# Characters in p-adic Heisenberg and Lattice Vertex Operator Algebras

Daniel Barake  and  Cameron Franc<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We study characters of states in $p$-adic vertex operator algebras. In
particular, we show that the image of the character map for both the $p$-adic
Heisenberg and $p$-adic lattice vertex operator algebras contain infinitely-
many non-classical $p$-adic modular forms which are not contained in the image
of the algebraic character map.

##  1\. Introduction

We study $p$-adic properties of certain vertex operator algebras. Motivated by
both physical and number-theoretical methods, the authors of [FM22] introduce
the study of $p$-adic VOAs which arise from a completion of the axioms for
usual (algebraic) VOAs. Though the existence of $p$-adic variants of known
VOAs such as the Virasoro, Monster and Heisenberg was central to this work,
the authors demonstrate in Sections 9 and 10 that the character map (i.e.
$1$-point correlation function or graded trace) on such VOAs has image in
Serre’s ring of $p$-adic modular forms $\mathfrak{M}_{p}$. This facet of the
theory is at the heart of our current discussion.

Following [FM22], and denoting by $S_{\text{alg}}$ the Heisenberg algebra, the
character map is first extended to a surjective $\mathbb{Q}_{p}$-linear map

(1.1) $\displaystyle\textbf{{f}}:S_{\text{alg}}$
$\displaystyle\to\mathbb{Q}_{p}[E_{2},E_{4},E_{6}]$ $\displaystyle a$
$\displaystyle\mapsto\eta(q)Z(a,q)$

where $q=e^{2\pi iz}$ and where $E_{2}$, $E_{4}$ and $E_{6}$ denote the
Eisenstein series of weights $2$, $4$ and $6$ respectively, normalized such
that they have constant term equal to $1$ (cf. [MT10, DMN01]). Note above that
f is a rescaling of the character map $Z(a,q)$ which ensures the
$\eta(q)^{-1}$ factor is removed from the image. The $p$-adic completion of
$\mathbb{Q}[E_{4},E_{6}]$ by the sup-norm yields $\mathfrak{M}_{p}$ (cf.
[Ser73]), and so motivated by this construction, states in the $p$-adic
Heisenberg VOA $S_{1}$ are formed as $p$-adic limits of convergent sequences
of states in $S_{\text{alg}}$. Theorem 9.7 of [FM22] then establishes that f
extends further to a map into $\mathfrak{M}_{p}$, thus obtaining $p$-adic
modular forms as characters of states in $S_{1}$ amounts to making the
following diagram commute:

${S_{1}}$${\mathfrak{M}_{p}}$${S_{\text{alg}}}$${\mathbb{Q}_{p}[E_{2},E_{4},E_{6}]}$ff

Here, the upward arrows denote the respective completion functors. Unlike the
algebraic case in which it is known that f is surjective on the ring of quasi-
modular forms, it is yet undetermined whether every $p$-adic modular form
arises as the character of some state from a $p$-adic VOA. It was established
in Section 10 of [FM22] that the image of f on $S_{1}$ contains the $p$-adic
Eisenstein series of weight $2$ whose $q$-expansion is given by

(1.2) $\displaystyle G_{2}^{\star}(q)=\frac{p-1}{24}+\sum_{n\geq
1}\sigma^{\star}(n)q^{n}$

where $\sigma^{\star}(n)$ denotes the sum of all divisors of $n$ which are
coprime to a prime $p$. It is then natural to speculate whether the entire
$p$-adic Eisenstein family lies in the image of the character map on $S_{1}$.
Indeed, by considering a broader set of states and extending the methods of
[FM22], in Section 3 of this discussion we prove:

######  Theorem 1.1.

The image of the character map on the $p$-adic Heisenberg VOA $S_{1}$ contains
infinitely-many non-classical $p$-adic modular forms. In particular, the image
contains

$\displaystyle 2^{(t+1)/2}t(t-2)!!G_{2}^{(t-1)/2}(q)G_{l+1}^{\star}(q)$

which is not quasi-modular of level one, and where $t,l\geq 1$ are odd and
$l\not\equiv-1\mod p-1$.

The appearance of the factors $G_{2}^{(t-1)/2}$ above is also new and provides
some clarity on the image of f on $S_{1}$. Note that setting $t=1$ above and
normalizing returns the classical $p$-adic Eisenstein series $G_{k}^{\star}$
where $p-1\nmid k$.

In a similar direction, the character map on VOAs based on lattice theories
$V_{\Lambda}$ (cf. [FLM88, LL04]) is comparable to that of $S_{\text{alg}}$
albeit with the emergence of theta series (cf. [DMN01]), and so it is feasible
to suspect that results such as Theorem 1.1 are not unique to $S_{1}$. From
[FM22], one may deduce the existence of $p$-adic variants of $V_{\Lambda}$
formed from the $p$-adic completion with respect to the sup-norm. Such VOAs
will be denoted by $\widehat{V}_{\Lambda}$, and we have an analogue of the
character map seen in eq. 1.1:

(1.3) $\displaystyle\textbf{{f}}:V_{\Lambda}$
$\displaystyle\to\mathbb{Q}_{p}[E_{2},E_{4},E_{6}]$ $\displaystyle a$
$\displaystyle\mapsto\eta(q)^{d}Z(a,q)$

where $d$ is the rank of the lattice $\Lambda$. It will be clear in our
discussion when working over $S_{\text{alg}}$ or $V_{\Lambda}$, and so the use
of f for both characters should cause no confusion. In this case, concrete
examples of $p$-adic characters have yet to be constructed prior to this work.
In Section 4, we build infinite families of states in $\widehat{V}_{\Lambda}$
which give rise to $p$-adic modular forms now in the sense of Katz [Kat73].
Our second theorem is:

######  Theorem 1.2.

Let $\Lambda$ be an even unimodular lattice. The image of the character map on
the $p$-adic lattice VOA $\widehat{V}_{\Lambda}$ contains infinitely-many non-
classical Katz $p$-adic modular forms. In particular, the image contains the
sum

$\displaystyle 2tG_{l+1}^{\star}(q)\sum_{k\geq
0}\binom{t-1}{2k}\frac{(2k)!}{k!}G_{2}(q)^{k}\Theta_{\Lambda,t,k}(q)$

where $t,l\geq 1$ are odd with $l\not\equiv-1\mod p-1$, and each
$\Theta_{\Lambda,t,k}$ is quasi-modular.

The precise definition of $\Theta_{\Lambda,t,k}$ is given in the statement of
Proposition 4.3. In this case, by setting $t=1$ and normalizing, we obtain the
product $G_{k}^{\star}\Theta_{\Lambda}$ where $\Theta_{\Lambda}$ is the theta
series for $\Lambda$ which is well-known to be modular since $\Lambda$ is even
and unimodular.

Both Theorems 1.1 and 1.2 demonstrate the need for an ameliorated theory
better suited to the study of the $p$-adic character. The methods currently
employed are analogous to those which were used by Serre in [Ser73] and
generalized in a geometric way by Katz in [Kat73]. Hence a geometric
reformulation of the axioms for p-adic VOAs in view of Katz’ theory of
$p$-adic modular forms could serve as such a device.

Another possibility would be to develop a theory of Hecke operators arising
from $S_{\text{alg}}$ or $V_{\Lambda}$. For example, a realization of the
operator $U_{p}$ as in [Ser73] directly within a $p$-adic VOA could prove
useful in assessing the surjectivity of the $p$-adic character on
$\mathfrak{M}_{p}$. Though this is only hypothetical, the authors of [KSU89a],
[KSU89b], and [KSU91] have shown that such operators do arise algebraically in
the context of conformal field theory, namely out of the Fock space of the
Fermionic algebra. It is then an open question whether similar structures
exist in $S_{\text{alg}}$, $V_{\Lambda}$, or in any of their various
incarnations.

##  2\. Background

###  2.1. Vertex Operator Algebras

For elementary notions in VOA theory, the reader is invited to consult [LL04].
A vertex operator algebra consists of a $\mathbb{Z}$-graded vector space, or
Fock space,

$\displaystyle V=\coprod_{n\in\mathbb{Z}}V_{(n)}$

where $\dim V_{(n)}<\infty$ for all $n$ and $V_{(n)}=0$ for $n\ll 0$. We say
the state $a\in V$ is homogeneous of weight $k$ if $a\in V_{(k)}$. The vertex
operator linear map

$\displaystyle Y:V$ $\displaystyle\to\text{End}(V)[[z,z^{-1}]]$ $\displaystyle
a$ $\displaystyle\mapsto Y(a,z)=\sum_{n\in\mathbb{Z}}a(n)z^{-n-1}$

packages the modes $a(n)\in\text{End}(V)$ of the state $a$ into a formal
Laurent series, and these satisfy the truncation condition $a(n)b=0$ for $n\gg
0$ for $a,b\in V$. The vacuum state $\textbf{1}\in V_{(0)}$ satisfies
$Y(\textbf{1},z)=1$ and $\lim_{z\to 0}Y(a,z)\textbf{1}=a$. These are the
vacuum and creation properties, respectively. The conformal state $\omega\in
V_{(2)}$ has vertex operator given by

$\displaystyle Y(\omega,z)=\sum_{n\in\mathbb{Z}}L(n)z^{-n-2}\hskip
5.0pt\left(=\sum_{n\in\mathbb{Z}}\omega(n)z^{-n-1}\right)$

where the $L(i)\in\text{End}(V)$ generate a copy of the Virasoro algebra with
central charge $c_{V}\in\mathbb{C}$. The $L(0)$-eigenspace decomposition of
$V$ coincides with the grading of $V$, that is, $L(0)a=ka$ where $a\in
V_{(k)}$, and the $L(-1)$-derivative property holds, i.e. $L(-1)\textbf{1}=0$.
Finally, the Jacobi identity is satisfied for any $a,b\in V$ and triple of
integers $(r,s,t)$:

(2.1) $\displaystyle\sum_{i\geq 0}\binom{r}{i}(-1)^{i}$ $\displaystyle
a(s+r-i)b(t+i)-(-1)^{r}\sum_{i\geq 0}\binom{r}{i}(-1)^{i}b(t+r-i)a(s+i)$
$\displaystyle=\sum_{i\geq 0}\binom{s}{i}(a(r+i)b)(s+t-i).$

We shall also require the notion of a $V$-module $M$ in arguments involving
lattice VOAs. Recall this is a $\mathbb{C}$-graded vector space

$\displaystyle M=\coprod_{h\in\mathbb{C}}M_{(h)}$

such that $\dim M_{(h)}<\infty$ for all $h$ and $M_{(h)}=0$ for $h\ll 0$. Here
also, the $L(0)$-eigenspace decomposition coincides with the grading above. We
call $a\in M$ homogeneous of weight $h$ if $a\in M_{(h)}$. We have an
analogous vertex operator linear map

$\displaystyle Y_{M}:V$ $\displaystyle\to\text{End}(M)[[z,z^{-1}]]$
$\displaystyle a$ $\displaystyle\mapsto
Y_{M}(a,z)=\sum_{n\in\mathbb{Z}}a(n)z^{-n-1}$

where the truncation and vacuum conditions as well as the Jacobi identity (eq.
2.1) hold true for the above modes.

Given $a\in V_{(k)}$ and $m\in\mathbb{Z}$, one shows that $a(n):V_{(m)}\to
V_{(m+k-n-1)}$ (cf. [MT10]). Denote then the zero mode $o(a)$ of $a$ as
$a(k-1)$ such that $o(a):V_{(m)}\to V_{(m)}$, and extend the definition to all
of $V$ additively. By tracing the zero mode over each subspace $V_{(n)}$
(recall $\dim V_{(n)}<\infty$ for all $n\in\mathbb{Z}$) and packaging these
into a generating function, we obtain the character

$\displaystyle
Z(a,q)=q^{-c_{V}/24}\sum_{n\in\mathbb{Z}}\text{Tr}_{V_{(n)}}o(a)q^{n}.$

In the case of $V$-modules, the character is defined in a natural way where
the zero mode $o(a)=a(k-1)\in\text{End}(M)$ for $a\in V_{(k)}$ preserves each
homogeneous space $M_{(h)}$.

###  2.2. p-adic Vertex Operator Algebras

The definition of a VOA given above can be easily generalized over any field
$k$ of characteristic zero. In particular, it is perfectly reasonable to set
$k=\mathbb{Q}_{p}$. All of the VOAs of interest in our discussion (Heisenberg
modules and lattice VOAs) have bases defined over the $p$-adic integers
$\mathbb{Z}_{p}$, such that the corresponding modes expressed in these bases
are all $p$-adically integral, satisfying some mild hypotheses on boundedness
of these operators. A $p$-adic VOA $\widehat{V}$ is then formed by completing
and inverting $p$. One description is to first consider the
$\mathbb{Z}_{p}$-module

(2.2) $\displaystyle\widehat{U}=\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}U/p^{m}U$

and then letting
$\widehat{V}=\widehat{U}\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p}$. Equivalently,
$\widehat{V}$ can be described in terms of a completion with respect to a sup-
norm arising from the integral basis in $U$. See Section 7 of [FM22] for
details on these constructions. For our purposes, it will be enough to think
of elements of $\widehat{V}$ as limits of convergent sequences of states in an
underlying algebraic VOA, though the full axiomatic description of
$\widehat{V}$ is given in [FM22].

For a concrete example, setting $V$ to be the Heisenberg algebra (which we
will describe momentarily below as polynomial ring in countably many
indeterminates $h(i)$) over $\mathbb{Q}_{p}$, let $S_{r}$ for
$r\in\mathbb{R}_{>0}$ denote the completion of $V$ with respect to the sup-
norm

(2.3)
$\displaystyle\left\lvert\sum_{I}a_{I}h^{I}\right\rvert_{r}=\sup_{I}|a_{I}|r^{I}$

where $I$ is a finite multi-subset of $\mathbb{Z}_{<0}$ with $|I|=-\sum_{i\in
I}i$, and $h^{I}=\prod_{i\in I}h(i)$. Then Proposition 9.1 in [FM22] shows
that elements of $S_{r}$ are of the form $\sum_{I}a_{I}h^{I}$ where

(2.4) $\displaystyle\lim_{|I|\to\infty}|a_{I}|r^{|I|}=0.$

In particular, with $r=1$, the ring $S_{1}$ has the structure of a $p$-adic
VOA, and there is a nested chain of subpaces $S_{1}\supseteq S_{2}\supseteq
S_{3}\supseteq\cdots$.

