Length is determined by calculating the average
duration of all fixation events within a specified time interval. The
Percentage of Road Center (PRC) is then defined as the proportion of fixation
duration that occurs within the road center relative to the total fixation
duration for any chosen time interval.

Stationary Gaze Entropy (SGE) measures the predictability of fixation
locations [22], indicating gaze dispersion during a viewing period and
quantifying attention distribution as follows:

$\mathcal{H}_{\text{SGE}}(x)=-\sum_{i=1}^{n}p_{i}\log_{2}p_{i},$ (1)

where $\mathcal{H}_{\text{SGE}}(x)$ represents the entropy value for a given
set $x$ (each time bin within each scenario), with $i$ denoting each
fixation’s location on a 2D plane within $x$, $n$ is the total number of
fixations in $x$, and $p_{i}$ is the proportion of fixations at each state
space within $x$.

Gaze Transition Entropy (GTE) extends this analysis to the dynamics of gaze
shift [23], applying conditional entropy to first-order Markov transitions
between fixations:

$\mathcal{H}_{\text{GTE}}(x)=-\sum_{i=1}^{n}p_{i}\sum_{j=1}^{n}p(i|j)\log_{2}p(i|j),i\neq
j.$ (2)

where $p_{i}$ is the stationary distribution across fixation locations, and
$p(i|j)$ represents the transition probability to location $j$ from $i$. This
equation captures the variability in gaze transitions, reflecting the
complexity of visual attention shifts. For the analysis under different
conditions, fixation coordinates were discretized by categorizing them into
spatial bins of $100\times 100$ pixels [23, 20].

##  III Results

###  III-A Speeds

Figure 2 presents the gaze density heat map across various speed profiles.
Visual analysis of the gaze heat map reveals that the low-speed scenario
exhibits a more dispersed distribution of gazes compared to the other
scenarios. In contrast, the high-speed scenario (>20 mph) demonstrates a
higher concentration of gazes towards the center of the viewing area, with the
medium-speed scenario (10-20 mph) showing a similar, albeit less concentrated
pattern.

Figure 2: Gaze density variation across three speed intervals. At lower speeds
(0-10 mph), gaze points are broadly distributed, indicating a wider visual
scan. Medium speeds (10-20 mph) start to show focused attention, which further
intensifies at speeds above 20 mph.

Figure 3 depicts a representative round trip on an e-scooter beginning and
ending at the University of Virginia (UVA) and passing through the downtown
area of Charlottesville. Speed variations along the route are visually
depicted using color gradients, with darker hues indicating higher speeds. The
figure includes detailed views of two segments characterized by high-speed
travel on downhill paths. Notably, these segments lack dedicated bike lanes,
compelling e-scooter riders to share the road with vehicles and pedestrians, a
factor that significantly escalates safety concerns for all road users.
Particularly concerning is that one of these high-speed areas aligns precisely
with the location of a documented accident, occurring after the data
collection period, where an e-scooter rider was struck by a car and
subsequently required hospitalization222https://www.wric.com/news/western-
virginia/person-hit-by-car-while-riding-scooter-on-uvas-campus/, highlighting
the real-world dangers present in these high-risk zones.

Figure 3: Visualization of one representative e-scooter trip on December 11,
2023, detailing a complete loop from UVA through downtown Charlottesville with
speed variations marked by color coding: deeper shades represent higher
speeds. Two zoom-in areas emphasize downhill sections without bike lanes where
higher speeds were reached, including a site of a documented accident.

###  III-B Infrastructures

The infrastructural analysis offers critical insights into the interaction
between e-scooter riders and the built environment. To comprehensively assess
this dynamic, we examined the effects of riding on various types of
infrastructure, including sidewalks, pedestrian trails (sidewalks situated
away from main roads), bike lanes, and roads without bike lanes.

