# Elliptic Calogero-Moser system,
crossed and folded instantons,
and bilinear identities

Andrei Grekov, Nikita Nekrasov  Simons Center for Geometry and Physicsn
Yang Institute for Theoretical Physicsg,n
Stony Brook University, Stony Brook NY 11794-3636, USA

(Date: June 2023)

###### Abstract.

Affine analogues of the $Q,{\tilde{Q}}$-functions are constructed using folded
instantons partition functions. They are shown to be the solutions of the
quantum spectral curve of the $N$-body elliptic Calogero-Moser (eCM) system,
the quantum Krichever curve. $Q,{\tilde{Q}}$ also solve the elliptic analogue
of the quantum Wronskian equation. In the companion paper we present the
quantum analogue of Krichever’s Lax operator for eCM. A connection to crossed
instantons on Taub-Nut spaces, and opers on a punctured torus is pointed out.

##  1\. Elliptic Calogero-Moser system

Let

(1.1) ${\tau}={\tau}_{1}+{\mathrm{i}}{\tau}_{2}\in{\mathbb{C}}\,,\
{\tau}_{2}>0\,,\ \mathfrak{q}=e^{2\pi\mathrm{i}\tau}\ .$

Let

(1.2)
$E_{\tau}={\mathbb{C}}/{\mathbb{Z}}+{\tau}{\mathbb{Z}}\approx{\mathbb{C}}^{\times}/{\mathfrak{q}}^{\mathbb{Z}}$

be the corresponding elliptic curve.

Fix an integer $N\geq 1$, and $n,\hbar\in{\mathbb{C}}$. Quantum elliptic
Calogero-Moser system (eCM) can be defined in many ways. At simplified level
it is a collection of commuting differential operators
${\hat{H}}_{1},{\hat{H}}_{2},\ldots,{\hat{H}}_{N}$ in $N$ variables
$z_{1},\ldots,z_{N}$,

(1.3) $\displaystyle{\hat{H}}_{1}=\sum_{i=1}^{N}\frac{\partial}{\partial
z_{i}}\,,$
$\displaystyle{\hat{H}}_{2}=-\frac{1}{2}\sum_{i=1}^{N}\frac{\partial^{2}}{\partial
z_{i}^{2}}\,+\,n(n-1)\sum_{i<j}{\wp}(z_{i}-z_{j}|{\tau})$
$\displaystyle\ldots$

invariant under the action of the double affine Weyl group

(1.4)
$\left(z_{i}\right)_{i=1}^{N}\mapsto\left(z_{{\sigma}(i)}+a_{i}+{\tau}b_{i}\right)_{i=1}^{N}\,,\
{\sigma}\in S(N)\,,\ a_{k},b_{k}\in{\mathbb{Z}}$

The elliptic Calogero-Moser system admits several interesting degenerations.
One such limit, ${\tau}\to{\mathrm{i}\infty}$, produces the
trigonometric/hyperbolic CM system. Another limit is more subtle: let
$z_{k}=y_{k}+{\tau}k/N$, $k=1,\ldots,N$,
${\Lambda}^{2N}={\mathfrak{q}}\left(-({\hbar}n)^{2}\right)^{N}$. Send
$\tau\to\mathrm{i}\infty,n\to\infty$ while keeping ${\bf y}=(y_{k})_{k=1}^{N}$
and $\Lambda,\hbar$ finite. The result is the $N$-body periodic Toda chain
[22]:

(1.5) $\displaystyle{\hat{H}}_{1}=\sum_{i=1}^{N}\frac{\partial}{\partial
y_{i}}\,,$
$\displaystyle{\hat{H}}_{2}=-\frac{1}{2}\sum_{i=1}^{N}\frac{\partial^{2}}{\partial
y_{i}^{2}}\,+\,({\Lambda}/{\hbar})^{2}\sum_{i=1}^{N}e^{y_{i}-y_{i+1}}\,,$
$\displaystyle\ldots$ $\displaystyle y_{N+1}\equiv y_{1}$

###  1.1. Spectral curve of eCM

In the classical limit $\hbar\to 0$, $n\to\infty$, ${\hbar n}={\nu}$ fixed,
the operators (1.3) go over to the Poisson-commuting functions
$H_{1}(p,z),\ldots,H_{N}(p,z)$ on
${\mathcal{P}}_{\mathbb{C}}=\left(T^{*}E_{\tau}\right)^{N}/S(N)$, or on its
smooth resolution

(1.6) ${\tilde{\mathcal{P}}}_{\mathbb{C}}=Hilb^{[N]}(S)\,,$

the Hilbert scheme of $N$ points on the complex symplectic surface $S$, which
is an affine bundle over the cotangent bundle to $E_{\tau}$ (which can be
analytically identified with the algebraic torus
${\mathbb{C}}^{\times}\times{\mathbb{C}}^{\times}$, or, non-canonically,
mapped to $T^{*}E_{\tau}$, isomorphically outside a fiber over one point in
$E_{\tau}$). In [28] the functions $H_{k}$ are constructed with the help of
the Lax operator

(1.7) ${\bf L}(u)=\|L_{ij}(u)\|_{i,j=1}^{N}\,,\
L_{ij}(u)=p_{i}{\delta}_{ij}+{\nu}(1-{\delta}_{ij})\frac{{\theta}(z_{i}-z_{j}+u|{\tau}){\theta}^{\prime}(0|{\tau})}{{\theta}(z_{i}-z_{j}|{\tau}){\theta}(u|{\tau})}$

where

(1.8)
${\theta}(u|{\tau})=\sum_{l\in{\mathbb{Z}}}(-1)^{l}e^{2\pi\mathrm{i}lu}{\mathfrak{q}}^{\frac{l(l-1)}{2}}\
$

obeys

(1.9)
${\theta}(u+1|{\tau})={\theta}(u|{\tau})=-e^{2\pi\mathrm{i}u}{\theta}(u+{\tau}|{\tau})$

One defines the spectral curve ${\mathcal{C}}\subset
T^{*}\left(E_{\tau}\backslash\\{u=0\\}\right)$ via:

(1.10) $R({\lambda},u)\equiv{\rm Det}\left({\lambda}\cdot{\rm Id}_{N}-{\bf
L}(u)\right)=0$

As a function of $\lambda$, $R$ is a degree $N$ polynomial, with coefficients
meromorphic double-periodic functions of $u$, with poles only at $u=0$.

An innocent looking change of variables (used in [28])

(1.11)
${\lambda}=x-{\nu}\frac{{\theta}^{\prime}(u|{\tau})}{{\theta}(u|{\tau})}$

makes (1.10) into

(1.12) ${\mathcal{R}}(x,u)={\rm Det}\left(x\cdot{\rm
Id}_{N}-{\bf{\tilde{L}}}(u)\right)=0$

with ${\tilde{L}}(u)$ have a first order pole at
$u\in{\mathbb{Z}}+{\tau}{\mathbb{Z}}$ with rank one residue. Therefore, the
function ${\mathcal{R}}(x,u)$ has, at fixed $x$ only a first order pole at
$u=0$. At fixed $u$ it is still a degree $N$ polynomial in $x$. However, it is
not a double-periodic function of $u$ at fixed $x$, rather:

(1.13)
${\mathcal{R}}(x-2{\pi}{\mathrm{i}}{\nu},u+{\tau})={\mathcal{R}}(x,u+1)={\mathcal{R}}(x,u)\,,$

Thus ${\mathcal{R}}=0$ is an equation for a curve ${\mathcal{C}}$ in the total
space $S_{\nu}$ of affine bundle over $T^{*}E_{\tau}$, again with the fiber
over $u=0$ removed. Note that $S_{\nu}$ has a rich set of holomorphic
functions, e.g. any Laurent polynomial in

(1.14) $U_{1}={\exp}\,\left(\frac{x}{\nu}\right)\,,\
U_{2}={\exp}\,\left(2\pi\mathrm{i}u+{\tau}\frac{x}{\nu}\right)$

Thus, for $\nu\neq 0$, outside the fiber over $u=0$ the surface $S_{\nu}$ is
isomorphic to the algebraic torus
${\mathbb{C}}^{\times}\times{\mathbb{C}}^{\times}$. The curve ${\mathcal{C}}$,
however, is not algebraic, only analytic, in the $\left(U_{1},U_{2}\right)$
coordinate system.

##  2\. Quantum Krichever curve

In a very naive way the equation ${\mathcal{R}}(x,u)=0$ can be quantized, for
example, by making $x,u$ the generators ${\hat{x}},{\hat{u}}$ of a non-
commutative algebra, obeying

(2.1) $[{\hat{u}},{\hat{x}}]={\hbar}\cdot 1$

where ${\hbar}\in{\mathbb{C}}$ is a parameter. It is not obvious how to order
${\hat{x}}$ and ${\hat{u}}$ in the complicated function such as
${\mathcal{R}}(x,u)$, or, say, $R(x,u)$. However, ${\mathcal{R}}(x,u)$ is much
better suited for quantization than $R(x,u)$ for the following reason. As we
pointed out earlier, the operator ${\bf\tilde{L}}(u)$ has a first order pole
at $u=0$ (modulo the lattice ${\mathbb{Z}}+{\mathbb{Z}}{\tau}$) with the rank
one residue. Thus,

(2.2) ${\mathcal{R}}(x,u)=\sum_{i=0}^{N}x^{N-i}f_{i}(u)$

with the coefficients $f_{i}(u)$ having only a first order pole at
$u\in{\mathbb{Z}}+{\mathbb{Z}}{\tau}$. Of course, the shifted periodicity
(1.13) means that $f_{i}(u)$’s are not double-periodic meromorphic functions
of $u$ (with the exception of $f_{0}(u)\equiv 1$). Now, let us multiply
${\mathcal{R}}(x,u)$ by ${\theta}(u|{\tau})$, and define the Fourier
coefficients ${\tilde{\mathfrak{R}}}_{l}(x)$ by:

(2.3)
${\mathfrak{R}}(x,u)={\theta}(u|{\tau}){\mathcal{R}}(x,u)=\sum_{l\in{\mathbb{Z}}}(-1)^{l}e^{2\pi\mathrm{i}lu}{\mathfrak{q}}^{\frac{l(l-1)}{2}}{\tilde{\mathfrak{R}}}_{l}(x)$

It is clear from the definition and (1.13) that ${\mathfrak{R}}(x,u)$ is an
entire function of $x,u$, degree $N$ polynomial in $x$ at fixed $u$, obeying

(2.4)
${\mathfrak{R}}(x,u+1)=-e^{2\pi\mathrm{i}u}\,{\mathfrak{R}}(x-2\pi\mathrm{i}\nu,u+\tau)={\mathfrak{R}}(x,u)\
.$

Thus ${\tilde{\mathfrak{R}}}_{l}(x)$ are all degree $N$ (monic) polynomials in
$x$, obeying

(2.5)
${\tilde{\mathfrak{R}}}_{l}(x-2{\pi}{\mathrm{i}}{\nu})={\tilde{\mathfrak{R}}}_{l+1}(x)$

In other words

(2.6) ${\tilde{\mathfrak{R}}}_{l}(x)={\mathcal{Y}}({\sf w}-{\nu}l)$

where $x=2\pi\mathrm{i}{\sf w}$, and
${\mathcal{Y}}={\tilde{\mathfrak{R}}}_{0}$ is a degree $N$ monic polynomial.
Now we quantize (2.3) by replacing $u$ in the exponent by

(2.7) ${\hat{u}}={\hbar}\frac{\partial}{\partial
x}=\frac{\hbar}{2\pi\mathrm{i}}\frac{\partial}{\partial\sf w}$

We can place $e^{2\pi\mathrm{i}l{\hat{u}}}$ to the right of
${\tilde{\mathfrak{R}}}_{l}(x)$, or we can place it to the left, amounting to
a renormalization $\nu\to\nu\pm\hbar$. We can also imagine a dual
quantization, where $e^{2\pi\mathrm{i}l{\hat{u}}}$ is to the right of
${\tilde{\mathfrak{R}}}_{l}(x)$ but acts on the left. We shall see both
versions realized below.