###  2.3. p-adic modular forms

Following [Ser73], for a prime $p\geq 2$ we denote the $p$-adic weight space
as

$\displaystyle X=\lim_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}\mathbb{Z}/p^{m}(p-1)\mathbb{Z}\cong\mathbb{Z}_{p}\times\mathbb{Z}/(p-1)\mathbb{Z}.$

Note that when $p=2$, we have $X\cong\mathbb{Z}_{2}$. We say $k\in X$ is even
if $k\in 2X$, and that $k$ is odd if it is not even. A $p$-adic modular form
is then a formal series

$\displaystyle f=\sum_{n=0}^{\infty}a_{n}q^{n}\in\mathbb{Q}_{p}[[z]]$

such that $f$ is the $p$-adic limit of modular forms $f_{i}$ of weight $k_{i}$
having rational coefficients. The ring $\mathfrak{M}_{p}$ is then defined as
the completion of $\mathbb{Q}_{p}[E_{4},E_{6}]$ by the sup-norm on the Fourier
coefficients $a_{n}$, and $f\in\mathfrak{M}_{p}$ has weight equal to the
$p$-adic limit of the $k_{i}$, which is an element of $X$.

The ring $\mathfrak{M}_{p}$ contains $p$-adic analogues of classical
Eisenstein series, and these take the form

$\displaystyle
G_{k}^{\star}(q)=\frac{1}{2}\zeta_{p}(1-k)+\sum_{n=1}^{\infty}\sigma_{k-1}^{\star}(n)q^{n}$

where $\sigma_{k-1}^{\star}(n)$ is defined as in eq. 1.2 and $\zeta_{p}$
denotes the Kubota-Leopoldt $p$-adic zeta function (cf. [KL64]). For even
integers $k\geq 2$, $G_{k}^{\star}$ is a modular form on the congruence
subgroup $\Gamma_{0}(p)$. For example, taking the sequence
$\\{G_{2+p^{m}(p-1)}\\}_{m\geq 0}$ of classical Eisenstein series as
$m\to\infty$ for some $p\geq 2$ yields $G_{2}^{\star}\in\mathfrak{M}_{p}$ seen
in eq. 1.2. The images in Theorems 1.1 and 1.2 will be obtained in a similar
manner.

##  3\. p-adic characters in the Heisenberg VOA

###  3.1. The Heisenberg VOA and the square-bracket formalism

We briefly outline the construction of the Heisenberg VOA $S_{\text{alg}}$. A
rigorous treatment can be found in many texts such as [FLM88, LL04, MT10,
FBZ04]. For generalities on Lie algebras alongside their representations, we
follow Chapter 9 of [Car05]. Let $\mathfrak{h}$ be a finite-dimensional vector
space viewed as an abelian Lie algebra. Then associated the (untwisted) affine
Lie algebra

$\displaystyle\widehat{\mathfrak{h}}=\left(\mathfrak{h}\otimes\mathbb{C}[t,t^{-1}]\right)\oplus\mathbb{C}\textbf{k}$

where k is central. There are two graded subalgebras which we write with
respect to the notation given in [LL04] (by weights) as:

$\displaystyle\widehat{\mathfrak{h}}_{+}=\mathfrak{h}\otimes
t^{-1}\mathbb{C}[t^{-1}]$
$\displaystyle\widehat{\mathfrak{h}}_{-}=\mathfrak{h}\otimes t\mathbb{C}[t].$

Define $\widehat{\mathfrak{h}}_{\leq
0}=\widehat{\mathfrak{h}}_{-}\oplus\left(\mathfrak{h}\oplus\textbf{k}\right)$,
and denote by $\mathcal{U}(\cdot)$ the universal enveloping algebra. Let
$\mathbb{C}\textbf{1}$ be the $1$-dimensional $\widehat{\mathfrak{h}}_{\leq
0}$-module where $\widehat{\mathfrak{h}}_{-}$ acts trivially on $\mathbb{C}$
and k acts as the identity. Construct the induced module

$\displaystyle
S_{\text{alg}}=\text{Ind}_{\mathcal{U}(\widehat{\mathfrak{h}}_{\leq
0})}^{\mathcal{U}(\widehat{\mathfrak{h}})}\mathbb{C}\textbf{1}=\mathcal{U}(\widehat{\mathfrak{h}})\otimes_{\mathcal{U}(\widehat{\mathfrak{h}}_{\leq
0})}\mathbb{C}\textbf{1}$

where we write the action of $h\otimes t^{n}$ on $S_{\text{alg}}$ as $h(n)$
for $n\in\mathbb{Z}$. Note that $S_{\text{alg}}\cong
S(\widehat{\mathfrak{h}}_{+})$, the symmetric algebra. That is,
$\\{h(-1),h(-2),h(-3),\ldots\\}$ form a basis for $S_{\text{alg}}$. Also note
the $h(n)$ for $n\geq 1$ act on $S_{\text{alg}}$ as $n\partial_{h(-n)}$, and
$h(0)$ acts as zero. The Heisenberg vertex operator (or Heisenberg field) is
defined as

$\displaystyle
h(z)=\sum_{n\in\mathbb{Z}}h(n)z^{-n-1}\in\text{End}(S_{\text{alg}})[[z,z^{-1}]].$

This is no more than a generating function for the operators
$h(n)\in\text{End}(S_{\text{alg}})$. Hence we may view $S_{\text{alg}}$ as
being generated by the Fourier coefficients of the state $h(-1)\textbf{1}$
(see [FBZ04]).

Instrumental to establishing the connection between VOAs and modular forms is
the notion of transforming a VOA by way of a map $z\mapsto\phi(z)$, introduced
first in [Zhu96]. An overview of this technique can be found in detail in
[MT10] and in Chapter 5 of [FBZ04]. In particular, setting $\phi(z)=e^{z}-1$
formulates the vertex operators as living on a torus rather than a sphere, and
for $S_{\text{alg}}$ these are given by

(3.1) $\displaystyle
Y(h(-1)\textbf{1},e^{z}-1)e^{z}=\sum_{n\in\mathbb{Z}}h[n]z^{-n-1}.$

This resulting “square-bracket" VOA is equipped with a new grading which
respects the grading on the space of quasi-modular forms. Multiplying eq. 3.1
by $z^{n}$ we obtain

(3.2) $\displaystyle h[n]$ $\displaystyle=\textup{Res}_{z}\hskip
3.0ptY(h(-1)\textbf{1},e^{z}-1)(e^{z})(z^{n}).$

Note that $Y(h(-1)\textbf{1},e^{z}-1)$ truncates when applied to any state in
$S_{\text{alg}}$, and that we can use a change of variables theorem (cf. eq.
(1.1.3) of [Zhu96]) with $w=e^{z}-1$ to show

(3.3) $\displaystyle h[n]=\textup{Res}_{w}\hskip
3.0pth(w)\left(\log(w+1)\right)^{n}$

Observe that $h[0]=h(0)$. For our purposes, we will only require the explicit
expression where $n=-1$ above. If we let $c_{0}=1$, $c_{i}=0$ for $i<0$, and

$\displaystyle c_{k}=\sum_{i\geq 1}\frac{(-1)^{i+1}}{i+1}c_{k-i}$

so that

$\displaystyle c_{1}=\frac{1}{2},\hskip 5.0ptc_{2}=-\frac{1}{12},\hskip
5.0ptc_{3}=\frac{1}{24},\hskip 5.0ptc_{4}=-\frac{19}{720},\hskip 5.0pt\cdots$

then $c_{k}=\text{Coeff}_{w^{k-1}}(\log(w+1))^{-1}$ and we see that

(3.4) $\displaystyle
h[-1]=h(-1)+\frac{1}{2}h(0)-\frac{1}{12}h(1)+\frac{1}{24}h(2)-\frac{19}{720}h(3)+\cdots$

###  3.2. Some square-bracket states

We construct as in [FM22] a family of square-bracket states in
$S_{\text{alg}}$ which map under f (recall eq. 1.1) to an appropriate subspace
of the ring of quasi-modular forms. In order to assess $p$-adic convergence in
the subsequent Lemmas 3.5 and 3.6, we must write these square-bracket states
in the round-bracket formalism, that is, in terms of the basis
$\\{h(-1),h(-2),h(-3)\ldots\\}$ of $S_{\text{alg}}$.

######  Lemma 3.1.

For an odd integer $t\geq 1$, we have

(3.5) $\displaystyle h[-1]^{t}\textbf{1}=\sum_{k\geq
0}\binom{t}{2k}\frac{(2k)!}{k!(-24)^{k}}h(-1)^{t-2k}\textbf{1}.$

###### Proof.

We proceed by induction. Recall that in $S_{\text{alg}}$, eq. 3.4 shows that
$h[-1]\textbf{1}=h(-1)\textbf{1}$ which agrees with the statement. Suppose the
claim holds up to some $t>1$. Then applying $h[-1]$ to the right-hand side of
eq. 3.5 gives

$\displaystyle\sum_{k\geq
0}\binom{t}{2k}\frac{(2k)!}{k!(-24)^{k}}h(-1)^{t-2k+1}\textbf{1}+\sum_{k\geq
0}\binom{t}{2k}\frac{2(2k)!(t-2k)}{k!(-24)^{k+1}}h(-1)^{t-2k-1}\textbf{1}$
$\displaystyle=\sum_{k\geq
0}\binom{t}{2k}\frac{(2k)!}{k!(-24)^{k}}h(-1)^{t-2k+1}\textbf{1}+\sum_{k\geq
0}\binom{t}{2k+1}\frac{(2k+2)!}{(k+1)!(-24)^{k+1}}h(-1)^{t-2k-1}\textbf{1}.$
Re-indexing the second sum, we get $\displaystyle=h(-1)^{t+1}+\sum_{k\geq
1}\binom{t}{2k}\frac{(2k)!}{k!(-24)^{k}}h(-1)^{t-2k+1}\textbf{1}+\sum_{k\geq
1}\binom{t}{2k-1}\frac{(2k)!}{k!(-24)^{k}}h(-1)^{t-2k+1}\textbf{1},$

and by using Pascal’s rule this equals $h[-1]^{t+1}\textbf{1}$ which completes
the induction. ∎

######  Remark.

It is interesting to note that the expression given in Lemma 3.1 is related to
generalized Hermite polynomials

(3.6)
$\displaystyle\text{He}_{n}^{\alpha}(x)=\left(x-\alpha\partial_{x}\right)^{n}\cdot
1=e^{-\frac{\alpha(\partial_{x})^{2}}{2}}x^{n}$

for $n\geq 0$ and where $\alpha>0$ (cf. [Rom84, AAR99]). Expanding the
rightmost side of eq. 3.6:

$\displaystyle\text{He}_{n}^{\alpha}(x)=\sum_{k\geq
0}\frac{(-\alpha)^{k}}{2^{k}(k)!}\partial_{x}^{2k}x^{n}=\sum_{k\geq
0}\frac{(n)!(-\alpha)^{k}}{2^{k}(k)!(n-2k)!}x^{n-2k}=\sum_{k\geq
0}\binom{n}{2k}\frac{(-\alpha)^{k}(2k)!}{2^{k}(k)!}x^{n-2k}.$

By comparing eq. 3.6 to repeated application of $h[-1]$ on 1, setting
$x=h(-1)$, $\alpha=1/12$ and $1=\textbf{1}$ above gives the expression of
Lemma 3.1. Expressions of the form $h[-r]^{t}\textbf{1}$ for $r\geq 2$ in the
round-bracket formalism are much more difficult to derive, however would prove
extremely useful in computing further $p$-adic characters.

######  Lemma 3.2.

For an odd integer $t\geq 1$, we have

$\displaystyle h(1)h[-1]^{t}\textbf{1}=\sum_{k\geq
0}\binom{t}{2k+1}\frac{(2k+1)!}{k!(-24)^{k}}h(-1)^{t-2k-1}\textbf{1}.$

###### Proof.

The proof is similar to the computation of the second sum in Lemma 3.1. ∎

The following proposition is a slight generalization of Lemma 10.2 of [FM22]
which outlines the family of square-bracket states which we will be
considering throughout.

######  Proposition 3.3.

For odd integers $r,t\geq 1$, we have

$\displaystyle(r-1)!h[-r]h[-1]^{t}\textbf{1}=\sum_{n\geq 0}\sum_{k\geq 0}$
$\displaystyle\binom{t}{2k}\frac{n!S_{r}^{(n+1)}(2k)!}{k!(-24)^{k}}h(-n-1)h(-1)^{t-2k}\textbf{1}$
$\displaystyle-\sum_{k\geq
0}\binom{t}{2k+1}\frac{B_{r+1}(2k+1)!}{k!(r+1)(-24)^{k}}h(-1)^{t-2k-1}\textbf{1}$

where

$\displaystyle S_{r}^{(n+1)}=\frac{1}{n!}\sum_{j\geq
0}\binom{n}{j}(-1)^{n+j}(j+1)^{r-1}$

are the Stirling numbers of the second kind.

###### Proof.

First, using Lemma 3.1 we compute

$(r-1)!h[-r]h[-1]^{t}\textbf{1}\\\ =(r-1)!\textup{Res}_{z}\hskip
3.0ptz^{-r}e^{z}\sum_{n\in\mathbb{Z}}h(n)\left(\sum_{k\geq
0}\binom{t}{2k}\frac{(2k)!}{k!(-24)^{k}}h(-1)^{t-2k}\textbf{1}\right)\left(e^{z}-1\right)^{-n-1}.$

Applying Lemma 3.2 gives

$=(r-1)!\textup{Res}_{z}\hskip 3.0ptz^{-r}e^{z}\left(\sum_{n\geq 0}\sum_{k\geq
0}\binom{t}{2k}\frac{(2k)!}{k!(-24)^{k}}h(-n-1)h(-1)^{t-2k}\textbf{1}\left(e^{z}-1\right)^{n}\right)\\\
+(r-1)!\textup{Res}_{z}\hskip 3.0ptz^{-r}e^{z}\sum_{k\geq
0}\binom{t}{2k+1}\frac{(2k+1)!}{k!(-24)^{k}}h(-1)^{t-2k-1}\textbf{1}\left(e^{z}-1\right)^{-2}.$

It was shown in [FM22] that

$\displaystyle(r-1)!\textup{Res}_{z}\hskip
3.0ptz^{-r}e^{z}(e^{z}-1)^{n}=n!S_{r}^{(n+1)}$
$\displaystyle(r-1)!\textup{Res}_{z}\hskip
3.0ptz^{-r}e^{z}(e^{z}-1)^{-2}=-\frac{B_{r+1}}{r+1}$

and by comparing with the claim, this concludes the proposition. ∎

###  3.3. Proof of Theorem 1.1

We may now take the character of the family of states considered in
Proposition 3.3 by using the following reformulation of equation (44) of
[MT10] which gives an explicit description for the character of the state
$a=h[-k_{1}]h[-k_{2}]\cdots h[-k_{r}]\textbf{1}$ with $k_{i}\geq 1$ under the
map f encountered in eq. 1.1:

(3.7)
$\displaystyle\textbf{{f}}(a)=\sum_{(\ldots\\{s,t\\}\ldots)\in\mathcal{P}(\Phi,2)}\prod_{\\{s,t\\}}\frac{2(-1)^{s+1}}{(s-1)!(t-1)!}G_{s+t}(q).$

Here, $\mathcal{P}(\Phi,2)$ denotes all possible partitions of the set $\Phi$
into parts $\\{s,t\\}$ of size 2. Then for each such partition, the product is
taken over each part.

