Séminaire Lotharingien de Combinatoire 84 (2022), Article B84c

QUIVERS WITH POTENTIALS ASSOCIATED TO TRIANGULATIONS OF CLOSED SURFACES WITH AT MOST TWO PUNCTURES

JAN GEUENICH, DANIEL LABARDINI-FRAGOSO, AND JOSÉ LUIS MIRANDA-OLVERA

Abstract. We tackle the classification problem of non-degenerate potentials for quivers arising from triangulations of surfaces in the cases left open by Geiss, Labardini-Fragoso and Schröer. Namely, for once-punctured closed surfaces of positive genus, we show that the quiver of any triangulation admits infinitely many non-degenerate potentials that are pairwise not weakly right-equivalent; we do so by showing that the potentials obtained by adding the 3-cycles coming from triangles and a fixed power of the cycle surrounding the puncture are well behaved under flips and QP-mutations. For twice-punctured closed surfaces of positive genus, we prove that the quiver of any triangulation admits exactly one non-degenerate potential up to weak right-equivalence, thus confirming the veracity of a conjecture of the aforementioned authors.

Contents

1. Introduction

2. Preliminaries 3 3. Once-punctured surfaces 11 4. Twice-punctured surfaces 15 Acknowledgements 20 References 21

1. Introduction

Albeit technical in nature, the problem of classifying all non-degenerate potentials on a given 2-acyclic quiver is relevant in different interesting, seemingly unrelated, contexts. In cluster algebra theory, having only one weak right-equivalence class means, very roughly speaking, that Derksen, Weyman and Zelevinsky’s representation-theoretic approach to the corresponding cluster algebra can be performed in essentially only one way.

The classification problem of non-degenerate potentials plays also a role in algebraic geometry and in symplectic geometry (more precisely, in the subjects of Bridgeland stability conditions and Fukaya categories). In [1, Theorem 9.9], the uniqueness of non-degenerate

2010 Mathematics Subject Classification. Primary 16P10, 16G20; Secondary 13F60, 57N05, 05E99. Key words and phrases. Surface, marked points, punctures, triangulation, flip, quiver, potential, mu-

tation, non-degenerate potential.

potentials on the quivers arising from positive genus closed surfaces with at least three punctures is used by Bridgeland and Smith to prove that there is a short exact sequence

\xymatrix{1 \ar[r] & Sph_{△}(D(Σ, 𝕄)) \ar[r] & Aut_{△}(D(Σ, 𝕄)) \ar[r] & MCG^{±}(Σ, 𝕄) \ar[r] & 1,}

where D(Σ, 𝕄) is the 3-Calabi–Yau triangulated category associated to (Σ, 𝕄), defined as the full subcategory that the dg-modules with finite-dimensional cohomology determine inside the derived category of the Ginzburg dg-algebra of the quiver with potential of any^{1} tagged triangulation of (Σ, 𝕄), the group Aut_{△}(D(Σ, 𝕄)) is the quotient of the group of auto-equivalences of D(Σ, 𝕄) that preserve the distinguished connected component Tilt_{△}(D(Σ, 𝕄)) by the subgroup of auto-equivalences that act trivially on Tilt_{△}(D(Σ, 𝕄)), Sph_{△}(D(Σ, 𝕄)) is the subgroup of Aut_{△}(D(Σ, 𝕄)) generated by (the quotient images of) the twist functors at the simple objects of a heart A ∈ Tilt_{△}(D(Σ, 𝕄)), and MCG^{±}(Σ, 𝕄) = MCG(Σ, 𝕄) ⋉ ℤ_{2}^{𝕄} is the signed mapping class group.

In [14, Theorem 1.1], Smith shows that, if (Σ, 𝕄) is a positive genus closed surface with at least three punctures (i.e., |𝕄| ≥ 3), then there is a linear fully faithful embedding of the 3-Calabi–Yau triangulated category D(Σ, 𝕄) into a Fukaya category of a 3-fold that fibres over Σ (with poles of a quadratic differential removed from Σ). He explains that the reason behind the hypothesis |𝕄| ≥ 3 in [14, Theorem 1.1] arises from the fact that, for positive genus closed surfaces with at least three punctures, the quiver of any triangulation has exactly one non-degenerate potential up to weak right-equivalence (a fact shown by Geiss, Labardini-Fragoso and Schröer [3]). See [14, Sections 1.3 and 2.2].

Together with results from his work [1] with Bridgeland, the embeddings from the previous paragraph allow Smith to obtain non-trivial computations of spaces of stability conditions on Fukaya categories of symplectic six-manifolds.

In this paper we prove the following result.

Theorem 1.1. (1) For once-punctured closed surfaces of positive genus, the quiver of any triangulation admits infinitely many non-degenerate potentials that are pairwise not weakly right-equivalent, provided the underlying field has characteristic zero.

(2) For twice-punctured closed surfaces of positive genus, the quiver of any triangulation admits exactly one non-degenerate potential up to weak right-equivalence, provided the underlying field is algebraically closed.

Let (Σ, 𝕄) be a once-punctured closed surface, n a positive integer and x ∈ K any scalar. For a triangulation τ of (Σ, 𝕄) let S(τ, x, n) be the potential obtained by adding the 3-cycles of Q(τ) arising from triangles of τ and the x-multiple of the n^{th} power of the cycle of Q(τ) that runs around the puncture of (Σ, 𝕄). The following result of independent interest plays a central role in our proof of part (1) of Theorem 1.1:

Theorem 1.2. Let (Σ, 𝕄) be a once-punctured closed surface. If τ and σ are triangulations of (Σ, 𝕄) related by the flip of an arc k ∈ τ, then the quivers with potential (Q(τ), S(τ, x, n)) and (Q(σ), S(σ, x, n)) are related by the mutation of quivers with potential μ_{k}.

1That D(Σ, 𝕄) is independent of the tagged triangulation used follows after combining results of Keller and Yang [5] and Labardini [8], see [9, Section 5].

That the quivers associated to triangulations of once-punctured closed surfaces of positive genus admit more than one weak right-equivalence class of non-degenerate potentials has always been expected, since the articles [2] of Derksen, Weyman and Zelevinsky and [6, 7] of Labardini-Fragoso exhibit non-degenerate potentials for the Markov quiver^{2} that are not weakly right-equivalent.

In [3, Theorem 8.4], Geiss, Labardini-Fragoso and Schröer proved that every quiver associated to some triangulation of a positive-genus closed surface with at least three punctures admits exactly one weak right-equivalence class of non-degenerate potentials, and conjectured that the same result holds in the case of two punctures. The reason why their proof fails for twice-punctured closed surfaces is that these do not admit triangulations all of whose arcs connect distinct punctures. The fact that such triangulations do exist for closed surfaces with at least three punctures plays an essential role in the proof of [3, Theorem 8.4].

The structure of the paper is straightforward: in Section 2 we prove a few facts (some of them quite technical) about the form of cycles and non-degenerate potentials for quivers arising from combinatorially nice triangulations of surfaces with empty boundary (see conditions (2.1) and (2.2)). Section 3 is devoted to proving part (1) of Theorem 1.1, whereas Section 4 is devoted to showing part (2).

2. Preliminaries

Let K be any field. For a quiver Q, the vertex span is the K-algebra R defined as the K-vector space with basis {e_{j} | j ∈ Q_{0}}, with multiplication defined as the K-bilinear extension of the rule

e_{i}e_{j} := δ_{i,j}e_{j}, for all i, j ∈ Q_{0},

where δ_{i,j} ∈ K is the Kronecker delta of i and j. Thus, R is (a K-algebra isomorphic to) K^{Q_{0}} with both sum and multiplication defined componentwise. The complete path algebra of Q is the K-vector space

K⟨⟨Q⟩⟩ := ∏_{ℓ∈ℤ_{≥0}} A^{(ℓ)},

where A^{(0)} := R, and for ℓ > 0, A^{(ℓ)} is the K-vector space with basis all the paths of length ℓ on Q. The multiplication of K⟨⟨Q⟩⟩ is defined in terms of the concatenation of paths.