The quantum version of the spectral curve can, therefore, be represented as an
infinite order difference operator ${\hat{\mathfrak{R}}}$ with polynomial
coefficients, acting on functions of one variable $\sf w$, or its dual. The
corresponding solutions ${\bf\Psi}$, ${\bf\Psi}^{\vee}$ are the functions of
$\sf w$, obeying:

(2.8) ${\hat{\mathfrak{R}}}{\bf\Psi}({\sf
w})\,=\,\sum_{l\in{\mathbb{Z}}}(-1)^{l}{\mathfrak{q}}^{\frac{l(l-1)}{2}}{\mathcal{Y}}({\sf
w}-{\nu}l){\bf\Psi}({\sf w}+{\hbar}l)=0\\\
{\hat{\mathfrak{R}}}^{*}{\bf\Psi}^{\vee}({\sf
w})\,=\,\sum_{l\in{\mathbb{Z}}}(-1)^{l}{\mathfrak{q}}^{\frac{l(l-1)}{2}}{\mathcal{Y}}({\sf
w}-{\nu}l-{\hbar}l){\bf\Psi}^{\vee}({\sf w}-{\hbar}l)=0\\\ $

The quantum version of the spectral curve can, equivalently, be represented in
$u$-variable, with

(2.9) ${\widehat{\sf w}}=\frac{\hbar}{2\pi\mathrm{i}}\frac{\partial}{\partial
u}$

as the $N$’th order meromorphic differential operator on $E_{\tau}$, with a
regular singularity at $u=0$, a $PGL_{N}$-oper. The corresponding Fourier
transforms

(2.10) $\displaystyle{\chi}_{\alpha}(u)=\sum_{{\sf
w}\in{\alpha}+{\hbar}{\mathbb{Z}}}e^{\frac{2\pi\mathrm{i}{\sf
w}}{\hbar}u}\,{\bf\Psi}({\sf w})$
$\displaystyle{\chi}_{\alpha}^{\vee}(u)=\sum_{{\sf
w}\in{\alpha}+{\hbar}{\mathbb{Z}}}e^{-\frac{2\pi\mathrm{i}{\sf
w}}{\hbar}u}\,{\bf\Psi}^{\vee}({\sf w})$

where $\alpha$ is chosen so that the series on the right hand side of (2.10)
converge are the local solutions of the oper. They should play a rôle in the
quantum separation of variables for eCM and its spectral dual
$Y(\widehat{\mathfrak{gl}}(1))$-based spin chain, analogously to the genus
zero ($\mathfrak{sl}_{r+1}$ spin chain) case studied in [23]. Also, the
construction of [12] seems to generalize to genus one case, giving rise to the
$L^{2}$, or $z,{\bar{z}}$-version of the elliptic Calogero-Moser system, where
the wave-function of the $N$-body system is an $L^{2}$-function on
$E_{\tau}^{N}/S(N)$, which is a common eigenvector of the Hamiltonians of the
conventional, meromorphic, eCM and its complex conjugate. The parameters
$\alpha$ and the coefficients of the polynomial ${\mathcal{Y}}$ would have to
be fixed by a requirement of the $PGL_{N}({\mathbb{R}})$-holonomy of the oper.
The ${\mathcal{N}}=2^{*}$ counterpart of this construction seems to be placing
the theory on extremely elongated four dimensional ellipsoid with a surface
defect filling up a two dimensional equatorial section thereof.

We shall return to this picture elsewhere.

##  3\. $T-Q$-equation

Functional Bethe ansatz [41] (see [11, 14, 44, 42] for the reviews of the
foundational work) approach to $\mathfrak{sl}_{2}$-spin chains or quantum
periodic $N$-body Toda chain uses the following $T-Q$-equation [3, 40]:

(3.1) $P_{+}({\sf w}-2{\hbar}){\mathbf{Q}}({\sf
w}+\hbar)+{\mathfrak{q}}P_{-}({\sf w}+2{\hbar}){\mathbf{Q}}({\sf
w}-\hbar)=(1+{\mathfrak{q}}){\mathbf{T}}({\sf w}){\mathbf{Q}}({\sf w})$

with degree $N$ monic polynomial ${\mathbf{T}}({\sf w})$ and entire function
${\mathbf{Q}}({\sf w})$. The Eq. (3.1) describes the eigenvalues ${\bf T}({\sf
w})$ of the transfer matrix ${\hat{T}}({\sf w})$. There is a spectral dual Lax
operator ${\bf L}(z)$ of a particular case of $\mathfrak{sl}_{N}$ Gaudin-
Garnier model, whose eigenvalues belong to the ${\sf w}$ line. The equation
(3.1) can be derived from the Yangian version of the $q$-characters of
$\mathfrak{sl}_{2}$ [13], from the ${{\varepsilon}}_{2}\to 0$ limit of the
$qq$-characters of the $A_{1}$-type ${\mathcal{N}}=2$ superconformal gauge
theory (= $N_{f}=2N_{c}$ super-QCD) [32], it can also be viewed as a quantum
version of the Seiberg-Witten curve of the $A_{1}$ type gauge theory (for the
topological recursion approach to quantum curves of that type see [7], and for
the extension to the local CY geometries [9]). In this paper we define (and
solve) an ${\widehat{\mathfrak{gl}(1)}}$-analogue of (3.1).

##  4\. Partitions, characters, and contents

In what follows $\Lambda$ denotes the set of all partitions, i.e. non-
increasing sequences

(4.1)
${\Lambda}=\Biggl{\\{}\,{\lambda}\,|{\lambda}=\left({\lambda}_{1}\geq{{\lambda}}_{2}\ldots\geq{{\lambda}}_{{\ell}({\lambda})}>0={\lambda}_{{\ell}({\lambda})+1}\right)\Biggr{\\}}$

The size and length maps

(4.2) $|\cdot|\,,{\ell}\,:\,{\Lambda}\longrightarrow{\mathbb{Z}}_{\geq
0}\,,\qquad|{{\lambda}}|\equiv{{\lambda}}_{1}+\ldots+{{\lambda}}_{\ell({{\lambda}})}\
,$

and the “transposition” map: ${{\lambda}}\mapsto{{\lambda}}^{t}$,
${\ell}({{\lambda}}^{t})={{\lambda}}_{1}$, $|{{\lambda}}^{t}|=|{{\lambda}}|$,
best described by viewing partitions as finite subsets
$\lambda\subset{\mathbb{Z}}_{>0}\times{\mathbb{Z}}_{>0}$:

(4.3) ${\lambda}=\Biggl{\\{}{\square}=(i,j)\,|\,i,j\geq 1\,\ 1\leq
j\leq{{\lambda}}_{i}\,,\ 1\leq i\leq{{\lambda}}_{j}^{t}\,\Biggr{\\}}$

Define the contents of $\square$:

(4.4) ${\xi}_{\square}={\hbar}\left(i-j+n(j-1)\right)\,,\
{\upsilon}_{\square}={\hbar}\left(n(j-i)+1-j\right)\,,\
{\zeta}_{\square}={\hbar}\left(i-1-n(j-1)\right)$

Define the boundary sets ${\Gamma}^{+}_{\lambda}$, ${\Gamma}^{-}_{\lambda}$
by:

(4.5) $\sum_{(i,j)\in{\Gamma}^{+}_{\lambda}}\langle
i-1,j-1\rangle-\sum_{(i,j)\in{\Gamma}^{-}_{\lambda}}\langle
i,j\rangle=\sum_{i=1}^{{\ell}({\lambda})+1}\langle
i-1,{\lambda}_{i}\rangle-\sum_{i=1}^{{\ell}({\lambda})}\langle
i,{\lambda}_{i}\rangle$

where $\langle i,j\rangle$ is any linear function $\langle
i,j\rangle=(i-1)+f(j-1)$, for $f\in{\mathbb{C}}\backslash{\mathbb{Q}}$.

Finally, define the character

(4.6)
${\chi}_{ab}({\lambda})=\sum_{{\square}\in{{\lambda}}}q_{a}^{i-1}q_{b}^{j-1}$

for $a,b\in\\{1,2,3,4\\}$

##  5\. $Q$-functions, $\sf X,Y,Z$-observables, and
${\mathcal{X}},{\mathcal{Y}},{\mathcal{Z}}$-characters

Supersymmetric gauge theories produce [32] (see Appendix $\bf A$ for some
details) entire functions $Q({\sf w})$, which, in the limit
${{\varepsilon}}_{2}\to 0,{{\varepsilon}}_{1}={\hbar}$ of [31] solve Bethe
equations

(5.1) $\frac{Q({\sf w}+{\hbar})Q({\sf w}-{\hbar}n)Q({\sf
w}+{\hbar}(n-1))}{Q({\sf w}-{\hbar})Q({\sf w}+{\hbar}n)Q({\sf
w}+{\hbar}(1-n))}=-{\mathfrak{q}}\,,\ \forall{\sf w}\in Q^{-1}(0)\ .$

Define

(5.2) ${\mathsf{X}}({\sf w})=\frac{Q({\sf w})}{Q({\sf w}+{\hbar}n)}\,,\ {\sf
Y}({\sf w})=\frac{Q({\sf w})}{Q({\sf w}-{\hbar})}\,,\ {\sf Z}({\sf
w})=\frac{Q({\sf w})}{Q({\sf w}+{\hbar}(1-n))}\,.$

Then $U(N)$ ${{\mathcal{N}}}=2^{*}$ theory with adjoint hypermultiplet of mass
${\hbar}n$ on $\Omega$-deformed
${\mathbf{R}}^{2}_{\hbar}\times{\mathbb{R}}^{1,1}$ is characterized by the
property that

(5.3) ${\sf Y}({\sf w})\sim{\sf w}^{N}+\ldots\,{\rm at}\ {\sf w}\to\infty$

(it is called the $Y$-observable in [32]). Define the following triplet of
observables built out of $\mathsf{X},\mathsf{Y},\mathsf{Z}$, respectively:

(5.4) $\displaystyle{\mathcal{X}}({\sf
w})\,=\,\sum_{\lambda}{\mathfrak{q}}^{|{{\lambda}}|}\,\frac{\prod_{{\square}\in{\Gamma}^{+}_{\lambda}}{\sf
X}({\sf
w}-{\hbar}n+{\xi}_{\square})}{\prod_{{\square}\in{\Gamma}^{-}_{\lambda}}{\sf
X}({\sf w}+{\xi}_{\square})}\,,$ $\displaystyle{\mathcal{Y}}({\sf
w})\,=\,\sum_{\lambda}{\mathfrak{q}}^{|{{\lambda}}|}\,\frac{\prod_{{\square}\in{\Gamma}^{+}_{\lambda}}{\sf
Y}({\sf
w}+{\hbar}+{\upsilon}_{\square})}{\prod_{{\square}\in{\Gamma}^{-}_{\lambda}}{\sf
Y}({\sf w}+{\upsilon}_{\square})}\,,$ $\displaystyle{\mathcal{Z}}({\sf
w})\,=\,\sum_{\lambda}{\mathfrak{q}}^{|{{\lambda}}|}\,\frac{\prod_{{\square}\in{\Gamma}^{+}_{\lambda}}{\sf
Z}({\sf
w}+{\hbar}(n-1)+{\zeta}_{\square})}{\prod_{{\square}\in{\Gamma}^{-}_{\lambda}}{\sf
Z}({\sf w}+{\zeta}_{\square})}$

