# Harvey Lawson Manifolds and Dualities

Selman Akbulut and Sema Salur  Department of Mathematics, Michigan State
University, East Lansing, MI, 48824<EMAIL_ADDRESS>Department of
Mathematics, University of Rochester, Rochester, NY, 14627
<EMAIL_ADDRESS>
###### Abstract.

The purpose of this paper is to introduce Harvey-Lawson manifolds and review
the construction of certain “mirror dual” Calabi-Yau submanifolds inside a
$G_{2}$ manifold. More specifically, given a Harvey-Lawson manifold $HL$, we
explain how to assign a pair of tangent bundle valued 2 and 3-forms to a
$G_{2}$ manifold $(M,HL,\varphi,\Lambda)$, with the calibration 3-form
$\varphi$ and an oriented $2$-plane field $\Lambda$. As in [AS2] these forms
can then be used to define different complex and symplectic structures on
certain 6-dimensional subbundles of $T(M)$. When these bundles are integrated
they give mirror CY manifolds (related thru HL manifolds).

###### Key words and phrases:

mirror duality, calibration

###### 1991 Mathematics Subject Classification:

53C38, 53C29, 57R57

S. Akbulut is partially supported by NSF grant DMS 0505638, S.Salur is
partially supported by NSF grant 1105663

##  1\. Introduction

Let $(M^{7},\varphi)$ be a $G_{2}$ manifold with the calibration 3-form
$\varphi$. If $\varphi$ restricts to be the volume form of an oriented
3-dimensional submanifold $Y^{3}$, then $Y$ is called an associative
submanifold of $M$. In [AS2] the authors introduced a notion of mirror duality
in any $G_{2}$ manifold $(M^{7},\varphi)$ based on the
associative/coassociative splitting of its tangent bundle
$TM={\mathbb{E}}\oplus{\mathbb{V}}$ by the non-vanishing $2$-fame fields
provided by [T]. This duality initially depends on the choice of two non-
vanishing vector fields, one in ${\mathbb{E}}$ and the other in
${\mathbb{V}}$. In this article we give a natural form of this duality where
the choice of these vector fields are made more canonical, in the expense of
possibly localizing this process to the tubular neighborhood of the
$3$-skeleton of $(M,\varphi)$.

##  2\. Basic Definitions

Let us recall some basic facts about $G_{2}$ manifolds (e.g. [B1], [HL],
[AS1]). Octonions give an 8 dimensional division algebra
${\mathbb{O}}={\mathbb{H}}\oplus l{\mathbb{H}}={\mathbb{R}}^{8}$ generated by
$\langle 1,i,j,k,l,li,lj,lk\rangle$. The imaginary octonions
$im{\mathbb{O}}={\mathbb{R}}^{7}$ is equipped with the cross product operation
$\times:{\mathbb{R}}^{7}\times{\mathbb{R}}^{7}\to{\mathbb{R}}^{7}$ defined by
$u\times v=im(\bar{v}.u)$. The exceptional Lie group $G_{2}$ is the linear
automorphisms of $im{\mathbb{O}}$ preserving this cross product.
Alternatively:

(1) $G_{2}=\\{(u_{1},u_{2},u_{3})\in({\mathbb{R}}^{7})^{3}\;|\;\langle
u_{i},u_{j}\rangle=\delta_{ij},\;\langle u_{1}\times
u_{2},u_{3}\rangle=0\;\\}.$ (2) $G_{2}=\\{A\in
GL(7,{\mathbb{R}})\;|\;A^{*}\varphi_{0}=\varphi_{0}\;\\}.$

where $\varphi_{0}=e^{123}+e^{145}+e^{167}+e^{246}-e^{257}-e^{347}-e^{356}$
with $e^{ijk}=dx^{i}\wedge dx^{j}\wedge dx^{k}$. We say a $7$-manifold $M^{7}$
has a $G_{2}$ structure if there is a 3-form $\varphi\in\Omega^{3}(M)$ such
that at each $p\in M$ the pair $(T_{p}(M),\varphi(p))$ is (pointwise)
isomorphic to $(T_{0}({\mathbb{R}}^{7}),\varphi_{0})$. This condition is
equivalent to reducing the tangent frame bundle of $M$ from
$GL(7,{\mathbb{R}})$ to $G_{2}$. A manifold with $G_{2}$ structure
$(M,\varphi)$ is called a $G_{2}$ manifold (integrable $G_{2}$ structure) if
at each point $p\in M$ there is a chart $(U,p)\to({\mathbb{R}}^{7},0)$ on
which $\varphi$ equals to $\varphi_{0}$ up to second order term, i.e. on the
image of the open set $U$ we can write $\varphi(x)=\varphi_{0}+O(|x|^{2})$.

One important class of $G_{2}$ manifolds are the ones obtained from Calabi-Yau
manifolds. Let $(X,\omega,\Omega)$ be a complex 3-dimensional Calabi-Yau
manifold with Kähler form $\omega$ and a nowhere vanishing holomorphic 3-form
$\Omega$, then $X^{6}\times S^{1}$ has holonomy group $SU(3)\subset G_{2}$,
hence is a $G_{2}$ manifold. In this case $\varphi$= Re $\Omega+\omega\wedge
dt$. Similarly, $X^{6}\times\mathbb{R}$ gives a noncompact $G_{2}$ manifold.

######  Definition 1.

Let $(M,\varphi)$ be a $G_{2}$ manifold. A 4-dimensional submanifold $X\subset
M$ is called coassociative  if $\varphi|_{X}=0$. A 3-dimensional submanifold
$Y\subset M$ is called associative if $\varphi|_{Y}\equiv vol(Y)$; this
condition is equivalent to the condition $\chi|_{Y}\equiv 0$, where
$\chi\in\Omega^{3}(M,TM)$ is the tangent bundle valued 3-form defined by the
identity:

(3) $\langle\chi(u,v,w),z\rangle=*\varphi(u,v,w,z)$

The equivalence of these conditions follows from the ‘associator equality’ of
[HL]

$\varphi(u,v,w)^{2}+|\chi(u,v,w)|^{2}/4=|u\wedge v\wedge w|^{2}$

Similar to the definition of $\chi$ one can define a tangent bundle 2-form,
which is just the cross product of $M$ (nevertheless viewing it as a $2$-form
has its advantages).

######  Definition 2.

Let $(M,\varphi)$ be a $G_{2}$ manifold. Then $\psi\in\Omega^{2}(M,TM)$ is the
tangent bundle valued 2-form defined by the identity:

(4) $\langle\psi(u,v),w\rangle=\varphi(u,v,w)=\langle u\times v,w\rangle$

On a local chart of a $G_{2}$ manifold $(M,\varphi)$, the form $\varphi$
coincides with the form $\varphi_{0}\in\Omega^{3}({\mathbb{R}}^{7})$ up to
quadratic terms, we can express the tangent valued forms $\chi$ and $\psi$ in
terms of $\varphi_{0}$ in local coordinates. More generally, if
$e_{1},...e_{7}$ is any local orthonormal frame and $e^{1},...,e^{7}$ is the
dual frame, from definitions we get:

$\displaystyle\chi=$ $\displaystyle\;\;(e^{256}+e^{247}+e^{346}-e^{357})e_{1}$
$\displaystyle+(-e^{156}-e^{147}-e^{345}-e^{367})e_{2}$
$\displaystyle+(e^{157}-e^{146}+e^{245}+e^{267})e_{3}$
$\displaystyle+(e^{127}+e^{136}-e^{235}-e^{567})e_{4}$
$\displaystyle+(e^{126}-e^{137}+e^{234}+e^{467})e_{5}$
$\displaystyle+(-e^{125}-e^{134}-e^{237}-e^{457})e_{6}$
$\displaystyle+(-e^{124}+e^{135}+e^{236}+e^{456})e_{7}.$

$\displaystyle\psi=$ $\displaystyle\;\;(e^{23}+e^{45}+e^{67})e_{1}$
$\displaystyle+(e^{46}-e^{57}-e^{13})e_{2}$
$\displaystyle+(e^{12}-e^{47}-e^{56})e_{3}$
$\displaystyle+(e^{37}-e^{15}-e^{26})e_{4}$
$\displaystyle+(e^{14}+e^{27}+e^{36})e_{5}$
$\displaystyle+(e^{24}-e^{17}-e^{35})e_{6}$
$\displaystyle+(e^{16}-e^{25}-e^{34})e_{7}.$

Here are some useful facts :

######  Lemma 1.

([AS1]) To any $3$-dimensional submanifold $Y^{3}\subset(M,\varphi)$, $\chi$
assigns a normal vector field, which vanishes when $Y$ is associative.

######  Lemma 2.

([AS1]) To any associative manifold $Y^{3}\subset(M,\varphi)$ with a non-
vanishing oriented $2$-plane field, $\chi$ defines a complex structure on its
normal bundle (notice in particular that any coassociative submanifold
$X\subset M$ has an almost complex structure if its normal bundle has a non-
vanishing section).

###### Proof.

Let $L\subset{\mathbb{R}}^{7}$ be an associative $3$-plane, that is
$\varphi_{0}|_{L}=vol(L)$. Then for every pair of orthonormal vectors
$\\{u,v\\}\subset L$, the form $\chi$ defines a complex structure on the
orthogonal $4$-plane $L^{\perp}$, as follows: Define $j:L^{\perp}\to
L^{\perp}$ by

(5) $j(X)=\chi(u,v,X)$

This is well defined i.e. $j(X)\in L^{\perp}$, because when $w\in L$ we have:

$\langle\chi(u,v,X),w\rangle=*\varphi_{0}(u,v,X,w)=-*\varphi_{0}(u,v,w,X)=\langle\chi(u,v,w),X\rangle=0$

Also $j^{2}(X)=j(\chi(u,v,X))=\chi(u,v,\chi(u,v,X))=-X$. We can check the last
equality by taking an orthonormal basis $\\{X_{j}\\}\subset L^{\perp}$ and
calculating

$\displaystyle\langle\chi(u,v,\chi(u,v,X_{i})),X_{j}\rangle$ $\displaystyle=$
$\displaystyle*\varphi_{0}(u,v,\chi(u,v,X_{i}),X_{j})=-*\varphi_{0}(u,v,X_{j},\chi(u,v,X_{i}))$
$\displaystyle=$
$\displaystyle-\langle\chi(u,v,X_{j}),\chi(u,v,X_{i})\rangle=-\delta_{ij}$

The last equality holds since the map $j$ is orthogonal, and the orthogonality
can be seen by polarizing the associator equality, and by noticing
$\varphi_{0}(u,v,X_{i})=0$. Observe that the map $j$ only depends on the
oriented $2$-plane $\Lambda=<u,v>$ generated by $\\{u,v\\}$ (i.e. it only
depends on the complex structure on $\Lambda$).