The vertex span R is obviously a subring of K⟨⟨Q⟩⟩ (actually, a K-subalgebra), but it is often not a central subring. Despite of this, any ring automorphism φ : K⟨⟨Q⟩⟩ → K⟨⟨Q⟩⟩ such that φ|_{R} = 11_{R} will be said to be an R-algebra automorphism of K⟨⟨Q⟩⟩.

Definition 2.1. Let Q be a quiver and S, W ∈ K⟨⟨Q⟩⟩ be potentials on Q. We will say that:

2The Markov quiver arises as the quiver associated to any triangulation of the once-punctured torus.

(1) two cycles a_{1} ⋯ a_{ℓ} and b_{1} ⋯ b_{m} on Q are rotationally equivalent if a_{1} ⋯ a_{ℓ} = b_{1} ⋯ b_{m} or a_{1} ⋯ a_{ℓ} = b_{k} ⋯ b_{m}b_{1} ⋯ b_{k−1} for some k ∈ {2, …, m};

(2) S and W are rotationally disjoint if no cycle appearing in S is rotationally equivalent to a cycle appearing in W;

(3) S and W are cyclically equivalent if, with respect to the 𝔪-adic topology of K⟨⟨Q⟩⟩, the element S − W belongs to the topological closure of the vector subspace of K⟨⟨Q⟩⟩ spanned by all elements of the form a_{1} ⋯ a_{ℓ} − a_{2} ⋯ a_{ℓ}a_{1} with a_{1} ⋯ a_{ℓ} running through the set of all cycles on Q; notation: S ∼_{cyc} W;

(4) S and W are right-equivalent if there exists a right equivalence from S to W, i.e., an R-algebra automorphism φ : K⟨⟨Q⟩⟩ → K⟨⟨Q⟩⟩ that acts as the identity on the set of idempotents {e_{j} | j ∈ Q_{0}} and satisfies φ(S) ∼_{cyc} W; notation: S ∼_{r.e.} W;

(5) S and W are weakly right-equivalent if S and λW are right-equivalent for some non-zero scalar λ ∈ K.

Throughout the paper, (Σ, 𝕄) will be a punctured closed surface of positive genus. That is, Σ will be a compact, connected, oriented two-dimensional real differentiable manifold with positive genus and empty boundary, and 𝕄 will be a non-empty finite subset of Σ.

It is very easy to show that there exists at least one triangulation τ of (Σ, 𝕄) such that

(2.1) every puncture has valency at least 4 with respect to τ; (2.2) for any two arcs i and j of τ, the quiver Q(τ) has at most one arrow from j to i.

Throughout the paper, we will permanently suppose that τ satisfies (2.1) and (2.2).

Following Ladkani [10] we define two maps f, g : Q(τ)_{1} → Q(τ)_{1} as follows. Each triangle △ of τ gives rise to an oriented 3-cycle α_{△}β_{△}γ_{△} on Q(τ). We set f(α_{△}) = γ_{△}, f(β_{△}) = α_{△} and f(γ_{△}) = β_{△}. Now, given any arrow α of Q(τ), the quiver Q(τ) has exactly two arrows starting at the terminal vertex of α. One of these two arrows is f(α). We define g(α) to be the other arrow.

Note that the map f (respectively g) splits the arrow set of Q(τ) into f-orbits (respectively g-orbits). The set of f-orbits is in one-to-one correspondence with the set of triangles of τ. All f-orbits have exactly three elements. The set of g-orbits is in one-to-one correspondence with the set of punctures of (Σ, 𝕄). For every arrow α of Q(τ), we denote the size of the g-orbit of α by m_{α} (m_{α} ≥ 4 by (2.1)). Note that g^{m_{α}−1}(α)g^{m_{α}−2}(α) ⋯ g(α)α is a cycle surrounding the puncture p corresponding to the g-orbit of α. We denote this cycle by G(α) or G(p). On the other hand, for every arrow β of Q(τ) and any non-negative integer r, we use the notation G(r, β) to denote the path g^{r−1}(β)g^{r−2}(β) ⋯ g(β)β. Similarly, we use the notation F (r, β) to denote the path f^{r−1}(β)f^{r−2}(β) ⋯ f(β)β.

Let x = (x_{p})_{p∈𝕄} be a choice of a non-zero scalar x_{p} ∈ K for each puncture p ∈ 𝕄. For ideal triangulations which satisfy (2.1) and (2.2) the potential S(τ, x) defined by the second author [8] takes a simple form, namely,

S(τ, x) = T (τ) + ∑_{p∈ℙ} x_{p}G(p),

with T (τ) ∼_{cyc} ∑_{α∈Γ}(f^{2}(α)f(α)α) for any fixed set Γ containing exactly one arrow from each triangle of τ.

Lemma 2.2 (Types of cycles). Let (Σ, 𝕄) be a punctured surface with empty boundary, and let τ be a triangulation of (Σ, 𝕄) that satisfies (2.1) and (2.2). Then every cycle in Q(τ) is rotationally equivalent to a cycle of one of the following types:

(f-cycles) (f^{2}(α)f(α)α)^{n} for some n ≥ 1; (g-cycles) (g^{m_{β}−1}(β)g^{m_{β}−2}(β) ⋯ g(β)β)^{n} for some n ≥ 1; (fg-cycles) f^{2}(a)f(a)λ for some arrow a and some path λ, such that λ = g^{−1}f(a)λ′ with

λ′ of positive length.

Proof. Let ξ = α_{1} ⋯ α_{r} be any cycle on Q(τ). Write α_{r+1} = α_{1}, and notice that, for every ℓ = 1, …, r, we have either α_{ℓ} = f(α_{ℓ+1}) or α_{ℓ} = g(α_{ℓ+1}). Let s_{ξ} be the length-r sequence of fs and gs that has an f at the ℓ^{th} place if α_{ℓ} = f(α_{ℓ+1}) and a g otherwise.

If s_{ξ} consists only of fs, then ξ is rotationally equivalent to (f^{2}(α)f(α)α)^{n} for some arrow α and some n ≥ 1. Furthermore, if s_{ξ} consists only of gs, then ξ is rotationally equivalent to (g^{m_{β}−1}(β)g^{m_{β}−2}(β) ⋯ g(β)β)^{n} for some arrow β and some n ≥ 1. Therefore, if s_{ξ} involves only fs or only gs, then ξ is an f-cycle or a g-cycle.

Suppose that at least one f and at least one g appear in s_{ξ}. Rotating ξ if necessary, we can assume that s_{ξ} starts with an f followed by a g, i.e., s_{ξ} = (f, g, …). This means that, if we set a := f^{−1}(α_{2}), then α_{1} = f^{2}(a), α_{2} = f(a) and α_{3} = g^{−1}f(a). By (2.2), a is the only arrow in Q(τ)_{1} such that α_{1}α_{2}a is a cycle. Since α_{3} = g^{−1}f(a) ≠ a, this implies ξ = f^{2}(a)f(a)g^{−1}f(a)λ′ with λ′ of positive length. □

Remark 2.3. As in the case of cycles, every path falls within exactly one of three types of paths: f-paths, g-paths, and fg-paths.

By Lemma 2.2, up to cyclical equivalence we can write every potential S in Q(τ) as S = S_{f} + S_{g} + S_{fg}, where

S_{f} = ∑_{△} ∑_{n=1}^{∞} z_{△,n}(f^{2}(α_{△})f(α_{△})α_{△})^{n},

S_{g} = ∑_{p∈ℙ} ∑_{n=1}^{∞} ν_{p,n}(G(p))^{n},
S_{fg} = ∑_{a∈Q(τ)_{1}} f^{2}(a)f(a)ω_{a},

with each z_{△,n}, ν_{p,n} ∈ K, and ω_{a} a possibly infinite linear combination of paths of the form g^{−1}f(a)λ′ for each a ∈ Q(τ).

Lemma 2.4. Let (Σ, 𝕄) be a punctured surface with empty boundary, and let τ be a triangulation of (Σ, 𝕄) that satisfies (2.1) and (2.2). Every non-degenerate potential S on Q(τ) is right-equivalent to a potential of the form T (τ) + U for some U rotationally disjoint from T (τ).

