The Lagrangian cobordism group of $T^2$

Abstract

We compute the Lagrangian cobordism group of the standard symplectic 2-torus and show that it is isomorphic to the Grothendieck group of its derived Fukaya category. The proofs use homological mirror symmetry for the 2-torus.

Appendices

Appendix 1: Iterated cone decompositions and $K_0$

1.1 Triangulated categories

A triangulated category ${\fancyscript{D}}$ is an additive category equipped with an additive autoequivalence $S: {\fancyscript{D}} \rightarrow {\fancyscript{D}}$ called the shift functor, and a set of exact triangles $X \xrightarrow {f} Y \xrightarrow {g} Z \xrightarrow {h} S(X)$. These data are required to satisfy a list of axioms, for which we refer to [22]. The most relevant for us is that every morphism $f: X \rightarrow Y$ in ${\fancyscript{D}}$ can be completed to such an exact triangle. The object $Z$ is then determined up to isomorphism, and we call it a cone on the morphism $f$. Moreover, we write $X[n] = S^nX$ and $f[n] = S^n(f)$ for the effect of iterates of the shift functor $S$ on objects and morphisms. This is in reminiscence of the homotopy category of complexes $K({\fancyscript{A}})$ over an Abelian category ${\fancyscript{A}}$, which is the prototypical example of a triangulated category.

1.2 Generation and iterated cone decompositions

Given a full subcategory of a triangulated category ${\fancyscript{D}}$ with objects a collection $\{X_i ~|~ i \in I\}$, one can consider the full subcategory consisting of all objects that are cones on morphisms between the $X_i$. Iterating this construction, i.e., including in each step all cones on morphisms between the previously constructed objects, one ends up with the subcategory of ${\fancyscript{D}}$ generated by the $X_i$.

In the other direction, one can ask whether and how an object $X$ of ${\fancyscript{D}}$ can be constructed as an iterated cone on morphisms between other objects. The following notion is useful to formalize this.

Definition 9.1

Let ${\fancyscript{D}}$ be a triangulated category and let $X \in {\fancyscript{D}}$. An iterated cone decomposition of $X$ is a sequence of exact triangles

$$\begin{aligned} C_{i-1}[-1] \rightarrow X_i \rightarrow C_i \rightarrow C_{i-1}, \quad i = 1, \ldots , k, \end{aligned}$$

with objects $C_0,C_1,\ldots , C_k \in {\fancyscript{D}}$ that satisfy $C_0 = 0$ and $C_k = X$. The tuple $(X_1,\ldots ,X_k)$ is called the linearization of the cone decomposition.

Remark 9.2

This definition is adapted to the cohomological conventions we use and therefore differs from the one in [6, Section 2.6], where homological conventions are used.

Iterated cone decompositions can themselves be iterated and are well behaved with respect to that in the sense of the following lemma (cf. the composition in Biran–Cornea’s category $T^S{\fancyscript{D}}$ of iterated cone decompositions [6, Section 2.6]).

Lemma 9.3

Suppose that $X \in {\fancyscript{D}}$ admits an iterated cone decomposition with linearization $(X_1,\ldots ,X_k)$, and that one of the $X_h,\,1 \le h \le k$ admits an iterated cone decomposition with linearization $(X_h^1,\ldots ,X_h^\ell )$. Then, $X$ admits an iterated cone decomposition with linearization

$$\begin{aligned} (X_1,\ldots ,X_{h-1},X_h^1,\ldots ,X_h^\ell ,X_{h+1},\ldots ,X_k). \end{aligned}$$

1.3 Grothendieck groups

Let ${\fancyscript{D}}$ be a triangulated category. The Grothendieck group $K_0({\fancyscript{D}})$ is defined as the quotient $K_0({\fancyscript{D}}) = \langle \mathrm{Ob}\,{\fancyscript{D}} \rangle / R$ of the free Abelian group generated by the objects of ${\fancyscript{D}}$ by the subgroup $R$ generated by all expressions $X - Y + Z \in \langle \mathrm{Ob}\,{\fancyscript{D}} \rangle$ such that there exists an exact triangle $X \rightarrow Y \rightarrow Z \rightarrow X[1]$. The following lemma is straightforward.