Then ${\mathcal{Y}}({\sf w})$ is a degree $N$ polynomial in ${\sf w}$, as
(5.1) guarantees the absence of poles, and the asymptotics (5.3) fixes the
rest (it is a particular, ${\hat{A}}_{0}$, case of the main theorem of [32]).
In fact, ${\mathcal{Y}}({\sf w})$ is the ${{\varepsilon}}_{2}\to 0$ limit of
the the fundamental $qq$-character of ${\hat{A}}_{0}$-theory, introduced in
[32]. It can be related to the $q$-character of [13] for the
$\widehat{\mathfrak{gl}}(1)$-algebra. Our main _new_ result is the set of
three

##  6\. Bilinear equations

(6.1) $\displaystyle{\mathcal{Y}}\star{\mathcal{X}}({\sf
w})\equiv\sum_{l\in{\mathbb{Z}}}(-1)^{l}{\mathfrak{q}}^{\frac{l(l+1)}{2}}\,{\mathcal{Y}}({\sf
w}-{\hbar}n(l+1)){\mathcal{X}}({\sf w}-{\hbar}l)=0$
$\displaystyle{\mathcal{Y}}\star{\mathcal{Z}}({\sf
w})\equiv\sum_{l\in{\mathbb{Z}}}(-1)^{l}{\mathfrak{q}}^{\frac{l(l+1)}{2}}\,{\mathcal{Y}}({\sf
w}+{\hbar}(n-1)(l+1)){\mathcal{Z}}({\sf w}-{\hbar}l)=0$
$\displaystyle{\mathcal{X}}\star{\mathcal{Z}}({\sf
w})\equiv\sum_{l\in{\mathbb{Z}}}(-1)^{l}{\mathfrak{q}}^{\frac{l(l+1)}{2}}\,{\mathcal{X}}({\sf
w}+{\hbar}(n-1)(l+1)){\mathcal{Z}}({\sf w}+{\hbar}nl)=0$

A quick comparison with (LABEL:eq:qceq) reveals ${\mathcal{X}}$ and
${\mathcal{Z}}$ are the dual solutions of the quantum spectral curve (1.12).
Having these solutions is a step towards the quantum separation of variables
[43] for the elliptic CM system.

##  7\. Discussion: gauge theory and integrable system

Gauge theory approach to elliptic Calogero-Moser system was started in [18].
It was shown there that the phase space and the Hamiltonians of the many-body
system can be obtained by Hamiltonian reduction from an infinite-dimensional
phase space, which is the space of two-dimensional gauge fields and adjoint-
valued one-forms (mathematically called Higgs fields, although they were
introduced and studied by N. Hitchin in [21], the construction of [18] being a
degenerate version [30]). In this realization the classical and quantum
evolution of the many-body system is identified with that of some $2+1$
dimensional gauge theory. The proposal [10] identified the family of spectral
curves found in [18] (which thanks to [28] are the spectral curves of elliptic
Calogero-Moser system) with the family of Seiberg-Witten curves of
${\mathcal{N}}=2^{*}$ theory with gauge group $SU(N)$ (an earlier paper [17]
pointed out a connection between the Seiberg-Witten proposal for the geometry
of vacua of ${\mathcal{N}}=2$ theories and integrable systems). The analysis
[29] allows to extract the low-instanton number non-perturbative corrections
to the low-energy effective action from the careful expansion of the periods
of a degenerating family of curves. The bottom up approach [31], starting with
supersymmetric gauge theory, computing the effects of the non-perturbative
dynamics on the geometry of its moduli space of vacua, produced a two-
parameter generalization of the Seiberg-Witten geometry. It was pointed out
that these generalizations might lead to the quantization of the integrable
system, as well as other deformations. In [33, 34] important steps were made
in showing that indeed, for ${\mathcal{N}}=2^{*}$ theory (among others) the
gauge theory computation produces the spectral curve of the elliptic Calogero-
Moser system.

Our present achievement is to find, within gauge theory, the observables whose
expectation values play the role of the single-particle wave-functions for the
quantum version of the spectral curve. We build them using the technique of
folded instantons, introduced in [32]. Again, we stress that our results are
derived, not conjectured. In this way we made independent checks of many
string duality based claims e.g. [2, 1, 15, 24, 6]. The proper embedding of
folded instanton observables in gauge theory remains an interesting open
problem (see [35] for string theory considerations).

Likewise, the meaning of the sums over $l$ in (6.1) begs for a gauge theory
explanation. Our conjecture is that just like the ${\mathcal{Y}}$-observable
can be interpreted as a partition function of an auxiliary four dimensional
theory living on a copy of ${\mathbf{R}}^{4}$ transverse to the physical
spacetime ${\mathbf{R}}^{4}$ in some ambient eight-dimensional special
holonomy geometry, the sum in ${\mathfrak{R}}(x,u)$ is the partition function
of a similar $U(1)$ theory placed in the Taub-Nut background. The sum over
$l\in{\mathbb{Z}}$ is the sum over the fluxes of the abelian gauge field,
which is supported by the $L^{2}$-normalizable self-dual harmonic two-form.
One needs to find a Taub-Nut $\times{\mathbb{R}}^{4}$ analogue of the moduli
space of crossed instantons. The ADHM construction for instantons on Taub-Nut
spaces has been beautifully generalized in [8].

Of course, the relation between (LABEL:eq:qceq) and the quantum spectral curve
of elliptic Calogero-Moser system is not very direct. We simply match the
families of curves, and their non-commutative versions. However we can claim
the match of the coordinate systems, the coordinates $z_{1},\ldots,z_{N}$ of
the particles having gauge-theoretic significance. We know from [32] the
regular surface defect of ${\mathcal{N}}=2^{*}$ theory obeys eCM Schrödinger
equation, with $z_{i}-z_{i+1}$ being the complex Kähler moduli of the surface
defect. As we said in the beginning, we are discussing a simpler version of
eCM, defined in complex-analytic terms, so there are separate “in” and “out”
wave-functions, related by ${\hbar}\mapsto-{\hbar}$ transformations. A short
form of (6.1) reads

(7.1)
$0=\left(\sum_{l\in{\mathbb{Z}}}(-1)^{l}{\mathfrak{q}}^{\frac{l(l+1)}{2}}\,{\mathcal{Y}}({\sf
w}-{\hbar}n(l+1))e^{-{\hbar}l{\partial}_{\sf w}}\right){\Psi}\,,\\\
0=\left(\sum_{l\in{\mathbb{Z}}}(-1)^{l}{\mathfrak{q}}^{\frac{l(l+1)}{2}}\,e^{{\hbar}l{\partial}_{\sf
w}}{\mathcal{Y}}({\sf w}-{\hbar}n(l+1))\right){\Psi}^{\vee}\,,$

where ${\Psi}({\sf w})={\mathcal{X}}({\sf w})\,,\ {\Psi}^{\vee}({\sf
w})={\mathcal{Z}}({\sf w}+{\hbar}(1-n))$. The quantum separation of variables
which would express the regular surface defect as an integral transform of a
product of the single-particle wave-functions, which should be the
${\Psi}\sim{\mathcal{X}}$ or ${\Psi}^{\vee}\sim{\mathcal{Z}}$-observables.

The solutions ${\mathcal{X}}$ and ${\mathcal{Z}}$ are the ${\hat{A}}_{0}$
analogues of the two independent (over the quasiconstants, i.e.
$\hbar$-periodic functions) solutions of the $T-Q$ equations. The third
bilinear equation ${\mathcal{X}}\star{\mathcal{Z}}=0$ is the analogue of the
very important _quantum Wronskian relation_ (see the Appendix $\bf B$).

In the forthcoming publication [20] we shall establish another detail in the
gauge theory-integrable system package dictionary: the Lax operators as gauge
theory observables. Our main equations (6.1) will get a matrix-vector dress
explaining the origin of so many beautiful _zero-curvature representations_ in
the theory of integrable systems. The manipulation (2.3) leading from
$\mathcal{R}$ to $\mathfrak{R}$ will also get a matrix version, generalizing
that of [19, 45]. The story of the ${\mathcal{N}}=2^{*}$ (${\hat{A}}_{0}$)
theory we discussed generalizes to the ${\hat{A}}_{r}$-type quiver theories,
with (6.1) mapping, upon a Fourier transform, to $PGL_{N}$-opers on elliptic
curve $E_{\tau}$ with $r+1$ regular singularities with minimal monodromies
around the poles (another limit, ${{\varepsilon}}_{1}+{{\varepsilon}}_{2}\to
0$ of this theory was studied recently in [5]).

The quantum separation of variables for eCM is still around the corner…

Our equations should also play a rôle in the solution of the quantum
intermediate long-wave equation, a $N\to\infty$ counterpart of eCM system,
related to the six dimensional gauge theories and sigma models on instanton
moduli spaces [38], as well as to the integrable structure of two dimensional
conformal field theory [4], and quantum hydrodynamics [37, 27].

##  8\. Acknowledgments

We enormously benefited from discussions and collaboration with I. Krichever
in 1995-2022. We also thank S. Cherkis and S. Grushevsky for discussions.
Research was partly supported by NSF PHY Award 2310279. Any opinions expressed
are solely our own and do not represent the views of the National Science
Foundation.