Proof. By (2.2) the hypotheses of [3, Corollary 2.5] are satisfied. So, if S is a nondegenerate potential, then every f-cycle f^{2}(α)f(α)α appears in S. Hence,

S ∼_{cyc} ∑_{△} z_{△,1}f^{2}(α_{△})f(α_{△})α_{△} + U′,

with all z_{△,1} ≠ 0 and U′ rotationally disjoint from T (τ).

We define an R-algebra automorphism φ : K⟨⟨Q(τ)⟩⟩ → K⟨⟨Q(τ)⟩⟩ by means of the rule

φ(α_{△}) = \frac{1}{z_{△,1}} α_{△}.

We see that φ(S) ∼_{cyc} T (τ)+U, for some potential U rotationally disjoint from T (τ). □

Lemma 2.5 (Replacing f-potentials and fg-potentials by longer ones). Let (Σ, 𝕄) be a punctured surface with empty boundary, and let τ be a triangulation of (Σ, 𝕄) that satisfies (2.1) and (2.2). Let ϕ be one of the symbols f and fg, and let ν be the other symbol, so that {ϕ, ν} = {f, fg} as sets of symbols. If W, A ∈ K⟨⟨Q(τ)⟩⟩ are potentials rotationally disjoint from T (τ), and if A_{ϕ} ≠ 0, then there exists a potential B ∈ K⟨⟨Q(τ)⟩⟩ which is rotationally disjoint from T (τ) and satisfies the following four conditions:

short(B_{ϕ}) > short(A_{ϕ});
short(B_{g}) ≥ min(short(A_{g}), short(A_{ϕ}) + 1);
short(B_{ν}) ≥ min(short(A_{ν}), short(A_{ϕ}) + 1);
(Q(τ), T (τ) + W + A) ∼_{r.e.} (Q(τ), T (τ) + W + B).

Proof. Let us deal with the case ϕ = f. Write

A_{f} = ∑_{△} ∑_{n≥ \frac{short(A_{f})}{3}} z_{△,n} (f^{2}(α_{△})f(α_{△})α_{△})^{n} ,

and define an R-algebra homomorphism φ : K⟨⟨Q(τ)⟩⟩ → K⟨⟨Q(τ)⟩⟩ by means of the rule

φ(α_{△}) = α_{△} − ∑_{n≥ \frac{short(A_{f})}{3}} z_{△,n}α_{△} (f^{2}(α_{△})f(α_{△})α_{△})^{n−1} .

Then φ is a unitriangular automorphism of depth short(A_{f}) − 3, and

φ(T (τ) + W + A) = T (τ) − A_{f} + W + A + (φ(W + A) − (W + A)).

Consequently, if we set B = A_{g} + A_{fg} + (φ(W + A) − (W + A)), then:

• φ(T (τ) + W + A) = T (τ) + W + B;

• short(φ(W + A) − (W + A)) ≥ depth(φ) + short(W + A) ≥ short(A_{f}) − 3 + 4 =

short(A_{f}) + 1;
• short(B_{f}) = short((φ(W + A) − (W + A))_{f}) ≥ short(A_{f}) + 1;
• short(B_{g}) ≥ min(short(A_{g}), short((φ(W + A) − (W + A))_{g})
≥ min(short(A_{g}), short(A_{f}) + 1); and
• short(B_{fg}) ≥ min(short(A_{fg}, short((φ(W + A) − (W + A))_{fg}))
≥ min(short(A_{fg}), short(A_{f}) + 1).

Now we deal with the case ϕ = fg. Write

A_{fg} = ∑_{a∈Q(τ)_{1}} f^{2}(a)f(a)ω_{a},

with ω_{a} ∈ e_{h(a)}K⟨⟨Q(τ)⟩⟩e_{t(a)} for each a ∈ Q(τ)_{1}. Furthermore, we define an R-algebra homomorphism φ : K⟨⟨Q(τ)⟩⟩ → K⟨⟨Q(τ)⟩⟩ by means of the rule φ(a) = a − ω_{a} for a ∈ Q(τ)_{1}. Then φ is a unitriangular automorphism of depth short(A_{fg}) − 3, and

φ(T (τ) + W + A) = ∑_{△} (f^{2}(α_{△}) − ω_{f^{2}(α_{△})}) (f(α_{△}) − ω_{f(α_{△})}) (α_{△} − ω_{α_{△}})

+ W + A + (φ(W + A) − (W + A))

∼_{cyc} T (τ) + W + A_{f} + A_{g} + (φ(W + A) − (W + A))

+ ∑_{a∈Q(τ)_{1}} f^{2}(a)ω_{f(a)}ω_{a} − ∑_{△} ω_{f^{2}(a_{△})}ω_{f(a_{△})}ω_{a_{△}}.

Consequently, if we set

B = A_{f} + A_{g} + (φ(W + A) − (W + A)) + ∑_{a∈Q(τ)_{1}} f^{2}(a)ω_{f(a)}ω_{a} − ∑_{△} ω_{f^{2}(a_{△})}ω_{f(a_{△})}ω_{a_{△}},

then:

• φ(T (τ) + W + A) ∼_{cyc} T (τ) + W + B;

• short (φ(W + A) − (W + A)) ≥ depth(φ) + short(W + A) ≥ short(A_{fg}) − 3 + 4 =

short(A_{fg}) + 1;

• short (∑_{a∈Q(τ)_{1}} f^{2}(a)ω_{f(a)}ω_{a}) ≥ 2 short(A_{fg}) − 3 ≥ short(A_{fg}) + 4 − 3

= short(A_{fg}) + 1 (since short(A_{fg}) ≥ 4);

• short (∑_{△} ω_{f^{2}(a_{△})}ω_{f(a_{△})}ω_{a_{△}}) ≥ 3 short(A_{fg}) − 6 ≥ short(A_{fg}) + 8 − 6

≥ short(A_{fg}) + 1;

• short(B_{fg}) ≥ min (short (φ(W + A) − (W + A)) , short (∑_{a∈Q(τ)_{1}} f^{2}(a)ω_{f(a)}ω_{a}) ,

short (∑_{△} ω_{f^{2}(a_{△})}ω_{f(a_{△})}ω_{a_{△}})) ≥ short(A_{fg}) + 1;

• short(B_{g}) ≥ min (short(A_{g}), short (φ(W + A) − (W + A)) ,

short (∑_{a∈Q(τ)_{1}} f^{2}(a)ω_{f(a)}ω_{a}) , short (∑_{△} ω_{f^{2}(a_{△})}ω_{f(a_{△})}ω_{a_{△}}))
≥ min (short(A_{g}), short(A_{fg}) + 1); and

• short(B_{f}) ≥ min (short(A_{g}), short (φ(W + A) − (W + A)) ,

short (∑_{a∈Q(τ)_{1}} f^{2}(a)ω_{f(a)}ω_{a}) , short (∑_{△} ω_{f^{2}(a_{△})}ω_{f(a_{△})}ω_{a_{△}}))
≥ min (short(A_{f}), short(A_{fg}) + 1).

Lemma 2.5 is proved. □

Proposition 2.6 (Replacing potentials by sums of powers of g-cycles). Let (Σ, 𝕄) be a punctured surface with empty boundary, and let τ be a triangulation of (Σ, 𝕄) satisfying (2.1) and (2.2). If U, Z ∈ K⟨⟨Q(τ)⟩⟩ are potentials rotationally disjoint from T (τ), then there exist a unitriangular automorphism φ : K⟨⟨Q(τ)⟩⟩ → K⟨⟨Q(τ)⟩⟩ of depth at least short(U) − 3 and a potential W ∈ K⟨⟨Q(τ)⟩⟩ involving only positive powers of gcycles, such that short(W) ≥ short(U) and φ is a right-equivalence (Q(τ), T (τ)+Z+U) → (Q(τ), T (τ) + Z + W).

Proof. Set W_{0} = U_{g} and U_{0} = U−U_{g}. We obviously have short(U_{0}), short(W_{0}) ≥ short(U).