Lemma 9.4

Suppose that $X \in {\fancyscript{D}}$ admits an iterated cone decomposition with linearization $(X_1,\ldots ,X_k)$. Then, $[X] = [X_1] + \cdots + [X_k]$ in $K_0({\fancyscript{D}})$.

One can also define the Grothendieck group $K_0({\fancyscript{A}})$ of an Abelian category ${\fancyscript{A}}$, by starting with the free Abelian group on objects and imposing a relation $[A] + [C] = [B]$ for every short exact sequence $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$. Recall the canonical inclusion ${\fancyscript{A}} \hookrightarrow D^b({\fancyscript{A}})$, which on objects is given by viewing $X \in \mathrm{Ob}\,{\fancyscript{A}}$ as a complex concentrated in degree zero. The following statement is well known.

Proposition 9.5

The canonical inclusion ${\fancyscript{A}} \hookrightarrow D^b({\fancyscript{A}})$ induces an isomorphism $K_0({\fancyscript{A}}) \cong K_0(D^b{\fancyscript{A}})$.

Appendix 2: Exact triangles from SESs of local systems

Consider a symplectic manifold $(M,\omega )$ for which we can define the Fukaya category ${\fancyscript{F}}^\sharp (M)$ with gradings and signs as outlined in Sect. 3 such that the objects are Lagrangian branes with local systems. There appears to be no reference in the literature for the following statement.

Proposition 10.1

Let $L^{}$ be a Lagrangian brane and let $0 \rightarrow E' \rightarrow E \rightarrow E'' \rightarrow 0$ be a short exact sequence of local systems on $L$. Then, there exists an exact triangle

$$\begin{aligned} (L^{},E'')[-1] \rightarrow (L^{},E') \rightarrow (L^{},E) \rightarrow (L^{},E'') \end{aligned}$$

in $D {\fancyscript{F}}^\sharp (M)$.

We view local systems as assignments of vector spaces and parallel transport maps as described in Sect. 3.1.3. By a short exact sequence of local systems $0 \rightarrow E' \xrightarrow {i} E \xrightarrow {p} E'' \rightarrow 0$, we mean a family of short exact sequences of vector spaces

$$\begin{aligned} 0 \rightarrow E'_x \xrightarrow {i_x} E_x \xrightarrow {p_x} E''_x \rightarrow 0 \end{aligned}$$

for every $x \in L$ such that the $i_x$ and $p_x$ define morphisms of local systems, i.e., such that they commute with parallel transport maps. For the following proof, we choose a splitting

for every $x \in L$, that is, maps $q_x: E_x \rightarrow E'_x$ and $j_x: E''_x \rightarrow E_x$ such that $q_x \circ i_x = \mathrm{id}_{E'_x},\,p_x \circ j_x = \mathrm{id}_{E''_x}$ and $i_x\circ q_x + j_x \circ p_x = \mathrm{id}_{E_x}$. Note that the $j_x$ and $q_x$ do generally not define morphisms of local systems, unless the short exact sequence splits globally.

In the proof of Proposition 10.1, it will be convenient to model the relevant morphism spaces in ${\fancyscript{F}}^\sharp (M)$ as spaces of Morse cochains with coefficients in local systems. This is possible under certain conditions on $M$ and $L$, the fundamental example being that $M$ is a cotangent bundle and $L$ is an exact Lagrangian (from which the case of interest in this paper follows immediately by a covering argument).

We will adapt a construction used in [2] (which goes back to [7]) and consider an $A_\infty$-category ${\fancyscript{M}}(L)$ defined as follows: The objects of ${\fancyscript{M}}(L)$ are all those objects of ${\fancyscript{F}}^\sharp (M)$ whose underlying Lagrangian is $L$. For the morphism spaces, we fix a Morse function $f: L \rightarrow {\mathbb {R}}$ and define

$$\begin{aligned} \mathrm{hom}_{{\fancyscript{M}}(L)}^i (E_0,E_1) = \bigoplus _{\begin{array}{c} x \in \mathrm{Crit}\,f \\ \vert x \vert = i \end{array}} \mathrm{Hom}(E_{0,x},E_{1,x}), \end{aligned}$$

where $\vert \cdot \vert$ denotes the Morse index; here, we denote objects of ${\fancyscript{M}}(L)$ simply by their local system. The $A_\infty$-operations $\mu ^d_{{\fancyscript{M}}(L)}$ are defined by considering rigid perturbed gradient flow trees with vertices at critical points and summing up parallel transport maps in the relevant local systems along the edges of these trees. See [2] for the description of the relevant moduli spaces.

