joined along _seams_ where facets of thickness $a$ and $b$ merge into a facet
of thickness $a+b$, subject to compatibility of orientations. A foam may
contain _vertices_ , which are singular points that connect pairs of seams
between two different ways of merging three facets of thicknesses $a,b,c$ into
a facet of thickness $a+b+c$. A facet of thickness $k$ of a foam may contain
dots labelled by symmetric functions in $k$ variables. A foam $F$ evaluates to
$\langle F\rangle$ which is a symmetric polynomial in $N$ variables. We refer
to [RW2, KK] and references in [KK] for details.

It’s useful to label the set of variables by $X$ with $|X|=N$ and view
$\langle F\rangle$ as a symmetric function in these variables, denoting the
corresponding ring of symmetric functions by $\mathrm{Sym}(X)$.

When $F$ is a connected surface (a single facet) of maximal thickness $N$,
with a dot on it labelled by $f(X)\in\mathrm{Sym}(X)$, the evaluation $\langle
F\rangle=f(X)$ does not depend on the genus of the surface, see Figure 6.7. Of
course, for most other foams, including surfaces of thickness less that
$N=|X|$, the evaluation will strongly depend on the genii of components of the
foam.

$f$$=$$f$$=$$f(X)$$X$$X$$|X|$$|X|$ Figure 6.7. When $F$ is a single facet
foam, of maximal thickness $N=|X|$, it evaluates to the product of symmetric
functions over all the dots on $F$. In particular, the evaluation does not
depend on the genus of the surface $F$. The figure shows the case of a single
dot and surface $F$ having genus $1$ or $0$. Dashed circle on the sphere is
there to depict the sphere schematically (it’s not a seam circle on a sphere
separating its into two facets of complementary thickness). The latter seam
lines appear in the next few figures.

In [Kh4, Section 3] an extension of this evaluation is proposed when several
foams for disjoint sets of variables overlap in $\mathbb{R}^{3}$. Foam
evaluation $\langle F\rangle$, which is a sum of evaluations $\langle
F,c\rangle$ over all colorings $c$, is modified by scaling $\langle
F,c\rangle$ by $(x_{i}-y_{j})^{m(i,j,c)}$, where $m(i,j,c)$ is the number of
circles in the intersection of the union $F_{i}(c)$ of facets colored $i$,
$x_{i}\in X$ and the union $F_{j}(c)$ of facets colored $j$, $y_{j}\in Y$. The
product of these terms is taken over all pairs $X,Y$ and $x_{i}\in X,y_{j}\in
Y$. Ordering of each pair of sets $(X,Y)$ of foam labelings is fixed to have a
well-defined term $x_{i}-y_{j}$ versus $y_{j}-x_{i}$.

For a closely related notion of an $\mathsf{SL}(N)$ foam and its evaluation,
other seam lines are allowed as well, where oriented facets of thickness
$a,b,c$ with $a+b+c=N$ or $a+b+c=2N$ meet along seams. Case $N=3$ and foams
with $(a,b,c)=(1,1,1)$ seam lines have been treated in details in the
literature, but for $N>3$ foam evaluation is mostly considered for
$\mathsf{GL}(N)$ foams. See Section 2.3.1 in [RW1] for a brief discussion on
modifying evaluation from $\mathsf{GL}(N)$ to $\mathsf{SL}(N)$ foams, with the
caveat that what call $\mathsf{GL}(N)$ foams is referred to as
$\mathfrak{sl}_{N}$ foams in [RW1], and our $\mathsf{SL}(N)$ foams are called
_generalized foams_ in [RW1].

Given finite sets of variables $Y$ and $Z$, define

(69) ${\mathcal{R}}(Y,Z)=\prod_{y\in Y,z\in
Z}(y-z),\qquad{\mathcal{R}}(Y,Z)=1\quad\mathrm{if}\ Y=\emptyset\ \mathrm{or}\
Z=\emptyset.$

Note that ${\mathcal{R}}(Y,Z)$ is a polynomial that’s symmetric in variables
in $Y$ and symmetric in ‘variables in $Z$, thus

${\mathcal{R}}(Y,Z)\ \in\ \mathrm{Sym}(Y)\otimes\mathrm{Sym}(Z),$

where $\mathrm{Sym}(Y)$ stands for the ring of symmetric polynomials in $Y$
with coefficients in $\mathbb{Z}$ or in a field $\mathbf{k}$, likewise for
$\mathrm{Sym}(Z)$. Polynomial ${\mathcal{R}}(Y,Z)$ equals the evaluation as in
[Kh4, Section 3] of overlapping connected surfaces (foams with one facet)
labelled by $Y$ and $Z$ and of maximal thickness $|Y|$ and $|Z|$,
respectively, see Figure 6.8.