##  3\. Calabi-Yau’s hypersurfaces in $G_{2}$ manifolds

In [AS2] authors proposed a notion of mirror duality for Calabi-Yau
submanifold pairs lying inside of a $G_{2}$ manifold $(M,\varphi)$. This is
done first by assigning almost Calabi-Yau structures to hypersurfeces induced
by hyperplane distributions. The construction goes as follows. Suppose $\xi$
be a nonvanishing vecor fileld $\xi\in\Omega^{0}(M,TM)$, which gives a
codimension one integrable distribution $V_{\xi}:=\xi^{\perp}$ on $M$. If
$X_{\xi}$ is a leaf of this distribution, then the forms $\chi$ and $\psi$
induce a non-degenerate $2$-form $\omega_{\xi}$ and an almost complex
structure $J_{\xi}$ on $X_{\xi}$ as follows:

(6)
$\omega_{\xi}=\langle\psi,\xi\rangle\;\;\;\mbox{and}\;\;J_{\xi}(u)=u\times\xi.$
(7)
$\textup{Re}\;\Omega_{\xi}=\varphi|_{V_{\xi}}\;\;\mbox{and}\;\;\;\textup{Im}\;\Omega_{\xi}=\langle\chi,\xi\rangle.$

where the inner products, of the vector valued differential forms $\psi$ and
$\chi$ with vector field $\xi$, are performed by using their vector part. So
$\omega_{\xi}=\xi\lrcorner\;\varphi$, and
$\textup{Im}\;\Omega_{\xi}=\xi\lrcorner\;*\varphi$. Call
$\Omega_{\xi}=\textup{Re}\;\Omega_{\xi}+i\;\textup{Im}\;\Omega_{\xi}$. These
induce almost Calabi-Yau structure on $X_{\xi}$, analogous to Example 1.

######  Theorem 3.

([AS2]) Let $(M,\varphi)$ be a $G_{2}$ manifold, and $\xi$ be a unit vector
field such that $\xi^{\perp}$ comes from a codimension one foliation on $M$,
then $(X_{\xi},\omega_{\xi},\Omega_{\xi},J_{\xi})$ is an almost Calabi-Yau
manifold such that $\varphi|_{X_{\xi}}=Re\;\Omega_{\xi}$ and
$*\varphi|_{X_{\xi}}=*_{3}\;\omega_{\xi}$. Furthermore, if
$\mathcal{L}_{\xi}(\varphi)|_{X_{\xi}}=0$ then $d\omega_{\xi}=0$, and if
$\mathcal{L}_{\xi}(*\varphi)|_{X_{\xi}}=0$ then $J_{\xi}$ is integrable; when
both conditions are satisfied $(X_{\xi},\omega_{\xi},\Omega_{\xi},J_{\xi})$ is
a Calabi-Yau manifold.

Here is a brief discussion of [AS2] with explanation of its terms: Let
${\xi}^{\\#}$ be the dual $1$-form of $\xi$, and $e_{\xi^{\\#}}$ and
$i_{\xi}=\xi\lrcorner$ denote the exterior and interior product operations on
differential forms, then

$\varphi=e_{\xi^{\\#}}\circ i_{\xi}(\varphi)+i_{\xi}\circ
e_{\xi^{\\#}}(\varphi)=\omega_{\xi}\wedge\xi^{\\#}+Re\;\Omega_{\xi}.$

This is the decomposition of the form $\varphi$ with respect to
$\xi\oplus\xi^{\perp}$. The condition that the distribution $V_{\xi}$ to be
integrable is $d{\xi}^{\\#}\wedge{\xi}^{\\#}=0$. Also it is clear from
definitions that $J_{\xi}$ is an almost complex structure on $X_{\xi}$, and
the $2$-form $\omega_{\xi}$ is non-degenerate on $X_{\xi}$, because

$\omega_{\xi}^{3}=(\xi\lrcorner\;\varphi)^{3}=\xi\lrcorner\;[\;(\xi\lrcorner\;\varphi)\wedge(\xi\lrcorner\;\varphi)\wedge\varphi\;]=\xi\lrcorner\;(6|\xi|^{2}\mu)=6\mu_{\xi}$

where $\mu_{\xi}=\mu|_{V_{\xi}}$ is the induced orientation form on $V_{\xi}$.
For $u,v\in V_{\xi}$.

$\displaystyle\omega_{\xi}(J_{\xi}(u),v)$ $\displaystyle=$
$\displaystyle\omega_{\xi}(u\times\xi,v)=\langle\psi(u\times\xi,v),\xi\rangle=\varphi(u\times\xi,v,\xi)$
$\displaystyle=$ $\displaystyle-\varphi(\xi,\;\xi\times
u,\;v)=-\langle\;\xi\times(\xi\times u),v\;\rangle$ $\displaystyle=$
$\displaystyle-\langle\;-|\xi|^{2}u+\langle\xi,u\rangle\xi,v\;\rangle=|\xi|^{2}\langle
u,v\rangle-\langle\xi,u\rangle\langle\xi,v\rangle$ $\displaystyle=$
$\displaystyle\langle u,v\rangle.$

implies $\langle
J_{\xi}(u),J_{\xi}(v)\rangle=-\omega_{\xi}(u,J_{\xi}(v))=\langle u,v\rangle$.
By a calculation of $J_{\xi}$, one checs that the $3$-form $\Omega_{\xi}$ is a
$(3,0)$ form, furthermore it is non-vanishing because

$\displaystyle\frac{1}{2i}\;\Omega_{\xi}\wedge\overline{\Omega}_{\xi}=Im\;\Omega_{\xi}\wedge
Re\;\Omega_{\xi}$ $\displaystyle=$
$\displaystyle(\xi\lrcorner*\varphi)\wedge[\;\xi\lrcorner\;(\xi^{\\#}\wedge\varphi)\;]$
$\displaystyle=$
$\displaystyle-\xi\lrcorner\;[\;(\xi\lrcorner*\varphi)\wedge(\xi^{\\#}\wedge\varphi)\;]$
$\displaystyle=$
$\displaystyle\xi\lrcorner\;[*(\xi^{\\#}\wedge\varphi)\wedge(\xi^{\\#}\wedge\varphi)\;]\;\;\;\;\;\;\;$
$\displaystyle=$
$\displaystyle|\xi^{\\#}\wedge\varphi|^{2}\;\xi\lrcorner\;vol(M)$
$\displaystyle=$ $\displaystyle
4|\xi^{\\#}|^{2}\;(*\xi^{\\#})=4\;vol(X_{\xi}).\;\;\;\;\;\;\;\;$

It is easy to see $*Re\;\Omega_{\xi}=-Im\;\Omega_{\xi}\wedge\xi^{\\#}$ and
$*Im\;\Omega_{\xi}=Re\;\Omega_{\xi}\wedge\xi^{\\#}$.

$*_{3}Re\;\Omega_{\xi}=Im\;\Omega_{\xi}.$

Notice that $\omega_{\xi}$ is a symplectic structure on $X_{\xi}$ when
$d\varphi=0$ and $\mathcal{L}_{\xi}(\varphi)|_{V_{\xi}}=0$,
($\mathcal{L}_{\xi}$ is the Lie derivative along $\xi$), since
$\omega_{\xi}={\xi}\lrcorner\;\varphi$ and:

$d\omega_{\xi}=\mathcal{L}_{\xi}(\varphi)-\xi\lrcorner\;d\varphi=\mathcal{L}_{\xi}(\varphi)$

$J_{\xi}$ is integrable complex structure if $d^{*}\varphi=0$ and
$\mathcal{L}_{\xi}(*\varphi)|_{V_{\xi}}=0$ since

$d(Im\Omega_{\xi})=d(\xi\lrcorner*\varphi)=\mathcal{L}_{\xi}(*\varphi)-\xi\lrcorner\;d(*\varphi)=0$

Also notice that $d\varphi=0\implies$
$d(Re\;\Omega_{\xi})=d(\varphi|_{X_{\xi}})=0$.

##  4\. HL manifolds and Mirror duality in $G_{2}$ manifolds

By [T] any $7$-dimensional Riemanninan manifold admits a non-vanishing
orthonormal $2$-frame field $\Lambda=<u,v>$, in particular $(M,\varphi)$
admits such a field. $\Lambda$ gives a section of the bundle of oriented
$2$-frames $V_{2}(M)\to M$, and hence gives an associative/coassociative
splitting of the tangent bundle $TM={\bf E}\oplus{\bf V}$, where ${\bf E}={\bf
E}_{\Lambda}=<u,v,u\times v>$ and ${\bf V}={\bf V}_{\Lambda}={\bf E}^{\perp}$.
When there is no danger of confusion we will denote the $2$-frame fields and
the $2$-planes fields which they induce by the same symbol $\Lambda$. Also,
any unit section $\xi$ of ${\bf E}\to M$ induces a complex structure $J_{\xi}$
on the bundle ${\bf V}\to M$ by the cross product $J_{\xi}(u)=u\times\xi$.

In [AS2] any two almost Calabi-Yau’s $X_{\xi}$ and $X_{\xi^{\prime}}$ inside
$(M,\varphi)$ were called dual if the defining vector fields $\xi$ and
$\xi^{\prime}$ are chosen from ${\bf V}$ and ${\bf E}$, respectively. Here we
make this correspondence more precise, in particular showing how to choose
$\xi$ and $\xi^{\prime}$ in a more canonical way.

######  Definition 3.

A $3$-dimensional submanifold $Y^{3}\subset(M,\varphi)$ is called Harvey-
Lawson manifold (HL in short) if $\varphi|_{Y}=0$.

######  Definition 4.

([AS1]) Call any orthonormal $3$-frame field $\Gamma=<u,v,w>$ on
$(M,\varphi)$, a $G_{2}$-frame field if $\varphi(u,v,w)=<u\times v,w>=0$,
equivalently $w$ is a unit section of ${\bf V}_{\Lambda}\to X$, with
$\Lambda=<u,v>$ (see (1)).

Now pick a nonvanishing $2$-frame field $\Lambda=<u,v>$ on $M$ and let
$TM={\bf E}\oplus{\bf V}$ be the induced splitting with ${\bf E}=<u,v,u\times
v>$. Let $w$ be a unit section of the bundle ${\bf V}\to M$. Such a section
$w$ may not exist on whole $M$, but by obstruction theory it exists on a
tubular neighborhood $U$ of the $3$-skeleton $M^{(3)}$ of $M$ (which is a
complement of some $3$-complex $Z\subset M$). So $\varphi(u,v,w)=0$, and hence
$\Gamma=<u,v,w>$ is a $G_{2}$ frame field. Next consider the non-vanishing
vector fields:

  * •

$R=\chi(u,v,w)=-u\times(v\times w)$

  * •

$R^{\prime}=\frac{1}{\sqrt{3}}(u\times v+v\times w+w\times u)$

  * •

$R^{\prime\prime}=\frac{1}{\sqrt{3}}(u+v+w)$

If the $6$-plane fields $R^{\perp}$, $R^{\prime\perp}$, and
$R^{\prime\prime\perp}$, are integrable we get almost Calabi-Yau manifolds
$(X_{R},w_{R},\Omega_{R},J_{R})$,
$(X_{R^{\prime}},w_{R^{\prime}},\Omega_{R^{\prime}},J_{R^{\prime}})$, and
$(X_{R^{\prime\prime}},w_{R^{\prime\prime}},\Omega_{R^{\prime\prime}},J_{R^{\prime\prime}})$.
Let us use the convention that $a,b,c$ are real numbers, and $[u_{1},..u_{n}]$
is the distribution generated by the vectors $u_{1},..,u_{n}$.