Claim 1. There exist sequences (U_{n})_{n≥1} and (W_{n})_{n≥1} of potentials on the quiver Q(τ), and a sequence (φ_{n})_{n≥1} of unitriangular automorphisms of K⟨⟨Q(τ)⟩⟩, such that the following properties are satisfied for every n ≥ 1:

• φ_{n} is a right-equivalence (Q(τ), T (τ) + Z + U_{n−1} + W_{n−1}) → (Q(τ), T (τ) + Z + U_{n} + W_{n});

• depth(φ_{n}) = short(U_{n−1}) − 3;

• each of U_{n} and W_{n} is rotationally disjoint from T (τ), U_{n} does not involve powers of g-cycles, and W_{n} involves only powers of g-cycles;

• short(W_{n} − W_{n−1}) ≥ short(U_{n−1}) + 1;

• short(U_{n+1}) ≥ short(U_{n−1}) + 1.

We shall produce the three sequences (U_{n})_{n≥1}, (W_{n})_{n≥1} and (φ_{n})_{n≥1} recursively. Fix a positive integer n. If U_{n−1} = 0, we set U_{n} to be U_{n−1}, W_{n} to be W_{n−1} and φ_{n} to be the identity of K⟨⟨Q(τ)⟩⟩. Otherwise, let ϕ_{n−1}, ν_{n−1} ∈ {f, fg} be symbols such that {ϕ_{n−1}, ν_{n−1}} = {f, fg} and short((U_{n−1})_{ϕ_{n−1}}) ≤ short((U_{n−1})_{ν_{n−1}}). By the proof of Lemma 2.5, there exist a potential V_{n} ∈ K⟨⟨Q(τ)⟩⟩ rotationally disjoint from T (τ) and a unitriangular automorphism φ_{n} : K⟨⟨Q(τ)⟩⟩ → K⟨⟨Q(τ)⟩⟩ such that

• depth(φ_{n}) = short(U_{n−1}) − 3;

• φ_{n} is a right-equivalence (Q(τ), T (τ) + Z + W_{n−1} + U_{n−1}) → (Q(τ), T (τ) + Z + W_{n−1} + V_{n});

• short((V_{n})_{ϕ_{n−1}}) > short((U_{n−1})_{ϕ_{n−1}});

• short((V_{n})_{g}) ≥ min(short((U_{n−1})_{g}), short((U_{n−1})_{ϕ_{n−1}})+1) = short((U_{n−1})_{ϕ_{n−1}})+1;

• short((V_{n})_{ν_{n−1}}) ≥ min(short((U_{n−1})_{ν_{n−1}}), short((U_{n−1})_{ϕ_{n−1}}) + 1). We set U_{n} = V_{n} − (V_{n})_{g} and W_{n} = W_{n−1} + (V_{n})_{g}. It is clear that the first four properties stated in the claim are satisfied. For the fifth property, note that, if ϕ_{n} = ϕ_{n−1}, then short((U_{n})_{ϕ_{n}}) > short((U_{n−1})_{ϕ_{n−1}}), whereas, if ϕ_{n} ≠ ϕ_{n−1}, then

short((U_{n+1})_{ϕ_{n}}) > short((U_{n})_{ϕ_{n}}) ≥ short(U_{n−1})

and

short((U_{n+1})_{ϕ_{n−1}}) ≥ min(short((U_{n})_{ϕ_{n−1}}), short((U_{n})_{ϕ_{n}}) + 1)
≥ min ( short((U_{n−1})_{ϕ_{n−1}}) + 1,
min ( short((U_{n−1})_{ν_{n−1}}), short((U_{n−1})_{ϕ_{n−1}}) + 1) + 1)
> short((U_{n−1})_{ϕ_{n−1}}).

These facts, together with the observation that for each n ≥ 0 we have short((U_{n})_{ϕ_{n}}) = short(U_{n}), allow us to deduce that short(U_{n+1}) ≥ short(U_{n−1}) + 1 for all n ≥ 1, thus completing the proof of Claim 1.

From the claim, we see that

lim_{n→∞} short(U_{n}) = ∞, lim_{n→∞} short(W_{n} − W_{n−1}) = ∞ and lim_{n→∞} depth(φ_{n}) = ∞.

Hence, if we set W = lim_{n→∞} W_{n}, then φ := lim_{n→∞} φ_{n} ∘ ⋯ ∘ φ_{1} is a right-equivalence (Q(τ), T (τ) + Z + U) → (Q(τ), T (τ) + Z + W). Proposition 2.6 follows. □

Lemma 2.7. Let (Σ, 𝕄) be a punctured surface with empty boundary, let τ be a triangulation of (Σ, 𝕄) satisfying (2.1) and (2.2), and let x = (x_{p})_{p∈𝕄} be any choice of non-zero scalars. Suppose that m and t are positive integers and U, W ∈ K⟨⟨Q(τ)⟩⟩ are potentials rotationally disjoint from S(τ, x) that satisfy the following properties:

(1) short(U) ≥ m;

(2) 2 short(W) − 3 > m;

(3) W = λf(a)aG(t, g^{−t}(a))c for some non-zero scalar λ ∈ K, some arrow a, and some path c.

Then there exists a unitriangular R-algebra automorphism ζ : K⟨⟨Q(τ)⟩⟩ → K⟨⟨Q(τ)⟩⟩ of depth short(W) − 3 that serves as a right-equivalence between the QPs (Q(τ), S(τ, x) + U + W) and (Q(τ), S(τ, x) + U + U′ + W′) for some potentials U′, W′ ∈ K⟨⟨Q(τ)⟩⟩ that satisfy:

(1) short(U′) > m;

(2) short(W′) > short(W);

(3) W′ = λ′f(b)bG(t − 1, g^{−(t−1)}(b))c′ for some non-zero scalar λ′, some arrow b, and some path c′.

Proof. Let ζ : K⟨⟨Q(τ)⟩⟩ → K⟨⟨Q(τ)⟩⟩ be the R-algebra homomorphism given by the rule

ζ(f^{−1}(a)) = f^{−1}(a) − λG(t, g^{−t}(a))c.

Since τ satisfies (2.2), short(W) − 3 is a positive integer by Lemma 2.2, and hence ζ is actually a unitriangular automorphism of K⟨⟨Q(τ)⟩⟩. The depth of ζ is obviously short(W) − 3.

The arrow f^{−1}(a) connects two arcs of τ. Let p_{f^{−1}(a)} be the puncture at which these arcs are incident. Direct computation shows that

ζ(S(τ, x) + U + W) ∼_{cyc} S(τ, x) − W − λx_{p_{f^{−1}(a)}}G(m_{f^{−1}(a)} − 1, gf^{−1}(a))G(t, g^{−t}(a))c
+ U + W + (ζ(U + W) − (U + W))

= S(τ, x) − λx_{p_{f^{−1}(a)}}G(m_{f^{−1}(a)} − 2, g^{2}f^{−1}(a))gf^{−1}(a)g^{−1}(a)G(t − 1, g^{−t}(a))c

+ U + (ζ(U + W) − (U + W))

∼_{cyc} S(τ, x) − λx_{p_{f^{−1}(a)}}gf^{−1}(a)g^{−1}(a)G(t − 1, g^{−t}a)cG(m_{f^{−1}(a)} − 2, g^{2}f^{−1}(a))

+ U + (ζ(U + W) − (U + W)) .

So, the lemma follows if we remember that gf^{−1}(a) = fg^{−1}(a) and set

U′ := ζ(U + W) − (U + W),
λ′ := −λx_{p_{f^{−1}(a)}},
b := g^{−1}(a),
c′ := cG(m_{f^{−1}(a)} − 2, g^{2}f^{−1}(a))
and W′ := λ′f(b)bG(t − 1, g^{−(t−1)}(b))c′.

Indeed, property (3) is obviously satisfied, whereas the inequalities short(U) ≥ m, depth(ζ) > 0 and 2 short(W) − 3 > m imply that

short(U′) ≥ min(short(ζ(U) − U), short(ζ(W) − W))

≥ min(depth(ζ) + short(U), depth(ζ) + short(W)) = min(depth(ζ) + short(U), 2 short(W) − 3) > m.