##  9\. Appendix $\bf A$. Derivation of the main bilinear equations

In this section we use the notations of [32]. Let
${{\varepsilon}}_{1},{{\varepsilon}}_{2},{{\varepsilon}}_{3},{{\varepsilon}}_{4}\in{\mathbb{C}}$
sum to zero,

(9.1)
${{\varepsilon}}_{1}+{{\varepsilon}}_{2}+{{\varepsilon}}_{3}+{{\varepsilon}}_{4}=0$

and let $q_{a}=e^{{{\varepsilon}}_{a}}$, $P_{a}=1-q_{a},P_{ab}=P_{a}P_{b}$,
$a,b,=1,\ldots,4$. The ${\mathcal{Y}}\star{\mathcal{X}}$ of (6.1) is the
${{\varepsilon}}_{2}\to 0$ limit of the sum over $l$ of

(9.2) $(-1)^{l}{\mathfrak{q}}^{\frac{l(l+1)}{2}}\times
Z_{{{\varepsilon}}_{1},{{\varepsilon}}_{2}=0,{{\varepsilon}}_{3},{{\varepsilon}}_{4}}(N_{ab})\,,$

the partition function of the gauge origami theory with

(9.3) ${{\varepsilon}}_{1}={\hbar}\,,\ {{\varepsilon}}_{3}=-{\hbar}n\,,\
{{\varepsilon}}_{4}={\hbar}(n-1)$

and the Chan-Paton characters

(9.4) $N_{12}=\sum_{\alpha=1}^{N}e^{a_{\alpha}}\,,\ N_{34}=e^{{\sf
w}+{{\varepsilon}}_{3}(l+1)}\,,\ N_{14}=e^{{\sf w}-{{\varepsilon}}_{1}l}\,,$

the ${\mathcal{Y}}\star{\mathcal{Z}}$ corresponds to the origami with Chan-
Paton characters

(9.5) $N_{12}=\sum_{\alpha=1}^{N}e^{a_{\alpha}}\,,\ N_{34}=e^{{\sf
w}+{{\varepsilon}}_{4}(l+1)}\,,\ N_{13}=e^{{\sf w}-{{\varepsilon}}_{1}l}\,,$

finally ${\mathcal{X}}\star{\mathcal{Z}}$ corresponds to the origami with

(9.6) $N_{12}=\sum_{\alpha=1}^{N}e^{a_{\alpha}}\,,\ N_{14}=e^{{\sf
w}+{{\varepsilon}}_{4}(l+1)}\,,\ N_{13}=e^{{\sf w}-{{\varepsilon}}_{3}l}\,,$

We prove the ${\mathcal{Y}}\star{\mathcal{X}}=0$ equation from (6.1) by
constructing a involution

(9.7)
${\Upsilon}:{\Lambda}\times{\Lambda}\times{\mathbb{Z}}\longrightarrow{\Lambda}\times{\Lambda}\times{\mathbb{Z}}\
,$

sending $({\lambda},{\mu},l)$ to
$({\tilde{\lambda}},{\tilde{\mu}},{\tilde{l}})$, such that:

(9.8) $\displaystyle 1)$ $\displaystyle\qquad|l-{\tilde{l}}|=1$ $\displaystyle
2)$ $\displaystyle\qquad
P_{1}S_{34}(l,{\lambda})+P_{3}S_{14}(l,{\mu})=P_{1}S_{34}({\tilde{l}},{\tilde{\lambda}})+P_{3}S_{14}({\tilde{l}},{\tilde{\mu}})$

the latter equation implying

(9.9)
$\frac{l(l+1)}{2}+|{\lambda}|+|{\mu}|=\frac{{\tilde{l}}({\tilde{l}}+1)}{2}+|{\tilde{\lambda}}|+|{\tilde{\mu}}|$

where

(9.10) $S_{34}(l,{\lambda})=e^{{\sf
w}}q_{3}^{l+1}\left(1-P_{34}{\chi}_{34}({\lambda})\right)\,,\\\
S_{14}(l,{\mu})=e^{{\sf
w}}q_{1}^{-l}\left(1-P_{14}{\chi}_{14}({\mu})\right)\,,$

The involution acts by:

(9.11)
${\Upsilon}({{\lambda}},{\mu},l)=\begin{cases}\left({\tilde{\lambda}},{\tilde{\mu}},l-1\right)\,&,\qquad{\tilde{\lambda}}_{1}={\mu}_{1}+l\,,\
{\tilde{\lambda}}_{i+1}={{\lambda}}_{i}\,,\ {\mu}_{i+1}={\tilde{\mu}}_{i},\
i\geq 1\ ,\\\ &\qquad\qquad{\rm when}\ \ {\mu}_{1}+l\geq{{\lambda}}_{1}\ ,\\\
\left({\hat{\lambda}},{\hat{\mu}},l+1\right)\,&,\qquad{\hat{\mu}}_{1}={{\lambda}}_{1}-l-1\,,\
{\hat{\lambda}}_{i}={{\lambda}}_{i+1}\,,\ {\hat{\mu}}_{i+1}={\mu}_{i}\,,\
i\geq 1\ ,\\\ &\qquad\qquad{\rm when}\ \ {\mu}_{1}+l<{{\lambda}}_{1}\
.\end{cases}$

Let us check (9.8) in the case ${\mu}_{1}+l\geq{{\lambda}}_{1}$, i.e.
${\tilde{l}}=l-1$. In that case, the second equation in (9.8) reads

(9.12)
$\frac{1-q_{4}^{l}}{1-q_{4}}+q_{3}{\chi}_{34}({\lambda})+q_{4}^{l}{\chi}_{14}({\mu})={\chi}_{34}({\tilde{\lambda}})+q_{4}^{l}q_{1}{\chi}_{14}({\tilde{\mu}})$

Let us assume $l\geq 1$ (the case $l\leq 0$ is analogous) and expand the left
hand side of (9.12) in $q_{1},q_{4}$:

(9.13)
$\sum_{i=1}^{{\ell}({\lambda})}q_{1}^{-i}\sum_{j=1}^{{{\lambda}}_{i}}q_{4}^{j-i-1}+\sum_{j=1}^{{\mu}_{1}+l}q_{4}^{j-1}+\sum_{i=2}^{{\ell}({\mu})}q_{1}^{i-1}\sum_{j=l+1}^{{\mu}_{i}+l}q_{4}^{j-1}$

while the right hand side of (9.12) expands, similarly, as

(9.14)
$\sum_{i=1}^{{\ell}({\tilde{\mu}})}q_{1}^{i}\sum_{j=l+1}^{{\tilde{\mu}}_{i}+l}q_{4}^{j-1}+\sum_{i=1}^{{\ell}({\tilde{\lambda}})}q_{1}^{1-i}\sum_{j=1}^{{\tilde{\lambda}}_{i}}q_{4}^{j-i}$

Comparing the $q_{1}^{0}$, $q_{1}^{<0},q_{1}^{\geq 0}$ terms, respectively, we
get the precise match, given (9.11).

##  10\. Appendix $B$. Limit to periodic Toda and relation to earlier work

The Toda limit, i.e.

(10.1) $n\to\infty\,,\mathfrak{q}\to 0\,,$

while keeping finite

(10.2) ${\Lambda}^{2N}={\mathfrak{q}}(-1)^{N}\left({\hbar}n\right)^{2N}$

is a bit delicate, as far as the ${\mathcal{X}},{\mathcal{Z}},{\sf X},{\sf Z}$
functions are concerned. The $Q$-function has a smooth limit, with the Bethe
equations (5.1) going over to

(10.3) $\frac{Q({\sf w}+{\hbar})}{Q({\sf w}-{\hbar})}=-{\Lambda}^{2N}\,,\
\forall{\sf w}\in Q^{-1}(0)\ .$

More precisely, this is the limit of $Q({\sf w})$ at fixed $\sf w$. On the
other hand, $Q$ with the shifted arguments have somewhat singular limits,
which we need to account for:

(10.4) $Q({\sf w}-{\hbar}nl)\sim\
(nl)^{\frac{N}{2}}\,e^{Nnl}\,\left(-{\hbar}nl\right)^{\frac{N{\sf
w}}{\hbar}-Nnl}\times\left(1+O(n^{-1})\right)$

Specifically, we shall need (10.4) for $l=-1,1$ and a limit of the product

(10.5) ${\mathfrak{q}}^{\frac{\sf w}{\hbar}+p}Q({\sf w}+{\hbar}(p-n))Q({\sf
w}+{\hbar}(n-1+p))\sim\frac{(-1)^{N\left(n+\frac{1}{2}\right)}}{\hbar^{N}}{\Lambda}^{2N\left(\frac{\sf
w}{\hbar}+p\right)}$

The series for ${\mathcal{X}}({\sf w}),{\mathcal{Z}}({\sf w})$, in our limit,
receive contributions only from partitions ${\lambda}=(1^{p})$, for
$p=0,1,\ldots$, giving:

(10.6) $\displaystyle{\mathcal{X}}({\sf w})\sim{\tilde{Q}}({\sf
w}-{\hbar})\,,$ $\displaystyle{\mathcal{Z}}({\sf w})\sim{\tilde{Q}}({\sf
w}-{\hbar})\,,$

where $\sim$ means up to a factor $e^{c_{1}(n){\sf w}+c_{0}(n)}$,

(10.7) ${\tilde{Q}}({\sf w})={\Lambda}^{2N\frac{\sf w}{\hbar}}\,Q({\sf
w})\sum_{p=0}^{\infty}\frac{{\Lambda}^{2Np}}{Q({\sf w}+{\hbar}p)Q({\sf
w}+{\hbar}(p+1))}\ .$

The ${\hbar}n$-shifted ${\mathcal{X}}({\sf w}),{\mathcal{Z}}({\sf w})$
functions have $Q$-function as its principal asymptotics:

(10.8) ${\mathcal{X}}({\sf w}+{\hbar}n)\sim Q({\sf w})\,,\ {\mathcal{Z}}({\sf
w}+{\hbar}(1-n))\sim Q({\sf w})$

Finally, the series for ${\mathcal{Y}}({\sf w})$ retains only two terms in the
limit (10.1):

(10.9) ${\mathcal{Y}}({\sf w})={\mathsf{Y}}({\sf
w}+{\hbar})+\frac{{\Lambda}^{2N}}{{\mathsf{Y}}({\sf w})}$

Either by recalling the relation between $Q({\sf w})$ and $\mathsf{Y}({\sf
w})$, or by taking the limit of (6.1) we obtain the _T-Q equation_ (cf. [23,
39, 32]) , obeyed both by $Q({\sf w})$ and ${\tilde{Q}}({\sf w})$:

(10.10) $\displaystyle Q({\sf w}+{\hbar})+{\Lambda}^{2N}Q({\sf
w}-{\hbar})={\mathcal{Y}}({\sf w})Q({\sf w})$ $\displaystyle{\tilde{Q}}({\sf
w}+{\hbar})+{\Lambda}^{2N}{\tilde{Q}}({\sf w}-{\hbar})={\mathcal{Y}}({\sf
w}){\tilde{Q}}({\sf w})$

in complete agreement with (10.3). Note that (10.10) can also be obtained by a
limit of (3.1).

The Toda limit (10.1) of (6.1) gives (10.10) and the Wronskian relation (cf.
[26])

(10.11) ${\tilde{Q}}({\sf w})Q({\sf w}+{\hbar})-Q({\sf w}){\tilde{Q}}({\sf
w}+{\hbar})={\Lambda}^{\frac{2N{\sf w}}{\hbar}}$

Both $Q$ and ${\tilde{Q}}$ functions prominently feature (in disguise) in the
separation of variables for the quantum Toda system (the Ref. [25] used
determinant representation of Gaudin-Pasquier [16], which can be related to
our $Q$, ${\tilde{Q}}$’s).

We leave the verification to the reader as an exercise in application of
(10.6), (10.5), (10.4).

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# Low-Level Physiological Implications of
End-to-End Learning of Speech Recognition

###### Abstract

Current speech recognition architectures perform very well from the point of
view of machine learning, hence user interaction. This suggests that they are
emulating the human biological system well. We investigate whether the
inference can be inverted to provide insights into that biological system; in
particular the hearing mechanism. Using SincNet, we confirm that end-to-end
systems do learn well known filterbank structures. However, we also show that
wider band-width filters are important in the learned structure. Whilst some
benefits can be gained by initialising both narrow and wide-band filters,
physiological constraints suggest that such filters arise in mid-brain rather
than the cochlea. We show that standard machine learning architectures must be
modified to allow this process to be emulated neurally.

Index Terms: speech recognition, cochlear models, end-to-end architectures,
filterbanks, SincNet

##  1 Introduction

Advances in automatic speech recognition (ASR) have led to performance that is
very good in terms of word error rate (WER), but perhaps at the expense of our
own understanding of how they function. End-to-end (E2E) techniques [1] have
removed the need for knowledge of the hearing mechanism. Self-supervised
training [2] has done the same for phonetics. More generally, large pre-
trained models are available [3] removing the need for even the machine
learning (ML) know-how. Given that these systems work well, the question
arises: “what have they learned?” This is difficult to answer because their
component parts cannot readily be mapped to biological ones.