######  Lemma 4.

By definitions, the following hold

  * (a)

$Y:=[u,v,w]=[au+bv+cw\;|\;a+b+c=0]\oplus[R^{\prime\prime}]$

  * (b)

${\mathbb{V}}=[u,v,w,R]$, is a coassociative $4$-plane field.

  * (c)

${\mathbb{E}}:=[u\times v,v\times w,w\times u]$ is an associative $3$-plane
field.

  * (d)

${\mathbb{E}}\perp{\mathbb{V}}$

######  Theorem 5.

For $(a,b,c)\in{\mathbb{R}}^{3}$ with $a+b+c=0$, then

  * (a)

$TX_{R}=[au+bv+cw,R^{\prime\prime},R^{\prime},a(v\times w)+b(w\times
u)+c(u\times v)]$

  * $J_{R}(au+bv+cw)=-a(v\times w)-b(w\times u)-c(u\times v)$

  * $J_{R}(R^{\prime\prime})=-R^{\prime}$

  * (b)

$TX_{R^{\prime}}=[au+bv+cw,R^{\prime\prime},R,a(v\times w)+b(w\times
u)+c(u\times v)]$

  * $J_{R^{\prime}}(au+bv+cw)=-((b-c)u+(c-a)v+(a-b)w)/\sqrt{3}$

  * $J_{R^{\prime}}(a(v\times w)+b(w\times u)+c(u\times v))=\\\ ((b-c)(v\times w)+(c-a)(w\times u)+(a-b)(u\times v))/\sqrt{3}$

  * $J_{R^{\prime}}(R^{\prime\prime})=R$

  * (c)

$TX_{R^{\prime\prime}}=[au+bv+cw,R,R^{\prime},(a(v\times w)+b(w\times
u)+c(u\times v)]$

  * $J_{R^{\prime\prime}}(au+bv+cw)=\\\ ((b-a)(u\times v)+(c-b)(v\times w)+(a-c)(w\times u))/\sqrt{3}$
$J_{R^{\prime\prime}}(R)=R^{\prime}$

  * (d)

$\\{u,v,w,R,u\times v,v\times w,w\times u\\}$ is an orthonormal frame field.

###### Proof.

To show (a) by using (4) we calculate:

$\displaystyle R\times u$ $\displaystyle=$ $\displaystyle\chi(u,v,w)\times
u=-[u\times(v\times w)]\times u=u\times[u\times(v\times w)]$ $\displaystyle=$
$\displaystyle-\chi(u,u,v\times w)-<u,u>(v\times w)+<u,v\times w>u$
$\displaystyle=$ $\displaystyle-(v\times w)+\varphi(u,v,w)u$ (8)
$\displaystyle\mbox{Therefore}\;\;R\times u$ $\displaystyle=$
$\displaystyle-(v\times w)$

Similarly, $R\times v=-(w\times u)$ and $R\times w=-(u\times v)$. Therefore we
have $J_{R}(au+bv+cw)=-a(v\times w)-b(w\times u)-c(u\times v)$, and
$J_{R}(R^{\prime\prime})=-R^{\prime}$.

$\displaystyle\sqrt{3}\;R^{\prime}\times u$ $\displaystyle=$
$\displaystyle(u\times v+v\times w+w\times u)\times u$ $\displaystyle=$
$\displaystyle-u\times(u\times v)-u\times(v\times w)-u\times(w\times u)$
$\displaystyle=$ $\displaystyle<u,u>v-<u,v>u$
$\displaystyle+\;\chi(u,v,w)+<u,v>w-<u,w>v$ $\displaystyle+<u,w>u-<u,u>w$ (9)
$\displaystyle\mbox{Therefore}\;\;\sqrt{3}\;R^{\prime}\times u$
$\displaystyle=$ $\displaystyle R+(v-w)$

Similarly $\sqrt{3}\;R^{\prime}\times v=R+(w-u)$, and
$\sqrt{3}\;R^{\prime}\times w=\;R+(u-v)$, which implies the first part of (b),
and $J_{R^{\prime}}(R^{\prime\prime})=R$.

For the second part of (b) we need the compute the following:

(10) $\sqrt{3}R^{\prime}\times[a(v\times w)+b(w\times u)+c(u\times v)]=$
$(u\times v+v\times w+w\times u)\times[a(v\times w)+b(w\times u)+c(u\times
v)]$

For this first by repeatedly using (4) and $\varphi(u,v,w)=0$ we calculate:

$\displaystyle(v\times u)\times(w\times v)$ $\displaystyle=$
$\displaystyle-\chi(v\times u,w,v)-<v\times u,w>v+<v\times u,v>w$
$\displaystyle=$ $\displaystyle-\chi(v\times u,w,v)=-\chi(w,v,v\times u)$
$\displaystyle=$ $\displaystyle w\times(v\times(v\times u))+<w,v>(v\times
u)-<w,v\times u>v$ $\displaystyle=$ $\displaystyle w\times(v\times(v\times
u))$ $\displaystyle=$ $\displaystyle
w\times(-\chi(v,v,u)-<v,v>u+<v,u>v)=-(w\times u)$

Then by plugging in (9) gives (b). Proof of (c) is similar to (a)

In particular from the above calculations we get can express $\varphi$ as:

######  Corollary 6.

$\varphi=u^{\\#}\wedge v^{\\#}\wedge(u^{\\#}\times v^{\\#})+v^{\\#}\wedge
w^{\\#}\wedge(v^{\\#}\times w^{\\#})+w^{\\#}\wedge u^{\\#}\wedge(w^{\\#}\times
u^{\\#})$ $\;\;\;\;\;\;\;+\;u^{\\#}\wedge R^{\\#}\wedge(v^{\\#}\times
w^{\\#})+(v^{\\#}\wedge R^{\\#})\wedge(w^{\\#}\times u^{\\#})+w^{\\#}\wedge
R^{\\#}\wedge(u^{\\#}\times v^{\\#})$ $-\;(u^{\\#}\times
v^{\\#})\wedge(v^{\\#}\times w^{\\#})\wedge(w^{\\#}\times u^{\\#})$

Recall that in an earlier paper we proved the following facts:

######  Proposition 7.

[AS2] Let $\\{\alpha,\beta\\}$ be orthonormal vector fields on $(M,\varphi)$.
Then on $X_{\alpha}$ the following hold

  * (i)

$Re\;\Omega_{\alpha}=\omega_{\beta}\wedge\beta^{\\#}+Re\;\Omega_{\beta}$

  * (ii)

$Im\;\Omega_{\alpha}=\alpha\lrcorner\;(\star\omega_{\beta})-(\alpha\lrcorner\;Im\;\Omega_{\beta})\wedge\beta^{\\#}$

  * (iii)

$\omega_{\alpha}=\alpha\lrcorner\;Re\;\Omega_{\beta}+(\alpha\lrcorner\;\omega_{\beta})\wedge\beta^{\\#}$

###### Proof.

Since $Re\;\Omega_{\alpha}=\varphi|_{X_{\alpha}}$ (i) follows. Since
$Im\;\Omega_{\alpha}=\alpha\lrcorner*\varphi$ following gives (ii)

$\displaystyle\alpha\lrcorner\;(\star\omega_{\beta})$ $\displaystyle=$
$\displaystyle\alpha\lrcorner\;[\;\beta\lrcorner\;*(\beta\lrcorner\;\varphi)\;]$
$\displaystyle=$
$\displaystyle\alpha\lrcorner\;\beta\lrcorner\;(\beta^{\\#}\wedge*\varphi)$
$\displaystyle=$
$\displaystyle\alpha\lrcorner*\varphi+\beta^{\\#}\wedge(\alpha\lrcorner\;\beta\lrcorner*\varphi)$
$\displaystyle=$
$\displaystyle\alpha\lrcorner*\varphi+(\alpha\lrcorner\;Im\;\Omega_{\beta})\wedge\beta^{\\#}$

(iii) follows from the following computation

$\alpha\lrcorner\;Re\;\Omega_{\beta}=\alpha\lrcorner\;\beta\lrcorner\;(\beta^{\\#}\wedge\varphi)=\alpha\lrcorner\;\varphi+\beta^{\\#}\wedge(\alpha\lrcorner\;\beta\lrcorner\;\varphi)=\alpha\lrcorner\;\varphi-(\alpha\lrcorner\;\omega_{\beta})\wedge\beta^{\\#}$

Note that even though the identities of this proposition hold only after
restricting the right hand side to $X_{\alpha}$, all the individual terms are
defined everywhere on $(M,\varphi)$. Also, from the construction, $X_{\alpha}$
and $X_{\beta}$ inherit vector fields $\beta$ and $\alpha$, respectively.

######  Corollary 8.

[AS2] Let $\\{\alpha,\beta\\}$ be orthonormal vector fields on $(M,\varphi)$.
Then there are $A_{\alpha\beta}\in\Omega^{3}(M)$, and
$W_{\alpha\beta}\in\Omega^{2}(M)$ satisfying

$\displaystyle(a)\;\;\;\;\;\;\;\;\;\;\;\varphi|_{X_{\alpha}}=Re\;\Omega_{\alpha}$
and $\displaystyle\varphi|_{X_{\beta}}=Re\;\Omega_{\beta}$
$\displaystyle(b)\;\;\;\;\;\;A_{\alpha\beta}|_{X_{\alpha}}=Im\;\Omega_{\alpha}$
and $\displaystyle
A_{\alpha\beta}|_{X_{\beta}}=\alpha\lrcorner\;(\star\omega_{\beta})$
$\displaystyle(c)\hskip 36.135ptW_{\alpha\beta}|_{X_{\alpha}}=\omega_{\alpha}$
and $\displaystyle
W_{\alpha\beta}|_{X_{\beta}}=\alpha\lrcorner\;Re\;\Omega_{\beta}$

Now we can choose $\alpha$ as $R$ and $\beta$ as $R^{\prime}$ of the given
$HL$ manifold. That concludes that given a $HL$ submanifold of a $G_{2}$
manifold, it will determine a “canonical” mirror pair of Calabi-Yau manifolds
(related thru the HL manifold) with the complex and symplectic structures
given above.

## References

  * [AS1] S. Akbulut and S. Salur, Deformations in $G_{2}$ manifolds, Adv. in Math, vol 217, Issue 5 (2008) 2130-2140.
  * [AS2] S. Akbulut and S. Salur, Mirror duality via $G_{2}$ and Spin(7) manifolds Arithmetic and Geometry Around Quantization, Progress in Math., vol 279 (2010) 1-21.
  * [AS3] S. Akbulut and S. Salur, Calibrated manifolds and gauge theory, Jour. Reine Angew. Math. (Crelle’s J.) vol 2008, no.625 (2008) 187-214.
  * [B1] R.L. Bryant, Some remarks on $G_{2}$-structures, Proceedings of G kova Geometry-Topology Conference (2005), 75-109. 1–106.
  * [HL] F.R. Harvey, and H.B. Lawson, Calibrated geometries, Acta. Math. 148 (1982), 47–157.
  * [M] R.C. McLean, Deformations of calibrated submanifolds, Comm. Anal. Geom. 6 (1998), 705–747.
  * [T] E. Thomas, Postnikov invariants and higher order cohomology operations, Ann. of Math. vol 85 (1967), 184–217.