Furthermore, we also have

short(W′) = m_{f^{−1}(a)} − 2 + short(W) − 1 > short(W),

where the inequality follows from the fact that τ satisfies (2.1). □

Corollary 2.8 (Replacing certain cycles by sums of long g-cycles). Under the hypotheses of Lemma 2.7, if the path c is assumed to be an arrow, then there exists a unitriangular R-algebra automorphism Π : K⟨⟨Q(τ)⟩⟩ → K⟨⟨Q(τ)⟩⟩ of depth at least min(m − 3, short(W) − 3) that serves as a right-equivalence between the QPs (Q(τ), S(τ, x) + U + W) and (Q(τ), S(τ, x) + U + ξ) for some potential ξ that involves only positive powers of g-cycles and satisfies short(ξ) > m.

Proof. This corollary follows from an inductive use of Lemma 2.7. Set U_{0} = U, W_{0} = W, a_{0} = a, c_{0} = c, and λ_{0} = λ. Using Lemma 2.7, we obtain a unitriangular automorphism ζ_{1} : K⟨⟨Q(τ)⟩⟩ → K⟨⟨Q(τ)⟩⟩, potentials Z_{1}, W_{1} ∈ K⟨⟨Q(τ)⟩⟩, an arrow a_{1}, a path c_{1}, and a non-zero scalar λ_{1}, such that:

(1) depth(ζ_{1}) = short(W_{0}) − 3;

(2) ζ_{1} is a right-equivalence (Q(τ), S(τ, x)+U_{0}+W_{0}) → (Q(τ), S(τ, x)+U_{0}+Z_{1}+W_{1});

(3) short(Z_{1}) > m and short(W_{1}) ≥ short(W_{0}) + 1;

(4) W_{1} = λ_{1}f(a_{1})a_{1}G(t − 1, g^{−(t−1)}(a_{1}))c_{1}. Setting U_{1} = U_{0}+Z_{1}, we see that U_{1}, W_{1}, a_{1}, c_{1} and λ_{1} satisfy the hypotheses of Lemma 2.7 for the integers m and t − 1.

Assuming that for i ∈ {0, …, t − 1} we have U_{i}, W_{i}, a_{i}, c_{i} and λ_{i} satisfying the hypotheses of Lemma 2.7 for the integers m and t − i, we can produce a unitriangular automorphism ζ_{i+1} : K⟨⟨Q(τ)⟩⟩ → K⟨⟨Q(τ)⟩⟩, potentials Z_{i+1}, W_{i+1} ∈ K⟨⟨Q(τ)⟩⟩, an arrow a_{i+1}, a path c_{i+1}, and a non-zero scalar λ_{i+1}, such that:

(1) depth(ζ_{i+1}) = short(W_{i}) − 3; (2) ζ_{i+1} is a right-equivalence

(Q(τ), S(τ, x) + U_{i} + W_{i}) → (Q(τ), S(τ, x) + U_{i} + Z_{i+1} + W_{i+1});

(3) short(Z_{i+1}) > m and short(W_{i+1}) ≥ short(W_{i}) + 1;

(4) W_{i+1} = λ_{i+1}f(a_{i+1})a_{i+1}G(t − i − 1, g^{−(t−i−1)}(a_{i+1}))c_{i+1}.

Setting U_{i+1} = U_{i}+Z_{i+1}, we see that U_{i+1}, W_{i+1}, a_{i+1}, c_{i+1} and λ_{i+1} satisfy the hypotheses of Lemma 2.7 for the integers m and t − (i + 1).

The composition ζ = ζ_{t}∘ζ_{t−1}∘⋯∘ζ_{1} is a unitriangular automorphism of K⟨⟨Q(τ)⟩⟩ that has depth at least short(W)−3 and serves as a right-equivalence (Q(τ), S(τ, x)+U +W) → (Q(τ), S(τ, x) + U_{t} + W_{t}). Notice that U_{t} = U + ∑_{i=1}^{t} Z_{i}, that short (∑_{i=1}^{t} Z_{i}) > m, and that short(W_{t}) ≥ short(W) + t = 2 short(W) − 3 > m.

By Proposition 2.6, there exists a unitriangular automorphism φ : K⟨⟨Q(τ)⟩⟩ → K⟨⟨Q(τ)⟩⟩ of depth greater than m − 3 that makes (Q(τ), S(τ, x) + U + ∑_{i=1}^{t} Z_{i} + W_{t}) right-equivalent to (Q(τ), S(τ, x) + U + ξ) for some potential ξ ∈ K⟨⟨Q(τ)⟩⟩ that involves only powers of g-cycles and satisfies short(ξ) ≥ short (∑_{i=1}^{t} Z_{i} + W_{t}) > m.

From the two previous paragraphs we deduce that the automorphism Π := φ∘ζ satisfies the desired conclusion of Corollary 2.8. □

3. Once-punctured surfaces

In [6] and [8], the second author showed that the potentials S(τ, x) are well behaved with respect to flips and mutations, in the sense that, if two triangulations are related by a flip, then the associated QPs are related by the corresponding QP-mutation. In this section, we show that for once-punctured closed surfaces the same result is true for a wider class of potentials. Namely, given a triangulation τ of a once-punctured closed surface of positive genus (Σ, 𝕄), a scalar x ≠ 0 and a positive integer n, we define a potential S(τ, x, n) as

S(τ, x, n) = T (τ) + xG(p)^{n},

where p is the only puncture in (Σ, 𝕄).

Theorem 3.1. Let (Σ, 𝕄) be a once-punctured closed surface of positive genus, n be any positive integer, and x ∈ K be any scalar. If τ and σ are triangulations of (Σ, 𝕄) that are related by the flip of an arc k ∈ τ, then the QPs μ_{k}(Q(τ), S(τ, x, n)) and (Q(σ), S(σ, x, n)) are right-equivalent.

Proof. Let a_{i}, b_{i}, c_{i}, i = 1, 2, be the arrows in the two triangles with one side k as in Figure 1.

\begin{matrix}a_{2} \\ c_{2} \\ b_{2} \\ a_{1} \\ b_{1} \\ c_{1} \\ k \end{matrix}

Figure 1. The two triangles with one side k.

Up to rotation we can write G(p) = a_{1}Aa_{2}B. Notice that b_{2}c_{1}, b_{1}c_{2} are factors of G(p), but b_{1}c_{1} and b_{2}c_{2} are not. The potential μ̃_{k}(S(τ, x, n)) is cyclically equivalent to

[T (τ)] + x([a_{1}Aa_{2}B])^{n} + c_{1}^{∗}b_{2}^{∗}[b_{2}c_{1}] + c_{2}^{∗}b_{1}^{∗}[b_{1}c_{2}] + c_{1}^{∗}b_{1}^{∗}[b_{1}c_{1}] + c_{2}^{∗}b_{2}^{∗}[b_{2}c_{2}] = T (σ) + a_{1}[b_{1}c_{1}] + a_{2}[b_{2}c_{2}] + c_{1}^{∗}b_{1}^{∗}[b_{1}c_{1}] + c_{2}^{∗}b_{2}^{∗}[b_{2}c_{2}] + x(a_{1}[A]a_{2}[B])^{n},

where the paths [A], [B] are the result of replacing b_{2}c_{1}, b_{1}c_{2} in A, B by [b_{2}c_{1}], [b_{1}c_{2}], respectively.

a_{1} a_{2} B A Q(τ) b_{1}^{∗} c_{1}^{∗} c_{2}^{∗} b_{2}^{∗} [B] [A] Q(σ)

Figure 2. The cycle on Q(τ) and Q(σ) surrounding the puncture.

We define R-algebra homomorphisms φ_{1}, φ_{2} : K⟨⟨Q(τ)⟩⟩ → K⟨⟨Q(τ)⟩⟩ by means of the rules

φ_{1}(a_{1}) = a_{1} − c_{1}^{∗}b_{1}^{∗};
φ_{2}([b_{1}c_{1}]) = [b_{1}c_{1}] − x ∑_{j=0}^{n−1}(−1)^{j}[A]a_{2}[B]((a_{1} − c_{1}^{∗}b_{1}^{∗})[A]a_{2}[B])^{n−j−1}(c_{1}^{∗}b_{1}^{∗}[A]a_{2}[B])^{j}.