######  Remark.

Here and throughout, we employ the normalization

(3.8) $\displaystyle G_{k}(q)=-\frac{B_{k}}{2k}+\sum_{n\geq
1}\sigma_{k-1}(n)q^{n}$

seen in [Ser73] and so the results of the papers cited here have been adjusted
to this change.

######  Proposition 3.4.

For odd integers $r,t\geq 1$, we have

$\displaystyle\textbf{{f}}\hskip
2.0pt(h[-r]h[-1]^{t}\textbf{1})=\frac{2^{(t+1)/2}t(t-2)!!}{(r-1)!}G_{2}^{(t-1)/2}(q)G_{r+1}(q).$

Here, $n!!$ denotes the product of all odd integers no greater than $n$.

###### Proof.

We make use of eq. 3.7. In this case, $\Phi=\\{r,1_{1},1_{2},\ldots,1_{t}\\}$
where we have labelled the $1$s for clarity. Since $t$ is odd, $|\Phi|$ is
even and so $\mathcal{P}(\Phi,2)\neq 0$.

Suppose first that $r$ is paired with $1_{1}$, and the remaining
$1_{2},\ldots,1_{t}$ are paired amongst themselves. The $t-1$ remaining $1$s
then get put in $(t-1)/2$ parts of size $2$, hence

$\displaystyle\left(\frac{2}{(r-1)!}G_{r+1}(q)\right)\left(\frac{2}{(1-1)!}G_{2}(q)\right)^{(t-1)/2}=\frac{2^{(t+1)/2}}{(r-1)!}G_{2}^{(t-1)/2}(q)G_{r+1}(q)$

where we recall that $r$ is odd here. There are then $(t-2)!!$ distinct ways
of partitioning the remaining $1_{2},\ldots,1_{t}$ into $(t-1)/2$ parts of
size $2$. Each way yields the same expression as above and so we multiply by
$(t-2)!!$. Finally, we may repeat this process $t$ times, since $r$ can be
paired with $1_{2}$ then with $1_{3}$ and so on, until $r$ is paired with
$1_{t}$. This yields the expression

$\displaystyle\frac{2^{(t+1)/2}t(t-2)!!}{(r-1)!}G_{2}^{(t-1)/2}(q)G_{r+1}(q)$

which establishes the result. ∎

With this, we denote (in the notation of [FM22]) the rescaled state

$u_{r,t}=(1-p^{r})(r-1)!h[-r]h[-1]^{t}\textbf{1}$

for some prime $p>3$. By Proposition 3.4,

(3.9)
$\textbf{{f}}(u_{r,t})=(1-p^{r})(2^{(t+1)/2}t(t-2)!!)G_{2}^{(t-1)/2}(q)G_{r+1}(q).$

For each fixed odd $t,l\geq 1$ where $l\not\equiv-1\mod p-1$, we will assess
the convergence of the sequence

(3.10) $(u_{p^{a}(p-1)+l,t})_{a\geq
0}=\left((1-p^{p^{a}(p-1)+l})(p^{a}(p-1))!h[-p^{a}(p-1)-l]h[-1]^{t}\textbf{1}\right)_{a\geq
0}$

in the $p$-adic topology. In order to do this, we require that the square-
bracket states above be written in the round-bracket formalism. This was done
in Proposition 3.3. The following is then an extension of Lemma 10.4 of [FM22]
and establishes the $p$-adic convergence of the terms involving Stirling
numbers.

######  Lemma 3.5.

Fix $t,l\geq 1$ odd, and let $p$ be an odd prime with $r=p^{a}(p-1)+l$,
$s=p^{b}(p-1)+l$ and $a\leq b$. Then for any fixed $k$ within the range $0\leq
k\leq\lfloor t/2\rfloor$ and any $n\geq 0$ we have

$\displaystyle(1-p^{r})\binom{t}{2k}\frac{n!S_{r}^{(n+1)}(2k)!}{k!(-24)^{k}}\equiv(1-p^{s})\binom{t}{2k}\frac{n!S_{s}^{(n+1)}(2k)!}{k!(-24)^{k}}\mod
p^{a+x+1}.$

for some fixed integer $x$.

###### Proof.

Let $|\cdot|_{p}$ denote the $p$-adic absolute value. We have

$\displaystyle\left\lvert\binom{t}{2k}\frac{n!S_{r}^{(n+1)}(2k)!}{k!(-24)^{k}}-\binom{t}{2k}\frac{n!S_{s}^{(n+1)}(2k)!}{k!(-24)^{k}}\right\rvert_{p}=\left\lvert
n!S_{r}^{(n+1)}-n!S_{s}^{(n+1)}\right\rvert_{p}\left\lvert\binom{t}{2k}\frac{(2k)!}{k!(-24)^{k}}\right\rvert_{p}.$

The rightmost term is dependent only on $t$ and $k$ which are fixed, and so
denote by $x$ the $p$-adic valuation of this term, which is also fixed. We
show now that $n!S_{r}^{(n+1)}\equiv n!S_{s}^{(n+1)}\mod p^{a+1}$. From the
statement of Proposition 3.3, we have the formula

$\displaystyle
n!S_{r}^{(n+1)}=\sum_{j=0}^{n}\binom{n}{j}(-1)^{n+j}(j+1)^{r-1}.$

If $p\mid(j+1)$ then of course $(j+1)^{r-1}\equiv 0\mod p^{a+1}$. Suppose
then, that $p\nmid(j+1)$. This means that $p$ and $(j+1)$ are coprime, and
subsequently that $p^{a+1}$ and $(j+1)$ are coprime. Recall that
$\varphi(p^{a+1})=p^{a}(p-1)$ where $\varphi$ denotes the Euler totient
function. So by Euler’s theorem,

$\displaystyle(j+1)^{p^{a}(p-1)}\equiv 1\mod p^{a+1}.$

Note also that $(j+1)^{l-1}$ is an integer independent of $r$ and so by
multiplying both sides of the above congruence by this factor we get
$(j+1)^{r-1}\equiv(j+1)^{l-1}\mod p^{a+1}$. Thus

$\displaystyle n!S_{r}^{(n+1)}\equiv\sum_{\begin{subarray}{c}j=0\\\
p\nmid(j+1)\end{subarray}}^{n}\binom{n}{j}(-1)^{n+j}(j+1)^{l-1}\mod p^{a+1}$

and since the right hand side above does not depend on $r$, this establishes
that $n!S_{r}^{(n+1)}\equiv n!S_{s}^{(n+1)}\mod p^{a+1}$. Putting everything
together, we get

$\displaystyle\binom{t}{2k}\frac{n!S_{r}^{(n+1)}(2k)!}{k!(-24)^{k}}\equiv\binom{t}{2k}\frac{n!S_{s}^{(n+1)}(2k)!}{k!(-24)^{k}}\mod
p^{a+x+1}.$

Finally since $x$ is fixed, for sufficently large $a$ it is clear that

$\displaystyle 1-p^{p^{a}(p-1)+l}\equiv 1-p^{p^{b}(p-1)+l}\mod p^{a+x+1},$

and so by combining the above two congruences together, we obtain the desired
result. ∎

Next we establish the $p$-adic convergence of the terms involving the
Bernoulli numbers. This is done in the following lemma by use of the classical
Kummer congruences (cf. [Kum51] for the statement and proof).

######  Lemma 3.6.

Fix $t,l\geq 1$ odd where $l\not\equiv-1\mod p-1$, and let $p\geq 5$ be a
prime with $r=p^{a}(p-1)+l$, $s=p^{b}(p-1)+l$ and $a\leq b$. Then for $k$ in
the range $0\leq k\leq\lfloor(t-1)/2\rfloor$, we have

$\displaystyle(1-p^{r})\binom{t}{2k+1}\frac{B_{r+1}(2k+1)!}{k!(r+1)(-24)^{k}}\equiv(1-p^{s})\binom{t}{2k+1}\frac{B_{s+1}(2k+1)!}{k!(s+1)(-24)^{k}}\mod
p^{a+y+1}$

for some fixed integer $y$.

###### Proof.

First write

$\displaystyle\left\lvert(1-p^{r})\frac{B_{r+1}}{r+1}-(1-p^{s})\frac{B_{s+1}}{s+1}\right\rvert_{p}\left\lvert\binom{t}{2k+1}\frac{(2k+1)!}{k(-24)^{k}}\right\rvert_{p}.$

Once again the rightmost term is dependent only on $t$ and $k$ which are
fixed, and so denote by $y$ the $p$-adic valuation of this term, which is also
fixed. Notice that

$\displaystyle p^{a}(p-1)+l+1\equiv p^{b}(p-1)+l+1\mod p^{a}(p-1)$

and $p^{a}(p-1)=\varphi(p^{a+1})$. Since $l$ is odd, both $r+1$ and $s+1$ are
even. Since $l\not\equiv-1\mod p-1$, $p-1$ does not divide $l+1$ and so both
$r+1$ and $s+1$ are not divisible by $p-1$ (since $p>3$). Thus by Kummer’s
congruence we obtain

$\displaystyle\left\lvert(1-p^{r})\frac{B_{r+1}}{r+1}-(1-p^{s})\frac{B_{s+1}}{s+1}\right\rvert_{p}\left\lvert\binom{t}{2k+1}\frac{(2k+1)!}{k(-24)^{k}}\right\rvert_{p}=\frac{1}{p^{a+y+1}}$

which gives us the right congruence. ∎

We are now in a position to prove Theorem 1.1. The congruences established in
both Lemmas 3.5 and 3.6 imply that

$\displaystyle\lim_{a\to\infty}u_{p^{a}(p-1)+l,t}=u_{l,t}$

for some state $u_{l,t}$ in the $p$-adic Heisenberg VOA $S_{1}$. Then in the
$p$-adic topology and from eq. 3.9, we have

$\displaystyle\textbf{{f}}(u_{l,t})$
$\displaystyle=\lim_{a\to\infty}(1-p^{p^{a}(p-1)+l})2^{(t+1)/2}t(t-2)!!G_{2}^{(t-1)/2}(q)G_{p^{a}(p-1)+l+1}(q)$
$\displaystyle=2^{(t+1)/2}t(t-2)!!G_{2}^{(t-1)/2}(q)\lim_{a\to\infty}(1-p^{p^{a}(p-1)+l+1})G_{p^{a}(p-1)+l+1}(q).$

Since $\lim_{a\to\infty}1-p^{p^{a}(p-1)+l+1}=1$ and
$\lim_{a\to\infty}p^{a}(p-1)+l+1=l+1$,

$\displaystyle\textbf{{f}}(u_{l,t})=2^{(t+1)/2}t(t-2)!!G_{2}^{(t-1)/2}(q)G_{l+1}^{\star}(q)$

where $G_{l+1}^{\star}(q)$ is a $p$-adic Eisenstein modular form of weight
$l+1$. This proves Theorem 1.1.

##  4\. P-adic characters in lattice VOAs

###  4.1. Lattice VOAs

We recount relevant facts about lattice VOAs $V_{\Lambda}$ which are necessary
in the computation of characters in the subsection below. In the context of
[DMN01] or [LL04], let $\Lambda$ be an even (integral) unimodular lattice of
rank $d$ with associated symmetric $\mathbb{Z}$-bilinear form
$\langle\cdot,\cdot\rangle$, and let

$\displaystyle\mathfrak{h}=\Lambda\otimes_{\mathbb{Z}}\mathbb{C}.$

Following the process outlined in Section 3.1, construct $S_{\text{alg}}$
where we denote by $a(n)$ the action of $a\otimes t^{n}$ on $S_{\text{alg}}$
for $a\in\mathfrak{h}$, and where

$\displaystyle[a(m),b(n)]=\delta_{m+n,0}\langle a,b\rangle m.$

for integers $m,n\in\mathbb{Z}$. Denote by $\mathbb{C}\\{\Lambda\\}$ the group
algebra with basis $\\{e^{\alpha}\mid\alpha\in\Lambda\\}$ and form the space

$\displaystyle V_{\Lambda}=S_{\text{alg}}\otimes\mathbb{C}\\{\Lambda\\}.$

Vertex operators of states in $V_{\Lambda}$ with trivial
$\mathbb{C}\\{\Lambda\\}$ tensor factor are defined in the same way as in
$S_{\text{alg}}$. We identify $S_{\text{alg}}$ with $S_{\text{alg}}\otimes
e^{0}$ hence the vacuum of $V_{\Lambda}$ is the same as that of
$S_{\text{alg}}$. On $V_{\Lambda}$, states $a\in\mathfrak{h}$ with $n\neq 0$
act on $S_{\text{alg}}$ as before:

$\displaystyle a(n)(b\otimes e^{\alpha})=a(n)b\otimes e^{\alpha},$

whereas the operator $a(0)$ acts on $\mathbb{C}\\{\Lambda\\}$ as

(4.1) $\displaystyle a(0)(b\otimes e^{\alpha})=\langle
a,\alpha\rangle(b\otimes e^{\alpha}).$

Define the dual lattice
$\Lambda^{\circ}=\Set{\alpha\in\mathfrak{h}}{\langle\alpha,\Lambda\rangle\subset\mathbb{Z}}$.
Since $\Lambda$ is integral, $\Lambda\subset\Lambda^{\circ}$ and so we may
consider the coset decomposition

$\displaystyle\Lambda^{\circ}\bigcup_{i\in\Lambda^{\circ}/\Lambda}(\Lambda+\lambda_{i})$

From [Don93], each space

$\displaystyle
V_{\Lambda+\lambda_{i}}=S_{\text{alg}}\otimes\mathbb{C}\\{\Lambda+\lambda_{i}\\}$

constitute the complete set of non-isomorphic irreducible
$V_{\Lambda}$-modules, and furthermore, a theorem of Zhu shows that each
$Z_{\Lambda+\lambda_{i}}(a,q)$ is modular.