# A* shortest string decoding for non-idempotent semirings

Kyle Gorman & Cyril Allauzen
Google

###### Abstract

The single shortest path algorithm is undefined for weighted finite-state
automata over non-idempotent semirings because such semirings do not guarantee
the existence of a shortest path. However, in non-idempotent semirings
admitting an order satisfying a monotonicity condition (such as the plus-times
or log semirings), the notion of _shortest string_ is well-defined. We
describe an algorithm which finds the shortest string for a weighted non-
deterministic automaton over such semirings using the backwards shortest
distance of an equivalent deterministic automaton (DFA) as a heuristic for A*
search performed over a companion idempotent semiring, which is proven to
return the shortest string. While there may be exponentially more states in
the DFA, this algorithm needs to visit only a small fraction of them if
determinization is performed on the fly.

##  1 Introduction

Weighted finite-state automata provide a compact representation of hypotheses
in speech recognition and text processing applications (e.g., Mohri,, 1997;
Mohri et al.,, 2002; Roark and Sproat,, 2007; Gorman and Sproat,, 2021). Under
a wide set of assumptions, weighted finite-state lattices allow for efficient
polynomial-time decoding via shortest-path algorithms (Mohri,, 2002). However,
when weights of a lattice are defined over non-idempotent semirings, there is
no guarantee that a shortest path exists. However, in non-idempotent semirings
admitting an order satisfying a monotonicity condition, the notion of
_shortest string_ is well-defined. We show below that it is still possible to
efficiently determine the shortest string for such lattices. This study
proposes a simple algorithm which finds the shortest string of automata over
non-idempotent monotonic negative semirings such as the plus-times and log
semirings. The algorithm uses shortest-path search with the A* queue
discipline (Hart et al.,, 1968) and on the fly determinization (Mohri,, 1997).
After providing definitions and the algorithm, we describe an implementation
and evaluate it using speech recognizer word lattices.

##  2 Definitions

Before we introduce the algorithm we provide brief definitions of key notions.

###  2.1 Semirings

Weighted automata algorithms operate with respect to an algebraic system known
as a _semiring_ , defined by the combination of two _monoids_.

######  Definition 2.1.

A _monoid_ is a pair $(\mathbb{K},\bullet)$ where $\mathbb{K}$ is a set and
$\bullet$ is a binary operator over $\mathbb{K}$ with the following
properties:

  1. 1.

_closure_ : $\forall a,b\in\mathbb{K}:a\bullet b\in\mathbb{K}$,

  2. 2.

_associativity_ : $\forall a,b,c\in\mathbb{K}:(a\bullet b)\bullet
c=a\bullet(b\bullet c)$, and

  3. 3.

_identity_ : $\exists e\in\mathbb{K}:e\bullet a=a\bullet e=a$.

######  Definition 2.2.

A monoid is _commutative_ in the case that $\forall a,b\in\mathbb{K}:a\bullet
b=b\bullet a$.

######  Definition 2.3.

A semiring is a five-tuple $(\mathbb{K},\oplus,\otimes,\bar{0},\bar{1})$ such
that:

  1. 1.

$(\mathbb{K},\oplus)$ is a commutative monoid with identity element $\bar{0}$,

  2. 2.

$(\mathbb{K},\otimes)$ is a monoid with identity element $\bar{1}$,

  3. 3.

$\forall a,b,c\in\mathbb{K}:a\otimes(b\oplus c)=(a\otimes b)\oplus(a\otimes
c)$, and

  4. 4.

$\forall a\in\mathbb{K}:a\otimes\bar{0}=\bar{0}\otimes a=\bar{0}$.

######  Definition 2.4.

A semiring is _zero-sum-free_ if non-$\bar{0}$ elements cannot sum to
$\bar{0}$; that is, $\forall a,b\in\mathbb{K}:a\oplus b\implies a=b=\bar{0}$.

######  Definition 2.5.

A semiring is _idempotent_ if $\oplus$ is idempotent; that is, $\forall
a\in\mathbb{K}:a\oplus a=a$.

######  Definition 2.6.

A semiring has the _path property_ if $\forall a,b\in\mathbb{K}:a\oplus
b\in\\{a,b\\}$.

######  Remark 2.1.

If a semiring has the path property it is also idempotent.

######  Definition 2.7.

The _natural order_ of an idempotent semiring is a boolean operator $\preceq$
such that $\forall a,b\in\mathbb{K}:a\preceq b$ if and only if $a\oplus b=a$.

######  Remark 2.2.

In a semiring with the path property, the natural order is a _total_ order.
That is, $\forall a,b\in\mathbb{K}$, either $a\preceq b$ or $b\preceq a$.

######  Definition 2.8.

A semiring is _monotonic_ if $\forall a,b,c\in\mathbb{K}$, $a\preceq b$
implies:

  1. 1.

$a\oplus c\preceq b\oplus c$,

  2. 2.

$a\otimes c\preceq b\otimes c$, and

  3. 3.

$c\otimes a\preceq c\otimes b$.

######  Definition 2.9.

A semiring is _negative_ if $\bar{1}\preceq\bar{0}$.

######  Remark 2.3.

In a monotonic negative semiring, $\forall a,b\in\mathbb{K}:a\preceq\bar{0}$
and $a\oplus b\preceq b$.

Some examples of monotonic negative semirings are given in Table 1.

|  | $\mathbb{K}$ | $\oplus$ | $\otimes$ | $\bar{0}$ | $\bar{1}$ | $\preceq$
---|---|---|---|---|---|---|---
Plus-times |  | $\mathbb{R}_{+}$ | $+$ | $\times$ | $0$ | $1$ | $\geq$
Max-times |  | $\mathbb{R}_{+}$ | max | $\times$ | $0$ | $1$ | $\geq$
Log |  | $\mathbb{R}\cup\\{-\infty,+\infty\\}$ | $\oplus_{\textrm{log}}$ | $+$ | $+\infty$ | $0$ | $\leq$
Tropical |  | $\mathbb{R}\cup\\{-\infty,+\infty\\}$ | min | $+$ | $+\infty$ | $0$ | $\leq$
Table 1: Common monotonic negative semirings;
$a\oplus_{\textrm{log}}b=-\textrm{ln}(e^{-a}+e^{-b})$.

######  Definition 2.10.

The _companion semiring_ of a monotonic negative semiring
$(\mathbb{K},\oplus,\otimes,\bar{0},\bar{1})$ with a total order $\preceq$ is
the semiring $(\mathbb{K},\widehat{\oplus},\otimes,\bar{0},\bar{1})$ where
$\widehat{\oplus}$ is the minimum binary operator for $\preceq$:

$a~{}\widehat{\oplus}~{}b=\begin{cases}a&\textrm{if }a\preceq b\\\
b&\textrm{otherwise}\\\ \end{cases}$

######  Remark 2.4.

The max-times and tropical semirings are companion semirings to the plus-times
and log semirings, respectively.

######  Remark 2.5.

By construction a companion semiring has the path property and natural order
$\preceq$.

###  2.2 Weighted finite-state acceptors

Without loss of generality, we consider single-source $\epsilon$-free weighted
finite-state acceptors.111 The definition provided here can easily be
generalized to automata with multiple initial states, with initial or final
weights, or with $\epsilon$-transitions (e.g., Roark and Sproat,, 2007, ch. 1,
Mohri,, 2009, Gorman and Sproat,, 2021, ch. 1).

######  Definition 2.11.

A _weighted finite-state acceptor_ (WFSA) is defined by a five-tuple
$(Q,s,\Sigma,\omega,\delta)$ and a semiring
$(\mathbb{K},\oplus,\otimes,\bar{0},\bar{1})$ where:

  1. 1.

$Q$ is a finite set of states,

  2. 2.

$s\in Q$ is the _initial state_ ,

  3. 3.

$\Sigma$ is the _alphabet_ ,

  4. 4.

$\omega\subseteq Q\times\mathbb{K}$ is the _final weight function_ , and

  5. 5.

$\delta\subseteq Q\times\Sigma\times\mathbb{K}\times Q$ is the _transition
relation_.

######  Definition 2.12.

An WFSA is _acyclic_ if there exists a _topological ordering_ , an ordering of
the states such that if there is a transition from state $q$ to $r$ where
$q,r\in Q$, then $q$ is ordered before $r$. Otherwise, the WFSA is _cyclic_.

######  Definition 2.13.

A WFSA is _deterministic_ if, for each state $q\in Q$, there is at most one
transition with a given label $z\in\Sigma$ from that state, and _non-
deterministic_ otherwise.

######  Remark 2.6.

For every non-deterministic, acyclic WFSA (or NFA) over a zero-sum-free
semiring, there exists an equivalent deterministic WFSA (henceforth, a DFA).
However, a DFA may be exponentially larger than an equivalent NFA.

Figure 1 gives an example of an NFA and an equivalent DFA.

|  |
---|---|---
Figure 1:  State diagrams showing a weighted NFA (left) and an equivalent DFA
(right).

###  2.3 Shortest distance

######  Definition 2.14.

A state $q\in Q$ is _final_ if $\omega(q)\neq\bar{0}$.

######  Definition 2.15.

Let $F=\\{q\mid\omega(q)\neq\bar{0}\\}$ denote the set of final states.

######  Definition 2.16.

A _path_ through an acceptor $p$ is a triple consisting of:

  1. 1.

a state sequence $q[p]=q_{1},q_{2},\ldots,q_{n},\in Q^{n}$,

  2. 2.

a weight sequence $k[p]=k_{1},k_{2},\ldots,k_{n}\in\mathbb{K}^{n}$, and

  3. 3.

a string $z[p]=z_{1},z_{2}\ldots,z_{n}\in\Sigma^{n}$.

subject to the constraint that $\forall
i\in[1,n]:(q_{i},z_{i},k_{i},q_{i+1})\in\delta$.

That is, each transition from $q_{i}$ to $q_{i+1}$ must have label $z_{i}$ and
weight $k_{i}$.

######  Definition 2.17.

Let $P_{q\rightarrow r}$ be the set of all paths from $q$ to $r$ where $q,r\in
Q$.

######  Definition 2.18.

The _forward shortest distance_ $\alpha\subseteq Q\times\mathbb{K}$ is a
partial function from a state $q\in Q$ that gives the $\oplus$-sum of the
$\otimes$-product of the weights of all paths from the initial state $s$ to
$q$:

$\alpha(q)=\bigoplus_{p\in P_{s\rightarrow q}}\bigotimes_{k_{i}\in
k[p]}k_{i}.$

######  Definition 2.19.

The _backwards shortest distance_ $\beta\subseteq Q\times\mathbb{K}$ is a
partial function from a state $q\in Q$ that gives the $\oplus$-sum of
$\otimes$-product of the weights of all paths from $q$ to a final state,
including the final weight of that final state:

$\beta(q)=\bigoplus_{f\in F}\left(\bigoplus_{p\in P_{q\rightarrow
f}}\bigotimes_{k_{i}\in k[p]}k_{i}\otimes\omega(f)\right).$

######  Definition 2.20.

The _total shortest distance_ through an automaton is given by $\beta(s)$.