By adapting the arguments in [2], one can show (under certain conditions, as indicated above) that there is an $A_\infty$-quasi-isomorphism

$$\begin{aligned} {\fancyscript{F}}^\sharp (L) \rightarrow {\fancyscript{M}}(L), \end{aligned}$$

where ${\fancyscript{F}}^\sharp (L)$ denotes the full $A_\infty$-subcategory of ${\fancyscript{F}}^\sharp (M)$ consisting of objects whose underlying Lagrangian is $L$. It will therefore suffice to prove that there exists an exact triangle of the claimed form in $H^0(Tw {\fancyscript{M}}(L))$, where $Tw {\fancyscript{M}}(L)$ denotes the category of twisted complexes over ${\fancyscript{M}}(L)$, which we use to model the triangulated closure of that category (see [19, Section (3l)]).

Proof of Proposition 10.1

We assume that the Morse function $f: L \rightarrow {\mathbb {R}}$ defining the morphism spaces in ${\fancyscript{M}}(L)$ has a single local minimum $x_0 \in L$. As for notation, we write $\pi _\gamma ',\pi _\gamma , \pi _\gamma ''$ for the parallel transport in $E',E,E''$ along a path $\gamma$, and we denote by $\overline{\gamma }$ the path obtained by reversing $\gamma$.

Let $c_2 \in \mathrm{hom}^0(E',E) = \mathrm{Hom}(E_{x_0}',E_{x_0})$ and $c_3 \in \mathrm{hom}^0(E,E'') = \mathrm{Hom}(E_{x_0},E''_{x_0})$ be the morphisms in ${\fancyscript{M}}(L)$ determined by the short exact sequence, that is, $c_2 = i_{x_0}$ and $c_3 = p_{x_0}$. Then, define $c_1 \in \mathrm{hom}^1(E'',E') = \bigoplus _{y}\mathrm{Hom}(E''_{y},E'_{y})$ by

$$\begin{aligned} c_1 = \bigoplus _y \sum _{\gamma } \pm \, \pi _{\gamma }' \circ q_{x_0} \circ \pi _{\overline{\gamma }} \circ j_y, \end{aligned}$$

where the first sum runs over all critical points $y$ of $f$ with Morse index 1, and the second over all gradient flow lines of $f$ from $x_0$ to $y$, and where the sign $\pm$ is associated with $\gamma$ as indicated above. In fact, we view $c_1$ as living in $\mathrm{hom}^0(E''[-1],E')$, and $c_3$ as living in $\mathrm{hom}^{1}(E,E''[-1])$.

We claim that these morphisms fit into an exact triangle

$$\begin{aligned} E''[-1] \xrightarrow {c_1} E'\xrightarrow {c_2} E \xrightarrow {c_3} E'' \end{aligned}$$

in $H^0(Tw {\fancyscript{M}}(L))$. In this model, the cone $C = \mathrm{Cone}(c_1)$ is the twisted complex

$$\begin{aligned} C = \left( E'' \oplus E', \delta = \begin{pmatrix} 0 &{} 0 \\ -c_1 &{} 0 \end{pmatrix} \right) , \end{aligned}$$

see [19, Section 3(p)] (we suppress the shift of $c_1$). It comes together with morphisms $p_C = (e_{E''},0) \in \mathrm{hom}_{Tw}(C,E'')$ and $i_C = (0,e_{E'})^T \in \mathrm{hom}_{Tw}(E',C)$, where $e_{E''} = \mathrm{id}_{E_{x_0}''}$ and $e_{E'} = \mathrm{id}_{E_{x_0}'}$ denote the chain-level identity morphisms of $E'', E'$ in ${\fancyscript{M}}(L)$.