$Y$$Z$$|Y|$$|Z|$ Figure 6.8. Overlapped connected surfaces labelled $Y$ and
$Z$ of maximal thickness $|Y|$ and $|Z|$, respectively, evaluate to
${\mathcal{R}}(Y,Z)$. In the picture, the surfaces are a torus and a sphere.

Given sets of variables $A=\\{\alpha_{1},\dots,\alpha_{m}\\}$ and
$B=\\{\beta_{1},\dots,\beta_{n}\\}$, the Sylvester double sum [Sy, DHKS, KSV],
for $0\leq p\leq m$ and $0\leq q\leq n$, is given as follows:

(70) $\mathsf{Syl}_{p,q}(A,B)(x)\ :=\
\sum_{\begin{subarray}{c}A^{\prime}\subset A,\,B^{\prime}\subset B,\\\
|A^{\prime}|=p,\,|B^{\prime}|=q\end{subarray}}{\mathcal{R}}(A^{\prime},B^{\prime}){\mathcal{R}}(A\setminus
A^{\prime},B\setminus
B^{\prime})\frac{{\mathcal{R}}(x,A^{\prime}){\mathcal{R}}(x,B^{\prime})}{{\mathcal{R}}(A^{\prime},A\setminus
A^{\prime}){\mathcal{R}}(B^{\prime},B\setminus B^{\prime})}.$

The sum is over all subsets of $A$ and $B$ of cardinality $p$ and $q$. We
refer the reader to [LPr] and the above papers for applications of Sylvester
double sums and their relation to subresultants.

Sylvester double sum is a polynomial in $x$ of degree at most $d:=p+q$. When
$p=0$ or $q=0$, the expression is called a _single sum_. Function
$\mathsf{Syl}_{p,q}(A,B)(x)$ is a polynomial in $x$ with coefficients in the
ring $\mathrm{Sym}_{m,n}\cong\mathrm{Sym}(A)\otimes\mathrm{Sym}(B)$ which is
the tensor product of rings of symmetric functions in the $m$ variables in $A$
and $n$ variables in $B$, respectively.

To interpret this sum via foam evaluation, we observe that denominator terms
may come from seamed 2-spheres, since in foam evaluation their positive Euler
characteristics make the corresponding products go into the denominators.
These 2-spheres should have seam circles splitting the 2-spheres into pairs of
discs of complementary thickness $p,m-p$ for the $A$-variables sphere and
$q,n-q$ for the $B$-sphere. This would produce denominator terms
${\mathcal{R}}(A^{\prime},A\setminus A^{\prime})$ and
${\mathcal{R}}(B^{\prime},B\setminus B^{\prime})$ in the sum.

Furthermore, the 2-spheres should intersect, to account for the two other
terms in the product that do not contain $x$. Finally, to incorporate $x$, we
introduce a third group of variables $\\{x\\}$, in addition to $A$ and $B$,
and a connected surface of thickness one for $\\{x\\}$ that intersects the
2-spheres labelled $A$ and $B$ in one circle each, to account for the terms in
the product that contain $x$. These three components of the foam are shown in
Figure 6.9.

$x$$1$$(a)$

$p$$m-p$$A^{\prime}$$A\setminus A^{\prime}$$(b)$

$B^{\prime}$$B\setminus B^{\prime}$$q$$n-q$$(c)$

Figure 6.9. Three components of the double sum foam in Figure 6.10, from left
to right: (a) connected surface (genus is unimportant, chosen to be one) of
maximal thickness $1$ carrying variable set $X=\\{x\\}$, (b) seamed 2-sphere
glued from disks of thickness $p$ and $m-p$, respectively, with the variable
set $A$, (c) seamed 2-sphere glued from disks of thickness $q$ and $n-q$,
respectively, with the variable set $B$. Colorings of (b) are in bijections
with $A^{\prime}\subset A$, $|A^{\prime}|=p$, colorings of (c) are in
bijections with $B^{\prime}\subset B,$ $|B^{\prime}|=q$.

Figure 6.10 shows how these three foams can overlap in $\mathbb{R}^{3}$, with
the resulting evaluation equal to $\mathsf{Syl}_{p,q}(A,B)(x)$.