###  2.4 Shortest path

######  Definition 2.21.

A path is _complete_ if

  1. 1.

$(s,z_{1},k_{1},q_{1})\in\delta$, and

  2. 2.

$q_{n}\in F$.

That is, a complete path must also begin with an arc from the initial state
$s$ to $q_{1}$ with label $z_{1}$ and weight $k_{1}$, and halt in a final
state.

######  Definition 2.22.

The weight of a complete path is given by the $\otimes$-product of its weight
sequence and its final weight

$\bar{k}=\left(\bigotimes_{k_{i}\in k[p]}k_{i}\right)\otimes\omega(q_{n}).$

######  Definition 2.23.

A _shortest path_ through an automaton is a complete path whose weight is
equal to the total shortest distance $\beta(s)$.

Note that automata over non-idempotent semirings need not have a shortest path
as defined here (Mohri,, 2002, 322). Consider for example the NFA shown in the
left side of Figure 1. Let us assume that $k\oplus k\preceq k<k^{\prime}$.
Then, the total shortest distance is $k\oplus k$ but the shortest path is $k$.
Definitionally, a non-idempotent semiring does not guarantee these two weights
will be equal. In that case, no shortest path exists because no complete path
with the weight of the total shortest distance exists.

######  Remark 2.7.

It is generally impossible to find the shortest path efficiently over non-
monotonic semirings.222 See Mohri, (2002) for general conditions under which
the shortest path can be found in polynomial time.

###  2.5 Shortest string

######  Definition 2.24.

Let $P_{z}$ be a set of paths with string $z\in\Sigma^{*}$, and let the weight
of $P_{z}$ be

$\sigma(z)=\bigoplus_{p\in P_{z}}\bar{k}[p].$

######  Definition 2.25.

A _shortest string_ $z$ is one such that $\forall
z^{\prime}\in\Sigma^{*},\sigma(z)\preceq\sigma(z^{\prime})$.

######  Lemma 2.1.

In an idempotent semiring, a shortest path’s string is also a shortest string.

###### Proof.

Let $p$ be a shortest path. By definition,
$\bar{k}[p]\preceq\bar{k}[p^{\prime}]$ for all complete paths $p^{\prime}$. It
follows that $\forall z^{\prime}\in\Sigma^{*}:\sigma(z[p])=\bigoplus_{p\in
P_{z}}\bar{k}[p]\preceq\sigma(z^{\prime}[p^{\prime}])=\bigoplus_{p^{\prime}\in
P_{z}}\bar{k}[p^{\prime}]$ so $z[p]$ is the shortest string. ∎

######  Lemma 2.2.

In a DFA over a monotonic semiring, a shortest string is the string of a
shortest path in that DFA viewed as an WFSA over the corresponding companion
semiring.

###### Proof.

Determinism implies that for all complete path $p^{\prime}$,
$\bar{k}[p^{\prime}]=\sigma(z[p^{\prime}])$. Let $z$ be the shortest string in
the DFA and $p$ the unique path admitting the string $z$. Then
$\bar{k}[p]=\sigma(z)\preceq\sigma(z[p^{\prime}])=\bar{k}[p^{\prime}]$ for any
complete path $p^{\prime}$. Hence
$\bar{k}[p]=\widehat{\bigoplus}_{p^{\prime}\in P_{s\rightarrow
F}}\bar{k}[p^{\prime}]$. Thus $p$ is a shortest path in the DFA viewed over
the companion semiring. ∎

###  2.6 A* search

A* search (Hart et al.,, 1968) is a common _shortest-first_ search strategy
for computing the shortest path in a WFSA over an idempotent semiring. It can
be thought of as a variant of Dijkstra,’s (1959) algorithm, in which
exploration is guided by a shortest-first priority queue discipline. At every
iteration, the algorithm explores the state $q$ which minimizes $\alpha(q)$,
the shortest distance from the initial state $s$ to $q$, until all states have
been visited. In A* search, priority is instead a function of
$\digamma\subseteq Q\times\mathbb{K}$, known as the _heuristic_ , which gives
an estimate of the weight of paths from some state to a final state. At every
iteration, A* instead explores the state $q$ which minimizes
$\alpha(q)\otimes\digamma(q)$.333 One can thus consider Dijkstra,’s algorithm
a special case of A* search with the uninformative heuristic
$\digamma=\bar{1}$.

######  Definition 2.26.

An A* heuristic is _admissible_ if it never overestimates the shortest
distance to a state. That is, it is admissible if $\forall q\in
Q:\digamma(q)\preceq\beta(q)$.

######  Definition 2.27.

An A* heuristic is _consistent_ if it never overestimates the cost of reaching
a successor state. That is, it is consistent if $\forall q,r\in Q$ such that
$\digamma(q)\preceq k\otimes\digamma(r)$ if $(q,z,k,r)\in\delta$, i.e., if
there is a transition from $q$ to $r$ with some label $z$ and weight $k$.

######  Remark 2.8.

If $\digamma$ is _admissible_ and _consistent_ , A* search is guaranteed to
find a shortest path (if one exists) after visiting all states such that
$\digamma[q]\preceq\beta[s]$ (Hart et al.,, 1968, 104f.).

##  3 The algorithm

Consider an acyclic, $\epsilon$-free WFSA over a monotonic negative semiring
$(\mathbb{K},\oplus,\otimes,\bar{0},\bar{1})$ with total order $\preceq$ for
which we wish to find the shortest string. The same WFSA can also be viewed as
a WFSA over the corresponding companion semiring
$(\mathbb{K},\widehat{\oplus},\otimes,\bar{0},\bar{1})$, and we denote by
$\widehat{\beta}$ the backward shortest-distance over this companion semiring.
We prove two theorems, and then introduce an algorithm for search.

######  Theorem 3.1.

The backwards shortest distance of an WFSA over a monotonic negative semiring
is an admissible heuristic for the A* search over its companion semiring.

###### Proof.

In a monotonic negative semiring, the $\oplus$-sum of any $n$ terms is upper-
bounded by each of the $n$ terms and hence by the $\widehat{\oplus}$-sum of
these $n$ terms. It follows that:

$\digamma(q)=\beta(q)=\bigoplus_{p\in P_{q\rightarrow
F}}\bar{k}[p]\preceq\widehat{\bigoplus_{p\in P_{q\rightarrow
F}}}\bar{k}[p]=\widehat{\beta}(q),$

showing that $\digamma=\beta$ is an admissible heuristic for
$\widehat{\beta}$. ∎

######  Theorem 3.2.

The backwards shortest distance of an WFSA over a monotonic negative semiring
is a consistent heuristic for the A* search over its companion semiring.

###### Proof.

Let $(q,z,k,r)$ be a transition in $\delta$. Leveraging again the property
that an $\oplus$-sum of any $n$ terms is upper-bounded by any of these terms,
we show that

$\digamma(q)=\beta(q)=\bigoplus_{p\in P_{q\rightarrow
F}}\bar{k}[p]=\bigoplus_{(q,z^{\prime},k^{\prime},r^{\prime})\in\delta}k^{\prime}\otimes\beta(r^{\prime})\preceq
k\otimes\beta(r)=k\otimes\digamma(r)$ (1)

showing that $\digamma=\beta$ is a consistent heuristic. ∎

Having established that this is an admissible and consistent heuristic for A*
search over the companion semiring, a naïve algorithm then suggests itself
following Lemma 2.2 and Remark 2.8. Given a non-deterministic WFSA over the
monotonic negative semiring $(\mathbb{K},\oplus,\otimes,\bar{0},\bar{1})$,
apply determinization to obtain an equivalent DFA, compute $\beta_{d}$, the
backwards shortest distance over the resulting DFA over
$(\mathbb{K},\oplus,\otimes,\bar{0},\bar{1})$ and then perform A* search over
the companion semiring using $\beta_{d}$ as the heuristic. However,
determinization has an exponential worse-case complexity in time and space and
is often prohibitive in practice. Yet determinization—and the computation of
elements of $\beta_{d}$—only need to be performed for states actually visited
by A* search. Let $\beta_{n}$ denote backwards shortest distance over a non-
deterministic WFSA over the monotonic negative semiring
$(\mathbb{K},\oplus,\otimes,\bar{0},\bar{1})$. Then, the algorithm is as
follows.

  1. 1.

Compute $\beta_{n}$ over $(\mathbb{K},\oplus,\otimes,\bar{0},\bar{1})$.

  2. 2.

Lazily determinize the WFSA, lazily computing $\beta_{d}$ from $\beta_{n}$
over $(\mathbb{K},\oplus,\otimes,\bar{0},\bar{1})$.

  3. 3.

Perform A* search for the shortest string using $\beta_{d}$ as the heuristic
over $(\mathbb{K},\widehat{\oplus},\otimes,\bar{0},\bar{1})$.

##  4 Evaluation

We evaluate the proposed algorithm using non-idempotent speech recognition
lattices.

###  4.1 Data

We search for the shortest string in a sample of 700 word lattices derived
from Google Voice Search traffic. This data set was previously used by Mohri
and Riley, (2015) and Gorman and Sproat, (2021, ch. 4) for evaluating related
WFSA algorithms. Each path in these lattices is a single hypothesis
transcription produced by a production-grade automatic speech recognizer, here
treated as a black box. The exact size of each input lattice size is
determined by a probability threshold, so paths with probabilities below a
certain threshold have been pruned. These lattices are acyclic,
$\epsilon$-free, on-deterministic WFSAs over the log semiring, a non-
idempotent semiring.

###  4.2 Implementation

The above algorithm is implemented as part of OpenGrm-BaumWelch, an open-
source C++17 library released under the Apache-2.0 license.444
https://baumwelch.opengrm.org This toolkit includes baumwelchdecode, a
command-line tool which implements the above algorithm over the log semiring,
using the tropical semiring as a companion semiring. This implementation
depends in turn on implementations of determinization, shortest distance, and
shortest path algorithms provided by OpenFst (Allauzen et al.,, 2007). This
command-line tool, along with various OpenFst command-line utilities, were
used to conduct the following experiment.

###  4.3 Method

We compute the number of states in the non-deterministic (NFA) lattice, the
number of states in an equivalent DFA lattice—created by applying weighted
determinization as implemented by OpenFst’s fstdeterminize command—and the
number of DFA states visited during A* search as implemented by
baumwelchdecode.

###  4.4 Results

Results are shown in Figure 2. The relationship between the size of the NFA
and the number of DFA states visited by the proposed decoding method appears
roughly monomial (i.e., log-log linear). The experiment was repeated by
performing weighted determinization on each of the NFA lattices beforehand,
which produces roughly a factor-of-seven increase in the size of the lattice.
Since the increase in state size associated with full determinization is
itself linear, a monomial relation also holds between the size of the DFA and
the number of DFA states visited by A* search. From these results we infer
that the heuristic substantially reduces the number of DFA states visited, and
thus the degree of determinization required, compared to the naïve algorithm.