In this study, we are interested in the hearing mechanism. The biological
mechanism is quite well understood [4], with important parts being the
logarithmic response to both frequency and amplitude. For many years,
filterbank approaches were used as models of this process [5, 6]. Whilst many
variations have been studied, the authors’ ad-hoc experience suggests that the
details do not lead to big changes. Recent E2E approaches, however, have
clearly demonstrated that training the filterbanks can be beneficial [7]. A
(1D) convolution layer in the machine learning field is a filter in the signal
processing field. However, the only validations of which we are aware tend to
show that the component convolutions learn filters with a distribution similar
to a mel filterbank. This in turn tends to reinforce the above question rather
than answer it.

In SincNet, Ravanelli et al. [8] constrained the convolutions to be a sinc
($\sin(x)/x$) form, leading to a rectangular band-pass filter. The filter is
then defined by two trainable parameters: the lower frequency and the
bandwidth. Whilst not being biologically accurate, this approach has a
distinct attraction of being interpretable.

In the remainder of this mainly experimental paper, we describe SincNet and a
modest frame-based experimental scenario. We confirm that SincNet learns a mel
filterbank, but also show that wider bandwidth filters are important for
performance. We argue that such filters arise because of restrictions of
standard ML convolutional architectures, and conclude with what this infers
about how to construct a biologically plausible hearing model.

##  2 Background

The study of the human cochlea has interested many researchers since the
beginning of the 20th century. Von Békésy laid the groundwork of the research
on this topic in 1960 [9].

The basilar membrane in the cochlea can be interpreted as a natural filterbank
[10, 11]. Current understanding of the working of the cochlea is that wave
propagation is an active process [12] and works as an array of Hopf
oscillators [13, 14]. However, in this study, we limit ourselves to passive
analogues. The scaling of this natural filterbank has been analysed from
different points of view, leading to several scaling (or warping, spacing)
approaches. The Greenwood scaling [15] is the one that best represents the
scaling of the frequency sensitive hair cells in the cochlea based on the
physical distance on the basilar membrane of the hair cells. The mel scale
[16] is based on frequencies judged to be equally spaced in human perceptual
tests. Bark [17, 18] and ERB (equivalent rectangular bandwidth) [19] are
somewhere between mel and Greenwood, but by contrast are derived from critical
_bands_ of hearing.

ASR frontends take either some preprocessed features as input or, more
recently, raw input waveforms. Filterbanks have been the basis of feature
extraction [20] for many years. As early as 2001, a study [21] showed that a
filterbank could be obtained from a mathematical derivation of a data driven
design. From the resulting filterbank, about half of the filters were then
kept to represent the filterbank motivated by the fact that those filters were
enough to cover the whole frequency range. For the E2E approaches, CNNs for
ASR were introduced by Hinton et al. [22, 23] and have been used for a decade.
Since 2018 some architectures propose a way to combine both the filterbanks
and the E2E architecture, where the filterbanks are trainable and part of the
convolution layers. Zeghidour et al. [24] proposed an implementation with with
Gabor filters and Ravanelli et al. [25] with rectangular filters (SincNet).
Other work on trainable filterbanks includes that of Seki et al. [26], who
proposed an architecture based on a filter layer combined with a DNN where the
filter features were directly computed with a log-compression after the filter
layer. In that study the gain, central frequency, bandwidth and filter shape
were free to train, whilst in SincNet only two parameters are free to train,
defining the filters in the first layer.

##  3 Initial Analysis

In an initial, quite basic analysis, the main motivation was to understand
what the trainable filters learn; i.e., which typical hyperparameters (e.g.,
the number of filters needed to describe the signal) can be derived from those
trainable filter models, which initializations are appropriate. For this
study, we focus on SincNet.

###  3.1 SincNet setup

The SincNet model [25] is built with a 4-layer CNN followed by a 5-layer DNN.
The first layer of the CNN is constructed with trainable filters. Those
filters are initialised as a rectangular bandwidth mel-scale filterbank, an
easily computable type of filter in the time domain. Since the inverse Fourier
transform of a rectangular low-pass filter is a sinc function, a rectangular
bandwidth filter can be written as the difference of two low-pass filters as
in equation 1.

$h[n]=\operatorname{sinc}(2\pi f_{2}n)-\operatorname{sinc}(2\pi f_{1}n),$ (1)

where $h[n]$ represents a typical filter of the first convolutional layer. The
trainable parameters of the SincNet filters are the lower frequency ($f_{1}$)
and the bandwidth ($f_{2}-f_{1}$), i.e., linear combinations of $f_{1}$ and
$f_{2}$. Moreover, in the time domain, filtering a signal is mathematically
equivalent to the convolution of this signal with the filter kernel.

Between the different convolution layers the following operations are used:
maxpooling for downsampling, layernorm, ReLU and dropout before passing
through a five-layer DNN. The input of signal is a raw waveform of 200 ms at
16 kHz. For this research the experiments are performed on TIMIT [27] and to
verify that the observations are not database related, the baseline
experiments have been double checked with the mini-Librispeech database [28].

###  3.2 Baseline

Figure 1: Evolution of the baseline implementation of SincNet: the graph on
the top shows the initial filter distribution and the bottom plot show the
filter distribution after training. The x-axis represents the frequency range
and the y-axis the amplitude of the filters. The filters themselves are
represented by their central frequency (dot) and their bandwidth (bar).

In the default implementation, the number of filters is initialised to 128
followed by 3 CNN layers of 60 filters each. The filter distribution for a
similar experiment (60 filters on the first layer) is illustrated on figure 1.
We observe that some filters with a comparatively narrow bandwidth train
towards a filterbank. The others train towards wider bandwidth filters.
Concerning the frequency range, the wide-band filters could in principle be
reconstructed with a linear combination of narrow band filters. In this paper,
those two types of filters will be refered to as narrow and wide-band filters.
The first part of this study focuses on the narrow band filters, since a large
number of the wide-band filters seem to overfit the data.

###  3.3 Number of filters

Some filters in the first convolutional layer stay narrow-band while the
others train towards wider bandwidths. Table 1 gives an overview of the number
of narrow band filters as well as the phone error rate (PER) on TIMIT. To
determine the number of narrow band filters an ad-hoc pruning operation has
been applied after the filter training: the filters with wide bandwidths
covering parts of the spectrum that are already covered by smaller bandwidth
filters are discarded and only one filter is taken into account around the
Nyquist frequency.

Table 1: Filter pruning experiment: numbers of narrow band filters and related PER in function of the initialization. Sinc-Layer num. filters | CNN-layers |  narrow band filters | PER (%)
---|---|---|---
128 | 60 | 60 | 60 |  39 | 17.1
100 | 60 | 60 | 60 |  45 | 17.1
80 | 60 | 60 | 60 |  38 | 17.2
60 | 60 | 60 | 60 |  32 | 17.4
40 | 60 | 60 | 60 |  27 | 17.5
30 | 60 | 60 | 60 |  24 | 17.5

The number of filters needed by the model to build a filterbank covering the
whole frequency range can be determined by the number of narrow band filters.
From table 1 we can deduce that above 30 filters, the number of narrow-band
filters that describe the frequency range is around 30 - 40 filters, this
correlates with the results obtained by Zeghidour [24] using Gabor filters.

We also notice that when the first layer is initialized to 30 or 40 filters
(corresponding to the number of narrow-band filters of other layers), some of
those filters still train toward larger band filters. To analyse whether
keeping the initilization to the initial scale performs as well as the
combination of narrow and wide-band filters that the model learns, experiments
have been made on 30 and 40 filters for fixed and non-fixed filters, the
results are given in table 2. This raises the hypothesis that the wide-band
filters are bringing some information not provided by the narrow band filters.

Table 2: SincNet experiment: compare the performance of the training with the
filters fixed and the filters that are free to train.

|  fixed filters |  trained filters
---|---|---
num. filters |  loss |  PER |  loss |  PER
40 |  2.35 |  18.3 |  2.31 |  17.6
30 |  2.37 |  18.0 |  2.33 |  17.5

.

###  3.4 Scale after training

Given that there are several frequency warpings that can be justified from a
physiological point of view, it is informative (and simple) to investigate
which one is preferred by an E2E system. It is clear by inspection that it is
the narrow band filters that learn the warped filterbank. In [29] a
convolutional variational autoencoder (CVAE) architecture that learns
convolutional filters from raw waveforms using unsupervised learning was
proposed. However, the analysis was only based on the central frequencies
learned by those filters, not the narrow/wide-band distinction. In the present
paper the central frequencies of only the narrow band filters are taken into
account.

####  3.4.1 Experiment

The experiment consisted simply of analysing which filterbank the narrow-band
filters trained above were learning. The experiment was repeated for several
models with different initializations. The metric used to compute the distance
between the initial and trained scale is the mean of the Euclidean distance:

$d(x,s)=\frac{1}{N}\sqrt{\sum_{i=0}^{N}(x_{i}-s_{i})^{2}}$ (2)

Table 3: Mean Euclidean distance between narrow bandfilter’s normalized central frequency distribution and different scalings for different amount of filters (Mel filterbank) and different initial scalings (30 filters). Initialized to | Compared to
---|---
scale - filters | Mel | Bark | ERB | Greenwood
Mel - 128 | 0.0023 | 0.0047 | 0.0070 | 0.0086
Mel - 60 | 0.0018 | 0.0044 | 0.0070 | 0.0088
Mel - 40 | 0.0022 | 0.0039 | 0.0065 | 0.0084
Mel - 30 | 0.0020 | 0.0043 | 0.0071 | 0.0091
Bark - 30 | 0.0025 | 0.0037 | 0.0062 | 0.0082
ERB - 30 | 0.0030 | 0.0029 | 0.0055 | 0.0076
Greenwood -30 | 0.0037 | 0.0068 | 0.0095 | 0.0116

The narrow band filters of a filterbank initialized to the mel scale remain
mel-scale distributed. When initializing 30 – 40 filters as starting point to
different scalings, other scalings also train towards mel scale. It follows
that the mel scaling is an appropriate choice for filterbank initializations.

###  3.5 Corollary

The results so far have reinforced that E2E approaches do indeed learn what
has been known for many years about cochlear models: that 30 to 40 filters are
sufficient and that, regardless of physical measurements, the mel scale is the
one that is perceptually important. However, from figure 1 it is clear that
SincNet filters train towards a mixture of narrow _and wide-band_ filters.
Moreover from table 1, in all the experiments done for this section, the model
always learns wide-band filters. It follows that these wide-band structures
are important. Two questions arise:

  1. 1.

Can the filters be initialized to wide-band, removing or reducing the need to
train them?

  2. 2.

Why do wide-band filters appear at all given that they are, to a first
approximation, just linear combinations of narrow-band filters?

These are addressed in the following section.

##  4 Wide-band filter analysis

###  4.1 Wide-band initialization

Figure 2: Filter repartition of superimposed filterbanks before (top plot) and
after (bottom plot) training. In the initialization, the red filterbank is a
narrow-band filterbank composed of 30 filters. The rest are filterbanks of 10,
5 and 1 filters capturing information that could in principle be reconstructed
by a combination of the narrow band filters.

####  4.1.1 Hypothesis

Wide-band filters are important; it follows that the training can be optimized
by initializing a narrow band filterbank as before and adding wide-band
filters in addition of those filters. This hypothesis can be confirmed by an
experiment comprising initializing several frozen superimposed filterbanks
where the wide-band filters are combinations of several narrow band filters.