It is easy to show ([DMN01], Lemma 4.2.1) that
$Z_{\Lambda+\lambda_{i}}(a,q)=0$ for any $a=S_{\text{alg}}\otimes
e^{\alpha}\in V_{\Lambda}$ for some non-zero $\alpha\in\Lambda$. The proof
follows from the fact that any state in $V_{\Lambda}$ with non-zero
$\mathbb{C}\\{\Lambda\\}$ tensor factor has no mode which preserves grading.
As such we will consider only characters of states of the form
$S_{\text{alg}}\otimes e^{0}\in V_{\Lambda}$, and we henceforth omit writing
the trivial $\mathbb{C}\\{\Lambda\\}$ tensor factor for such states. For each
$\alpha\in\Lambda+\lambda_{i}$, we have

(4.2) $\displaystyle
V_{\Lambda+\lambda_{i}}=\bigoplus_{\alpha\in\Lambda+\lambda_{i}}S_{\text{alg}}\otimes
e^{\alpha}$

and so following [DMN01], denote for $a\in V_{\Lambda}$ the functions

$\displaystyle
Z_{\alpha}(a,q)=Z_{S_{\text{alg}}\otimes\iota(e_{\alpha})}(a,q)$
$\displaystyle Z_{i}(a,q)=Z_{\Lambda+\lambda_{i}}(a,q).$

It is then clear from eq. 4.2 that

(4.3) $\displaystyle
Z_{i}(a,q)=\sum_{\alpha\in\Lambda+\lambda_{i}}Z_{\alpha}(a,q).$

Computing characters in $V_{\Lambda}$ is similar to that in $S_{\text{alg}}$,
however one must be cautious due to discrepancies caused by the behaviour of
$a(0)$ exposed in eq. 4.1 above. In particular, equation eq. 3.7 of [MT10]
does not apply for most states in $V_{\Lambda}$. We resort then to using the
following general recursive expression given first in [Zhu96] from which eq.
3.7 is based: Given any $V$-module $M$ and for $a,b\in V$, we have

(4.4) $\displaystyle
Z_{M}(a[-n]b,q)=\delta_{n,1}\text{Tr}_{M}\left(o(a)o(b)q^{L_{0}-c_{V}/24}\right)+\sum_{m\geq
1}\frac{2(-1)^{m+1}}{m!(n-1)!}G_{n+m}(q)Z_{M}(a[m]b,q)$

where we have again applied the normalization seen in eq. 3.8.

###  4.2. Proof of Theorem 1.2

We take $a\in V_{\Lambda}$ such that $\langle a,a\rangle=1$ and
$\alpha\in\Lambda^{\circ}$ throughout. We will consider as in Section 3.2 the
states

$\displaystyle v_{r,t}=(r-1)!a[-r]a[-1]^{t}\textbf{1}\in S_{\text{alg}}\otimes
e^{0}$

where $r,t$ are odd integers, and where $t\geq 1$ and $r\geq 3$.

In order to assess convergence later, we must re-write $v_{r,t}$ in the round-
bracket formalism. This will be similar to what was done in Proposition 3.3,
however from eq. 4.1, we must be careful with the product $a(0)$. Recall that
from eq. 3.4 we have

$\displaystyle a[-1]=a(-1)+\frac{1}{2}a(0)-\frac{1}{12}a(-1)+\cdots.$

We wish to find expressions for $a[-1]^{t}\textbf{1}$ and
$a(1)a[-1]^{t}\textbf{1}$. Since both are elements of $S_{\text{alg}}\otimes
e^{0}$ however, the product $a(0)$ in the expansion above acts as zero since
$\langle a,0\rangle=0$. We will thus obtain the same expressions for
$a[-1]^{t}\textbf{1}$ and $a(1)a[-1]^{t}\textbf{1}$ as the ones in the
Heisenberg case given in Lemmas 3.1 and 3.2 with $h=a$. For the same reason,
the statement of Proposition 3.3 holds in in this case as well:

######  Proposition 4.1.

Let $a\in V_{\Lambda}$ where $\langle a,a\rangle=1$ and
$\alpha\in\Lambda^{\circ}$. For $r,t$ odd integers and $t\geq 1$ and $r\geq
3$, we have

$\displaystyle v_{r,t}=\sum_{n\geq 0}\sum_{k\geq 0}$
$\displaystyle\binom{t}{2k}\frac{n!S_{r}^{(n+1)}(2k)!}{k!(-24)^{k}}a(-n-1)a(-1)^{t-2k}\textbf{1}$
$\displaystyle-\sum_{k\geq
0}\binom{t}{2k+1}\frac{B_{r+1}(2k+1)!}{k!(r+1)(-24)^{k}}a(-1)^{t-2k-1}\textbf{1}$

We now take the character of $v_{r,t}$ over $S_{\text{alg}}\otimes
e^{\alpha}$. The following is an extension of Lemma 4.2.2 of [DMN01] and is
proven similarly.

######  Lemma 4.2.

Let $a\in V_{\Lambda}$ where $\langle a,a\rangle=1$ and
$\alpha\in\Lambda^{\circ}$. For $t\geq 0$ we have

$\displaystyle Z_{\alpha}(a[-1]^{t}\textbf{1},q)=\left(\sum_{k\geq
0}\binom{t}{2k}\frac{(2k)!}{k!}\langle
a,\alpha\rangle^{t-2k}G_{2}(q)^{k}\right)q^{\frac{1}{2}\langle\alpha,\alpha\rangle}/\eta(q)^{d}.$

###### Proof.

We proceed by induction on $t$. The case $t=0$ is simply the graded dimension
over $S_{\text{alg}}\otimes e^{0}$, and so let $t=1$. Then using eq. 4.4,

$\displaystyle Z_{\alpha}(a[-1]\textbf{1},q)$ $\displaystyle=\langle
a,\alpha\rangle Z_{\alpha}(\textbf{1},q)=\langle a,\alpha\rangle
q^{\frac{1}{2}}/\eta(q)^{d}$

where we used the fact that $o(a)=\langle a,\alpha\rangle$ since we work over
$S_{\text{alg}}\otimes e^{\alpha}$. Suppose the claim holds up to $t-1$. Then
using eq. 4.4 again, we obtain

$\displaystyle Z_{\alpha}(a[-1]^{t}\textbf{1},q)$
$\displaystyle=Z_{\alpha}(a[-1]a[-1]^{t-1}\textbf{1},q)$ (4.5)
$\displaystyle=\langle a,\alpha\rangle
Z_{\alpha}(a[-1]^{t-1}\textbf{1},q)+2(t-1)G_{2}(q)Z_{\alpha}(a[-1]^{t-2}\textbf{1},q).$

By induction, we write the first term of eq. 4.5 as

(4.6) $\displaystyle\langle a,\alpha\rangle^{t}+\left(\sum_{k\geq
1}\binom{t-1}{2k}\frac{(2k)!}{k!}\langle
a,\alpha\rangle^{t-2k}G_{2}^{k}(q)\right)q^{\frac{1}{2}\langle\alpha,\alpha\rangle}/\eta(q)^{d},$

and similarly the second term of eq. 4.5 equates to

$\displaystyle\left(\sum_{k\geq 0}\binom{t-2}{2k}\frac{2(t-1)(2k)!}{k!}\langle
a,\alpha\rangle^{t-2k-2}G_{2}(q)^{k+1}\right)q^{\frac{1}{2}\langle\alpha,\alpha\rangle}/\eta(q)^{d}$
$\displaystyle=\left(\sum_{k\geq
0}\frac{(k+1)(2k+1)!(2)(t-1)!}{(k+1)(2k+1)!(t-2k-2)!(k)!}\langle
a,\alpha\rangle^{t-2k-2}G_{2}(q)^{k+1}\right)q^{\frac{1}{2}\langle\alpha,\alpha\rangle}/\eta(q)^{d}$
$\displaystyle=\left(\sum_{k\geq
0}\binom{t-1}{2k+1}\frac{(2k+2)!}{(k+1)!}\langle
a,\alpha\rangle^{t-2k-2}G_{2}(q)^{k+1}\right)q^{\frac{1}{2}\langle\alpha,\alpha\rangle}/\eta(q)^{d}$
(4.7) $\displaystyle=\left(\sum_{k\geq
1}\binom{t-1}{2k-1}\frac{(2k)!}{k!}\langle
a,\alpha\rangle^{t-2k}G_{2}(q)^{k}\right)q^{\frac{1}{2}\langle\alpha,\alpha\rangle}/\eta(q)^{d}.$

Summing eqs. 4.6 and 4.7 and using Pascal’s rule as in Lemma 3.1, we obtain

$\displaystyle Z_{\alpha}(a[-1]^{t}\textbf{1},q)=\left(\sum_{k\geq
0}\binom{t}{2k}\frac{(2k)!}{k!}\langle
a,\alpha\rangle^{t-2k}G_{2}(q)^{k}\right)q^{\frac{1}{2}\langle\alpha,\alpha\rangle}/\eta(q)^{d}$

which proves the result. ∎

Following [DMN01], we will be denoting

$\displaystyle f_{\alpha,t}(q)=\sum_{k\geq
0}\binom{t}{2k}\frac{(2k)!}{k!}G_{2}(q)^{k}\langle a,\alpha\rangle^{t-2k}$

in the following analogue of Proposition 3.4:

######  Proposition 4.3.

Let $a\in V_{\Lambda}$ where $\langle a,a\rangle=1$. For $r,t$ odd integers
and $t\geq 1$ and $r\geq 3$ we have

$\displaystyle Z(v_{r,t},q)=2tG_{r+1}(q)\left(\sum_{k\geq
0}\binom{t-1}{2k}\frac{(2k)!}{k!}G_{2}(q)^{k}\Theta_{\Lambda,t-2k-1}(q)\right)/\eta(q)^{d}$

where

$\displaystyle\Theta_{\Lambda,t-2k-1}(q)=\sum_{\alpha\in\Lambda}\langle
a,\alpha\rangle^{t-2k-1}q^{\frac{1}{2}\langle\alpha,\alpha\rangle}.$

###### Proof.

We use eq. 4.4. This gives

$\displaystyle Z_{\alpha}(v_{r,t},q)$ $\displaystyle=(r-1)!\sum_{m\geq
1}\frac{2(-1)^{m+1}}{m!(r-1)!}G_{r+m}(q)Z_{\alpha}(a[m]a[-1]^{t}\textbf{1},q)$
$\displaystyle=2G_{r+1}(q)Z_{\alpha}(a[1]a[-1]^{t}\textbf{1},q)$
$\displaystyle=2tG_{r+1}(q)f_{\alpha,t-1}(q)q^{\frac{1}{2}\langle\alpha,\alpha\rangle}/\eta(q)^{d}.$

where we have made use of Lemma 4.2 in the last equality. Using eq. 4.3,

$\displaystyle Z_{i}(v_{r,t},q)$
$\displaystyle=2tG_{r+1}(q)\left(\sum_{\alpha\in\Lambda+\lambda_{i}}f_{\alpha,t-1}(q)q^{\frac{1}{2}\langle\alpha,\alpha\rangle}\right)/\eta(q)^{d}$
$\displaystyle=2tG_{r+1}(q)\left(\sum_{k\geq
0}\binom{t-1}{2k}\frac{(2k)!}{k!}G_{2}(q)^{k}\sum_{\alpha\in\Lambda+\lambda_{i}}\langle
a,\alpha\rangle^{t-2k-1}q^{\frac{1}{2}\langle\alpha,\alpha\rangle}\right)/\eta(q)^{d}$
$\displaystyle=2tG_{r+1}(q)\left(\sum_{k\geq
0}\binom{t-1}{2k}\frac{(2k)!}{k!}G_{2}(q)^{k}\Theta_{\Lambda+\lambda_{i},t-2k-1}(q)\right)/\eta(q)^{d}.$

The proof follows from the fact that specializing $i=0$ gives
$Z_{0}(v_{r,t},q)=Z(v_{r,t},q)$. ∎

The theta series $\Theta_{\Lambda,t,k}(q)$ in the statement of Proposition 4.3
appears troublesome since it is not quite the regular theta series
$\Theta_{\Lambda}(q)$ of $\Lambda$. It was shown in [DM01] and exposed in
Theorem 4.2.6 of [DMN01] that $\Theta_{\Lambda,t,k}(q)$ in fact lies in the
space of quasi-modular forms on the congruence subgroup $\Gamma_{0}(N)$ for
some $N\geq 1$. Hence the image of the character map for $V_{\Lambda}$ is
indeed quasi-modular, and we may consider $p$-adic limits.

With the assumptions of Proposition 4.3 in place, we proceed as in Section 3.3
by denoting

$\displaystyle
u_{r,t}=(1-p^{r})v_{r,t}=(1-p^{r})(r-1)!a[-r]a[-1]^{t}\textbf{1}$

and considering the sequence of states $\left(u_{p^{b}(p-1)+l,t}\right)_{b\geq
0}$ for fixed odd $t\geq 1$ where $p\geq 3$ is a prime and $l\not\equiv-1\mod
p-1$. Since the round-bracket states obtained in Proposition 4.1 are identical
to those in the Heisenberg case, both Lemmas 3.5 and 3.6 establish the
$p$-adic convergence of this sequence to a state $u_{t,l}$ in the $p$-adic
lattice VOA $\widehat{V}_{\Lambda}$. Then by taking the limit in the $p$-adic
topology as in Section 3.3 we obtain

$\displaystyle\textbf{{f}}(u_{t,l})=2tG_{l+1}^{\star}(q)\left(\sum_{k\geq
0}\binom{t-1}{2k}\frac{(2k)!}{k!}G_{2}(q)^{k}\Theta_{\Lambda,t-2k-1}(q)\right).$

where f is the character map in eq. 1.3. This proves Theorem 1.2.

## References

  * [AAR99] George E. Andrews, Richard Askey, and Ranjan Roy, _Special functions_ , Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958
  * [Car05] R. W. Carter, _Lie algebras of finite and affine type_ , Cambridge Studies in Advanced Mathematics, vol. 96, Cambridge University Press, Cambridge, 2005\. MR 2188930
  * [DM01] Chongying Dong and Geoffrey Mason, _Transformation laws for theta functions_ , Proceedings on Moonshine and related topics (Montréal, QC, 1999), CRM Proc. Lecture Notes, vol. 30, Amer. Math. Soc., Providence, RI, 2001, pp. 15–26. MR 1877753
  * [DMN01] Chongying Dong, Geoffrey Mason, and Kiyokazu Nagatomo, _Quasi-modular forms and trace functions associated to free boson and lattice vertex operator algebras_ , Internat. Math. Res. Notices (2001), no. 8, 409–427. MR 1827085
  * [Don93] Chongying Dong, _Vertex algebras associated with even lattices_ , J. Algebra 161 (1993), no. 1, 245–265. MR 1245855
  * [FBZ04] Edward Frenkel and David Ben-Zvi, _Vertex algebras and algebraic curves_ , second ed., Mathematical Surveys and Monographs, vol. 88, American Mathematical Society, Providence, RI, 2004. MR 2082709
  * [FLM88] Igor Frenkel, James Lepowsky, and Arne Meurman, _Vertex operator algebras and the Monster_ , Pure and Applied Mathematics, vol. 134, Academic Press, Inc., Boston, MA, 1988. MR 996026
  * [FM22] Cameron Franc and Geoffrey Mason, _p-adic vertex operator algebras_ , 2022, preprint, URL: https://arxiv.org/abs/2207.07455.
  * [Kat73] Nicholas M. Katz, _$p$ -adic properties of modular schemes and modular forms_, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Lecture Notes in Math., Vol. 350, Springer, Berlin, 1973, pp. 69–190. MR 0447119
  * [KL64] Tomio Kubota and Heinrich-Wolfgang Leopoldt, _Eine $p$-adische Theorie der Zetawerte. I. Einführung der $p$-adischen Dirichletschen $L$-Funktionen_, J. Reine Angew. Math. 214(215) (1964), 328–339. MR 163900
  * [KSU89a] Toshiyuki Katsura, Yuji Shimizu, and Kenji Ueno, _Formal groups and conformal field theory over ${\bf Z}$_, Integrable systems in quantum field theory and statistical mechanics, Adv. Stud. Pure Math., vol. 19, Academic Press, Boston, MA, 1989, pp. 347–366. MR 1048600
  * [KSU89b] by same author, _New bosonization and conformal field theory over ${\bf Z}$_, Comm. Math. Phys. 121 (1989), no. 4, 603–627. MR 990994
  * [KSU91] by same author, _Complex cobordism ring and conformal field theory over ${\bf Z}$_, Math. Ann. 291 (1991), no. 3, 551–571. MR 1133349
  * [Kum51] E. E. Kummer, _Über eine allgemeine Eigenschaft der rationalen Entwickelungscoefficienten einer bestimmten Gattung analytischer Functionen_ , J. Reine Angew. Math. 41 (1851), 368–372. MR 1578727
  * [LL04] James Lepowsky and Haisheng Li, _Introduction to vertex operator algebras and their representations_ , Progress in Mathematics, vol. 227, Birkhäuser Boston, Inc., Boston, MA, 2004. MR 2023933
  * [MT10] Geoffrey Mason and Michael Tuite, _Vertex operators and modular forms_ , A window into zeta and modular physics, Math. Sci. Res. Inst. Publ., vol. 57, Cambridge Univ. Press, Cambridge, 2010, pp. 183–278. MR 2648364
  * [Rom84] Steven Roman, _The umbral calculus_ , Pure and Applied Mathematics, vol. 111, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1984\. MR 741185
  * [Ser73] Jean-Pierre Serre, _Formes modulaires et fonctions zêta $p$-adiques_, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Lecture Notes in Math., Vol. 350, Springer, Berlin, 1973, pp. 191–268. MR 0404145
  * [Zhu96] Yongchang Zhu, _Modular invariance of characters of vertex operator algebras_ , J. Amer. Math. Soc. 9 (1996), no. 1, 237–302. MR 1317233


# Polynomial generators of ${\mathbf{M}\mathbf{S}\mathbf{U}}^{*}[1/2]$ related
to classifying maps of certain formal group laws

Malkhaz Bakuradze  Faculty of exact and natural sciences, A. Razmadze Math.
Institute, Iv. Javakhishvili Tbilisi State University, Georgia
<EMAIL_ADDRESS>
###### Abstract.