Figure 2: Word lattice decoding with the proposed algorithm. The $x$-axis
shows the number of states in each word lattice NFA; the $y$-axis shows the
number of states visited by A* decoding. Both axes are log scale.

##  5 Related work

Mohri and Riley, (2002) describe a related algorithm for finding the $n$-best
strings over an idempotent finite-state automaton. Much like the algorithm
proposed here, they use A* search and on-the-fly determinization; however,
they do not consider decoding over non-idempotent semirings.

##  6 Conclusions

We propose a algorithm which allows for efficient shortest string decoding of
weighted automata over non-idempotent semirings using A* search and on-the-fly
determinization. We find that A* search results in a substantial reduction in
the number of DFA states visited during decoding, which in turn minimizes the
degree of determinization required to find the shortest path.

We envision many applications for the proposed algorithm. For instance, it can
be used exact decoding of lattices scored with interpolated language models
(e.g., Jelinek and Mercer,, 1980).

## Acknowledgments

Thanks to Michael Riley, Brian Roark, and Jeffrey Sorensen for discussion and
code review.

## References

  * Allauzen et al., (2007) Allauzen, C., Riley, M., Schalkwyk, J., Skut, W., and Mohri, M. (2007).  OpenFst: a general and efficient weighted finite-state transducer library.  In Implementation and Application of Automata: 12th International Conference (CIAA 2007), pages 11–23.
  * Dijkstra, (1959) Dijkstra, E. W. (1959).  A note on two problems in connexion with graphs.  Numerische Mathematik, 1(1):269–271.
  * Gorman and Sproat, (2021) Gorman, K. and Sproat, R. (2021).  Finite-State Text Processing.  Morgan & Claypool.
  * Hart et al., (1968) Hart, P. E., Nilsson, N. J., and Raphael, B. (1968).  A formal basis for the heuristic determination of minimal cost paths.  IEEE Transactions on Systems Science and Cybernetics, 4(2):100–107.
  * Jelinek and Mercer, (1980) Jelinek, F. and Mercer, R. L. (1980).  Interpolated estimation of Markov source parameters from sparse data.  In In Proceedings of the Workshop on Pattern Recognition in Practice, pages 381–397.
  * Mohri, (1997) Mohri, M. (1997).  Finite-state transducers in language and speech processing.  Computational Linguistics, 23(2):269–311.
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Dark energy due to quantum corrections to effective
potential

V.A. Filippov1,4, R.M. Iakhibbaev1, D.I. Kazakov 1,2
and D.M. Tolkachev1,3

1Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear
Research, Dubna, Russia.
2Moscow Institute of Physics and Technology, Dolgoprudny, Russia
3 B.I. Stepanov Institute of Physics National Academy of Sciences of Belarus,
Minsk, Belarus
and
4Sarov branch of Moscow State University, Sarov, Russia

###### Abstract

In this paper we calculate quantum corrections to the effective potential in
different models of inflationary cosmology. We show that quantum corrections
lead to a modification of the initial potential uplifting its value at the
minimum, which can be interpreted as a cosmological constant/dark energy. We
concentrate on the models of $\alpha$-attractors and show that one can
naturally get small values of the cosmological constant. In the model of
quintessence, the same mechanism modifies the value of dark energy at the
present time.

##  1 Introduction

Inflationary expansion is one of the most promising models allowing one to
describe the properties of the early Universe and its later development [1, 2,
3]. Though the nature of early and late acceleration is still a significant
theoretical problem, the simple assumption that the potential energy of a
scalar field can be the source of accelerated expansion has been successful in
terms of investigation, deriving predictions, and agreement with current
observational data [4, 5]. In the simplest realisation of the inflationary
scenario, it is generally assumed that after the Universe leaves the
inflationary stage and post-inflationary reheating, it remains at minimum
potential with a value of $\Lambda\sim 10^{-47}$ GeV4 or $10^{-120}$ in Planck
units [6]. One can assume that an acceptable explanation of the current cosmic
acceleration is related to the introduction of a field that currently has the
value of $\Lambda$ [7]. In fact, shortly after the discovery of the current
cosmic acceleration at the end of the last century, a class of cosmological
models was presented that tries to unify accelerating expansion in early and
late times [8, 9, 10].

A small value of the cosmological constant $\Lambda$ can be explained by the
adoption of the anthropic principle, which removes the question of the meaning
of such a small value and refers this question to the problem of landscape in
string theory. Then it is enough to have a theoretical possibility of
obtaining diverse values of primordial quantum fluctuations in a viable
Universe with a small value of the dark energy density selected by the
anthropic principle. In this paper, however, we follow another point of view
and show that quantum corrections to some cosmological models[11, 12] can lead
to a significant modification of the behaviour of the initial potential and
the appearance of a non-zero ground state energy of the Universe which can be
interpreted as a cosmological constant.

In what follows, we apply the formalism of the effective potential to
cosmological models known as $\alpha$-attractors [13, 14, 15]. In this paper,
we consider the simplest forms of $\alpha$-attractors which can be represented
by the so-called $T$-models with the potential

$V(\varphi)=g\tanh^{n}\left(\frac{\varphi}{\sqrt{6\alpha}M}\right)=gV_{T},$
(1)

(we will mainly focus on the cases $n=2,4$, and call them the $T^{2}$, $T^{4}$
models, respectively) and $E$-models with the potential

$V(\varphi)=g\left(1-e^{-\sqrt{\frac{2}{3\alpha}}\frac{\varphi}{M}}\right)^{n}=gV_{E}$
(2)

(we call the cases $n=2,4$ $E^{2}$\- and $E^{4}$-models, respectively)[16].
Here $\varphi$ is the inflationary scalar field, $M_{Pl}=(8\pi G)^{-1/2}$ is
the Planck mass, $g=m_{inf}^{2}M_{Pl}^{2}$ is the scale of inflation, where
$m_{inf}\approx 10^{-5}M_{Pl}$ is the inflaton mass and $\alpha$ is a positive
number. These models are characterised by an exponentially fast reaching of a
plateau at large values of the inflaton field and are successfully used to
study the stage of early accelerated expansion of the Universe as well as the
dark energy dominated phase [17, 15, 9]. The predictions derived from these
models are consistent with the PLANCK and BICEP/Keck [4, 5] observational
data. Moreover, introducing a number of e-folding of inflation $N$ (in the
slow-roll approximation), one can obtain that all these models have a point of
attraction in the leading approximation in $1/N$ [13, 3]

$r\simeq 12\alpha/N^{2},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ n_{s}\simeq
1-2/N,$ (3)

which for $N=50-60$ and $\alpha\sim 1$ fits the observed values of the slow-
roll parameters [5].

Another type of models known as the quintessence inflation models of
$\alpha$-attractors can be represented by the T-model with $n=1$. In this
case, the cosmological constant is not really a constant but a function of the
field $\varphi$ at the present time [15, 9, 10].

The paper is organized as follows. In the first part, we recall the key ideas
for deriving the renormalizable group equation (RG-equation) for effective
potentials for non-renormalizable interactions generalizing the approach of
[18]. The RG-equation allows one to sum up all the leading contributions
(leading logs) to the effective potential in all orders of perturbation
theory. Then we consider the application of this approach to a class of $T$
and $E$ cosmological models and demonstrate how a non-zero cosmological
constant appears as a result of radiative corrections. We discuss the
properties of the the obtained quantum potentials from the point of view of
the inflation scenario. Finally, we consider a special case of the $T$ model,
which is usually used to study the quintessence model.

##  2 Effective potential for a general scalar model

The effective potential $V_{eff}(\varphi)$ is defined as a constant part of
the effective action [19]. It can be seen as supplement for the classical
potential by quantum loop corrections. The effective potential can be obtained
by summing all 1PI vacuum diagrams with propagators containing a mass term
depending on the classical field $\varphi$: $m^{2}(\varphi)=gv_{2}(\varphi)$,
where $v_{2}(\varphi)\equiv\frac{d^{2}V(\varphi)}{d\varphi^{2}}$ and each
vertex corresponds to
$v_{n}(\varphi)\equiv\frac{d^{n}V(\varphi)}{d\varphi^{n}}$, where $n$ is equal
to the number of propagators at each vertex. The above Feynman rules for
vertices can be found from the expansion of the potential
$V(\varphi+\widehat{\varphi})$ over the quantum field $\widehat{\varphi}$. As
a result, the effective potential is constructed as a perturbative expansion
over the coupling constant $g$

$V_{eff}(\varphi,g)=g\sum_{n=0}^{\infty}(-g)^{n}V_{n}(\varphi).$ (4)

First of all, we compute the one-loop diagram corresponding to $V_{1}$. To
regularize the UV divergence, we use dimensional regularization and take
$D=4-2\epsilon$. Then one has

$V_{1}=\frac{1}{16\pi^{2}}\frac{1}{4}\frac{v_{2}^{2}}{\epsilon}\left(\frac{\mu^{2}}{m^{2}}\right)^{\epsilon}\to\frac{1}{16\pi^{2}}\frac{v_{2}^{2}}{4}\left(\frac{1}{\epsilon}-\log\frac{m^{2}}{\mu^{2}}\right),\
\ \ m^{2}=gv_{2}(\varphi).$ (5)

The divergent part $\sim 1/\epsilon$ is usually absorbed into the
corresponding counterterm which in renormalizable theories leads to
renormalization of the coupling. In the non-renormalizable case, this
counterterm does not repeat the original Lagrangian and thus leads to the
appearance of a new structure. This situation is then continued in the
subsequent orders of perturbation theory so that one has an infinite series of
new terms. This means that in the non-renormalizable case, one has infinite
arbitrariness in the choice of counterterms, which is not reduced to the
renormalization of a finite number of couplings. Fortunately, this
arbitrariness does not influence the leading log terms that we discuss. For
this reason, in what follows we will drop the singular terms and take care of
the log dependence only.

Note that the coefficient of the singular term $\sim 1/\epsilon$ coincides
with the coefficient of the logarithm $\sim\log(gv_{2}(\varphi))$ giving the
contribution to the effective potential. This is true in all orders of PT.
Hence, to calculate the leading log terms $\sim\log^{n}\frac{m^{2}}{\mu^{2}}$,
one has to calculate the leading poles $1/\epsilon^{n}$ and, replacing the
pole by the log, get the leading contribution to the effective potential. In
its turn, to find the leading poles, one can use the BPHZ $\cal{R}$-operation
[20, 21, 22] and the Bogoliubov-Parasiuk theorem [23] which states that after
subtracting the subdivergences the singular part is local in coordinate space
or has a polynomial structure in momentum space. This allows one to get
recurrent relations for the subsequent leading divergences. Denoting the
singular part of the effective potential (the leading pole term $\sim
1/\epsilon^{n}$) in the n-th order of perturbation theory by $\Delta V_{n}$,
we can write the following recurrence equation [12]:

$n\Delta V_{n}=\frac{1}{4}\sum_{k=0}^{n-1}D_{2}\Delta V_{k}D_{2}\Delta
V_{n-1-k},\ \ \,n\geq 1,\ \ \Delta V_{0}=V_{0},$ (6)

where $D_{2}\equiv\partial^{2}/\partial\varphi^{2}$ is the second derivative
with respect to the field $\varphi$.