$x$$1$$p$$m-p$$q$$n-q$$A^{\prime}$$A\setminus A^{\prime}$$B\setminus
B^{\prime}$$B^{\prime}$ Figure 6.10. Foam evaluating to
$\mathsf{Syl}_{p,q}(A,B)(x)$. Four intersection circles of three components
are shown schematically, as pairs of points of four different colors (blue,
red, brown, cyan). The two seamed 2-spheres intersect along two circles
(indicated as pairs of blue and brown points), and the third surface (shown as
a torus, but its genus is irrelevant for the evaluation) intersects each
seamed 2-sphere along a circle (indicated as red and cyan pairs of points). A
coloring of this foam consist of assigning a subset $A^{\prime}\subset A$ of
cardinality $p$ to the left disk of the $A$ sphere and a subset
$B^{\prime}\subset B$ of cardinality $q$ to the left disk of the $B$ sphere.
$x$$1$$p$$m$$m-p$$q$$n$$n-q$$A^{\prime}$$A\setminus A^{\prime}$$B\setminus
B^{\prime}$$B^{\prime}$ Figure 6.11. $\mathsf{GL}$ version of the foam in
Figure 6.10. $A$-foam (shown in grey) and $B$-foam (shown in red) are theta-
foams, with one disk of maximal thickness in each (shaded disks labelled $m$
and $n$). Intersection circles are schematically depicted by pairs identically
colored points.

######  Remark 6.4.

In our evaluation of 2-spheres with a seam line separating disks with
complementary thickness we are tacitly considering $\mathsf{SL}$ evaluation.
To convert to $\mathsf{GL}$ evaluation, 2-spheres should be changed into
theta-foams with one disk facet of maximal thickness. The relation is shown in
Figure 6.12.

For each of the foam configurations in this section, it’s easy to find an
embedding into $\mathbb{R}^{3}$ that extends to an embedding of the
corresponding $\mathsf{GL}$ foam, with seamed 2-spheres becoming theta-foams
with the new disk facet of maximal thickness, while preserving the evaluation.

$A$$p$$m-p$

$A$$p$$m-p$$m$

Figure 6.12. $\mathsf{SL}(m)$ vs $\mathsf{GL}(m)$ foams. Left: An
$\mathsf{SL}(m)$ foam 2-sphere made of two disks with complementary
thicknesses $p$ and $m-p$. Right: A $\mathsf{GL}(m)$ theta-foam obtained by
adding a disk of maximal thickness $m$ to the 2-sphere. There are two ways to
orient the seam edge in the foams and the two evaluations differ by
$(-1)^{p(m-p)}$, see [RW1, KK].

Figure 6.11 shows the $\mathsf{GL}$ version of the foam that evaluates to the
Sylvester double sum.

$X$$m-d$$e_{m-d}$$=$$X$$A$$m-d$$d$$m-d$variables$X:\\{x_{1},\ldots,x_{m-d}\\}$$A:\\{\alpha_{1},\ldots,\alpha_{m}\\}$$e_{m-d}$
Figure 6.13. Foams for the relation (71). The thickness of $X$ is $m-d$. The
thickness of the left portion of $A$ is $d$, the thickness of its right
portion is $m-d$. The two components on the right hand side overlap along a
circle.
$X$$m-d$$f$$=$$X$$A$$m-d$$d$$m-d$variables$X:\\{x_{1},\ldots,x_{m-d}\\}$$A:\\{\alpha_{1},\ldots,\alpha_{m}\\}$$f$
Figure 6.14. Foams for the relation (72), a generalization of Figure 6.13
foam. Genus of the $X$ component is unimportant.
$x_{k}$$\alpha_{i}$$\alpha_{j}$$X$$A^{\prime}$$A\setminus
A^{\prime}$contributes $(x_{k}-\alpha_{i})$dot $e_{m-d}$ contributes
$\displaystyle{\prod_{r\in A\setminus A^{\prime}}}\alpha_{r}$$x_{k}\in
X,\quad\alpha_{i}\in A^{\prime},\quad\alpha_{j}\in A\setminus
A^{\prime}$$2$-sphere colored $(i,j)$ contributes
$(\alpha_{j}-\alpha_{i})^{-1}$ Figure 6.15. Foam for the RHS of the identity
(71). The 2-torus $X$ has maximal thickness $m-d=|X|$ and a unique coloring,
by $X$. It contributes $1$ to the product. Left disk of 2-sphere is colored by
$A^{\prime}\subset A$, right disk by its complement $A^{\prime}\setminus A$.
The denominator term on the RHS is the product of $\alpha_{j}-\alpha_{i}$,
over $\alpha_{i}\in A^{\prime}$ and $\alpha_{j}\in A\setminus A^{\prime}$. The
intersection circle contributes the product of $x_{k}-\alpha_{i}$, over all
$k=1,\dots,m-d$ and $\alpha_{i}\in A^{\prime}$. Dots on the LHS and RHS are
labelled by the elementary symmetric function of the degree equal to the
thickness $m-d$ of the facets and contribute $x_{1}\cdots x_{m-d}$,
respectively product of $\alpha_{j}\in A\setminus A^{\prime}$, to the terms.