This paper presents a commutative complex oriented cohomology theory that
realizes the Buchstaber formal group law $F_{B}$ localized away from 2. It is
shown that the restriction of the classifying map of $F_{B}$ on special
unitary cobordism ring localized away from 2 defines a four parameter genus,
studied by Hoehn and Totaro.

###### Key words and phrases:

Complex bordism, $SU$-bordism, Formal group law, Complex elliptic genus

###### 2010 Mathematics Subject Classification:

55N22; 55N35

The author was supported by Shota Rustaveli NSF grant FR-21-4713

##  1\. Introduction

The ring of complex cobordism ${\mathbf{M}\mathbf{U}}_{*}$ and the ring
${\mathbf{M}\mathbf{S}\mathbf{U}}_{*}$ of special unitary cobordism has been
studied by many authors. We refer the reader to [20], [17] for details. In
particular the ring ${\mathbf{M}\mathbf{S}\mathbf{U}}_{*}$, localized away
from 2, is torsion free

${\mathbf{M}\mathbf{S}\mathbf{U}}_{*}[1/2]=\mathbb{Z}[1/2][x_{2},x_{3},\cdots],\,\,\,\,|x_{i}|=2i$

and $SU$-structure forgetful homomorphism is the inclusion in complex
cobordism ring

${\mathbf{M}\mathbf{S}\mathbf{U}}_{*}[1/2]\subset{\mathbf{M}\mathbf{U}}_{*}[1/2]=\mathbb{Z}[1/2][x_{1},x_{2},x_{3},\cdots].$

In this paper we construct a commutative complex oriented cohomology theory
(Theorem 5.1) such that the coefficient ring is the scalar ring of the
Buchstaber formal group law $F_{B}$ with inverted 2, and show (Proposition
5.1) that after restricted to ${\mathbf{M}\mathbf{S}\mathbf{U}}_{*}[1/2]$, the
classifying map of $F_{B}$ can become a genus

(1.1)
${\mathbf{M}\mathbf{S}\mathbf{U}}_{*}[1/2]\to\mathbb{Z}[1/2][x_{2},x_{3},x_{4}],\,\,\,\,|x_{i}|=2i,$

studied by Hoehn [13] and Totaro [21].

Since ${\mathbf{M}\mathbf{S}\mathbf{U}}^{*}$ is not complex oriented, it is
difficult to compute the genus (1.1) on specific explicit elements. Using the
polynomial generators of the spherical cobordism ring $W^{*}[1/2]$ given by G.
Chernykh and T. Panov [12], we derive certain polynomial generators of
${\mathbf{M}\mathbf{S}\mathbf{U}}^{*}[1/2]$ in terms of the universal formal
group law. This gives a new understanding of the genus (1.1).

In particular, the classifying map $f_{B}$ of $F_{B}$ is a surjection on some
infinitely generated ring $\Lambda_{B}$, with kernel generated by some
explicit elements (Proposition 2.1). After it is tensored with rationals it is
identical (Proposition 2.2) to the complex elliptic genus

(1.2)
${\mathbf{M}\mathbf{U}}_{*}\otimes\mathbb{Q}\to\mathbb{Q}[x_{1},x_{2},x_{3},x_{4}],$

defined in Hoehn’s thesis [13](Section 2.5). Here $x_{1}$ is the image of
complex projective plane ${\mathbf{C}\mathbf{P}}_{1}$ and $x_{2},x_{3},x_{4}$
are the images of any first three generators of the polynomial ring
${\mathbf{M}\mathbf{S}\mathbf{U}}_{*}\otimes\mathbb{Q}=\mathbb{Q}[x_{2},x_{3},x_{4},\cdots]$.

By Hoehn [13], for $X$ an $SU$-manifold of complex dimension $n$, the
exponential characteristic class $\phi(X)$ is in fact a Jacobi form of weight
$n$. Jacobi forms are generalizations of modular forms. See details in [21].
Hoehn showed that the Jacobi forms $x_{2},x_{3},x_{4}$ arise as the elliptic
genera of certain explicit $SU$-manifolds, of complex dimensions 2, 3, 4, so
that the homomorphism

${\mathbf{M}\mathbf{S}\mathbf{U}}_{*}\to(\text{ Jacobi forms over
}\mathbb{Z})$

becomes surjective after it is tensored with $\mathbb{Z}[1/2]$.

In [21] (Theorem 4.1) Totaro proved that the Krichever-Hoehn complex elliptic
genus on complex cobordism viewed as a homomorphism (1.2) is surjective and
the kernel is equal to the ideal of complex flops.

Then Totaro proved (Theorem 6.1) that the kernel of the complex elliptic genus
on ${\mathbf{M}\mathbf{S}\mathbf{U}}_{*}\otimes\mathbb{Z}[1/2]$ is equal to
the ideal $I$ of $SU$-flops. Also, the quotient ring is a polynomial ring:

(1.3)
${\mathbf{M}\mathbf{S}\mathbf{U}}_{*}[1/2]/I=\mathbb{Z}[1/2][x_{2},x_{3},x_{4}].$

Unfortunately ${\mathbf{M}\mathbf{S}\mathbf{U}}_{*}$ is not complex oriented.
It would be nice to develop a method for calculating (1.3) explicitly in terms
of the universal formal group law using some generators of
${\mathbf{M}\mathbf{S}\mathbf{U}}_{*}[1/2]$ treated as explicit elements in
${\mathbf{M}\mathbf{U}}_{*}[1/2]$.

This goal can be achieved as follows: in Section 5 we replace the ideal of
complex flops with a more explicit ideal by considering the integral
Buchstaber genus which is identical to the Krichever-Hoehn complex elliptic
genus over ${\mathbf{M}\mathbf{U}}_{*}\otimes\mathbb{Q}$.

In Section 6 we use the polynomial generators of the spherical cobordism ring
$W_{*}[1/2]$ constructed in [12] and define certain polynomial generators of
${\mathbf{M}\mathbf{S}\mathbf{U}}_{*}[1/2]$. In particular, we use fact that
$W_{*}[1/2]$ is generated by the coefficients of the corresponding formal
group law. Given Novikov’s criteria and that $W_{*}[1/2]$ is a free
${\mathbf{M}\mathbf{S}\mathbf{U}}_{*}[1/2]$ module generated by $1$ and
${\mathbf{C}\mathbf{P}}^{1}$, we define some generators in
${\mathbf{M}\mathbf{S}\mathbf{U}}_{*}[1/2]$ ( Proposition 6.3) in terms of the
universal formal group law. Finally the explicit quotient map of the
Buchstaber formal group law (Proposition 2.1) gives the decompositions of
constructed generators ${\mathbf{M}\mathbf{S}\mathbf{U}}_{*}$ of dimensions
$>10$ in polynomial ring $\mathbb{Z}[1/2][x_{2},x_{3},x_{4}]$. It is a way to
calculate the genus (1.3) in terms of the universal formal group law.

In Section 7 we consider the restriction of classifying map of the universal
abelian formal group law on ${\mathbf{M}\mathbf{S}\mathbf{U}}_{*}[1/2]$ to
define a genus with one parameter.

##  2\. Preliminaries

The theory $W_{*}$ of $c_{1}$-spherical bordism is defined geometrically in
[20] (Chapter VIII). The closed manifolds $M$ with a $c_{1}$-spherical
structure, consist of

– a stably complex structure on the tangent bundle $TM$;

– a ${\mathbf{C}\mathbf{P}}^{1}$-reduction of the determinant bundle, that is,
a map $f:M\to{\mathbf{C}\mathbf{P}}^{1}$ and an equivalence
$f^{*}(\eta)\simeq\det TM$, where $\eta$ is the tautological bundle over
${\mathbf{C}\mathbf{P}}^{1}$.

This is a natural generalization of an $SU$-structure, which can be thought of
as a trivialization of the determinant bundle. The corresponding bordism
theory is called $c_{1}$ -spherical bordism and is denoted $W_{*}$. The
unitary and special unitary bordism rings are denoted by
${\mathbf{M}\mathbf{U}}_{*}$ and ${\mathbf{M}\mathbf{S}\mathbf{U}}_{*}$
respectively. We refer to [17] and [12] for details on
${\mathbf{M}\mathbf{S}\mathbf{U}}_{*}$ and $W_{*}$ that will be used
throughout the paper.

Motivated by string theory in [14], [13], [21] the universal Krichever-Hoehn
complex elliptic genus $\phi_{KH}$ is defined as the ring homomorphism

(2.1)
$\phi_{KH}:{\mathbf{M}\mathbf{U}}^{*}\to\mathbb{Q}[q_{1},q_{2},q_{3},q_{4}]$

associated to the Hirzebruch characteristic power series
$Q(x)=\frac{x}{f(x)}$, where

$h(x)=\frac{f^{\prime}(x)}{f(x)}$

is the solution of the differential equation in
${\mathbf{M}\mathbf{U}}^{*}\otimes\mathbb{Q}$

(2.2) $(h^{\prime})^{2}=S(h),$

where

$S(x)=x^{4}+q_{1}x^{3}+q_{2}x^{2}+q_{3}x+q_{4},$

for some formal parameters $|q_{i}|=2i$.

One consequence of Krichever-Hoehn’s rigidity theorem [14], [13], [21], is
that ([13], Kor 2.2.3) if $F\to E\to B$ is a fiber bundle of closed connected
weakly complex manifolds, with structure group a compact connected Lie group
$G$, and if $F$ is an $SU$-manifold, then the elliptic genus $\phi_{KH}$
satisfies $\phi_{KH}(E)=\phi_{KH}(F)\phi_{KH}(B)$. In fact, the elliptic genus
is the universal genus with the above multiplicative property

In ([21], Theorem 6.1) Totaro gave a geometric description of the kernel ideal
$I$ of the complex elliptic genus restricted to
${\mathbf{M}\mathbf{S}\mathbf{U}}[1/2]$, the ideal of $SU$ flops. This kernel
is equal to the ideal in ${\mathbf{M}\mathbf{S}\mathbf{U}}_{*}[1/2]$ generated
by twisted projective bundles $\tilde{{\mathbf{C}\mathbf{P}}}(A\oplus B)$ over
weakly complex manifolds $Z$ such that the complex vector bundles $A$ and $B$
over $Z$ have rank $2$ and $c_{1}Z+c_{1}A+c_{1}B=0$; in this case, the total
space is an $SU$-manifold. Then Totaros’s result says that the
$I\in{\mathbf{M}\mathbf{U}}_{*}[1/2]$ contains a polynomial generator of
${\mathbf{M}\mathbf{S}\mathbf{U}}_{*}[1/2]$ in real dimension $2n$ for all
$n\geq 5$ and

(2.3)
${\mathbf{M}\mathbf{S}\mathbf{U}}_{*}[1/2]/I\simeq\mathbb{Z}[1/2][x_{2},x_{3},x_{4}].$

In [13] by using of characteristic classes, Hoehn constructed a base sequence
$W_{1},W_{2},W_{3},\cdots$ of the rational cobordism ring
${\mathbf{M}\mathbf{U}}^{*}\otimes\mathbb{Q}$ on which $\phi_{KH}$ has the
values $A,B,C,D$, and $0$ for $W_{i}$ with $i>4$.

As another generalization of Oshanin’s elliptic genus Schreieder in [19]
studied a genus $\psi$ with logarithmic series

$\log_{\psi}(x)=\int_{0}^{x}\frac{dt}{R(t)},\,\,\,R(t)=\sqrt{1+q_{1}t+q_{2}t^{2}+q_{3}t^{3}+q_{4}t^{4}}.$

The genus $\psi$ is easily calculable on cobordism classes of complex
projective spaces ${\mathbf{C}\mathbf{P}}_{i}$, the generators of the domain

${\mathbf{M}\mathbf{U}}^{*}\otimes\mathbb{Q}=\mathbb{Q}[{\mathbf{C}\mathbf{P}}_{1},{\mathbf{C}\mathbf{P}}_{2},\cdots].$

This is because of the equation
$(\log_{\psi}(x)^{\prime})^{2}=1/R^{2}(x)=\sum_{i\geq
1}\psi({\mathbf{C}\mathbf{P}}_{i})x^{i}$ we need only the Taylor expansion of
$(1+y)^{-1/2}$.

It is natural to ask whether one can calculate $\phi_{KH}$ in an elementary
manner, different from that relying on the formulas in [13] and [8].

Viewed as a classifying map $\psi$ is strongly isomorphic to genus $\phi$ by
the series $\mu(x)=\sum_{i\geq 0}{\mathbf{C}\mathbf{P}}_{i}x^{i+1}$ [4]. This
gives a method for explicit calculation of $\phi_{KH}$.