This recurrent relation can be converted into the differential equation for
the sum

$\Sigma(z,\varphi)=\sum_{n=0}^{\infty}(-z)^{n}\Delta V_{n}(\varphi),$ (7)

where $z=g/\epsilon$. This is just the RG equation for the function
$\Sigma(z,\varphi)$

$\frac{\partial\Sigma}{\partial
z}=-\frac{1}{4}\left(\frac{\partial^{2}\Sigma}{\partial\varphi^{2}}\right)^{2},\
\ \Sigma(0,\varphi)=V_{0}(\varphi).$ (8)

This partial differential equation generalises the well-known RG-equation in
renormalizable scalar theory for the non-renormalizable case [12]. It is
universal and does not depend on the arbitrariness of subtracting the UV
divergences, as it was mentioned earlier. Unfortunately, except for the case
of the renormalizable $\varphi^{4}$ potential, one cannot be solved
analytically and one has to resort to numerical solution.

To obtain an effective potential, one only needs to replace in the obtained
solution the argument $z$ by the corresponding logarithm:

$V_{eff}(g,\varphi)=g\Sigma(z,\varphi)|_{z\to-\frac{g}{16\pi^{2}}\log{gv_{2}/\mu^{2}}}.$
(9)

The effective potential obtained this way contains the leading log
contributions from all orders of perturbation theory. The full solution
usually has smoother behaviour than the one-loop approximation and can include
both poles and discontinuities and restore the original symmetry of the
classical potential [11].

We now apply the formalism described above to study the global properties of
effective potentials for the cosmological models of chaotic inflation.

##  3 Cosmological constant from quantum corrections

Consider the following general action of the single-field inflationary
cosmology:

$S=\int
d^{4}x\sqrt{-{\textsl{g}}}\left[\frac{M_{Pl}^{2}}{2}R({\textsl{g}})+\frac{1}{2}{\textsl{g}}^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)\right],$
(10)

where $\phi$ is the inflaton field, ${\textsl{g}}_{\mu\nu}$ is the space-time
metric, $R({\textsl{g}})$ is the scalar curvature,
${\textsl{g}}=\det({\textsl{g}}_{\mu\nu})$ and $V(\phi)$ is the inflaton
potential. In [24] it was shown that there exists a wide class of so-called
$\alpha$-attractor models with hyperbolic geometry of the following form

$S=\int
d^{4}x\sqrt{-{\textsl{g}}}\left[\frac{M_{Pl}^{2}}{2}R({\textsl{g}})+\frac{1}{2}\frac{\partial_{\mu}\phi\partial^{\mu}\phi}{1-\frac{\phi^{2}}{6\alpha}}-V(\phi)\right],$
(11)

where $\alpha$ is a free parameter, that is in excellent agreement with the
observational data. Such geometries are natural in extended supergravity,
which is one of the justifications of this action [24].

One can get the canonical form of the kinetic term in eq.(11) by solving the
equation
$\partial\phi/\left(1-\frac{\phi^{2}}{1-6\alpha}\right)=\partial\varphi$ with
the solution
$\phi=\sqrt{6\alpha}\tanh\left(\frac{\varphi}{\sqrt{6\alpha}}\right)$. Then
the action (11) can be written in terms of the canonical field $\varphi$ with
the canonical kinetic term as

$S=\int
d^{4}x\sqrt{-{\textsl{g}}}\left[\frac{M_{Pl}^{2}}{2}R({\textsl{g}})+\frac{1}{2}\partial_{\mu}\varphi\partial^{\mu}\varphi-V\left(\sqrt{6\alpha}\tanh\left(\frac{\varphi}{\sqrt{6\alpha}}\right)\right)\right].$
(12)

Therefore the previously mentioned potentials
$V_{T}=\tanh^{n}\left(\frac{\varphi}{\sqrt{6\alpha}M}\right)$ correspond to
the power like potentials for the field $\phi$, namely $\phi^{n}$.

As stated earlier, quantum corrections in general can significantly affect the
behaviour of the potential, in particular, they can "uplift" the minimum
introducing the cosmological constant [12]. Below we consider how this
mechanism works in the case of $T$ and $E$ models of chaotic inflation
introduced above.

###  3.1 The $T$-model

To start with, we derive the generalised RG-equation for the general $T$-model
with the potential
$gV_{T}(\varphi)=g\tanh^{n}\left(\frac{\varphi}{\sqrt{6\alpha}M_{Pl}}\right)$
for an arbitrary power $n$. To transform the equation (8) into an equation for
the chosen potential, it is convenient to rewrite the function $\Sigma$ in
terms of the dimensionless variables $x=z/M_{Pl}^{4}$ and
$y=\tanh^{n}(\varphi/\sqrt{6\alpha}M_{Pl})$. This substitution becomes obvious
if one considers the loop decomposition of the $\Sigma$ function, which can be
represented as polynomials over $\tanh\varphi$. Then one has

$\Sigma(z/M_{Pl}^{4},\tanh^{n}(\varphi/\sqrt{6\alpha}M_{Pl}))\equiv S(x,y).$
(13)

This is somewhat similar to the arbitrary function
$F(\tanh(\frac{\varphi}{\sqrt{6\alpha}M_{Pl}}))$ of the $\alpha$-attractor-
type models [13]; however, this function is now constrained by the RG-
equation.

Changing the variables in the original equation (8), the generalised RG-
equation for the function $S(x,y)$ can be written as

$S_{x}=-\frac{n^{2}y^{2-\frac{4}{n}}\left(y^{2/n}-1\right)^{2}}{144\alpha^{2}}\left(\left(y^{2/n}+n\left(y^{2/n}-1\right)+1\right)S_{y}+ny\left(y^{2/n}-1\right)S_{yy}\right)^{2}.$
(14)

The initial conditions are chosen from the requirement that the expansion of
the effective potential starts from the classical potential in the limit of
$\varphi\to\infty$, i.e. $\tanh(\varphi/\sqrt{6\alpha M_{Pl}})\to 1$ the
potential reaches the plateau with a zero derivative, namely

$S(0,y)=y,\leavevmode\nobreak\ S(x,1)=1,\leavevmode\nobreak\ S_{y}(x,1)=0.$
(15)

Consider now the case when $n=2,4$, which we denote as $T^{2}$\- and $T^{4}$\-
models. The RG equations for theses cases were first derived in [12]. One gets
for $n=2$

$S_{x}=-\frac{(y-1)^{2}\left((3y-1)S_{y}+2(y-1)yS_{yy}\right)^{2}}{36\alpha^{2}},$
(16)

and for $n=4$

$S_{x}=-\frac{y\left(4y\left(\sqrt{y}-1\right)^{2}S_{yy}+\left(5y-8\sqrt{y}+3\right)S_{y}\right)^{2}}{9\alpha^{2}}.$
(17)

Now one has to solve these partial differential equations numerically for the
function of two variables and then to choose on the surface $(x,y)$ the line
corresponding to $x=-\frac{g}{16\pi^{2}}\log{gv_{2}(\varphi)/\mu^{2}}$. This
line has an additional parameter, the scale $\mu$. In the renormalizable case,
the explicit dependence on this scale is compensated by implicit dependence of
the coupling $g$. However, in the non-renormalizable case, $\mu$ remains a
free parameter. For various values of $\mu$ one can have different behaviour
of the potential. In what follows we will choose the value of $\mu$ to satisfy
the slow-roll inflation conditions.

The behaviour of the effective potential for the $T^{2}$-model in comparison
with the one-loop case is shown in Figure 1.

Figure 1: Comparison of the classical $T^{2}$-model potential (blue dashed
line), the one-loop correction (orange dashed line), and the RG effective
potential (black solid line) for illustration at $g\sim 1,\leavevmode\nobreak\
\mu<M_{Pl}$, $\tilde{\varphi}=\varphi/(\sqrt{6}M_{Pl})$

One can see how the one-loop quantum correction leads to a change of the
vacuum state of the potential and to the appearance of additional minima (this
is referred to as the Coleman-Weinberg mechanism [18]) when choosing some
values of the parameter $\mu$. However the sum of all leading quantum
corrections smoothes out the behaviour of the potential and restores the
original symmetry. Note that the plateau-like behaviour of the potential is
not changed whichever approximation of the effective potential we consider
[14]. We have studied various Hubble flow parameters and found out that
quantum corrections do not change the slow-roll behaviour of the potential for
a wide range of parameters $\alpha,g$ and $\mu$ [12].

We show also in Fig.2 a more detailed view of the full effective potential for
various values of $\mu$. Our aim is to get the uplifting of the potential to
obtain non-zero value of energy at the minimum.

Figure 2: $T^{2}$-model potential: variation of $\mu$. The classical potential
(blue dashed line), solid lines denotes all-loop effective potential
$\mu<M_{Pl}$ (black line), $\mu\ll M_{Pl}$ (red line), and $\mu>M_{Pl}$ (green
line). The scales are chosen for illustrative purposes as follows
$\tilde{\varphi}=\varphi/(\sqrt{6}M_{Pl})$ and $g=2$, $\alpha=1$

As can be seen from Fig.2, there are mainly two viable possibilities to get a
positive value of $\Lambda$. First, when the effective potential does not have
additional minima, but has only some uplift. And the second possibility is
when the potential, in addition to the uplift, has additional minima, and the
uplift occurs in these minima. The realisation of the first and second
variants depends only on the value of $\mu$.

In the case when the full result does not differ qualitatively from the one-
loop correction, it is convenient to use explicit formulas for the potential
to study its properties. The one-loop effective potential has the form

$V_{eff}=V_{0}+\frac{g^{2}}{16\pi^{2}}\frac{v_{2}^{2}}{4}\log\frac{gv_{2}(\varphi)}{\mu^{2}},$
(18)

where $v_{2}(\varphi)$ is the second derivative of the classical potential
$V_{0}$.

The non-zero value of the potential at the minimum may be associated with the
cosmological constant given by the relation

$\Lambda=\left.\left[\frac{g^{2}}{16\pi^{2}}\frac{v_{2}^{2}}{4}\log\left(\frac{gv_{2}}{\mu^{2}}\right)\right]\right|_{\varphi=\varphi_{vac}}.$
(19)

Applying this condition to the $T^{2}$-model, one can get the expression for
the parameter $\mu$

$\mu^{2}=\frac{g}{3\alpha
M_{Pl}^{2}}e^{-576\pi^{2}\frac{\alpha^{2}}{g^{2}}\Lambda M_{Pl}^{4}}.$ (20)

Choosing the following set of parameters corresponding to modern observational
data $g=10^{-10}M_{Pl}^{4}$, $M_{Pl}=(8\pi G)^{-\frac{1}{2}}$, $\alpha=1$, and
$\Lambda\sim 10^{-120}M_{Pl}^{4}$, we obtain that $\mu\approx 10^{-6}M_{Pl}$,
which is reasonably close to the inflaton mass. One can invert this argument
and take the value of $\mu$ of the order of the inflaton mass and get a
reasonably small value of the cosmological constant.