Chen and Louck in [CL, Theorem 2.1] give a certain polynomial identity for a
finite set of variables $A=\\{\alpha_{1},\dots,\alpha_{m}\\}$ and set of
variables $X=\\{x_{1},\dots,x_{m-d}\\}$. This is an identity in the ring of
rational functions
$\mathbb{Q}(\alpha_{1},\dots,\alpha_{m},x_{1},\dots,x_{m-d})$:

(71) $x_{1}\cdots x_{m-d}=\sum_{A^{\prime}\subset
A,\>|A^{\prime}|=d}\left(\prod_{\alpha_{j}\notin
A^{\prime}}\alpha_{j}\right)\frac{\displaystyle{\prod_{x_{j}\in
X,\>\alpha_{i}\in
A^{\prime}}}(x_{j}-\alpha_{i})}{\displaystyle{\prod_{\alpha_{j}\notin
A^{\prime},\>\alpha_{i}\in A^{\prime}}}(\alpha_{j}-\alpha_{i})}.$

More generally, they have the formula

(72) $f(X)=\sum_{A^{\prime}\subset A,\>|A^{\prime}|=d}f(A\setminus
A^{\prime})\frac{\displaystyle{\prod_{x_{j}\in X,\>\alpha_{i}\in
A^{\prime}}}(x_{j}-\alpha_{i})}{\displaystyle{\prod_{\alpha_{j}\notin
A^{\prime},\>\alpha_{i}\in A^{\prime}}}(\alpha_{j}-\alpha_{i})}$

for a symmetric polynomial $f$ in $m-d$ variables such that the degree of $f$
in any of its variables is at most $d$. When $m=d+1$, so that $X=\\{x_{1}\\}$,
their formula specializes to the classical Lagrange interpolation formula for
a one-variable polynomial of degree at most $d$, see [CL].

Foam equivalents of formulas (71) and (72) are depicted in Figures 6.13 and
6.14, correspondingly. $X$ foams there have maximal thickness $m-d=|X|$ and a
surface of any genus can be chosen in place of a torus for that component.
Figure 6.15 shows in detail why the foam in the RHS of Figure 6.13 evaluates
to the RHS of formula (71).

An important role in [KSV] and several related papers is played by the
_Exchange Lemma_. To state it, following [KSV], take $A$ and $B$ to be
disjoint sets of cardinalities $m$ and $n$, respectively. Then

(73) $\sum_{A^{\prime}\subset A,\>|A^{\prime}|=d}{\mathcal{R}}(A\setminus
A^{\prime},B)\frac{{\mathcal{R}}(X,A^{\prime})}{{\mathcal{R}}(A\setminus
A^{\prime},A^{\prime})}=\sum_{B^{\prime}\subset
B,\>|B^{\prime}|=d}{\mathcal{R}}(A,B\setminus
B^{\prime})\frac{{\mathcal{R}}(X,B^{\prime})}{{\mathcal{R}}(B^{\prime},B\setminus
B^{\prime})},$

$X$$r$$A$$d$$m-d$$B$$n$$A^{\prime}$$A\setminus
A^{\prime}$$r=|X|$$m=|{\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color@cmyk@fill{1}{0}{0}{0}A}|$$n=|{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}B}|$$=$$X$$r$$B$$d$$n-d$$A$$m$$B^{\prime}$$B\setminus
B^{\prime}$$m$$n$ Figure 6.16. Exchange Relation written via foam evaluation.
Seam circles of blue and orange theta-foams are oriented oppositely, to
incorporate implicit sign in formula (73) that appears if in one of the
denominators the order of a set and its complement is reversed, see [KSV]. $X$
and $B$ foams on the LHS and $X$ and $A$ foams on the RHS may carry any genus;
we chose genus $1$ for all four.

Foam interpretation of the both sides of this identity is shown in Figure
6.16.

$X$$A$$B$$A_{1}$$A\setminus
A_{1}$$|X|$$|{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}B}|$$d$$m-d$$m$$=$$|E|$$X$$|X|$$d$$E_{1}$$m-d$$E_{2}$$|E|\setminus
m$$E_{3}$$A$$|{\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color@cmyk@fill{1}{0}{0}{0}A}|$$B$$|{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}B}|$
Figure 6.17. Foams for the formula (74), also see [DKSV, Proposition 2.1]. On
the RHS, generalized theta-foam in the middle for variable set $E$ consists of
a thickness $|E|$ disk with three adjacent disks of thicknesses $d$, $m-d$ and
$|E|-m$, respectively. Each of these three disks intersects one of the spheres
for variable sets $A,B,X$.