####  4.1.2 Experiment

Figure 2 shows the initialization and training of a model built with 4 ranges
of filters illustrated on the upper plot.

An estimation of the filter distribution after training of this new
initialisation is illustrated in the bottom plot. Four experiments using those
filters are summarized in table 4:

Table 4: Summary of experiments using narrow and/or wide-band filters filters | filter type | fixed | loss | PER
---|---|---|---|---
10 - 5 - 1 | wide | yes | 2.410 | 18.6%
30 | narrow | yes | 2.374 | 18.0%
30 - 10 - 5 - 1 | narrow & wide | yes | 2.335 | 17.7%
30 - 10 - 5 - 1 | narrow & wide | no | 2.306 |  17.5%

Using only the narrow band filters gives a final PER of 18%, the combination
of narrow and wide-band filters give for frozen filters a PER of 17.7% and for
trained filters a PER of 17.5%. A combination of narrow and wide-band frozen
filters already gives an improvement of 0.3% WER. The same effect is observed
on the loss: the loss for a combination of narrow and wide-band filters is
lower than for only narrow band filters. The new initialization is
consequently closer to what the model learns compared to the baseline
initialization. Aside, it is interesting to notice from figure 2 that when
trained most of the narrow band filters stay narrow band and most of the wide-
band filters stay wide-band.

Thus the hypothesis is demonstrated. We conclude that it can indeed be
beneficial to provide a mixture of narrow and wide-band filters in an ASR
front-end.

###  4.2 Why wide-band filters?

Wide-band filters are in principle linear combinations of several narrow band
filters; the network should be able to learn such a combination trivially,
much as we assume it is learning the energy feature that was commonly used in
HMM-based models. The most plausible explanation for the network’s failure to
do so arises from the interaction of harmonic (voiced) and aperiodic
(unvoiced) components. Harmonic components in the same filter add
constructively in the magnitude domain. Aperiodic components, however, add as
random variables; the variances add leading to a magnitude reduction by a
factor of $\sqrt{N}$ for $N$ discrete components. The wider band filters hence
tend to favour the voiced components.

In complex narrowband (e.g., Fourier transform based) filters, the squaring
operation leads to both harmonic and aperiodic components adding _in the same
ratio_ in the magnitude or power domain, inhibiting emphasis of the harmonic
component. SincNet comprises real-valued filters; however, the subsequent
network architecture can inhibit the behaviour. Each convolutional layer is
followed by four typical operations: max-pooling (downsampling), layernorm,
ReLU (activation function) and dropout that have some influence on the signal.
Of these, the maxpooling function and ReLU activation bring some distortions
to the signal.

####  4.2.1 Hypothesis

It is possible to design a simple experiment to examine whether the above non-
linearities inhibit simulation of wideband filters. The experiment encompasses
three intuitive hypotheses:

1\. Using average-pooling instead of maxpooling removes the noise artifacts
that are created by maxpooling on the filtered signal, but since we continue
to use a pooling function, aliasing still happens for the high frequencies. In
[30] some experiments showed that using average pooling reduced the WER but
without explaining the possible reasons.
2\.  Moving the first downsampling factor towards a further layer inhibits
downsampling just after the filtering of the signal, this removes both the
effect of aliasing and signal distortion (although it increases the data size
at several convolution layers).
3\. By inspection, using a tanh or sigmoid function removes some low frequency
artifacts created by the ReLU function. However, it is well known that the
cochlea contains a rectifier function [31], implying that ReLU is the right
physiologically plausible solution. It is not clear a-priori which of these
properties is more important.

####  4.2.2 Experiment

Table 5 summarises the performance of the experiments implied above after the
first convolution layer: use average pooling, move the first downsampling
factor to a later layer and check that ReLU is appropriate.

Table 5: The effects of modifying the downsampling and pooling schemes. The numbers in the second column refer to the downsampling rate at each of the pooling operations in the convolutional layers (1 implies no downsampling). filters | downsampling |  pooling 1st layer | act. function | PER
---|---|---|---|---
30 | 3-3-3-2 |  max | ReLU | 17.5%
30 | 3-3-3-2 |  avg | ReLU | 17.1%
30 | 1-3-3-6 |  - | ReLU |  16.8%
30 | 1-3-3-6 |  - | sigmoid | 18.1%

Replacing the max-pooling with average-pooling leads to an improvement in
performance. Displacing the downsampling by one layer, in principle allowing
the network to combine filters, leads to a further improvement. This broadly
demonstrates the first two points of our hypothesis. Changing the activation
function to sigmoid deteriorates the PER. This tends to confirm that the
physiologically-implied rectifier — yielding a simple spectral envelope — is
also the right solution in the artificial solution. We note that, even with
the best performing architecture, the system still learns some wide-band
filters. This implies that our solution is not perfect.

##  5 Conclusion

E2E training of filterbanks for ASR leads to filters that resemble a standard
filterbank. However, wider bandwidth filters are learned too, and are
important for good ASR performance. The central frequencies of the narrow-band
filters tend to a mel spacing, regardless of the initialisation. This confirms
a well understood mechanism, suggesting that it exists in the biological
system. There appears to be an optimal number of filters — around 30 to 40 —
that also correlates with acknowledged literature.

We suggest that wide-band filters are learned to distinguish voiced (harmonic,
coherent) components from either background noise or unvoiced (aperiodic,
stochastic) components. In principle, a network should be able to learn such
wide-band components by combining narrow-band ones. We argue that this
capability is precluded by the (otherwise standard) ML architecture; in
particular the phase will be lost by maxpooling. This argument is borne out by
experiments showing that a structure retaining a more formal down-sampling
mechanism can lead to better performance.

We are not aware of structures in the inner-ear or cochlea that can emulate
physical wide-band filters. However the phase information is retained in our
current best understanding of cochlear operation, retaining also the
possibility that such filters are emulated in mid-brain. Proving this would be
difficult, perhaps requiring some combination of selective stimulii with MRI
or EEG sensing. It remains as a hypothesis for the neuroscience community. Our
own future work will focus on more biologically plausible architectures. This
experimental study indicates that any such model will need to retain phase in
order for the subsequent network to take advantage of both narrow-band and
wide-band features.

##  6 Acknowledgements

This work was supported as a part of NCCR Evolving Language, a National Centre
of Competence in Research, funded by the Swiss National Science Foundation
(grant number 180888).

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11institutetext: 1 Department of Astronomy, Nanjing University, Nanjing
210093, China
2 Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University),
Ministry of Education, China
11email<EMAIL_ADDRESS>
# Early Rebrightenings of X-ray Afterglows from Ring-Shaped GRB Jets

M. Xu1,2 Y. F. Huang1,2

(Received 00 00, 0000; accepted 00 00, 0000)

###### Abstract

Aims. Early rebrightenings at a post-burst time of $10^{2}$ — $10^{4}$ s have
been observed in the afterglows of some gamma-ray bursts (GRBs). Unlike X-ray
flares, these rebrightenings usually last for a relatively long period. The
continuous energy injection mechanism usually can only produce a plateau in
the afterglow light curve, but not a rebrightening. Also, a sudden energy
injection can give birth to a rebrightening, but the rebrightening is a bit
too rapid.

Methods. Here we argued that the early rebrightenings can be produced by the
ring-shaped jet model. In this scenario, the GRB outflow is not a full cone,
but a centrally hollowed ring. Assuming that the line of sight is on the
central symmetry axis of the hollow cone, we calculate the overall dynamical
evolution of the outflows and educe the multiband afterglow light curves.

Results. It is found that the early rebrightenings observed in the afterglows
of a few GRBs, such as GRBs 051016B, 060109, 070103 and 070208 etc, can be
well explained in this framework.

Conclusions. It is suggested that these long-lasted early rebrightenings in
GRB afterglows should be resulted from ring-shaped jets.

###### Key Words.:

gamma rays: bursts - ISM: jets and outflows

##  1 Introduction

Gamma-ray bursts (GRBs) are powerful explosive events in the Universe. The
standard fireball model suggests that the prompt emission should result from
internal shocks and the afterglow should result from external shocks. Since
the launch of the $Swift$ satellite (Gehrels et al. 2004), enormous
improvements have been achieved in understanding the nature of GRBs (for
recent reviews, see: Zhang 2007, Gehrels et al. 2009). Generally, some
interesting components have been identified in the X-ray afterglow light
curves of GRBs, i.e., the steep decay phase, the shallow decay phase, the
normal decay phase, the post jet-break phase, and X-ray flares.

Rebrightening behavior is also a very interesting feature among a few GRBs.
Unlike X-ray flares which are usually characterized by a rapid rise and fall
of the flux (with the mean ratio of width to peak time $\langle\Delta
t/t\rangle=0.13\pm 0.1$, Chincarini et al. 2007), these rebrightenings are
generally very gentle, with $\Delta t/t>1$. Rebrightenings can occur either in
the early afterglow stage ($t\leq 10^{4}$ s) or in the late afterglow stage
($t\gg 10^{4}$ s). Rebrightening in the late stage was first discovered in the
X-ray and optical afterglow of GRB 970508, at a post-burst time of about
$t\sim 2$ day (Piro et al. 1998). More examples of late rebrightenings
(typically occuring at $t\geq 1$ — 2 day) can be found in GRBs 030329 (Lipkin
et al. 2004), 031203 (Ramirez-Ruiz et al. 2005), 050408 (de Ugarte Postigo et
al. 2007), and 081028 (Margutti et al. 2009). Interpretations for these late
rebrightenings include off-axis jet model, and microphysics variation
mechanism (Kong et al. 2010).

In this study, we will mainly concentrate on the early rebrightenings that
typically occur at a post-burst time of $t\leq 10^{4}$ s. Examples of such
early rebrightenings can be found in GRBs 051016B, 060109, 070103, 070208 etc.
Most GRBs, like GRB 061007 (Kocevski & Butler 2008), do not exhibit early
rebrightenings. Among more than 500 Swift GRBs (Gehrels et al. 2009), we note
that only less than 1% events show early X-ray rebrightenings. The energy-
injection mechanism is a natural explanation for rebrightenings, but it is
unlikely to take effect here. For example, the continuous energy-injection
mechanism usually is more likely to produce a plateau-like structure in the
afterglow light curve, but not an obvious rebrightening (Dai & Lu 1998; Zhang
& Mészáros 2001; Yu & Huang 2007). Also, although a sudden energy-injection
can give birth to a rebrightening, the rebrightening is generally a bit too
rapid (Huang et al. 2006) as compared with the early rebrightenings considered
here.

We suggest that this kind of early rebrightenings may be produced by ring-
shaped GRB jets. A ring-shaped jet is not a full cone, but a centrally
hollowed ring (Granot 2005; Fargion & Grossi 2006; Xu et al. 2008). In this
paper, we numerically investigate the afterglow features of such outflows and
use this mechanism to explain a few observed afterglows. The structure of our
paper is as follows. The geometry of ring-shaped jets is sketched in Section
2. The dynamical evolution and afterglow of ring-shaped jets are studied in
Section 3, together with a detailed comparison with a few observations.
Finally, Section 4 is our discussion and conclusion.