To prove our claim, we will make use of Lemma 3.27 in [19], which gives a criterion for exactness of triangles in $H^0(Tw {\fancyscript{M}}(L))$. According to that, we have to find a cocyle $b \in \mathrm{hom}_{Tw}(E,C)$ such that $[b]$ is an isomorphism and $[\mu _{Tw}^2(p_C, b)] = [c_3],\,[\mu _{Tw}^2(b,c_2)] = [i_C]$ in $H^0(Tw {\fancyscript{M}}(L))$. (Again, we suppress some shifts.)

We claim that $b = (b'',b')$ with $b'' = p_{x_0} \in \mathrm{hom}^0(E,E''),\,b' = q_{x_0}\in \mathrm{hom}^0(E,E')$ satisfies these requirements. The necessary computations are straightforward. We start by verifying that $b$ is a cocycle, i.e., that $\mu _{Tw}^1(b) = 0$. Unravelling the definition of $\mu _{Tw}^1$, cf. [19, Section (3l)], we obtain

$$\begin{aligned} \mu _{Tw}^1(b) = \begin{pmatrix} \mu ^1 (b'')\\ \mu ^1(b') - \mu ^2(c_1,b'') \end{pmatrix}, \end{aligned}$$

where the $\mu ^d$’s are those of ${\fancyscript{M}}(L)$. Now, $\mu ^1(b'') = \mu ^1(p_{x_0})$ vanishes because the $p_x$ form a morphism of local systems. To compute $\mu ^2(c_1,b'')$, note that for every critical point $y$, there is a unique perturbed $Y$-shaped gradient tree with outgoing edges converging to $x_0$ and $y$ that contributes to the count, and that the incoming edge of this tree also converges to $y$ (recall that $x_0$ is the unique local minimum). In view of this and recalling the definition of $c_1$, we obtain

$$\begin{aligned} \begin{aligned} \mu ^2(c_1,b'')&= \bigoplus _y\sum _\gamma \pm \, \pi _{\gamma }' \circ q_{x_0} \circ \pi _{\overline{\gamma }} \circ j_y \circ p_y\\&= \bigoplus _y\sum _\gamma \pm \, \pi _{\gamma }' \circ q_{x_0} \circ \pi _{\overline{\gamma }} \circ \left( \mathrm{id}_{E_y} - i_y \circ q_y\right) \\&=\bigoplus _y \left( \sum _\gamma \pm \, \pi _{\gamma }' \circ q_{x_0} \circ \pi _{\overline{\gamma }} - \sum _\gamma \pm \,q_y \right) \\&=\bigoplus _y\sum _\gamma \pm \, \pi _{\gamma }' \circ q_{x_0} \circ \pi _{\overline{\gamma }}. \end{aligned} \end{aligned}$$

Here, we use that the $p_x$ give a morphism of local systems, i.e., commute with parallel transport maps (which makes the additional parallel transport maps disappear that would appear in the first line). Moreover, we use that $\pi '_\gamma \circ q_{x_0} \circ \pi _{\overline{\gamma }} \circ i_y = \pi '_\gamma \circ q_{x_0} \circ i_{x_0} \circ \pi _{\overline{\gamma }} = \mathrm{id}_{E_y'}$, and that $\sum _\gamma \pm q_y = 0$ as $x_0$ is a cocyle in the usual Morse complex. The result of the computation is equal to $\mu ^1(b')$, which shows that also the second component of $\mu ^1_{Tw}(b)$ vanishes.

It remains to check that $[\mu _{Tw}^2(p_C, b)] = [c_3],\,[\mu _{Tw}^2(b,c_2)] = [i_C]$ in $H^0(Tw {\fancyscript{M}}(L))$. After unravelling again definitions, the first identity follows immediately (on the chain level). As for the second, we obtain

$$\begin{aligned} \mu _{Tw}^2(b,c_2) = \begin{pmatrix} \mu ^2(b'',c_2)\\ \mu ^2(b',c_2) - \mu ^3(c_1,b'',c_2) \end{pmatrix}. \end{aligned}$$

The first component is $\mu ^2(b'',c_2) = p_{x_0} i_{x_0} = 0$, as required. As for the second component, we have $\mu ^2(b',c_2) = q_{x_0}i_{x_0} = \mathrm{id}_{E_{x_0}'} = i_C$, and hence, we are done if we can show that $\mu ^3(c_1,b'',c_2) = 0$. Recall that $b'' = p_{x_0}$ and $c_2 = i_{x_0}$, and that $p_{x_0} i_{x_0} = 0$; this together with the fact that the $i_x$ commute with parallel transport maps suffices to conclude that the term vanishes. Hence, the second required identity also holds on the chain level.