As another example, consider the formula in [DKSV, Proposition 2.1]. To state
it, let $A,B$ be finite sets with $|A|=m$ and $|B|=n$, and choose $0\leq d\leq
m$. Let $X,E$ be finite sets such that

$|E|\geq\max\\{|X|+d\,,\>m+n-d\,,\>m\\}.$

Then

(74) $\begin{split}&\sum_{\begin{subarray}{c}A_{1}\sqcup A_{2}=A\\\
|A_{1}|=d,\>\>|A_{2}|=m-d\end{subarray}}\frac{{\mathcal{R}}(A_{2},B){\mathcal{R}}(X,A_{1})}{{\mathcal{R}}(A_{1},A_{2})}=\\\
=&\sum_{\begin{subarray}{c}E_{1}\sqcup E_{2}\sqcup E_{3}=E\\\
|E_{1}|=d,\>\>|E_{2}|=m-d,\>\>|E_{3}|=|E|-m\end{subarray}}\frac{{\mathcal{R}}(A,E_{3}){\mathcal{R}}(E_{2},B){\mathcal{R}}(X,E_{1})}{{\mathcal{R}}(E_{1},E_{2}){\mathcal{R}}(E_{1},E_{3}){\mathcal{R}}(E_{2},E_{3})}.\end{split}$

Foam interpretation of this formula is shown in Figure 6.17. In the evaluation
of overlapping foams we assume that variable sets (in this example, $A,B,X,E$)
are disjoint, but perhaps this condition can be relaxed (formula (74) holds as
well when these sets have non-empty intersections, see [DKSV]).

We leave it to the reader to write a similar foam interpretation of the
relation between Sylvester double and single sums, see formula (2) in [DKSV].

From the present examples and those in [Kh4, Section 3] one can make a natural
guess that there exists a meaningful theory of overlapping foams, but it is
not clear to the authors how to develop it. One possible direction is to use
an extension of Sylvester’s subresultants to polynomials with multiple roots
constructed in [DKS, DKSV] to search for the symmetric analogue of the Robert-
Wagner foam evaluation [RW1]. Robert-Wagner work and many prior papers (see
[KK] for an imcomplete survey) deal with _exterior foams_ that are used to
categorify networks on intertwiners between quantum exterior powers of the
fundamental $U_{q}(\mathfrak{sl}_{N})$ representations. Papers [Ca, QRS, RW2]
indicate that a similar theory should exists for _symmetric foams_ that would
categorify networks of quantum symmetric powers of the fundamental
representation, but a definition and evaluation of symmetric foams is unknown
as of today.

##  7\. Appendix (joint with Lev Rozansky): Comparison with matrix
factorizations

Each finite degree field extension $\mathbf{k}\subset F$ is Frobenius. Any
nonzero $\mathbf{k}$-linear map $\varepsilon:F\longrightarrow\mathbf{k}$ is a
non-degenerate trace making $F$ a commutative Frobenius algebra over
$\mathbf{k}$. For separable extensions, there’s a canonical trace
${\mathrm{tr}}_{F/\mathbf{k}}$ used above.

Matrix factorizations deliver a supply of commutative Frobenius algebras and
2-dimensional TQFTs with corners [KRz, CM, DM]. A nondegenerate potential
$w\in\mathbf{k}[x_{1},\dots,x_{k}]$ defines the Jacobi algebra

(75) $J(w)\ :=\
\mathbf{k}[x_{1},\dots,x_{k}]/(\partial_{1}w,\dots,\partial_{k}w),\quad\partial_{i}w:=\partial
w/\partial x_{i}$

(a potential is called nondegenerate when this quotient algebra is finite-
dimensional). The Jacobi algebra is commutative Frobenius and carries a
canonical trace $\mathrm{tr}_{\mathsf{Gr}}$, given by the Grothendieck
residue, i.e., see [AGV, GH]. When $\mathbf{k}$ is a subfield of $\mathbb{C}$,
the trace may be written as a complex integral

(76) $\mathrm{tr}_{\mathsf{Gr}}(p(\underline{x}))=\frac{1}{(2\pi
i)^{k}}\int_{|\partial_{i}w|=R}\frac{p(\underline{x})}{\partial_{1}w\cdots\partial_{k}w}dx_{1}\cdots
dx_{k},\qquad p(\underline{x})\in\mathbf{k}[x_{1},\dots,x_{k}]$

over a contour that contains all roots of the system of equations
$\partial_{1}w=\ldots=\partial_{k}w=0$.