##  2 Ring-Shaped Jet

A ring-shaped jet can be produced by the central engine during the prompt GRB
phase. The most natural mechanism may be via precessing (Fargion & Grossi
2006; Zou & Dai 2006). In almost all GRB progenitor models, the central engine
is a rapidly rotating compact objects, such as a black hole or a neutron star.
The outflows accounting for the GRB are usually launched along the rotating
axis or the magnetic axis. If the nozzle of the central engine is precessing
while ejecting the outflow, then a ring-shaped jet will be naturally produced.
As an example, if GRBs are associated with the kick of neutron stars (Huang et
al. 2003), then the production of ring-shaped jet will be quite likely.
Alternatively, the apparent ring-shaped topology of the outflow can also arise
from a structured jet as suggested by van Putten & Levinson (2003) and van
Putten & Gupta (2009). This mechanism can explain the correlation between the
observed luminosity and variability in the light curves of long GRBs reported
by Reichert et al. (2001).

A detailed sketch of the ring-shaped jets has been presented in Xu et al.
(2008). Here we will adopt the notations there. The half opening angle of the
inner edge of the ring is denoted as $\theta_{\rm c}$, and the width of the
ring is $\Delta\theta$. Both the inner and outer edges may expand laterally,
so the inner edge will converge at the central axis and the ring-shaped jet
will finally become a conical jet after a period of time. In our calculations,
for simplicity, we assume that the line of sight is on the symmetry axis of
the centrally hollowed cone. Therefore, this scenario is more or less similar
to that of a normal off-axis jet at early stages, i.e. the line of sight is
not on the emitting outflow. A rebrightening is then expected to present in
the early afterglow phase, thanks to the deceleration and lateral expansion of
the ring-shaped jet.

Since our line of sight is “off-axis” initially, a natural problem is whether
the ring-shaped jet could produce the intensive $\gamma$-ray radiation in the
prompt GRB phase, similar to the problem confronted by a normal off-axis jet
model. Below, we show that the ring-shaped jet model has more advantages
against the off-axis jet model as long as this problem is concerned. The key
point is that all the material on a ring can contribute equally to the flux on
the central axis.

Assuming that the bulk Lorentz factor of a highly relativistic point-like jet
is $\gamma\gg 1$, and that it is viewed from an angle $\Theta$ relative to its
motion, then the observed flux will be amplified by a factor of $\sim D^{3}$,
due to the effect of relativistic beaming and Doppler boosting. Here the
Doppler factor is $D=[\gamma(1-\beta cos\Theta)]^{-1}$, with
$\beta=\sqrt{1-1/\gamma^{2}}$. For an extended outflow considered in our
framework, we should integrate over all the surface of the ring-shaped jet to
get the exact factor of amplification for the prompt GRB emission.

We first consider a normal conical jet with a full opening angle of
$\Delta\theta$ and a viewing angle of $\theta_{\rm c}+\frac{1}{2}\Delta\theta$
(similar to the viewing angle of our ring-shaped jet scenario). The observed
emission should be amplified by a factor of $f_{\rm j}=\int_{\theta_{\rm
c}}^{\theta_{\rm c}+\Delta\theta}D^{3}\phi(\Theta)d\Theta$ relative to that in
the jet rest frame, where $\phi(\Theta)$ is the toroidal angle given by

$\phi(\Theta)=2\arccos[\frac{\cos(\Delta\theta/2)-\cos(\theta_{\rm
c}+\Delta\theta/2)\cos\Theta}{\sin(\theta_{\rm
c}+\Delta\theta/2)\sin\Theta}].$ (1)

For simplicity, for a normal conical GRB jet with a full opening angle of
$\Delta\theta$ but with the viewing angle being $\sim 1/\gamma$, we define the
corresponding amplification factor as $f_{\rm o}$. Actually, $f_{\rm o}$ is a
measure of the amplification factor for an on-axis observer.

Now return to the ring-shaped jet considered in our framework. The
amplification factor for an observer on the central symmetry axis is $f_{\rm
r}=\int_{\theta_{\rm c}}^{\theta_{\rm
c}+\Delta\theta}D^{3}\phi(\Theta)d\Theta$, where $\phi(\Theta)=2\pi$. To get a
direct impression on this matter, let us simply take $\gamma=50,\theta_{\rm
c}=0.04$, $\Delta\theta=0.04$ as an example, and calculate the final
amplification factors. In this example, the viewing angle, i.e. the angle
between the jet axis and the line of sight, equals $3/\gamma$. We can easily
find that $f_{\rm o/r}\equiv f_{\rm o}/f_{\rm r}\approx 68.5$. It means that
the observed flux of our ring-shaped jet is only tens of times less than that
of an on-axis jet in the prompt GRB phase. At the same time, we can also find
that $f_{\rm r/j}\equiv f_{\rm r}/f_{\rm j}\approx 12.4$. It tells us that the
observed flux of our ring-shaped jet is more than 12 times larger than that of
an off-axis conical jet. So, the ring-shaped jet model is superior over the
normal off-axis conical jet model when considering the production of the
intensive $\gamma$-ray radiation in the prompt GRB phase.

##  3 Numerical Results

The overall dynamical evolution of a ring-shaped jet has been studied by Xu et
al. (2008) in detail. Generally, if the line of sight is on the central
symmetry axis of the hollow cone, a rebrightening phase should present in the
early afterglow light curve. Here, for completeness, we simply outline a few
important ingredients of the calculation. First, the dynamical equations
should be appropriate for both the ultra-relativistic and non-relativistic
stages. Recently, van Eerten et al. (2010) developed an accurate and also
correspondingly complex code for the dynamical evolution of GRB afterglows.
Here, we will use the simple and convenient equations proposed by Huang et al.
(1999). Secondly, for a ring-shaped jet, both the inner and outer edges may
expand laterally. When the inner edge converges at the central symmetry axis,
the ring-shaped jet will become a normal conical jet, and then we only need to
consider the sideways expansion of the outer edge. We suppose that the lateral
expansion is at the comoving sound speed ($c_{\rm s}$) approximately given by

$c_{\rm
s}^{2}=\hat{\gamma}(\hat{\gamma}-1)(\gamma-1)\frac{1}{1+\hat{\gamma}(\gamma-1)}c^{2},$
(2)

where $\hat{\gamma}\approx(4\gamma+1)/(3\gamma)$ is the adiabatic index (Dai
et al. 1999). We have $\hat{\gamma}\sim 4/3$ and $c_{s}=c/\sqrt{3}$ in
ultrarelativistic limit, and $\hat{\gamma}\sim 5/3$ and $c_{s}=\sqrt{5/9}\beta
c$ in nonrelativistic limit. The effect of $c_{\rm s}$ on the light curve of a
ring-shaped jet has been discussed by Xu et al. (2008) in detail. Generally
speaking, lateral expansion tends to make the light curve steeper and leads to
an earlier jet break. Thirdly, to calculate the afterglow flux, we integrate
the emission over the equal arrival time surface defined by

$\int\frac{1-\beta\cos\Theta}{\beta c}dR\equiv t,$ (3)

within the jet boundaries (Huang et al. 2000).

In this section, we will calculate the overall dynamical evolution and
multiband afterglow of ring-shaped jets, and compare the numerical results
with the observations of some $Swift$ GRBs with early rebrightenings, such as
GRBs 051016B, 060109, 070103, and 070208. In our calculations, we assume a
standard cosmology with $\Omega_{\rm M}=0.27$, $\Omega_{\Lambda}=0.73$, and
with the Hubble constant of $H_{0}=71\leavevmode\nobreak\ {\rm
km}\leavevmode\nobreak\ {\rm s}^{-1}{\rm Mpc}^{-3}$.

###  3.1 GRB 051016B

GRB 051016B was triggered and located as a soft burst by $Swift$-BAT at
18:28:09 UT on October 16, 2005 (Parsons et al. 2005). The light curve of its
prompt emission is shown in Figure 1a. Its duration is $T_{90}=4.0\pm 0.1$s.
The photon index in $15-150$ keV is $2.38\pm 0.23$, and the fluence is
$(1.7\pm 0.2)\times 10^{-7}{\rm erg\leavevmode\nobreak\ cm}^{-2}$ (Barbier et
al. 2005). Lying at a redshift of $z=0.9364$ (Soderberg et al 2005), the
isotropic-equivalent $\gamma$-ray energy release is $E_{\rm\gamma,iso}\sim
7.6\times 10^{50}$ erg. The light curve of early X-ray afterglow shows a
rising phase beginning at about 200s, and the rebrightening lasts for almost
one thousand seconds.

Using the ring-shaped jet model, we have fit the observed X-ray and optical
afterglow light curves of GRB 051016B numerically. The result is presented in
Figure 2, and the parameter values are given in Table 1. The observed X-ray
afterglow data are relatively abundant. We see that our model can reproduce
the rebrightening and the overall X-ray afterglow light curve satisfactorily.
Note that in the very early stage ($t\leq 200$ s), the afterglow may still be
in the steep decay phase and the flux should be dominant by the contribution
from the prompt tail emission. We thus have excluded these early data points
from our modeling. In Figure 2, the observed R-band optical afterglow light
curve is built up only by two data points, but they are in good agreement with
our prediction. From Table 1, we see that the ratio of $f_{\rm r/j}$ is about
5.8. It means that in the prompt GRB phase, the $\gamma$-ray flux of our model
is about six times more than that of a corresponding off-axis conical jet. So,
the ring-shaped jet model is better than the off-axis conical jet model when
considering the production of the intensive $\gamma$-ray radiation in the main
burst phase.

Figure 1: Prompt $\gamma$-ray light curves of GRBs 051016B, 060109, 070103,
and 070208 as observed by $Swift-BAT$ (15—350keV). The data are taken from the
$Swift$ website111$http://astro.berkeley.edu/\sim nat/swift/$ (Butler &
Kocevski 2007; Butler et al. 2007).  Figure 2: Our best fit to the X-ray
(solid line) and R-band optical (dashed line) afterglow light curves of GRB
051016B by using the ring-shaped jet model. The square data points are
observed X-ray afterglow by $Swift$-XRT (see the $Swift$ website) and the
triangle data points are observed R-band afterglow (Chen et al. 2005; Sharapov
et al. 2005).

###  3.2 GRB 060109

At 16:54:41 UT on January 9, 2006, $Swift$-BAT triggered and located GRB
060109 (De Pasquale et al. 2006). The light curve of its prompt emission is
shown in Figure 1b. Its duration in 15 — 350 keV is $T_{90}=116\pm 3$ s, and
the fluence in 15 — 150 keV band is $(6.4\pm 1.0)\times 10^{-7}{\rm
erg\leavevmode\nobreak\ cm}^{-2}$ (Palmer et al. 2006). No redshift is
measured for this event. If assuming a typical redshift of $z=1$, then the
isotropic-equivalent $\gamma$-ray energy release is $E_{\rm\gamma,iso}\sim
3.74\times 10^{51}$ erg. In the early X-ray afterglow, the light curve shows a
rebrightening that begins in less than 1000 s and lasts for several thousand
seconds.

Figure 3 illustrates our fit to the observed X-ray afterglow of GRB 060109 by
using the the ring-shaped jet model. The parameters involved are given in
Table 1. In our modeling, again we have omitted the observed steep decaying
phase. We see that the theoretical light curve matches with the observational
data very well. The observed flux of the ring-shaped jet is about 10 times
larger than that of a corresponding off-axis conical jet ($f_{\rm r/j}\sim
10.4$, see Table 1). The ratio of $f_{\rm o/r}$ is $\sim 48.5$. It rougly
equals to $E_{\rm K,iso}/E_{\gamma,iso}$, which is about 18 as can be derived
from Table 1. This proves that our modeling is self-consistent.