Appendix 3: Vector bundles on elliptic curves

For the convenience of the reader, we collect here a couple of facts from Atiyah’s classification [4] of vector bundles over an elliptic curve $X$, which are used in Sects. 7 and 8. We denote by ${{\mathcal {V}}}(r,d)$ the set of isomorphism classes of vector bundles on $X$ of rank $r$ and degree $d$.

Theorem 11.1

(Cf. Theorem 3 in [4]) There exists an integer $N(r,d)$ such that for every $n \ge N(r,d)$, every ${\mathcal {E}} \in {{\mathcal {V}}}(r,d)$ fits into a short exact sequence

$$\begin{aligned} 0 \rightarrow {{\mathcal {O}}}_X^{\oplus (r-1)}(-n) \rightarrow {\mathcal {E}} \rightarrow \mathrm{det}\,{\mathcal {E}}((r-1)n)\rightarrow 0. \end{aligned}$$

(Here, $-(k)$ denotes tensoring by the $k^{th}$ power of the hyperplane bundle.)

Theorem 11.2

(Cf. Theorem 5 in [4]) (i) There is a unique ${{\mathcal {F}}}_r \in {{\mathcal {V}}}(r,0)$ such that $H^0(X;{{\mathcal {F}}}_r) \ne 0$, and there is a short exact sequence $0 \rightarrow {\fancyscript{O}}_X \rightarrow {{\mathcal {F}}}_r \rightarrow {{\mathcal {F}}}_{r-1} \rightarrow 0$ for every $r > 1$. (ii) Every ${\mathcal {E}} \in {{\mathcal {V}}}(r,0)$ is of the form ${{\mathcal {F}}}_r \otimes {\mathcal {L}}$ for a unique ${\mathcal {L}} \in {{\mathcal {V}}}(1,0)$.

Theorem 11.3

(Cf. Theorems 6 and 7 in [4]) The choice of a line bundle ${\mathcal {L}} \in {{\mathcal {V}}}(1,1)$ determines natural 1–1 correspondences $\alpha _{r,d}: {{\mathcal {V}}}(h,0) \rightarrow {{\mathcal {V}}}(r,d)$, where $h = \mathrm{gcd}(r,d)$. These are such that $\mathrm{det}\,\alpha _{r,d}({\mathcal {E}}) = \mathrm{det}\,{\mathcal {E}} \otimes {\mathcal {L}}^{\otimes d}$, which implies that $\mathrm{det}:{{\mathcal {V}}}(r,d) \rightarrow {{\mathcal {V}}}(1,d)$ is an $h$-1 map.

Given ${\mathcal {L}} \in {{\mathcal {V}}}(1,1)$ and the corresponding map $\alpha _{r,d}: {{\mathcal {V}}}(h,0) \rightarrow {{\mathcal {V}}}(r,d)$ from Theorem 11.3, we write $E_{{\mathcal {L}}}(r,d) := \alpha _{r,d}({{\mathcal {F}}}_h).$

Lemma 11.4

(Cf. Lemma 24 in [4]) Suppose that $\mathrm{gcd}(r,d) = 1$. Then, $E_{{\mathcal {L}}}(r,d) \otimes {{\mathcal {F}}}_h \cong E_{{\mathcal {L}}}(hr,hd)$.

Lemma 11.5

(Cf. Lemma 26 in [4]) Let ${\mathcal {L}}_1 \in {{\mathcal {V}}}(1,1)$. Then, for every ${\mathcal {E}} \in {{\mathcal {V}}}(r,d)$, there exists some ${\mathcal {L}}_0 \in {{\mathcal {V}}}(1,0)$ such that ${\mathcal {E}} = E_{{\mathcal {L}}_1}(r,d) \otimes {\mathcal {L}}_0$.