Suppose that $F$ is a subfield of $\mathbb{C}$ (in particular,
$\mathrm{char}(\mathbf{k})=0$). Since $F/\mathbf{k}$ is a simple extension,
there is a generating element $\alpha\in F,$ $\mathbf{k}(\alpha)=F$, and

(77) $F\cong\mathbf{k}[x]/(f(x)),$

where $f$ is the minimal polynomial of $\alpha$ over $\mathbf{k}$,

(78) $f(x)=x^{n}+a_{n-1}x^{n-1}+\ldots+a_{0},\qquad a_{i}\in\mathbf{k}.$

We can realize $F$ as the Jacobi algebra of the singularity with the potential
$w(x)$ in a single variable $x$ such that $w^{\prime}(x)=f(x)$,

(79) $w(x)=\frac{1}{n+1}x^{n+1}+\frac{a_{n-1}}{n}x^{n}+\ldots+a_{0}x.$

The polynomial $f(x)$ is irreducible over $\mathbf{k}$ and can be fully
factored in the algebraic closure $\overline{\mathbf{k}}\subset\mathbb{C}$:

(80) $f(x)\ =\
(x-\lambda_{1})\cdots(x-\lambda_{n}),\qquad\lambda_{i}\in\overline{\mathbf{k}},\quad\lambda_{i}\not=\lambda_{j}.$

The Hessian

(81)
$w^{\prime\prime}(x)=f^{\prime}(x)=\sum_{i=1}^{n}\prod_{j\not=i}(x-\lambda_{j}),$

and

(82) $w^{\prime\prime}(\lambda_{i})\ =\
\prod_{j\not=i}(\lambda_{i}-\lambda_{j}).$

For a single variable $x$ and a potential $w(x)$ with $w^{\prime}(x)=f(x)$
having simple roots only, the Grothendieck trace is given by

(83) $\mathrm{tr}_{\mathsf{Gr}}(p(x))=\frac{1}{2\pi
i}\int_{|f(x)|=R}\frac{p(x)}{f(x)}dx=\sum_{i=1}^{n}\frac{p(\lambda_{i})}{\displaystyle{\prod_{j\not=i}}(\lambda_{i}-\lambda_{j})},\qquad
p(x)\in\mathbf{k}[x],\ \ R\gg 0.$

To compare the two traces, note that the canonical trace
${\mathrm{tr}}_{F/\mathbf{k}}$ in a finite separable field extension can also
be characterized as follows. The tensor product
$F\otimes_{\mathbf{k}}\overline{\mathbf{k}}$ of $F$ with the algebraic closure
$\overline{\mathbf{k}}$ of $\mathbf{k}$ is isomorphic to the direct product of
$n$ copies of $\overline{\mathbf{k}}$, where $n$ is the degree
$[F:\mathbf{k}]$,

(84) $F\otimes_{\mathbf{k}}\overline{\mathbf{k}}\ \cong\
\overline{\mathbf{k}}\times\dots\times\overline{\mathbf{k}}.$

This algebra contains $n$ minimal idempotents $e_{1},\dots,e_{n}$, one for
each term in the product. Trace ${\mathrm{tr}}_{F/\mathbf{k}}$ extends
$\overline{\mathbf{k}}$-linearly to a trace

$\overline{{\mathrm{tr}}}_{F/\mathbf{k}}\ :\
F\otimes_{\mathbf{k}}\overline{\mathbf{k}}\longrightarrow\overline{\mathbf{k}}$

that is characterized uniquely by its taking value $1$ on each minimal
idempotent, $\overline{{\mathrm{tr}}}_{F/\mathbf{k}}(e_{i})=1.$

Any other nondegenerate trace $\varepsilon:F\longrightarrow\mathbf{k}$ extends
likewise to a $\overline{\mathbf{k}}$-linear trace

$\overline{\varepsilon}:F\otimes_{\mathbf{k}}\overline{\mathbf{k}}\longrightarrow\overline{\mathbf{k}}$

taking a nonzero value on each idempotent $e_{i}$, with at least one of these
values different from $1$.

Minimal idempotents $e_{k}(x)\in\overline{\mathbf{k}}[x]/(f(x))$ are given by

(85) $e_{k}(x)\ =\
\prod_{j\not=k}\frac{x-\lambda_{j}}{\lambda_{k}-\lambda_{j}}.$

Indeed, $e_{k}(\lambda_{i})=\delta_{i,k}$, so these are delta functions when
evaluated on the roots of $f(x)$. Evaluating the Grothendieck trace on them
gives

(86)
$\mathrm{tr}_{\mathsf{Gr}}(e_{k}(x))=\sum_{i=1}^{n}\frac{e_{k}(\lambda_{i})}{\displaystyle{\prod_{j\not=i}}(\lambda_{i}-\lambda_{j})}=\frac{1}{\displaystyle{\prod_{j\not=k}}(\lambda_{k}-\lambda_{j})}.$