In [2, 3] we introduced the formal power series

(2.4) $A(x,y)=\sum
A_{ij}x^{i}y^{j}=F(x,y)(x\omega(y)-y\omega(x))\in{\mathbf{M}\mathbf{U}}^{*}[[x,y]],$

where

$F=F(x,y)=\sum\alpha_{ij}x^{i}y^{j}$

is the universal formal group law over complex cobordism ring
${\mathbf{M}\mathbf{U}}^{*}$ and

$\omega(x)=\frac{\partial F(x,y)}{\partial y}(x,0)=1+\sum_{i\geq 1}w_{i}x^{i}$

is the invariant differential of $F$.

The series $A(x,y)$ has proven to be interesting for the following reasons.

######  Proposition 2.1.

[3]. i) The obvious quotient map

$f_{B}:{\mathbf{M}\mathbf{U}}^{*}\to{\mathbf{M}\mathbf{U}}^{*}/(A_{ij},\,i,j\geq
3)$

classifies a formal group law which is identical to the universal Buchstaber
formal group law $F_{B}$, the universal formal group law of the form

$\frac{x^{2}A(y)-y^{2}A(x)}{xB(y)-yB(x)},$

where $A(0)=B(0)=1$.

(ii) If $A^{\prime}(0)=B^{\prime}(0)$ then $B(x)$ is identical to the image of
$\omega(x)$ under the classifying map $f_{B}$.

######  Proposition 2.2.

[4]. After it is tensored with rationals the classifying map $f_{B}$ of the
Buchstaber formal group law is identical to the Krichever-Hoehn complex
elliptic genus

$\phi:{\mathbf{M}\mathbf{U}}^{*}\otimes\mathbb{Q}\to\Lambda_{B}\otimes\mathbb{Q}=\mathbb{Q}[{\mathbf{C}\mathbf{P}}_{1},{\mathbf{C}\mathbf{P}}_{2},{\mathbf{C}\mathbf{P}}_{3},{\mathbf{C}\mathbf{P}}_{4}],$

where ${\mathbf{C}\mathbf{P}}_{i}$ are complex projective spaces.

For explicit calculation of the Krichever-Hoehn genus the following
observation is helpful.

######  Proposition 2.3.

[4]. Over the ring ${\mathbf{M}\mathbf{U}}^{*}\otimes\mathbb{Q}$ the series
$\frac{x}{\omega(x)}$ is the strong isomorphism from the formal group law with
logarithm series

$\int_{0}^{x}\frac{dt}{\sqrt{1+p_{1}t+p_{2}t^{2}+p_{3}t^{3}+p_{4}}}$

in [19] to the formal group law classified by $f$.

##  3\. Some auxiliary combinatorial definitions

By Euclid’s algorithm for the natural numbers $m_{1},m_{2},\cdots,m_{k}$ one
can find integers $\lambda_{1},\lambda_{2},\cdots\lambda_{k}$ such that

(3.1)
$\lambda_{1}m_{1}+\lambda_{2}m_{2}+\cdots+\lambda_{k}m_{k}=gcd(m_{1},m_{2},\cdots,m_{k}).$

Let

(3.2)
$d(m)=gcd\bigg{\\{}\binom{m+1}{1},\binom{m+1}{2},\cdots,\binom{m+1}{m-1}\big{|}\,\,m\geq
1\bigg{\\}}.$

By [15] one has

$d(m)=\begin{cases}p,&\mbox{if $m+1=p^{s}$ for some prime $p$,}\\\
1,&\mbox{otherwise}.\end{cases}$

For the coefficients of universal formal group law
$F(x,y)=\sum\alpha_{i\,j}x^{i}y^{j}$ the elements

(3.3)
$e_{m}=\lambda_{1}\alpha_{1\,m}+\lambda_{2}\alpha_{2\,m-1}+\cdots+\lambda_{m}\alpha_{m\,1}$

are multiplicative generators in ${\mathbf{M}\mathbf{U}}_{*}$.

By [9], Theorem 9.9, or [22]

(3.4) $\displaystyle\frac{D(m)}{d(m)}=\begin{cases}d(m-1)&\mbox{if }m\neq
2^{k}-2,\\\ 2&\mbox{if }m=2^{k}-2.\end{cases}$

where

(3.5) $D(m)=gcd\bigg{\\{}\binom{m+1}{i}-\binom{m+1}{i-1}\big{|}\,\,2<i\leq
m-1,m\geq 5\bigg{\\}}.$

Let $m\geq 4$ and let $\lambda_{2},\cdots,\lambda_{m-2}$ are such integers
that

(3.6)
$d_{2}(m):=\sum_{i=2}^{m-2}\lambda_{i}\binom{m+1}{i}=gcd\bigg{\\{}\binom{m+1}{2},\cdots,\binom{m+1}{m-2}\bigg{\\}}.$

Then by [9] Lemma 9.7 one has for $m\geq 3$

(3.7) $\displaystyle d_{2}(m)=d(m)d(m-1).$

Note $d(n)$ are the Chern numbers of the generators in complex cobordism of
dimension $2n$.

For the generators

$A_{ij},\,\,i,j\geq 3,\,\,i+j-2=m$

of the quotient ideal corresponding to $\Lambda_{B}$, the scalar ring of the
universal Buchstaber formal group law in Proposition 2.1 and the integers
$\lambda_{3},\cdots,\lambda_{m-1}$ corresponding to (3.5) consider the linear
combinations

(3.8)
$T_{m}=\lambda_{3}A_{3\,m-1}+\lambda_{4}A_{4\,m-2}+\cdots\lambda_{m-1}A_{m-1\,3}.$

The elements $T_{p^{s}}$, where $p$ is a prime number, and $e_{i}$ in (3.3)
for $i\neq p^{s}$ will play a major role in Section 4.

##  4\. Realization of the universal Buchstaber formal group law localized
away from 2.

Let $e_{i}$ and $T_{i}$ be as in (3.3) and (3.8) respectively.

Let $J_{B}$ be the ideal of
${\mathbf{M}\mathbf{U}}_{*}=\mathbb{Z}[e_{1},e_{2},\cdots]$

$J_{B}=\\{A_{ij},i,j\geq 3\\},$

the quotient ideal of the universal Buchstaber formal group law $F_{B}$
classified by

$f_{B}:{\mathbf{M}\mathbf{U}}_{*}\to{\mathbf{M}\mathbf{U}}_{*}/J_{B}=\Lambda_{B}.$

The ideal $J_{B}$ is not prime as the quotient ring $\Lambda_{B}$ is not an
integral domain: it has $2$-torsion element of degree $12$ [3]. Here we use
the results of [9], that $\Lambda_{B}$ is generated by $f_{B}(e_{j})$,
$j=1,2,3,4$ and $j=p^{r}$, $r\geq 1$, $p$ is prime, and $j=2^{k}-2$, $k\geq
3$. Then the ideal $Tor(\Lambda_{B})$ is generated by the elements of order 2,
namely $f_{B}(e_{j}),\,\,j=2^{k}-2$, $k\geq 3$.

The ideal $Tor(\Lambda_{B})$ is prime as $\Lambda_{\mathcal{B}}$ in

(4.1)
$f_{\mathcal{B}}:{\mathbf{M}\mathbf{U}}_{*}\to\Lambda_{B}\to\Lambda_{B}/Tor(\Lambda_{B})=\Lambda_{\mathcal{B}}.$

is an integral domain and so is
$J_{\mathcal{B}}=f_{B}^{-1}(Tor(\Lambda_{B}))$, the preimage ideal in
${\mathbf{M}\mathbf{U}}_{*}$. Then the ideal

(4.2) $J_{\mathcal{B}}=J_{B}+(e_{2^{k}-2}),k\geq 3$

is the kernel of the composition (4.1). Denote by $F_{\mathcal{B}}$ the formal
group law classified by $f_{\mathcal{B}}$.

Let

(4.3) $\displaystyle J=\big{(}\mathcal{T}_{l},l\geq 5\big{)},\text{ where
}\mathcal{T}_{l}=\begin{cases}T_{l},&\mbox{$n=p^{s}$, $p$ is a prime}\\\
e_{l},&\mbox{otherwise}.\end{cases}$

Let $J_{\mathcal{B}}(l)\subset J_{\mathcal{B}}$, generated by those elements
whose degree is greater or equal $-2l$, and let $J(l)\subset J$ be generated
by $\mathcal{T}_{5},\cdots,\mathcal{T}_{l}$.

######  Remark 4.1.

We note that those polynomials in $\mathbb{Z}[e_{1},e_{2},\cdots,e_{l}]$ that
are in the kernel of $f_{\mathcal{B}}$ can be viewed as the elements of
$J_{\mathcal{B}}(l)$.

######  Proposition 4.2.

$J(l)=J_{\mathcal{B}}(l)$ for any natural $l\geq 5$.

Proof. It is clear that $J(l)\subset J_{\mathcal{B}}(l)$.

Let us prove $J_{\mathcal{B}}(l)\subset J(l)$ by induction on $l$. It is
obvious for $l=5$ as $T_{5}=A_{34}$.

To prove $J_{\mathcal{B}}(l)=J_{\mathcal{B}}(l-1)+T_{l}$ note

$s_{i+j-2}(A_{ij})=\binom{i+j-1}{j-1}-\binom{i+j-1}{j}.$

Indeed, modulo decomposable elements

$A_{ij}=\alpha_{i-1j}-\alpha_{ij-1}$

and $s_{i+j-1}(\alpha_{ij})=-\binom{i+j}{i}$. Now apply Euclid’s algorithm for
$m_{i}=s_{l}(A_{i,l+2-i})$, fix the integers $\lambda_{i}$ and consider the
elements $T_{l}$ in (3.8).

The combinatorial identities in Section 3 implies that

(4.4) $s_{l}(T_{l})=D(l)$

is the greatest common divisor of the integers $s_{l}(A_{ij})$ for $A_{ij}\in
J_{B}$. $i+j-2=l$.

It follows that

$A_{ij}=\frac{s_{n}(A_{ij})}{D(l)}T_{l}+P(e_{1},e_{2},\cdots,e_{l-1}),$

for some polynomial $P$, i.e.,

$A_{ij}=P(e_{1},e_{2},\cdots,e_{l-1})\text{ modulo }T_{l}$

Therefore $P(e_{1},e_{2},\cdots,e_{l-1})$ is in the kernel of
$f_{\mathcal{B}}$, i.e., is in $J_{\mathcal{B}}(l-1)=J(l-1)$ by above Remark
4.1. ∎

Let $A_{l}=\mathbb{Z}[e_{1},e_{2},\cdots,e_{l}]$ and
$A^{l+1}=\mathbb{Z}[e_{l+1},e_{l+2},\cdots]$ i.e.,
${\mathbf{M}\mathbf{U}}_{*}=A_{l}\otimes A^{l+1}$. Let $J(l)$ as above be
generated by $\mathcal{T}_{1},\cdots,\mathcal{T}_{l}$. The preimage of $J(l)$
by obvious inclusion defines the ideal of $A_{l}$ denoted by same symbol so
that

${\mathbf{M}\mathbf{U}}_{*}/J(l)=A_{l}/J(l)\otimes A^{l+1}.$

######  Proposition 4.3.

i) The ideal $J$ in (4.3) is regular;

ii) $A_{l}/J(l)$ and ${\mathbf{M}\mathbf{U}}_{*}/J(l)$ are integral domains,
or equivalently $J(l)$ is prime.

It is clear that ii) implies i): If ${\mathbf{M}\mathbf{U}}_{*}/J(l)$ is
integral domain for any $l\geq 5$, i.e., it has no zero divisors, then
multiplication by $\mathcal{T}_{l+1}$ is monorphism. Therefore the sequence
$\mathcal{T}_{5},\mathcal{T}_{6},\cdots$ of generators of $J$ is regular.

We will see that ii) follows from the proof of Proposition 6.5 in [9] and the
following

######  Lemma 4.4.

For $p^{r}\leq l<p^{r+1}$ the ring $A_{l}/J(l)\otimes\mathbb{F}_{p}$ is
additively generated by the following monomials

For $p=2$,

$\alpha_{1}^{m_{1}}\beta_{2}^{m_{2}}\alpha_{3}^{m_{3}}\beta_{2}^{k_{1}}\beta_{2^{2}}^{k_{2}}\cdots\beta_{2^{r-1}}^{k_{r-1}}\beta_{2^{r}}^{m},\,\,\,\,\,\,\,k_{1},k_{2},k_{r-1}=0,1;$

For $p=3$,

$\alpha_{1}^{m_{1}}\beta_{2}^{m_{2}}\beta_{4}^{m_{4}}\beta_{3}^{k_{1}}\beta_{3^{2}}^{k_{2}}\cdots\beta_{3^{r-1}}^{k_{r-1}}\beta_{3^{r}}^{m},\,\,\,\,\,\,\,k_{1},k_{2},k_{r-1}=0,1,2;$

For prime $p>3$,

$\alpha_{1}^{m_{1}}\beta_{2}^{m_{2}}\alpha_{3}^{m_{3}}\beta_{4}^{m_{4}}\beta_{p}^{k_{1}}\beta_{p^{2}}^{k_{2}}\cdots\beta_{p^{r-1}}^{k_{r-1}}\beta_{p^{r}}^{m},\,\,\,\,\,\,\,0\leq
k_{1},k_{2},k_{r-1}\leq p-1,$

not divisible by
$\alpha_{1}^{j_{1}}\beta_{2}^{j_{2}}\alpha_{3}^{j_{3}}\beta_{4}^{j_{4}}$,
where $(j_{1}j_{2}j_{3}j_{4})$ corresponds to leading lexicographical monomial
ordering for which $\lambda_{j_{1}j_{2}j_{3}j_{4}}\not\equiv 0\mod p$ in

$p\beta_{p}=\sum\lambda_{j_{1}j_{2}j_{3}j_{4}}\alpha_{1}^{j_{1}}\beta_{2}^{j_{2}}\alpha_{3}^{j_{3}}\beta_{4}^{j_{4}}.$

Proof. We follow the proof of Proposition 6.5 in [9]. To get the generating
monomials in Lemma 4.4 we need only to modify the generating monomials of
$\Lambda_{\mathcal{B}}\otimes\mathbb{F}_{p}$. In particular, in (6.17), (6.18)
and (6.20) there are extra factors for $A_{l}/J(l)$, $p^{r}\leq l<p^{r+1}$,
namely

$\beta_{p^{r+1}}^{k_{1}}\beta_{p^{r+2}}^{k_{2}}\cdots\beta_{p^{r+s}}^{k_{s}}\cdots,\,\,\,\,k_{1},k_{2},\cdots\leq
p-1.$

We have to replace these factors by

$\beta_{p^{r}}^{pk_{1}+p^{2}k_{2}+\cdots+p^{s}k_{s}+\cdots}$

This is because of $\beta_{p^{r+1}}\notin A_{l}$. In this way for each $m$ we
keep the total number of generating monomials of
$\Lambda^{-2m}\otimes\mathbb{F}_{p}$ since there is no relation $(6.2)$ of [9]
in our ring $A_{l}/J(l)\otimes\mathbb{F}_{p}$. ∎

Denote by $[\mathcal{T}_{i}]$ the cobordism class representing the generator
$\mathcal{T}_{i}$ of the ideal $J$. Consider the sequence

$\Sigma_{T}=([\mathcal{T}_{5}],[\mathcal{T}_{6}],\cdots).$

The Sullivan-Baas construction [1] of cobordism with singularities
$\Sigma_{T}$ gives a cohomology theory
${\mathbf{M}\mathbf{U}}^{*}_{\Sigma_{T}}(-)$ which by regularity of the ideal
$J$ has a scalar ring

${\mathbf{M}\mathbf{U}}^{*}_{\Sigma_{T}}(pt)={\mathbf{M}\mathbf{U}}_{*}/J=\Lambda_{\mathcal{B}}.$

By Mironov [16] (Theorem 4.3 and Theorem 4.5)
${\mathbf{M}\mathbf{U}}^{*}_{\Sigma_{T}}(-)$ admits an associate
multiplication and all obstructions to commutativity are in
$\Lambda_{\mathcal{B}}\otimes\mathbb{F}_{2}$. Therefore after localization
away from 2 all obstructions vanish and we get a commutative cohomology

$\emph{h}_{\mathcal{B}}^{*}(-):={\mathbf{M}\mathbf{U}}^{*}_{\Sigma_{T}}[1/2](-).$

Here we recall that $\Lambda_{B}[1/2]=\Lambda_{\mathcal{B}}[1/2]$ by
definition of $\Lambda_{\mathcal{B}}$.