Applying φ_{1} to μ̃_{k}(S(τ, x, n)), we get

φ_{1}(μ̃_{k}(S(τ, x, n))) ∼_{cyc} T (σ) + a_{1}[b_{1}c_{1}] + a_{2}[b_{2}c_{2}] + c_{2}^{∗}b_{2}^{∗}[b_{2}c_{2}] + x((a_{1} − c_{1}^{∗}b_{1}^{∗})[A]a_{2}[B])^{n}
∼_{cyc} T (σ) + a_{1}[b_{1}c_{1}] + a_{2}[b_{2}c_{2}] + c_{2}^{∗}b_{2}^{∗}[b_{2}c_{2}] + x(−1)^{n}(c_{1}^{∗}b_{1}^{∗}[A]a_{2}[B])^{n}

+ x ∑_{j=0}^{n−1}(−1)^{j}a_{1}[A]a_{2}[B]((a_{1} − c_{1}^{∗}b_{1}^{∗})[A]a_{2}[B])^{n−j−1}(c_{1}^{∗}b_{1}^{∗}[A]a_{2}[B])^{j}.

The potential φ_{2}φ_{1}(μ̃_{k}(S(τ, x, n))) is cyclically equivalent to

φ_{2}φ_{1}(μ̃_{k}(S(τ, x, n))) ∼_{cyc} T (σ) + a_{1}[b_{1}c_{1}] + a_{2}[b_{2}c_{2}] + c_{2}^{∗}b_{2}^{∗}[b_{2}c_{2}] + x(−1)^{n}(c_{1}^{∗}b_{1}^{∗}[A]a_{2}[B])^{n}.

In an analogous way, we define R-algebra homomorphisms φ_{3}, φ_{4} : K⟨⟨Q(τ)⟩⟩ → K⟨⟨Q(τ)⟩⟩ by means of the rules

φ_{3}(a_{2}) = a_{2} − c_{2}^{∗}b_{2}^{∗};
φ_{4}([b_{2}c_{2}]) = [b_{2}c_{2}]

− x(−1)^{n} ∑_{j=0}^{n−1}(−1)^{j}[B]c_{1}^{∗}b_{1}^{∗}[A]((a_{2} − c_{2}^{∗}b_{2}^{∗})[B]c_{1}^{∗}b_{1}^{∗}[A])^{n−j−1}(c_{2}^{∗}b_{2}^{∗}[B]c_{1}^{∗}b_{1}^{∗}[A])^{j}.

We obtain

φ_{4}φ_{3}φ_{2}φ_{1}(μ̃_{k}(S(τ, x, n))) ∼_{cyc} T (σ) + a_{1}[b_{1}c_{1}] + a_{2}[b_{2}c_{2}] + x(c_{1}^{∗}b_{1}^{∗}[A]c_{2}^{∗}b_{2}^{∗}[B])^{n}
∼_{cyc} S(σ, x, n) + a_{1}[b_{1}c_{1}] + a_{2}[b_{2}c_{2}].

Therefore, the QPs μ_{k}(Q(τ), S(τ, x, n)) and (Q(σ), S(σ, x, n)) are right-equivalent. □

Remark 3.2. (1) For once-punctured closed surfaces, Theorem 3.1 constitutes a gen-

eralization of the second author’s results [6, Theorem 30] and [8, Theorem 8.1].

(2) It was observed by Ladkani [11, Proposition 3.1] that the proof of [6, Theorem 30] can be applied without change to produce a proof of Theorem 3.1 above for x = 0.

(3) In his Master thesis [4], the first author of this paper proved Theorem 3.1 for the once-punctured torus and x ≠ 0.

(4) Motivated by the first author’s Master thesis, the third author proved Theorem 3.1 in his Undergraduate thesis [12].

Proposition 3.3. Let (Σ, 𝕄) be a once-punctured closed surface of positive genus, τ a triangulation of (Σ, 𝕄), and x ∈ K a non-zero scalar. If the characteristic of the field K is zero, then

dim_{K}(P(Q(τ), S(τ, x, n))) < ∞ and lim_{n→∞} dim_{K}(P(Q(τ), S(τ, x, n))) = ∞.

Proof. For the proof of finite-dimensionality we follow ideas suggested by Ladkani in his proof of [10, Proposition 4.2], whereas our proof that the limits of the dimensions are ∞ follows ideas that appear in the first author’s Master thesis.

First, note that when we compute the cyclic derivative of S(τ, x, n) with respect to an arrow α, we get

(3.1)  ∂_{α}(S(τ, x, n)) = f^{2}(α)f(α) + xnG(nm_{α} − 1, g(α)).

So, f^{2}(α)f(α) and −xnG(nm_{α} − 1, g(α)) become equal in the Jacobian algebra P(Q(τ), S(τ, x, n)).

Every fg-path of length three has the form f^{2}(α)f(α)g^{−1}f(α) or gf^{2}(α)f^{2}(α)f(α) for some arrow α, and it is hence equal to

−xnG(nm_{α} − 1, g(α))g^{−1}f(α) = −xnG(nm_{α} − 3, g^{3}(α))g^{2}(α)g(α)f^{−1}g(α)

or

−xngf^{2}(α)G(nm_{α} − 1, g(α)) = −xnfg^{−1}(α)g^{−1}(α)g^{−2}(α)G(nm_{α} − 3, g(α))

in P(Q(τ), S(τ, x, n)). Thus, in P(Q(τ), S(τ, x, n)), every fg-path of length three is equal to another fg-path of length greater than three. In the same vein, an easy inductive argument shows that, in the Jacobian algebra, every fg-path is equal to an arbitrarily long fg-path, and therefore equal to 0 ∈ P(Q(τ), S(τ, x, n)).

Any f-path F (r, f(β)) = F (r −2, β)f^{2}(β)f(β) of length r greater than three is equal to the fg-path −xnF (r −2, β)G(nm_{β} −1, g(β)) in P(Q(τ), S(τ, x, n)), and, in this way, to 0. Furthermore, any g-path of the form G(r, g(β)) = G(r − nm_{β} + 1, β)G(nm_{β} − 1, g(β)) with length greater than nm_{β} is equal to the fg-path x^{−1}n^{−1}G(r − nm_{β} + 1, β)f^{2}(β)f(β), hence equal to 0 in the Jacobian algebra. Notice that, here, we have used that K is a field of characteristic zero.

Thus far, we have shown that every path of length greater than nm_{α} is equivalent to 0 in the Jacobian algebra P(Q(τ), S(τ, x, n)), and therefore the latter has finite dimension.

On the other hand, as the cyclic derivative of S(τ, x, n) with respect to any arrow α is equal to the sum of an f-path of length two and a scalar multiple of a g-path of length nm_{α} − 1 (cf. (3.1)), and since no g-path is a multiple of any f-path of length greater than one, we conclude that for any a, b ∈ K⟨⟨Q(τ)⟩⟩, no g-path of length smaller than nm_{α} − 1 appears in the expression of the element a∂_{α}(S(τ, x, n))b as a possibly infinite sum of paths on the quiver Q(τ). From this, it follows that no finite linear combination of g-paths of lengths smaller than nm_{α} − 1 can be written as a limit of finite sums of elements of the form a∂_{α}(S(τ, x, n))b, i.e., the set of g-paths of length smaller than nm_{α} − 1 is linearly independent in the Jacobian algebra P(Q(τ), S(τ, x, n)). Therefore, dim_{K}(P(Q(τ), S(τ, x, n))) ≥ nm_{α} − 2. □

Corollary 3.4. Over a field of characteristic zero, the quiver of any triangulation of a once-punctured closed surface of positive genus admits infinitely many non-degenerate potentials up to weak right-equivalence.

Remark 3.5. (1) In the case of the once-punctured torus, Proposition 3.3 was proved by the first author in his Master thesis [4].

(2) In his Undergraduate thesis [12], the third author has computed an actual Kvector space basis of P(Q(τ), S(τ, x, n)) for each n ≥ 1, showing in particular that different values of n never yield Jacobian algebras with the same dimension. This implies that different values of n always yield potentials that are not weakly right-equivalent.