Thus, values of the two traces on minimal idempotents are

(87)
$\overline{{\mathrm{tr}}}_{F/\mathbf{k}}(e_{k})=1,\qquad\mathrm{tr}_{\mathsf{Gr}}(e_{k})=\prod_{j\not=k}\frac{1}{(\lambda_{k}-\lambda_{j})},\qquad
1\leq k\leq n,$

and the field extension trace can be written as

(88) $\mathrm{tr}_{\mathsf{Gr}}(p(x))=\frac{1}{2\pi
i}\int_{|f(x)|=R}\frac{w^{\prime\prime}(x)p(x)}{w^{\prime}(x)}dx=\sum_{i=1}^{n}p(\lambda_{i}),\qquad
p(x)\in\mathbf{k}[x].$

Notice that we added the Hessian $w^{\prime\prime}(x)$ to the numerator of the
integral and kept the denominator. We see that the two traces differ by
multiplication by the Hessian,

(89) ${\mathrm{tr}}_{F/\mathbf{k}}(p(x))\ =\
\mathrm{tr}_{\mathsf{Gr}}(w^{\prime\prime}(x)p(x)).$

The second and first derivatives $w^{\prime\prime}(x),w^{\prime}(x)$ have no
common roots, since all roots of $w^{\prime}(x)=f(x)$ are simple, and
$w^{\prime\prime}(x)$ is an invertible element of $\mathbf{k}[x]/(f(x))\cong
F$ (the latter ring is a field anyway). In the 2D TQFT of the Landau-Ginzburg
model for the potential $w(x)$ the value of a one-holed torus, as an element
of the Jacobi algebra (the state space of the circle), is the Hessian
$w^{\prime\prime}(x)$, see Figure 7.1.

$=$$\omega^{\prime\prime}(x)$ Figure 7.1. One-holed torus represents the
element $w^{\prime\prime}(x)$ in the Jacobi algebra of a one-variable
potential.

Consequently, the field extension trace (the map induced by the cap in the
TQFT associated to $(F,\mathbf{k},{\mathrm{tr}}_{F/\mathbf{k}})$) can be
written as the cap with the genus one surface (holed torus) in the Landau-
Ginzburg TQFT associated to a given generating element $\alpha\in F$, as
described earlier, see Figure 7.2.

${\mathrm{tr}}_{F/\mathbf{k}}$$=$$=$$\omega^{\prime\prime}(x)$MF/LG TQFT maps
Figure 7.2. Cap given by the field extension trace equals the genus one cap
trace in the matrix factorization (Landau-Ginzburg) TQFT, for any choice of
generator $x$ and the corresponding potential $w(x)$.

Choosing a different generator $\alpha$ for $F$ will, in general, change the
polynomial $f(x)$, potential $w(x)$ and the value of the trace on idempotents
of $F\otimes_{\mathbf{k}}\overline{\mathbf{k}}$, while the trace
${\mathrm{tr}}_{F/\mathbf{k}}$ is defined canonically. At the same time, it’s
given by capping off by the holed torus, in any one-variable matrix
factorization TQFT realization of $F$ as the Jacobi algebra.

This amusing relation between matrix factorizations and field extensions may
be worth a further exploration. Notice, in particular, that $F$ may be
realized as the Jacobi algebra, $F\cong J(w)$, for a multivariable potential
$w(\underline{x})\in\mathbf{k}[x_{1},\dots,x_{k}]$. Equivalently, $F$ is the
zero-dimensional complete intersection of hypersurfaces $\partial_{i}w=0$,
$i=1,\dots,k$. It should be interesting to find nontrivial presentations of
that kind for various $F$ with $k>1$ or locate them in the literature.

The Jacobi algebra $J(w)$ is the endomorphism ring of the canonical matrix
factorization

(90) $K(w)\ =\ \bigotimes_{i=1}^{k}K(x_{i}-y_{i},u_{i}),$

a Koszul factorization with the potential
$w_{12}=w(\underline{x})-w(\underline{y})$ in $2n$ variables
$x_{1},\dots,x_{n}$, $y_{1},\dots,y_{n}$. Here $u_{i}$ are any polynomials in
$x$’s and $y$’s such that

$w_{12}=\sum_{i=1}^{k}(x_{i}-y_{i})u_{i},$

and $K(v,u)$ is the factorization

$\mathbf{k}[\underline{x},\underline{y}]\stackrel{{\scriptstyle
v}}{{\longrightarrow}}\mathbf{k}[\underline{x},\underline{y}]\stackrel{{\scriptstyle
u}}{{\longrightarrow}}\mathbf{k}[\underline{x},\underline{y}],$

see [KRz]. Matrix factorization $K(w)$ represents the identity functor on the
triangulated category $\mathsf{MF}(w)$ of matrix factorizations with potential
$w$ and morphisms being homs of matrix factorizations modulo homotopies [KRz].