It is clear that $\emph{h}_{\mathcal{B}}^{*}(-)$ is complex oriented as the
Atiyah-Hirzebruch spectral sequence
$H^{*}(-,\emph{h}_{\mathcal{B}}^{*}(pt))\Rightarrow\emph{h}_{\mathcal{B}}^{*}(-)$
collapses for $\mathop{\mathrm{BU}}(1)\times\mathop{\mathrm{BU}}(1)$.

Thus we can state

######  Theorem 4.5.

There exist a commutative complex oriented cohomology $h_{\mathcal{B}}^{*}(-)$
with scalar ring isomorphic to $\Lambda_{\mathcal{B}}[1/2]$, the ring of
coefficients of the universal Buchstaber formal group law localized away from
2.

This result without a complete proof, is announced in short communications of
MMS [5].

##  5\. The restriction of the Buchstaber genus on
${\mathbf{M}\mathbf{S}\mathbf{U}}_{*}[1/2]$

Taking into account [18] that after localized away from 2, the forgetful map
from the special unitary cobordism
${\mathbf{M}\mathbf{S}\mathbf{U}}_{*}[1/2]=\mathbb{Z}[1/2][x_{2},x_{3},\cdots]$
to complex cobordism ${\mathbf{M}\mathbf{U}}_{*}[1/2]$ is an injection, define
the following ideal extensions in ${\mathbf{M}\mathbf{U}}_{*}][1/2]$:

$J_{SU}^{e}$, generated by any polynomial generators $x_{n}$ of
${\mathbf{M}\mathbf{S}\mathbf{U}}_{*}[1/2]$, $n\geq 5$ viewed as elements in
${\mathbf{M}\mathbf{U}}_{*}[1/2]$ by forgetful injection map;

$J_{T}^{e}$, generated by $SU$-flops [21] of dimension $\geq 10$ again viewed
as elements in ${\mathbf{M}\mathbf{U}}_{*}[1/2]$;

$J_{B}^{e}$, the contraction ideal by the obvious inclusion
${\mathbf{M}\mathbf{U}}_{*}\to{\mathbf{M}\mathbf{U}}_{*}[1/2]$ of the ideal
$J_{B}$ of ${\mathbf{M}\mathbf{U}}_{*}$ generated by the elements
$\\{A_{ij},i,j\geq 3\\}$, defined in Section 3.

######  Proposition 5.1.

i) $J^{e}_{B}=J_{T}^{e}=J_{SU}^{e}$; ii) When restricted on
${\mathbf{M}\mathbf{S}\mathbf{U}}_{*}[1/2]$ the classifying map of the
Buchstaber formal group law localized away from 2, gives a genus with the
scalar ring $\mathbb{Z}[1/2][x_{2},x_{3},x_{4}]$, $|x_{i}|=2i$.

One motivation is the restricted Krichever-Hoehn complex elliptic genus below,
studied in [13] and [21]. Another construction with the scalar ring
$\mathbb{Z}_{(2)}[a,b]$, $|a|=2$, $|b|=6$ see in [6].

Proof. $J_{T}^{e}=J_{SU}^{c}\subseteq J^{e}_{B}$: The homomorphism

${\mathbf{M}\mathbf{U}}_{*}\xrightarrow{f_{B}}\Lambda_{B}\xrightarrow{\subset}\Lambda_{B}\otimes\mathbb{Q}=\mathbb{Q}[x_{1},x_{2},x_{3},x_{4}]$

is a specialization of the complex elliptic genus

${\mathbf{M}\mathbf{U}}_{*}\xrightarrow{\subset}{\mathbf{M}\mathbf{U}}_{*}\otimes\mathbb{Q}\to\mathbb{Q}[x_{1},x_{2},x_{3},x_{4}]$

by [2], therefore vanishes on the kernel of the complex elliptic genus which
is the ideal $I$ of complex flops by [21] (Theorem 4.1). On the other hand the
ring $\Lambda_{B}[1/2]$ is torsion free and injected in
$\mathbb{Q}[x_{1},x_{2},x_{3},x_{4}]$ by [9]. So the ring homomorphism
${\mathbf{M}\mathbf{U}}_{*}[1/2]\xrightarrow{f_{B}}\Lambda_{B}[1/2]$ vanishes
on $I$, therefore it vanishes on $SU$-flops. Moreover by [21] the ideal of
$SU$-flops $J_{T}$ in ${\mathbf{M}\mathbf{S}\mathbf{U}}_{*}[1/2]$ contains the
polynomial generators $x_{n},\,\,\,n\geq 5$ constructed by using Euclid’s
algorithm and $SU$-flops. Therefore $J_{T}=(x_{5},\cdots)$ and
$J_{T}^{e}=J_{SU}^{e}\subseteq J_{B}$.

To prove $J^{e}_{B}\subseteq J_{SU}^{e}$ note by (4.4) $T_{n}$ satisfies the
criteria for the membership of the set of polynomial generators in

${\mathbf{M}\mathbf{S}\mathbf{U}}^{*}[1/2]=\mathbb{Z}[1/2][x_{2},x_{3},\cdots],\,\,|x_{i}|=2i,$

described by Novikov in [18]. In particular, an $SU$-manifold $M$ of real
dimension $2n$, $n\geq 2$ is a polynomial generator if and only if $s_{n}(M)$,
the main Chern characteristic number is as follows

(5.1) $\displaystyle s_{n}(M)=\begin{cases}\pm 2^{k}p&\mbox{if
}n=p^{l},\,\,\,p\,\,\,\mbox{is odd prime, }\\\ \pm 2^{k}p&\mbox{if
}n+1=p^{l},\,\,\,p\,\,\,\mbox{is odd prime, }\\\ \pm
2^{k}&\mbox{otherwise}.\end{cases}$

It follows that the generators $A_{ij}$ of $J_{B}$, with $i+j=n+2$ and the
generators $x_{n}$ of $J_{SU}^{e}\subset J^{e}_{B}$ are related as follows
$A_{ij}\equiv\frac{s_{n}(A_{ij})}{D(n)}T_{n}$, $T_{n}\equiv\pm 2^{k(n)}x_{n}$,
$\mod$ decomposables. Therefore we can proceed as in the proof of Proposition
4.2. ∎

Recall also that $\Lambda_{B}[1/2]=\Lambda_{\mathcal{B}}[1/2]$ is an integral
domain. Note that Proposition 4.3 implies

######  Proposition 5.2.

The sequence $\\{x_{n}\\},n\geq 5$ of any polynomial generators in
${\mathbf{M}\mathbf{S}\mathbf{U}}_{*}[1/2]$ viewed as elements in
${\mathbf{M}\mathbf{U}}_{*}[1/2]$ by forgetful map is regular.

######  Corollary 5.3.

After restriction on ${\mathbf{M}\mathbf{S}\mathbf{U}}_{*}[1/2]$ the
Buchsteber genus $f_{\mathcal{B}}$ gives a cohomology theory
${\mathbf{M}\mathbf{S}\mathbf{U}}^{*}_{\Sigma}[1/2]$ with singularities
$\Sigma=(x_{5},\cdots)$, with the scalar ring

$\mathbb{Z}[1/2][x_{2},x_{3},x_{4}],\,\,\,|x_{i}|=2i.$

##  6\. ${\mathbf{M}\mathbf{S}\mathbf{U}}_{*}[1/2]$

Let $F_{U}(x,y)=\sum\alpha_{ij}u^{i}v^{j}$ be the universal formal group law.
Recall the idempotent in [7], [12]

$\pi_{0}:{\mathbf{M}\mathbf{U}}_{*}\to{\mathbf{M}\mathbf{U}}_{*}:\,\,\,\pi_{0}=1+\sum_{k\geqslant
2}\alpha_{1k}\partial_{k}$

and the projection $\pi_{0}:{\mathbf{M}\mathbf{U}}_{*}\to W_{*}=Im\pi_{0}$.

Then $W_{*}$ is a ring with multiplication $*$ and $\pi_{0}(a\cdot b)=a*b$. By
[12] Proposition 2.15 the multiplication $*$ is given by

$a*b=ab+2[V]\partial a\partial b,$

where $[V]=\alpha_{12}\in{\mathbf{M}\mathbf{U}}_{4}$ is the cobordism class
${\mathbf{C}\mathbf{P}}_{1}^{2}-{\mathbf{C}\mathbf{P}}_{2}$.

$W_{*}$ is complex oriented and by [12] Proposition 3.12 the ring $W_{*}[1/2]$
is generated by the coefficients of the formal group law

$F_{W}=F_{W}(x,y)=\sum w_{ij}x^{i}y^{j}.$

Following [7], [12] one can calculate $w_{ij}$ in terms of $\alpha_{kl}$ as
follows. Consider the multiplicative cohomology theory $\Gamma$ with

$\pi_{*}(\Gamma)={\mathbf{M}\mathbf{U}}_{*}[t]/(t^{2}-\alpha_{11}t-2\alpha_{21}),$

the free ${\mathbf{M}\mathbf{U}}_{*}$ module generated by $1$ and $t$.

There is a natural multiplicative transformation $\phi:W\to\Gamma$ given by

$\phi_{*}(x)=x+t\partial x$

for $x\in W_{*}$. The restriction of $\phi_{*}[1/2]$ on
${\mathbf{M}\mathbf{S}\mathbf{U}}_{*}[1/2]$, the subring in $W_{*}[1/2]$ of
cycles of $\partial$, is the natural inclusion in
${\mathbf{M}\mathbf{U}}_{*}[1/2]$.

Then

$\phi_{*}F_{W}=u+v+\sum_{i,j\geq 1}(w_{ij}+t\partial w_{ij})u^{i}v^{j}$

is strongly isomorphic to $F_{U}$ (considered as a formal group low over
$\pi_{*}(\Gamma)$ via the natural inclusion) by a series $\gamma^{-1}$, i.e.,

(6.1) $u+v+\sum_{i,j\geq 1}(w_{ij}+t\partial w_{ij})u^{i}v^{j}=\gamma
F_{U}(\gamma^{-1}(u),\gamma^{-1}(v)).$

Finally, we need to apply for $\gamma$ in (6.1) Lemma 3 in [7], which says
that any orientation $w\in W_{2}$ gives the following identity in
$\pi_{*}(\Gamma)$

(6.2) $\gamma(u)=\phi_{*}(w)=\pi_{0}(u)+t\partial u=u+tu\bar{u}+\sum_{i\geq
2}\alpha_{i1}u\bar{u}^{i}.$

By [12] one can specify an orientation of $W$ such that

$gcd(w_{ij},\,i+j-1=k)=d(k)d(k-1)$

modulo a power of 2. This allows to construct the generators of $W_{*}[1/2]$.

In particular, one can calculate main Chern numbers $s_{k}(w_{ij})$ in terms
of main Chern numbers of $\alpha_{ij}$ as follows. By [12] Lemma 3.5 any
orientation $w\in W_{2}$ gives $w_{i}\in W_{2i}$ and
$\lambda\in{\mathbf{M}\mathbf{U}}_{2}=W_{2}$ such that one has in
$\pi_{*}(\Gamma)$

$\gamma(u)=u-(\lambda+(2l+1)t)u^{2}+\sum_{i\geq
2}\gamma_{i+1}u^{i+1}modJ^{2}+tJ,$

where $2l=\partial\lambda$, $l\in\mathbb{Z}$,
$\gamma_{i+1}=(-1)^{i}\alpha_{1i}+w_{i}$, $J$ is the ideal in
${\mathbf{M}\mathbf{U}}_{*}$ of elements of positive degree.

######  Lemma 6.1.

Let $k=i+j-1\geq 3$ is not of the form $k=2^{l}=p^{s}-1$ for some odd prime
$p$. There is a choice of complex orientation $w$ for the theory $W$ such that

$\displaystyle s_{k}(w_{ij})=\begin{cases}ps_{k}(\alpha_{ij})&\mbox{if
}k=p^{s},\,\,\,p\,\,\,\mbox{is any prime, $s>0$},\\\
s_{k}(\alpha_{ij})&\mbox{otherwise}.\\\ \end{cases}$

Proof. By using (6.2) it is proved in [12] (Lemma 3.9) that for $k\geq 3$ one
has modulo decomposable elements

(6.3)
$\displaystyle\phi_{*}(w_{1k})=\alpha_{1k}+(k+1)((-1)^{k}\alpha_{1k}+w_{k});$
(6.4)
$\displaystyle\phi_{*}(w_{ij})=\alpha_{ij}+(-1)^{k}\binom{k+1}{i}\alpha_{1k}+\binom{k+1}{i}w_{k};$

Choose a complex orientation $w$ for the theory $W$ such that elements $w_{k}$
satisfy the following conditions

(6.5) $1+(-1)^{k}(k+1)-s_{k}(w_{k})=d(k-1)$

The it is easily checked that $w_{k}$ indeed belongs to $W_{*}$, that is
$s_{k}(w_{k})$ is divisible by $d(k-1)d(k)$.