When $\Lambda$ is as small as it follows from experimental data, the value of
$\mu$ practically does not depend on the precise value of $\Lambda$. The
variation of $\Lambda$ corresponding to the observed data changes the value of
$\mu$ only by a few decimal points. However, if we start to increase
$\Lambda$, we find that starting from about $10^{-24}M_{Pl}^{4}$, the $\mu$
parameter starts to depend very strongly on $\Lambda$, and $\mu$ quickly tends
to zero at $\Lambda\approx 10^{-20}M_{Pl}^{4}$.

At the same time, it is possible to calculate the point where $\Lambda$ is
exactly zero, i.e. the point where it changes the sign. This point is found
from the condition of changing the sign of the logarithm in (19), namely, when
the argument passes through unity, i.e.
$\left.\frac{gv_{2}}{\mu_{c}^{2}}\right|_{\varphi=\varphi_{vac}}=1$ and we
have the critical point

$\mu_{c}=\frac{10^{-5}}{\sqrt{3\alpha}}M_{Pl}.$ (21)

When $\mu>\mu_{c}$, $\Lambda$ is negative, when $\mu<\mu_{c}$, $\Lambda$ is
positive. This scale is comparable with the scale of the inflaton mass. The
critical value of $\mu$ determines the geometry in the action (10), namely: de
Sitter space (dS) when $\mu>\mu_{c}$, Minkowski space when $\mu=\mu_{c}$ and
anti-de Sitter space (AdS) when $\mu<\mu_{c}$.

A qualitatively similar situation takes place when one takes not only the one-
loop contribution to the effective potential but also the full solution that
includes the leading logs from all loops. In this scenario, the effective
potential has the only minimum at $\varphi_{vac}=0$ and no additional ones. At
the minimum one has a tiny uplift for small values of $\mu$. For large values
of $\mu$ one can have additional minima, but it is difficult to get the
desired small values of $\Lambda$.

Regarding the potential $T^{4}$ and higher powers, they do not have the same
uplift of the potential as in the case of the $T^{2}$-model, which can be seen
from the numerical solution shown in Fig.3. This is because $v_{2}(0)=0$.
Moreover, at additional minima the value of $\Lambda$ happens to be negative.
So these potentials are not suitable for describing the cosmological constant.

Figure 3: Classical $T^{4}$-model potential (blue dashed line), solid lines
denote the all-loop effective potential: red one at $\mu^{2}\ll M_{Pl}^{2}$
and green one at $\mu^{2}\gg M_{Pl}^{2}$. The scales are chosen for
illustrative purposes as follows $\tilde{\varphi}=\varphi/(\sqrt{6}M_{Pl})$
and $g=1$, $\alpha=1$

###  3.2 The $E$-model

Here we repeat the same analysis of the potentials for the $E$-model (2). This
potential is interesting because it is asymmetric and plateau-like behaviour
is present only for positive values of the field, while for negative values
the potential grows exponentially fast to infinity. From qualitative
considerations it is not quite clear whether such asymptotic behaviour of the
potential will remain intact after taking into account quantum corrections.

As before, to simplify equation (8), we rewrite the function $\Sigma$ in the
dimensionless variables $x=z/M_{Pl}^{4}$ and
$y=(1-e^{\sqrt{\frac{2}{3\alpha}}\frac{\varphi}{M_{Pl}}})^{n}$. Then one has

$\Sigma\left(z/M_{Pl}^{4},(1-e^{\sqrt{\frac{2}{3\alpha}}\frac{\varphi}{M_{Pl}}})^{n}\right)\equiv
S(x,y).$ (22)

Changing the variables in the original equation (8), the generalised RG-
equation for these potentials can be written as

$S_{x}=-\frac{n^{2}}{9\alpha^{2}}y^{2-\frac{4}{n}}\left(y^{1/n}-1\right)^{2}\left(\left(n\left(y^{1/n}-1\right)+1\right)S_{y}+ny\left(y^{1/n}-1\right)S_{yy}\right)^{2}.$
(23)

Taking the cases $n=2,4$ we obtain for $n=2$:

$S_{x}=-\frac{4}{9\alpha^{2}}\left(2y\left(\sqrt{y}-1\right)^{2}S_{yy}+\left(2y-3\sqrt{y}+1\right)S_{y}\right)^{2},$
(24)

and for $n=4$:

$S_{x}=-\frac{16}{9\alpha^{2}}\left(\sqrt[4]{y}-1\right)^{2}y\left(\left(4\sqrt[4]{y}-3\right)S_{y}+4y\left(\sqrt[4]{y}-1\right)S_{yy}\right)^{2},$
(25)

with the initial conditions given by

$S(0,y)=y,S(x,1)=1,S_{y}(x,1)=0.$ (26)

As before, the cases of $n=4$ and higher are unsuitable for the description of
the cosmological constant, because there is no corresponding uplift due to
$v_{2}(0)=0$.

From Fig.4 it is easy to see that in the case at small values of $\mu$ the
behaviour of the potential repeats the classical case except for a small
modification of the slow rolling inflation region. One can notice some
similarity between the behaviour of the potential for the $E^{2}$-model and
the potential given in [16] with small changes of $g,\alpha$ and $\mu\ll
M_{Pl}$.

Figure 4: Classical $E^{2}$\- model potential (red dashed line) and its RG-
corrected versions: the red line is for $\mu^{2}<M_{Pl}^{2}$ and the green
line is for $\mu^{2}>M_{Pl}^{2}$. The scales are chosen for illustrative
purposes as follows $\tilde{\varphi}=\varphi/(\sqrt{6}M_{Pl})$ and $g=1$,
$\alpha=1$

As in the previous case, we can estimate the value of $\mu^{2}$ from the one-
loop expression for the $E^{2}$-model (Fig.4) and get

$\mu^{2}=\frac{4g}{3\alpha
M_{Pl}^{2}}e^{-36\pi^{2}\frac{\alpha^{2}}{g^{2}}\Lambda M_{Pl}^{4}},$ (27)

for $\alpha=1$ this gives $\mu\approx 10^{-5}M_{Pl}$ for small $\Lambda$. In
this case the minimum of the potential is only slightly shift at small values
of $\mu$. But from the numerical solution of equation (24) one gets
$\varphi_{vac}=0$. As in the previous case, no additional minima appear. As
before, one can calculate the critical point where $\Lambda$ changes its sign:

$\mu_{c}=2\frac{10^{-5}}{\sqrt{3\alpha}}M_{Pl}.$ (28)

which is again close to the inflaton mass.

###  3.3 The model of quintessence

Consider at last an even simpler potential $T^{1}$ corresponding to the
quintessence model [6]

$V=\gamma\sqrt{6\alpha}\left(\tanh\left(\frac{\varphi}{\sqrt{6\alpha}M}\right)+1\right),$
(29)

which has the shape shown in Fig.5 (plus 1 is added to get the asymptotics as
$\varphi\to-\infty$ equal to zero).

Figure 5: The potential of the quintessence model: classical potential (a),
potential with quantum corrections (b). The green line is for $\mu\gg M_{Pl}$
and the red line is for $\mu\ll M_{Pl}$. The scales are chosen for
illustrative purposes as follows $\tilde{\varphi}=\varphi/(\sqrt{6}M_{Pl})$
and $g=10$, $\alpha=1$

In the presence of the cosmological constant one should expect that
$V|_{\varphi=-\infty}=\Lambda$ [6]. We check whether this scenario is realized
when taking into account quantum corrections. The numerical solution of
eq.(14) for the case of $n=1$

$S_{x}=-\frac{(y^{2}-1)^{2}}{144\alpha^{2}y}(2yS_{y}+(y^{2}-1)S_{yy})^{2}$
(30)

is shown in Fig.5 (right). One can see that in this case, as in the other
$\alpha$-attractor models, the asymptotics of the classical potential is
preserved when taking into account all quantum corrections[12]: it goes to 1
when $\varphi\to+\infty$ and to zero when $\varphi\to-\infty$ without any
additional minima. The only influence of quantum corrections is a change in
the shape of the potential in the middle while preserving the limit as
$\varphi\to\pm\infty$. This can be interpreted as an additional contribution
of the effective field-dependent cosmological constant $\Lambda(\varphi)$
which decreases with time [6] to the value of the dark energy determined by
the classical potential (29).

To find the contribution of quantum corrections to $\Lambda(\varphi)$, one can
again use the one-loop potential given by the same formula

$\Lambda(\varphi)=\frac{g^{2}}{16\pi^{2}}\frac{v_{2}^{2}}{4}\log\left(\frac{gv_{2}}{\mu^{2}}\right),$
(31)

In this way, it is possible to fit the parameter $\mu$ to get a modification
of the original classical potential that might be required by, say,
observational data and restrictions on quintessential inflation. Therefore,
even such a simple $T^{1}$ model of quintessence, with taking into account
quantum corrections, can be quite flexible for description of observational
data.

##  4 Conclusions

In this paper, we have considered the influence of quantum loop corrections on
the potentials of the $\alpha$-attractor models that are often used in
inflationary cosmology. For this purpose, we derived the generalised RG
equations (14,16,17,23-25) that sum up the whole sequence of leading
logarithmic contributions to the effective potential. Solving these equations,
one can take into account all quantum corrections that usually smooth out the
behaviour of the one-loop contribution and, depending on the value of the free
scale parameter $\mu$, either preserve the minimum of the potential at the
origin or create new minima but in both cases lift the potential, thus
producing the cosmological constant.

We mainly focused our attention on the effect of uplifting the effective
potential due to quantum corrections. This uplift, which appears in the one-
loop approximation, remains intact even if all quantum corrections taken into
account. We interpreted this uplift as the appearance of the cosmological
constant $\Lambda$ for the $T^{2}$ and $E^{2}$ models and as an additional
contribution to the dark energy for the $T^{1}$ model. The obtained value of
the free parameter $\mu\approx 10^{-5}M_{Pl}$, which was calculated in the
one-loop approximation, lies in an interesting energy region being close to
the inflaton mass. Numerical evaluation of the equations only slightly changes
this value. It is also important to note that the value of the parameter $\mu$
plays a crucial role in determining the geometry of space-time: flat, de
Sitter space (dS) or anti-de Sitter space (AdS). The critical value of $\mu$
can be easily evaluated for the models under consideration.

Thus, we have found out that the cosmological constant $\Lambda$ may exist as
a consequence of quantum corrections to the effective potential with some
value of the scale parameter $\mu$ even in non-renormalizable models of
inflation. The value of the cosmological constant $\Lambda$ allows one to fix
the parameter $\mu$ which is a free parameter in the non-renormalizable
theory. One may hope that the mechanism of radiative corrections can shed
light on the existence of Dark energy in the early Universe.

##  5 Acknowledgements

The authors are grateful to I.Buchbinder, S.Fedoruk, and A.Baushev for
valuable discussions.

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