4. Twice-punctured surfaces

In this section we prove part (2) of Theorem 1.1, as formulated in the next theorem.

Theorem 4.1. Let (Σ, 𝕄) be a twice-punctured closed surface of positive genus, and let τ be any (tagged) triangulation of (Σ, 𝕄). Over an algebraically closed field, any two non-degenerate potentials on the quiver Q(τ) are weakly right-equivalent.

Since any two ideal triangulations of (Σ, 𝕄) are related by a finite sequence of flips (see [13]), the first paragraphs of the proof of [3, Lemma 8.5] imply that the mere exhibition of a single triangulation τ of (Σ, 𝕄), with Q(τ) having only one weak right equivalence class of non-degenerate potentials, suffices in order to prove Theorem 4.1.

Example 4.2. Figure 3 sketches a triangulation τ of a positive-genus twice-punctured surface with empty boundary. The triangulation is easily seen to satisfy (2.1) and (2.2). Note that the puncture p has valency 8g, and the other puncture q has valency 4g.

\begin{matrix}1 \\ 2 \\ 1 \\ 2 \\ 2g − 1 \\ 2g \\ 2g \\ q \\ p \\ p \\ p \\ p \\ p \\ p \\ p \\ p \\ 2g + 1 \\ 2g + 2 \\ 2g + 4 \\ 2g + 3 \\ 6g \\ 6g − 1 \\ 6g − 3 \\ 6g − 2 \end{matrix}

Figure 3. A triangulation τ of a twice-punctured closed surface (Σ, 𝕄) of positive-genus.

\begin{matrix}1 \\ 2 \\ 1 \\ 2 \\ 2g − 1 \\ 2g \\ 2g \\ q \\ p \\ p \\ p \\ p \\ p \\ p \\ p \\ p \\ a_{1} \\ b_{1} \\ c_{1} \\ a_{3} \\ c_{3} \\ b_{3} \\ a_{4g−1} \\ b_{4g−1} \\ c_{4g−1} \\ a_{2} \\ c_{2} \\ b_{2} \\ a_{4} \\ b_{4} \\ c_{4} \\ a_{4g} \\ b_{4g} \\ c_{4g} \\ a_{4g−2} \\ c_{4g−2} \\ b_{4g−2} \end{matrix}

Figure 4. The associated quiver Q(τ) to the triangulation τ.

Lemma 4.3. Let (Σ, 𝕄) be a twice-punctured closed surface of positive genus, and let τ be the triangulation of (Σ, 𝕄) shown in Figure 3. If V ∈ K⟨⟨Q(τ)⟩⟩ is a potential involving only ≥ 2-powers of g-cycles, then (Q(τ), S(τ, x) + V) is right-equivalent to (Q(τ), S(τ, x)) for any choice x = (x_{p}, x_{q}) of non-zero scalars.

Proof. Let g be the genus of (Σ, 𝕄). Then

V ∼_{cyc} ∑_{n=2}^{∞} ν_{p,n}(G(p))^{n} + ∑_{n=2}^{∞} ν_{q,n}(G(q))^{n}

for some scalars ν_{p,n} and ν_{q,n} for n ≥ 2. Note that short(V) ≥ 2 val_{τ}(q) = 8g.

Claim 2. There exist a sequence (V_{m})_{m=8g}^{∞} of potentials on Q(τ), and a sequence (φ_{m})_{m=8g}^{∞} of unitriangular R-algebra automorphisms of K⟨⟨Q(τ)⟩⟩, satisfying the following properties:

(1) V_{8g} = V;

(2) lim_{m→∞} depth(φ_{m}) = ∞;

(3) for every m ≥ 8g:

(a) φ_{m} is a right-equivalence (Q(τ), S(τ, x) + V_{m}) → (Q(τ), S(τ, x) + V_{m+1});

(b) V_{m} involves only ≥ 2-powers of g-cycles;

(c) short(V_{m}) ≥ m.

For the proof of the claim, we start by setting V_{8g} = V. Let a_{p} (respectively a_{q}) be an arrow lying in the g-orbit that surrounds p (respectively q). Suppose that for a fixed value of m ≥ 8g we have already defined a potential V_{m} involving only ≥ 2-powers of g-cycles and satisfying short(V_{m}) ≥ m. We shall use V_{m} to define V_{m+1} and φ_{m}. Write

V_{m} ∼_{cyc} ∑_{n=2}^{∞} λ_{p,n}(G(a_{p}))^{n} + ∑_{n=2}^{∞} λ_{q,n}(G(a_{q}))^{n}

with λ_{p,n}, λ_{q,n} ∈ K for n ≥ 2. Set r_{p,m} (respectively r_{q,m}) to be the first value of n for which λ_{p,n} ≠ 0 (respectively λ_{q,n} ≠ 0) if such an n exists, and ∞ if such an n does not exist. Note that short(V_{m}) = min(8gr_{p,m}, 4gr_{q,n}) ≥ 8g.

Define an R-algebra homomorphism Υ_{p,m} : K⟨⟨Q(τ)⟩⟩ → K⟨⟨Q(τ)⟩⟩ by means of the rule

Υ_{p,m} : a_{p} ↦ a_{p} − \frac{λ_{p,r_{p,m}}}{x_{p}} a_{p}(G(a_{p}))^{r_{p,m}−1}.

Since r_{p,m} − 1 > 0, Υ_{p,n} is a unitriangular automorphism, and its depth is 8g(r_{p,m} − 1). Direct computation shows that

Υ_{p,m}(S(τ, x) + V_{m}) ∼_{cyc} S(τ, x) + U + W,

where

U = −λ_{p,r_{p,m}}(G(a_{p}))^{r_{p,m}} + Υ_{p,m} ( ∑_{n=r_{p,m}}^{∞} λ_{p,n}(G(a_{p}))^{n}) + ∑_{n=r_{q,m}}^{∞} λ_{q,n}(G(a_{q}))^{n},

W = −\frac{λ_{p,r_{p,m}}}{x_{p}} f(a_{p})a_{p}(G(a_{p}))^{r_{p,m}−1}f^{2}(a_{p}).

Note that short(U) ≥ m and 2 short(W) − 3 = 2 · 8g(r_{p,m} − 1) + 3 ≥ 8gr_{p,m} + 3 > 8gr_{p,m} ≥ m. So, applying Corollary 2.8, we see that there exists a unitriangular Ralgebra automorphism Π_{p,m} of K⟨⟨Q(τ)⟩⟩ that has depth at least min(m − 3, 8g(r_{p,m} − 1)) and serves as a right-equivalence between S(τ, x) + U + W and S(τ, x) + U + ξ for some potential ξ that involves only positive powers of g-cycles and satisfies short(ξ) > m ≥ 8g. These last inequalities imply that, actually, ξ involves only ≥ 2-powers of g-cycles.

Now, we can definitely write

U ∼_{cyc} ∑_{n=r_{p,m}+1}^{∞} κ_{p,n}(G(a_{p}))^{n} + ∑_{n=r_{q,m}}^{∞} λ_{q,n}(G(a_{q}))^{n}

for some scalars κ_{p,n} ∈ K. Define an R-algebra homomorphism Υ_{q,m} : K⟨⟨Q(τ)⟩⟩ → K⟨⟨Q(τ)⟩⟩ by means of the rule

Υ_{q,m} : a_{q} ↦ a_{q} − \frac{λ_{q,n}}{x_{q}} a_{q}(G(a_{q}))^{r_{q,m}−1}.

Since r_{q,m} − 1 > 0, Υ_{q,m} is a unitriangular automorphism, and its depth is 4g(r_{q,m} − 1). Direct computation shows that

Υ_{q,m}(S(τ, x) + U + ξ) ∼_{cyc} S(τ, x) + U′ + W′,

where

U′ = −λ_{q,r_{q,m}}(G(a_{q}))^{r_{q,m}} + ∑_{n=r_{p,m}+1}^{∞} κ_{p,n}(G(a_{p}))^{n} + Υ_{q,n} ( ∑_{n=r_{q,m}}^{∞} λ_{q,n}(G(a_{q}))^{n}) + Υ_{q,m}(ξ),

W′ = −\frac{λ_{q,r_{q,m}}}{x_{q}} f(a_{q})a_{q}(G(a_{q}))^{r_{q,m}−1}f^{2}(a_{q}).