Figure 3: Our best fit to the X-ray afterglow of GRB 060109 by using the ring-
shaped jet model. The square data points are observed by $Swift$-XRT (see the
$Swift$ website).

###  3.3 GRB 070103

At 20:46:39.41 UT on 2007 January 3, $Swift$-BAT triggered and located GRB
070103 (Sakamoto et al. 2007). The light curve of its prompt emission is shown
in Figure 1c. The duration in 15 — 350 keV is $T_{90}=19\pm 1$ s, and the
fluence in 15 — 150 keV is $(3.4\pm 0.5)\times 10^{-7}{\rm
erg\leavevmode\nobreak\ cm}^{-2}$ (Barbier et al. 2007). The isotropic-
equivalent $\gamma$-ray energy release is $E_{\rm\gamma,iso}\sim 1.8\times
10^{51}$ erg, assuming that the redshift is $z=1$. In the early X-ray
afterglow, the light curve shows a rebrightening that begins at about 200 s
and lasts for more than one thousand seconds.

In Figure 4, we illustrate our best fit to the X-ray afterglow of GRB 070103
by using the ring-shaped jet model, with the involving parameters given in
Table 1. Generally, the observed rebrightening and the overall light curve can
be well explained. According to Table 1, the value of $f_{\rm r/j}$ is about
5.3 . Note that the parameter of $f_{\rm o/r}$ is about 52.2, and the ratio of
$E_{\rm K,iso}/E_{\gamma,iso}$ is about 22. Again these two values are roughly
consistent with each other.

Figure 4: Our best fit to the X-ray afterglow of GRB 070103 by using the ring-
shaped jet model. The square data points are observed by $Swift$-XRT (see the
$Swift$ website).

###  3.4 GRB 070208

GRB 070208 was triggered and located by $Swift$-BAT at 09:10:34 UT on February
8, 2007 (Sato et al. 2007). The light curve of its prompt emission is shown in
Figure 1d. The duration in 15 — 350 keV is $T_{90}=48\pm 2$ s (Markwardt et
al. 2007), and the fluence in 15 — 150 keV is $(4.3\pm 1.0)\times 10^{-7}{\rm
erg\leavevmode\nobreak\ cm}^{-2}$. Lying at a redshift of $z=1.165$ (Cucchiara
et al. 2007), the isotropic-equivalent $\gamma$-ray energy release is
$E_{\rm\gamma,iso}\sim 3.32\times 10^{51}$ erg. The X-ray afterglow light
curve shows a rebrightening that begins at about 200 s and lasts for more than
one thousand seconds (Conciatore et al. 2007).

Using the ring-shaped jet model, we have tried to fit the multiband afterglow
of GRB 070208. The result is presented in Figure 5, with the parameters given
in Table 1. The observed early rebrightening in the X-ray afterglow can be
well reproduced. The optical afterglow can also be simultaneously explained.
Table 1 indicates that the radiation intensity of the ring-shaped jet is about
8 times more than that of a corresponding off-axis conical jet (i.e., $f_{\rm
r/j}\sim 7.7$). Note that the value of $f_{\rm o/r}$ is about 61. It again
agrees well with the ratio of $E_{K,iso}/E_{\rm\gamma,iso}$, which is about 24
in our modeling.

Figure 5:  Our best fit to the X-ray (solid line) and R-band optical (dashed
line) afterglow light curves of GRB 070208 by using the ring-shaped jet model.
The square data points are observed X-ray afterglow by $Swift$-XRT (see the
$Swift$ website) and the triangle data points are observed R-band afterglow
(Cenko et al. 2009; Halpern & Miraba 2007).

##  4 Conclusion and Discussion

It is possible that GRBs may be produced by ring-shaped jets. Interestingly, a
few recent hydrodynamical and magnetohydrodymical simulations also give some
supports to this idea (Aloy & Rezzolla 2006; Mizuno et al. 2008). A notable
feature of the afterglow of a ring-shaped jet is that a rebrightening can be
observed in the early afterglow stage, assuming that the line of sight is
within the central hollow cone. In this study, we have clearly shown that the
early rebrightenings observed in a few GRBs, such as GRBs 051016B, 060109,
070103, and 070208, can be well explained by the ring-shaped jet model. In the
case of GRBs 060109 and 070103, of which only X-ray afterglow data are
available, the fit result is very good. For GRBs 051016B and 070208, of which
both the X-ray and optical afterglow data are available, the interpretation is
also satisfactory. We propose that these GRBs with long-lasting rebrightening
in the early afterglow should be produced by ring-shaped jets.

Comparing with a normal off-axis conical jet model, the advantage of our ring-
shaped jet model is obvious. For a normal conical jet, when the observer is
off-axis, the observed brightness will be notably reduced (as compared to the
on-axis case). It then has difficulty in explaining the intensive $\gamma$-ray
emission observed in the prompt GRB phase. On the contrary, in the case of a
ring-shaped jet, all the material on the whole ring can contribute to the
emission on the line of sight. The observed intensity then can be
significantly enhanced as compared to that of an off-axis conical jet. In
fact, for the 4 GRBs studied here, the amplification factor ($f_{\rm r/j}$) is
generally 5 — 10, comparing with the corresponding off-axis conical jet.

From Figure 1, we interestingly notice that all the four GRBs show similar
2-pulse behavior in the prompt $\gamma$-ray light curve. This behavior is also
observed in some other Swift GRBs (Butler et al. 2007). It is quite unclear
whether this 2-pulse behavior is intrinsic to the mechanism that produces the
ring-shaped jet or not. Huang et al. (2003) have argued that if the jet is
produced by rapid precessing process (with the precession period much less
than the GRB duration), then the prompt $\gamma$-ray light curve could be
highly variable and very complicated. It then seems that these four GRBs might
not be due to normal fast precession. However, a slow precession may still be
possible. The structured jet mechanism by van Putten & Gupta (2009) is of
course another choise. Other possibilities include the evolution of an extreme
Kerr Black Hole surrounded by a precessing disk (Lei et al. 2007), or
explanation of the second pulse as the tail emission from the first pulse.

From Table 1, we see that the initial angular radius of the inner edge of the
ring is generally very small ($\theta_{\rm c}\sim$ 0.01 — 0.02). The width of
the ring is also small ($\Delta\theta\sim$ 0.02 — 0.08). Taking typical
parameter values of $\theta_{c}=0.015$ and $\Delta\theta=0.04$, we can give a
rough estimate for the observed event rate of early rebrightenings. Among all
Swift GRBs, the fraction of well-monitored afterglows with potential jet-
breaks is around $60\%$ (Panaitescu 2007). For most of these jet candidates,
the outflows might be normal conical jets, but it is quite likely that a small
portion (here we take the ratio as 15%, see Xu et al. 2008) of the jets are
ring-shaped ones. For a ring-shaped jet with $\theta_{c}=0.015$ and
$\Delta\theta=0.04$, the possibility that our line of sight is within the
central hollow cone (as compared with the possibility that the line of sight
is just on the ring) is about 0.08 (calculated from
$(1-cos\theta_{c})/[(1-cos(\theta_{c}+\Delta\theta))-(1-cos\theta_{c})]$).
Final, we estimate that the fraction of Swift events predicted by our model to
display early rebrightening is $0.6\times 0.15\times 0.08\sim 0.7\%$. This
number is consistent with the observed fraction of Swift events that show
early X-ray rebrightening ($<1\%$).

The angular information may provide useful clues on the central engine of
GRBs. In many progenitor models of GRBs, the central engine is a rapidly
rotating compact star and the outflow is ejected along the magnetic axis. Our
study then indicates that the inclination angle of the magnetic axis with
respect to the rotating axis should be small ($\sim$ 0.02 — 0.06). In a few
other progenitor models, the central engine is also a rapidly rotating compact
star, but the outflow is ejected just along the rotating axis, then our
results suggest that the processing (angular) radius of the rotating axis
should be $\sim$ 0.02 — 0.06.

In our framework, since the rebrightening is mainly a geometric effect, it
should generally be achromatic, i.e. the rebrightening should appear
simultaneously in all the X-ray and optical bands. For the four GRBs studied
here, the X-ray afterglow data are relatively prolific. But the optical
afterglow data are generally so lacking that almost no rebrightening can be
discerned in the optical light curves. Anyway, we have shown that our model is
consistent with both the X-ray and optical observations. In the future, more
examples with abundant multiband afterglow data will be available, and tighter
constraints on the existence of ring-shaped jet could be derived.

In our model, we assume the ring-shaped jet is uniform. But note that the
actual structure of the GRB outflow may be very complicate. For example, it
may be two-component jet as predicted by some engine models (e.g. van Putten &
Levinson 2003). If these ingredients are included, then the afterglow behavior
of ring-shaped jets will be correspondingly much more complicated.

###### Acknowledgements.

We thank Fayin Wang and Yang Guo for helpful discussion. We also would like to
thank the anonymous referee for useful comments and suggestions that led to an
overall improvement of this manuscript. This work was supported by the
National Natural Science Foundation of China (Grant No. 10625313) and the
National Basic Research Program of China (973 Program, Grant No.
2009CB824800).

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Table 1: Parameters of the 4 GRBs

GRB Name | GRB 051016B | GRB 060109 | GRB 070103 | GRB 070208
---|---|---|---|---
z | $0.9364$ | $1$ | $1$ | $1.165$
$E_{\rm\gamma,iso}$ (erg) | $7.6\times 10^{50}$ | $3.37\times 10^{51}$ | $1.79\times 10^{51}$ | $3.32\times 10^{51}$
$E_{\rm K,iso}$ (erg) | $1.3\times 10^{52}$ | $6.0\times 10^{52}$ | $4.0\times 10^{52}$ | $8.0\times 10^{52}$
$\theta_{\rm c}$ (rad) | $0.01$ | $0.018$ | $0.01$ | $0.009$
$\Delta\theta$ (rad) | $0.07$ | $0.024$ | $0.08$ | $0.022$
$\gamma$ | $180$ | $90$ | $120$ | $150$
n $(cm^{-3})$ | $0.05$ | $0.08$ | $0.1$ | $0.1$
$\epsilon_{\rm e}$ | $0.1$ | $0.1$ | $0.1$ | $0.1$
$\epsilon_{\rm B}$ | $5.0\times 10^{-4}$ | $1.0\times 10^{-4}$ | $1.0\times 10^{-4}$ | $1.0\times 10^{-4}$
$p$ | $2.05$ | $2.2$ | $2.75$ | $2.02$
$f_{\rm r/j}$ | $5.8$ | $10.4$ | $5.3$ | $7.7$
$f_{\rm o/r}$ | $228.7$ | $48.5$ | $52.2$ | $61.0$

  * Notes: z: redshift; $E_{\rm\gamma,iso}$: the isotropic-equivalent $\gamma$-ray energy release observed by $Swift$; $E_{\rm K,iso}$: the isotropic-equivalent kinetic energy of the outflow used in our model; $\theta_{\rm c}$: the half opening angle of the inner edge of the ring; $\Delta\theta$: the width of the ring; $\gamma$: the bulk Lorentz factor; $n$: number density of the circum-burst medium; $\epsilon_{\rm e}$: the electron energy fraction; $\epsilon_{\rm B}$: the magnetic energy fraction; $p$: the power-law index of the energy distribution function of electrons; $f_{\rm r/j}$: $f_{\rm r}/f_{\rm j}$ (see Section 2 for details); $f_{\rm o/j}$: $f_{\rm o}/f_{\rm r}$ (see Section 2 for details).