Figures 4 show PRC values corresponding to these infrastructure types. Riding
on sidewalks and pedestrian trails exhibits varied PRC, indicating extensive
scanning due to the presence of pedestrians and obstacles. Conversely, riding
on dedicated bike lanes displays a narrower focus, suggesting a more
straightforward path of travel with less navigational stress. In the case of
roads without bike lanes, the PRC values are typically more closely grouped or
concentrated than those on sidewalks or pedestrian trails but more varied than
in bike lanes. This pattern indicates a higher level of concentrated attention
required by riders to safely share the road with motor vehicles. The presence
of vehicles introduces a significant element of risk, necessitating constant
awareness and quick responsiveness to dynamic traffic conditions. The
necessity for heightened awareness in scenarios without bike lanes is
quantitatively supported by the increase in gaze variability as depicted in
Figures 5. These variations in gaze patterns emphasize the rider’s need to
continuously monitor their surroundings for potential hazards, a task that is
evidently less demanding on roads with bike lanes, as shown by the lower gaze
variability scores.

Figure 4: Comparison of PRC across infrastructures. Sidewalks and pedestrian
trails show varied PRC, indicating extensive scanning due to pedestrian
presence. Bike lanes exhibit focused gazes, suggesting easier navigation,
while the tight PRC on roads without bike lanes reflects concentrated
attention. Figure 5: Gaze variability across different infrastructures
indicates increased alertness on roads without bike lanes and reduced visual
scanning on bike lanes, suggesting safer navigation conditions.

The elevated SGE and GTE observed on roads without bike lanes, as shown in
Figure 6, signify a more dynamic and unpredictable visual engagement by
riders, necessitated by the challenging and less structured road environment.
These higher entropy values imply that riders must continuously adapt their
focus and attention to navigate safely, reflecting the high cognitive load
required in these conditions. Conversely, the reduced entropy on roads with
bike lanes indicates a more straightforward riding experience, where riders
can maintain a consistent gaze with less need to frequently adjust their
focus. For sidewalks and pedestrian trails, the moderate levels of SGE and GTE
suggest a need for frequent scanning and attention adjustments due to the
varied presence of pedestrians and other obstacles, pointing to a moderately
complex navigation environment.

Figure 6: SGE and GTE across different infrastructures. Higher SGE and GTE on
roads without bike lanes imply varied gaze patterns due to complex riding
conditions, contrasting with more stable gaze behavior on bike-laned roads.

Similarly, shorter mean fixation lengths on sidewalks and pedestrian trails,
as shown in Figure 7, imply less cognitive strain in these predictable
environments. In contrast, longer fixations and a higher frequency of
fixations per second on roads without bike lanes indicate a more challenging
navigational context, demanding increased alertness.

Figure 7: Fixations across different infrastructures, revealing shorter
durations on sidewalks and walkways, suggesting ease of navigation, while
longer fixations and increased frequency on roads without bike lanes highlight
greater challenges. Figure 8: Gaze density heat maps for different traffic
scenarios illustrating riders’ visual attention. “Bike lane to crosswalk” and
“Intersection” show focused gazes for critical navigation. “Occupied
crosswalk” displays widespread attention, while “Downhill” and “Close
proximity” reveal concentration due to potential risks. “Passing bus” and
“Passing pedestrian” indicate careful monitoring of dynamic elements. Table I:
Statistics of visual attention across different traffic scenarios. Metrics
with higher values ($\uparrow$) suggest more focused attention or cognitive
load, while lower values ($\downarrow$) indicate less cognitive demand and
potentially safer riding. Noted that these metrics are context-dependent and
should be interpreted relative to the specific traffic situation.

Events | B.L. to cross. | Intersection | Occupied cross. | Downhill | Close pro. | Road to B.L. | Passing bus | Passing ped.
---|---|---|---|---|---|---|---|---
PRC ($\uparrow$) | 0.40 | 0.49 | 0.48 | 0.72 | 0.70 | 0.72 | 0.63 | 0.53
Mean fixation length (s) ($\uparrow$) | 0.21 | 0.22 | 0.22 | 0.25 | 0.25 | 0.25 | 0.26 | 0.27
Fixations per second ($\downarrow$) | 2.95 | 2.96 | 3.19 | 2.94 | 3.01 | 2.99 | 2.86 | 3.03
SGE ($\downarrow$) | 5.77 | 5.32 | 4.30 | 5.63 | 4.03 | 4.52 | 4.66 | 4.41
GTE ($\downarrow$) | 28.68 | 21.94 | 11.24 | 20.92 | 7.34 | 9.20 | 12.52 | 11.53
Gaze variability x ($\downarrow$) | 209.96 | 167.79 | 152.41 | 113.22 | 107.13 | 116.25 | 166.50 | 117.83
Gaze variability y ($\downarrow$) | 115.82 | 105.47 | 73.39 | 83.04 | 87.82 | 71.47 | 76.37 | 73.82