Then by (6.3) and (6.5) we have for $k=p^{s}$

$\displaystyle w_{1k}=$
$\displaystyle-k\alpha_{1k}+(k+1)w_{k}=-k\alpha_{1k}+\alpha_{1k}(p^{s}+p)=p\alpha_{1k};$
$\displaystyle w_{ij}=$
$\displaystyle\alpha_{ij}-\binom{k+1}{i}\alpha_{1k}+\binom{k+1}{i}w_{k}$
$\displaystyle=$
$\displaystyle\alpha_{ij}-\alpha_{ij}(k+1)+\alpha_{ij}(k+p)=p\alpha_{ij}.$

For $k=2^{l}$ one has

$\displaystyle w_{1k}=$
$\displaystyle(2+k)\alpha_{1k}+(k+1)w_{k}=(2+k)\alpha_{1k}-\alpha_{1k}k=2\alpha_{1k};$
$\displaystyle w_{ij}=$
$\displaystyle\alpha_{ij}+\binom{k+1}{i}\alpha_{1k}-\binom{k+1}{i}w_{k}=\alpha_{ij}+\alpha_{ij}(k+1)-k\alpha_{ij}=2\alpha_{ij}.$

Similarly for other cases.

∎

If $k=2^{l}=p^{s}-1\geq 3$ for some odd prime $p$, then by [12] Lemma 3.15 one
has $k=8$ or $k=2^{2^{n}}$. We have to replace $d(k-1)=2$ in (6.5) by $4$ , if
$k=8$ to get $\phi_{*}(w_{i\,9-i})=4\alpha_{i\,9-i}$ and by $-2^{2^{n}}$, if
$k=2^{2^{n}}$ to get
$\phi_{*}(w_{i\,2^{2^{n}}+1-i})=-2^{2^{n}}\alpha_{i\,2^{2^{n}}+1-i}$.

Together with Lemma 6.1, this implies

######  Corollary 6.2.

Let $k\geq 3$. By (3.1) and (3.2) let $\lambda_{i}$ be such that the linear
combination ${\bf a}_{k}=\sum_{i=1}^{k}\lambda_{i}\alpha_{ik+1-i}$ is a
polynomial generator in ${\mathbf{M}\mathbf{U}}_{2k}$. Then

${\bf b}_{k}=\sum_{i=1}^{k}\lambda_{i}w_{ik+1-i}.$

is a polynomial generator in $W_{2k}[1/2]$.

Then ${\mathbf{M}\mathbf{S}\mathbf{U}}_{*}[1/2]$ is a subring of cycles of the
boundary operation $\partial$ in $W_{*}[1/2]$ with multiplication $*$

$\partial(a*b)=a*\partial b+\partial a*b-{\mathbf{C}\mathbf{P}}_{1}\partial
a*\partial b.$

One has $\partial{\mathbf{C}\mathbf{P}}_{1}=2$ and $a*b=a\cdot b$ whenever
$a\in Im\partial$ or $b\in Im\partial$. Therefore

$\partial({\mathbf{C}\mathbf{P}}_{1}*\alpha)=2\alpha-{\mathbf{C}\mathbf{P}}_{1}\cdot\partial\alpha,\forall\alpha\in
W.$

As mentioned in [17] this implies that

(6.6)
$\alpha=1/2\partial({\mathbf{C}\mathbf{P}}_{1}*\alpha)+1/2{\mathbf{C}\mathbf{P}}_{1}\cdot\partial\alpha,$

and therefore $W[1/2]$ is generated by $1$ and ${\mathbf{C}\mathbf{P}}_{1}$ as
a ${\mathbf{M}\mathbf{S}\mathbf{U}}_{*}[1/2]$ module . It is easily checked
that this module is free.

######  Proposition 6.3.

Let ${\bf b}_{k}$ be as in Corollary 6.2. Then

${\mathbf{M}\mathbf{S}\mathbf{U}}_{*}[1/2]=\mathbb{Z}[1/2][x_{2},x_{k}:k\geq
3],$

where $x_{2}={\mathbf{C}\mathbf{P}}_{2}-9/8{\mathbf{C}\mathbf{P}}_{1}^{2}$ and
$x_{k}=\partial({\mathbf{C}\mathbf{P}}_{1}*{\bf b}_{k})$.

Proof. One has for the values of the Chern numbers

$\displaystyle c_{1}c_{1}[{\mathbf{C}\mathbf{P}}_{1}^{2}]=8,$ $\displaystyle
c_{1}c_{1}[{\mathbf{C}\mathbf{P}}_{2}]=9,$ $\displaystyle
c_{2}[{\mathbf{C}\mathbf{P}}_{1}^{2}]=4,$ $\displaystyle
c_{2}[{\mathbf{C}\mathbf{P}}_{2}]=3.$

This imply that that $c_{1}c_{1}[{\bf b}_{2}]=0$. There are no more Chern
numbers having $c_{1}$ as a factor and $s_{2}[{\bf
b}_{2}]=s_{2}[{\mathbf{C}\mathbf{P}}^{2}]=3$. Therefore ${\bf b}_{2}$ forms a
generator of ${\mathbf{M}\mathbf{S}\mathbf{U}}_{4}[1/2]$.

Apply (6.6). The main Chern number vanishes on the second (decomposable)
component of

${\bf b}_{k}=1/2\partial({\mathbf{C}\mathbf{P}}_{1}*{\bf
b}_{k})+1/2{\mathbf{C}\mathbf{P}}_{1}\cdot\partial{\bf b}_{k},$

i.e., the first component $x_{k}$ has the main Chern number $2s_{k}({\bf
b}_{k})$. ∎

##  7\. The restriction of the classifying map of $F_{Ab}$ on
${\mathbf{M}\mathbf{S}\mathbf{U}}^{*}[1/2]$.

As above let $F_{U}=\sum\alpha_{ij}x^{i}y^{j}$ be the universal formal group
law. By definition the coefficient ring of the universal abelian formal group
law $F_{Ab}$ is the quotient ring

(7.1) $\Lambda_{Ab}={\mathbf{M}\mathbf{U}}_{*}/I_{Ab},\text{ where
}I_{Ab}=(\alpha_{ij},i,j>1).$

Let us apply Euclid’s algorithm for the Chern numbers
$s_{m-1}(\alpha_{i,m-i})$ in (3.6)

Let

$z_{k}=\sum_{i=2}^{k-1}\lambda_{i}\alpha_{i\,k+1-i},\,\,\,k\geq 3.$

By [10], [11] one has $I_{Ab}=I_{AB}=(z_{k},\,\,k\geq 3)$.

Consider the composition

(7.2)
$r_{Ab}:{\mathbf{M}\mathbf{S}\mathbf{U}}_{*}[1/2]\xrightarrow{\subset}{\mathbf{M}\mathbf{U}}_{*}[1/2]\xrightarrow{f_{Ab}}\Lambda_{Ab}[1/2],$

where $\subset$ is forgetful map.

######  Proposition 7.1.

One has the following polynomial generators in
${\mathbf{M}\mathbf{S}\mathbf{U}}_{*}[1/2]$ viewed as the elements in
${\mathbf{M}\mathbf{U}}_{*}[1/2]$

$\displaystyle
x_{2}={\mathbf{C}\mathbf{P}}_{2}-\frac{9}{8}{\mathbf{C}\mathbf{P}}_{1}^{2},$
$\displaystyle x_{3}=-\alpha_{22},$ $\displaystyle
x_{4}=-\alpha_{23}-\frac{3}{2}x_{3}{\mathbf{C}\mathbf{P}}_{1}.$

To prove this we have to check that all Chern numbers of $x_{i}$ having factor
$c_{1}$ are zero. Then we have to check the main Chern number $s_{i}(x_{i})$
for Novikov’s criteria.

We already did this for $x_{2}$ in the proof of Proposition 6.3. Then by
definition $x_{3}$ is the coefficient $-\alpha_{22}\in I_{AB}$ of the
universal formal group law. In ${\mathbf{M}\mathbf{U}}_{*}$ one has

$\displaystyle
2\alpha_{22}=-3{\mathbf{C}\mathbf{P}}_{3}+8{\mathbf{C}\mathbf{P}}_{1}{\mathbf{C}\mathbf{P}}_{2}-5{\mathbf{C}\mathbf{P}}_{1}^{3},$
$\displaystyle\alpha_{23}=2{\mathbf{C}\mathbf{P}}_{1}^{4}-7{\mathbf{C}\mathbf{P}}_{1}^{2}*{\mathbf{C}\mathbf{P}}_{2}+3{\mathbf{C}\mathbf{P}}_{2}^{2}+4{\mathbf{C}\mathbf{P}}_{1}{\mathbf{C}\mathbf{P}}_{3}-2{\mathbf{C}\mathbf{P}}_{4}.$

Let us compute the Chern numbers of $x_{3}=-\alpha_{22}$. One has

$\displaystyle X$ $\displaystyle c_{3}(X)$ $\displaystyle c_{1}c_{2}(X)$
$\displaystyle c_{1}c_{1}c_{1}(X)$ $\displaystyle{\mathbf{C}\mathbf{P}}_{3}$
$\displaystyle 4$ $\displaystyle 24$ $\displaystyle 64$
$\displaystyle{\mathbf{C}\mathbf{P}}_{1}{\mathbf{C}\mathbf{P}}_{2}$
$\displaystyle 6$ $\displaystyle 24$ $\displaystyle 54$
$\displaystyle{\mathbf{C}\mathbf{P}}_{1}^{3}$ $\displaystyle 8$ $\displaystyle
24$ $\displaystyle 48.$

It follows all Chern numbers of $\alpha_{22}$ having factor $c_{1}$ are zero.
Then $\alpha_{22}$ forms a generator in
${\mathbf{M}\mathbf{S}\mathbf{U}}_{6}[1/2]$ as $s_{3}[-2\alpha_{22}]=4\cdot
3$.

Similarly for $x_{4}$: the main Chern number $s_{4}(x_{4})=2\cdot 5$ fits for
Novikov’s criteria and one has

$\displaystyle X$ $\displaystyle c_{1}c_{1}c_{1}c_{1}(X)$ $\displaystyle
c_{1}c_{1}c_{2}(X)$ $\displaystyle c_{1}c_{3}(X)$
$\displaystyle{\mathbf{C}\mathbf{P}}_{1}^{4}$ $\displaystyle 384$
$\displaystyle 192$ $\displaystyle 64$
$\displaystyle{\mathbf{C}\mathbf{P}}_{1}^{2}{\mathbf{C}\mathbf{P}}_{2}$
$\displaystyle 432$ $\displaystyle 204$ $\displaystyle 60$
$\displaystyle{\mathbf{C}\mathbf{P}}_{2}^{2}$ $\displaystyle 486$
$\displaystyle 216$ $\displaystyle 54$
$\displaystyle{\mathbf{C}\mathbf{P}}_{1}{\mathbf{C}\mathbf{P}}_{3}$
$\displaystyle 512$ $\displaystyle 224$ $\displaystyle 56$
$\displaystyle{\mathbf{C}\mathbf{P}}_{4}$ $\displaystyle 625$ $\displaystyle
250$ $\displaystyle 50.$

Note $F_{Ab}$ is a specialization of the Buchstaber formal group law $F_{B}$.
In particular one can put $A(x)=B(x)^{2}$ in Proposition (2.1) to specify
$F_{B}$ to $F_{Ab}$ over torsion free ring $\Lambda_{Ab}[1/2]$. Then
Proposition 5.2 implies

######  Proposition 7.2.

After restriction on ${\mathbf{M}\mathbf{S}\mathbf{U}}_{*}[1/2]$ the
classifying map of the universal abelian formal group law becomes the one-
parameter genus

$r_{Ab}:{\mathbf{M}\mathbf{S}\mathbf{U}}_{*}[1/2]\to\mathbb{Z}[1/2][x_{2}].$

## References

  * [1] N. Baas, _On bordism theory of manifolds with singularities,_ Math. Scand., 33(1973), 279-302.
  * [2] M. Bakuradze, _On the Buchstaber formal group law and some related genera_ , Proc. Steklov Math. Inst., 286(2014), 7-21.
  * [3] M. Bakuradze, _Formal group laws by Buchstaber, Krichever and Nadiradze coincide_ , Russian Math. Surveys, 68:3 (2013), 571–573.
  * [4] M. Bakuradze, _Computing the Krichever genus_ , J. Homotopy Relat. Struct, 9, 1(2014), 85-93.
  * [5] M. Bakuradze, _Cohomological realization of the Buchstaber formal group law_ , Uspekhi Mat. Nauk, 77:5(467) (2022), 189-190.
  * [6] V. M. Buchstaber, E. Yu. Netay, _$CP(2)$ -multiplicative Hirzebruch genera and elliptic cohomology_, Russian Math. Surveys, 69:4 (2014), 757-759.
  * [7] V. M. Buchstaber, _Projectors in unitary cobordisms that are related to SU-theory_ , Uspekhi Mat. Nauk, 27:6(168)(1972), 231-232.
  * [8] V. Buchstaber, T. Panov, N. Ray, _Toric genera_ , Intern. Math. Res. Notices, 16(2010), 3207–3262.
  * [9] V. Buchstaber, K. Ustinov, _Coefficient rings of Buchstaber formal group laws_ , Math. Notices, 206:11(2015), 19-60.
  * [10] Bukhshtaber V.M., Kholodov A.N., _Formal groups, functional equations, and generalized cohomology theories, (English. Russian original)_ , Math. USSR Sbornik 69:1 (1991), 77–97.
  * [11] Ph. Busato, _Realization of Abel’s universal formal group law_ , Math. Z., 239(2002), 527–561.
  * [12] G. Chernykh, T. Panov, _$SU$ -linear operations in complex cobordism and the $c_{1}$-spherical bordism theory_, Izvestiya Ross. Akad. Nauk Ser. Mat. 87 (2023), no. 4.
  * [13] G. Höhn, _Komplexe elliptische Geschlechter und $S^{1}$-äquivariante Kobordismustheorie_, 2004, https://arxiv.org/abs/math/0405232
  * [14] I. Krichever, _Generalized elliptic genera and Baker-Akhiezer functions_ , Math. Notes, 47(1990), 132–142.
  * [15] E. E. Kummer, _Uber die Erganzungssatze zu den allgemeinen Reciprocitatsgesetzen_ , J. Reine Angew. Math., 44 (1852), 93–146.
  * [16] O. K. Mironov, _Multiplications in cobordism theories with singularities, and Steenrod – tom Dieck operations_ , Math. USSR Izvestija 13(1979), 89–106;
  * [17] I. Yu. Limonchenko, T. E. Panov, and G. S. Chernykh _SU-bordism: structure results and geometric representatives_ , Russian Math. Surv., 74:3(2019), 461–524.
  * [18] S. P. Novikov, _Homotopy properties of Thom complexes (Russian)_ , Mat. Sb. 57(1962), 407–442.
  * [19] S. Schreieder, _Dualization invariance and a new complex elliptic genus_ , 2012, arXiv:1109.5394 [math.AT]
  * [20] R. E. Stong, _Notes On Cobordism Theorey_ , Princeton University Press and University of Tokyo Press, (1968)
  * [21] B. Totaro, _Chern numbers for singular varieties and elliptic homology_ , Annals of Mathematics, 151(2000), 757–792.
  * [22] Zhi Lu, T. Panov, _On toric generators in the unitary and special unitary bordism rings_ , Algebraic and Geometric Topology 16 no. 5(2016), 2865-2893; arXiv:1412.5084