Assume that $F/\mathbf{k}$ is a finite Galois extension in characteristic $0$
and consider the Galois group $G=\mathsf{Gal}(F/\mathbf{k})$. One can ask to
find presentations of $F$ as the Jacobi algebra $J(w)$ such that Galois
symmetries $\sigma\in G$ lift to endofunctors of $\mathsf{MF}(w)$ defining an
action of the Galois group on that category. Precisely, for each $\sigma$ we
would like to have a matrix factorization
$M(\sigma)=M_{\underline{x},\underline{y}}(\sigma)$ (using subindices to
specify sets of variables) with the potential
$w(\underline{x})-w(\underline{y})$ together with isomorphisms in the homotopy
category $\mathsf{MF}(\underline{x}-\underline{z})$ of matrix factorizations
with the potential $w(\underline{x})-w(\underline{z})$

(91)
$M_{\underline{x},\underline{y}}(\sigma)\otimes_{\underline{y}}M_{\underline{y},\underline{z}}(\tau)\cong
M_{\underline{x},\underline{z}}(\sigma\tau),\ \ \sigma,\tau\in G,$

such that $M_{\underline{x},\underline{y}}(1)\cong K(w)$ . One can further
require that these isomorphisms satisfy compatibility relations so that $G$
acts on $\mathsf{MF}(w)$ in a strong sense. Furthermore, factorization
$M(\sigma)$ should induce the symmetry $\sigma$ on $J(w)\cong F$ upon taking
the trace of the identity endomorphism of $M(\sigma)$. Diagrammatically,
following notations from [KRz], denote
$M(\sigma)=M_{\underline{x},\underline{y}}(\sigma)$ by a dot labelled $\sigma$
on an oriented line with endpoints labelled $\underline{x},\underline{y}$, see
Figure 7.3.

$x$$x$$y$$y$$\sigma$

$\tau$$y$$\sigma$$x$$x$$z$$z$$\tau\sigma$

$\tau$$y$$\sigma$$x$$x$$z$$z$$\tau\sigma$

Figure 7.3. Top left arc represents the identity factorization $K(w)$. Bottom
left arc carrying dot $\sigma$ represents factorization $M(\sigma)$. Trivalent
vertices on the middle and right pictures show mutually-inverse isomorphisms
(91).

The identity map of $M(\sigma)$ is depicted by a defect interval, shown as a
vertical interval in Figure 7.4 left. Taking the trace of the identity map
corresponds, on the diagrammatic side, to closing of the square into an
annulus with a defect circle on it, see Figure 7.4.

$y$$x$$\sigma$

$x$$\sigma$$y$$\simeq$$x$$\sigma$$y$

Figure 7.4. Left: the identity endomorphism of $\sigma$. Middle and right: its
trace is a defect circle on an annulus. Boundaries of the annulus correspond
to closures of the identity factorization $K(w)$, given by equating variables
$\underline{x}=\underline{y}$ in that factorizations and taking cohomology of
the resulting 2-periodic complex. Cohomology is precisely the Jacobi algebra
$J(w)$, and the annulus with the $\sigma$-circle defines a linear endomorphism
of it.

In general, a defect circle on an annulus would only give a linear
endomorphism of the Jacobi algebra, not an algebra homomorphism. For that, we
would additionally want the equality shown in Figure 7.5 left, which may come
from a more local relation in Figure 7.5 right.

$\sigma$$\sigma$$=$

$\sigma$

$\sigma$$\sigma$$=$$\sigma$$\sigma$

Figure 7.5. Left: $\sigma$-circle defining an algebra endomorphism of $J(w)$.
Right: a sufficient local relation for that.

For a general separable finite field extension $F/\mathbf{k}$, it seems hard
to impossible to pick a potential $\omega\in\mathbf{k}[x_{1},\dots,x_{n}]$
with the Jacobi algebra $J(\omega)\cong F$ and invertible factorizations
$M(\sigma),\sigma\in\mathsf{Gal}(F/\mathbf{k})$, giving an action of the
Galois group $\mathsf{Gal}(F/\mathbf{k})$ on the homotopy category of matrix
factorizations $\mathsf{HMF}_{\omega}$ with potential $\omega$ such that the
action induces the Galois group action on $F$.

Potentially related structures appear in the theory of Landau–Ginzburg
orbifolds, where a group $G$ acts on $\mathbf{k}[x_{1},\dots,x_{k}]$
preserving the potential $w$, leading to the category of $G$-equivariant
matrix factorizations. In those examples usually $\mathbf{k}=\mathbb{C}$, and
it’s unclear whether some version of LG orbifold theory may be adapted to
relate Galois extensions and matrix factorizations.

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