###  III-C Traffic scenarios

Figure 8 depicts the gaze distribution across various traffic scenarios,
showing distinct attentional patterns of e-scooter riders. In transition areas
like “Bike lane to crosswalk” and “Intersection”, a concentrated gaze suggests
riders’ focused navigation. Dispersed patterns in “Occupied crosswalks”
(crosswalks occupied by pedestrians or vehicles) indicate searching behavior,
while the tight clustering in “Downhill” and “Close proximity” scenarios
suggests heightened alertness due to speed and spatial constraints. The
pattern in “Passing bus” is somewhat scattered but centered, suggesting that
while the bus itself is a focus, riders must also monitor surrounding traffic
conditions closely. The “entry into a bike lane” and “passing pedestrians”
also demonstrate deliberate gaze strategies, emphasizing the cognitive
complexity of urban micromobility. These findings highlight the need for
infrastructure that accommodates the diverse visual demands of riders in
different traffic contexts.

Table I provides a detailed evaluation with quantitative metrics for each
scenario. Notably, the highest PRC (0.72) is observed in the “Downhill” and
“Road to bike lane” scenarios, suggesting that riders devote a significant
portion of their attention to the roads. Interestingly, “Passing pedestrian”
situations elicit the longest mean fixation length, potentially reflecting the
need for riders to monitor pedestrian movements closely. Fixations per second
are notably elevated in “Occupied crosswalks”, which could reflect the rapid
visual assessment required in dynamic environments. This metric, coupled with
the highest SGE and GTE observed in “Bike lane to crosswalk” scenarios,
underscores the cognitive complexity of navigating through these areas. Gaze
variability, measured both horizontally (x) and vertically (y), is found to be
the most significant in “Bike lane to crosswalk” scenarios, further
highlighting the unpredictable nature and visual demands of these transition
spaces.

##  IV Conclusion

In conclusion, this paper presented a comprehensive examination of the
behavioral and physiological responses of e-scooter users across various road
infrastructures and traffic scenarios, filling a notable gap in current
micromobility research. Through naturalistic studies, we captured and analyzed
detailed eye-tracking data, which revealed significant insights into the
visual attention and cognitive load experienced by riders. Our findings
highlighted the profound impact of infrastructure on rider safety, with bike
lanes promoting more focused and predictable gaze patterns, thereby reducing
cognitive demands. Conversely, roads without bike lanes result in increased
gaze dispersion and fixations per second, indicating the need for heightened
alertness and suggesting a more complex navigational environment.

This paper also explored different situational challenges faced by riders,
such as navigating intersections, downhills, and crosswalks. The results
showed that riders adjust their visual focus according to the complexity and
risk associated with these scenarios, highlighting the need for strategic
infrastructure planning to mitigate potential hazards.

As e-scooters become increasingly integrated into urban transport ecosystems,
the implications of this paper are extensive. The evidence supports the need
for urban infrastructure that accommodates e-scooter dynamics, advocating for
designs that enhance rider safety and promote harmonious coexistence with
other road users. Our research offers valuable data-driven insights that can
guide policymakers, urban planners, and e-scooter manufacturers in creating
safer, more efficient environments for micromobility devices.

Looking forward, this work lays the groundwork for future research to include
more participants to explore the complex interplay of human factors, such as
weight and age, on riding risks. Continued exploration in this area will
enhance our understanding of micromobility safety and strengthen the
foundation for evidence-based infrastructural and policy advancements in urban
transportation.

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