Note that short(U′) > m and 2 short(W′) − 3 = 2 ∗ 4g(r_{q,m} − 1) + 3 ≥ 4gr_{q,m} + 3 > 4gr_{q,m} ≥ m. So, applying Corollary 2.8, we see that there exists a unitriangular Ralgebra automorphism Π_{q,m} of K⟨⟨Q(τ)⟩⟩ that has depth at least min(m − 3, 4g(r_{q,m} − 1)) and serves as a right-equivalence between S(τ, x) + U′ + W′ and S(τ, x) + U′ + ξ′ for some potential ξ′ that involves only positive powers of g-cycles and satisfies short(ξ′) > m ≥ 8g. These last inequalities imply that, actually, ξ′ involves only ≥ 2-powers of g-cycles.

It is clear that U′ involves only positive powers of g-cycles; these powers are actually greater than 1 because short(U′) > m ≥ 8g. So, if we set V_{m+1} = U′ + ξ′ and φ_{m} = Π_{q,m}Υ_{q,m}Π_{p,m}Υ_{p,m}, we see that φ_{m} is a right-equivalence (Q(τ), S(τ, x) + V_{m}) → (Q(τ), S(τ, x) + V_{m+1}), that V_{m+1} involves only ≥ 2-powers of g-cycles, and that short(V_{m+1}) ≥ m + 1.

From the previous paragraph we deduce that the sequences (V_{m})_{m≥8g} and (φ_{m})_{m≥8g} satisfy the third condition stated in Claim 2. Moreover, since m ≤ short(V_{m}) = min(8gr_{p,m}, 4gr_{q,m}) for every m ≥ 8g, we deduce that lim_{m→∞} r_{p,m} = ∞ = lim_{m→∞} r_{q,m}. This and the inequalities

depth(φ_{m}) ≥ min(depth(Π_{q,m}), depth(Υ_{q,m}), depth(Π_{p,m}), depth(Υ_{p,m}))
≥ min ( min (m − 3, 4g(r_{q,m} − 1)), 4g(r_{q,m} − 1),
min (m − 3, 8g(r_{p,m} − 1)), 8g(r_{p,m} − 1))

imply that lim_{m→∞} depth(φ_{m}) = ∞.

This completes the proof of Claim 2.

Lemma 4.3 now follows from an obvious combination of Claim 2 and [8, Lemma 2.4]. □

Proposition 4.4. Let (Σ, 𝕄) be a twice-punctured closed surface of positive genus, and let τ be the triangulation of (Σ, 𝕄) shown in Figure 3. If W ∈ K⟨⟨Q(τ)⟩⟩ is a potential that involves only positive powers of g-cycles and such that (Q(τ), T (τ) + W) is a nondegenerate QP, then W involves each of the g-cycles that arise from the two punctures p and q of (Σ, 𝕄), that is, T (τ) + W = S(τ, x) + V for some choice x = (x_{p}, x_{q}) of non-zero scalars and some potential V involving only ≥ 2-powers of g-cycles.

Proof. With the notation of Figures 3 and 4, let us write

W = ya_{1}a_{2} ⋯ a_{4g} + A

+ z (∏_{j=0}^{g−1} b_{4(g−j)}c_{4(g−j)−2}b_{4(g−j)−3}c_{4(g−j)−1}b_{4(g−j)−2}c_{4(g−j)}b_{4(g−j)−1}c_{4(g−j)−3}) + B,

with A = ∑_{n=2}^{∞} y_{n}(a_{1}a_{2} ⋯ a_{4g})^{n} and

B = ∑_{n=2}^{∞} z_{n} (∏_{j=0}^{g−1} b_{4(g−j)}c_{4(g−j)−2}b_{4(g−j)−3}c_{4(g−j)−1}b_{4(g−j)−2}c_{4(g−j)}b_{4(g−j)−1}c_{4(g−j)−3})^{n} .

If we set I = {2g + 1, 2g + 2, …, 6g − 1, 6g}, then (Q(τ), T (τ) + W) and I satisfy the hypotheses of [3, Proposition 2.4]. We deduce that y ≠ 0.

Note that, for every k ∈ {1, …, 2g −1}, the quiver μ̃_{k}μ̃_{k−1} ⋯ μ̃_{2}μ̃_{1}(Q(τ)) does not have 2-cycles incident to the vertex labelled k + 1. Therefore, the QP μ_{2g}μ_{2g−1} ⋯ μ_{2}μ_{1}(Q(τ), T (τ)+W) is right-equivalent to the reduced part of the QP μ̃_{2g}μ̃_{2g−1} ⋯ μ̃_{2}μ̃_{1}(Q(τ), T (τ)+ W), whose underlying quiver and potential are μ̃_{2g} ⋯ μ̃_{1}(Q(τ)) and

μ̃_{2g} ⋯ μ̃_{1}(T (τ) + W) = (∑_{j=1}^{4g} a_{j}[b_{j}c_{j}]) + ya_{1} ⋯ a_{4g} + A + [B]

+ z (∏_{j=0}^{g−1}[b_{4(g−j)}c_{4(g−j)−2}][b_{4(g−j)−3}c_{4(g−j)−1}][b_{4(g−j)−2}c_{4(g−j)}][b_{4(g−j)−1}c_{4(g−j)−3}])

+ (∑_{j=1}^{2g} (c_{j}^{∗}b_{j}^{∗}[b_{j}c_{j}] + c_{j+2}^{∗}b_{j}^{∗}[b_{j}c_{j+2}] + c_{j}^{∗}b_{j+2}^{∗}[b_{j+2}c_{j}] + c_{j+2}^{∗}b_{j+2}^{∗}[b_{j+2}c_{j+2}])) .

Consider the QP (μ̃_{2g} ⋯ μ̃_{1}(Q(τ)), \overline{S}), where

\overline{S} = (∑_{j=1}^{4g} a_{j}[b_{j}c_{j}]) + ya_{1} ⋯ a_{4g} + A + (∑_{j=1}^{2g} (c_{j}^{∗}b_{j}^{∗}[b_{j}c_{j}] + c_{j+2}^{∗}b_{j+2}^{∗}[b_{j+2}c_{j+2}])) ,

and let (Q, S) be its reduced part, computed according to the limit process with which Derksen, Weyman and Zelevinsky [2, Theorem 4.6] prove their Splitting Theorem. Note the presence of the sum ∑_{j=1}^{4g} a_{j}[b_{j}c_{j}] in \overline{S}. Then Q = Q(σ), where σ is a triangulation that can be obtained from τ by applying an orientation-preserving homeomorphism of (Σ, 𝕄) that exchanges p and q (thus τ and σ have the same shape, sketched in Figure 4; see also Example 4.5 below). Moreover, since no arrow of the form a_{j} or [b_{j}c_{j}] appears in any of the terms of the potential

W′ := z (∏_{j=0}^{g−1}[b_{4(g−j)}c_{4(g−j)−2}][b_{4(g−j)−3}c_{4(g−j)−1}][b_{4(g−j)−2}c_{4(g−j)}][b_{4(g−j)−1}c_{4(g−j)−3}]) + [B]

+ (∑_{j=1}^{2g} c_{j+2}^{∗}b_{j}^{∗}[b_{j}c_{j+2}] + c_{j}^{∗}b_{j+2}^{∗}[b_{j+2}c_{j}]) ,

the QP (Q(σ), S + W′) is a reduced part of (μ̃_{2g} ⋯ μ̃_{1}(Q(τ)), μ̃_{2g} ⋯ μ̃_{1}(T (τ) + W)) and hence is (right-equivalent to) the mutation μ_{2g} ⋯ μ_{1}(Q(τ), T (τ)+W). Furthermore, from the fact that no arrow of the form [b_{j}c_{ℓ}] with j ≠ ℓ appears in any of the terms of \overline{S}, we deduce that the coefficient in S of any of the rotations of the cycle

(∏_{j=0}^{g−1}[b_{4(g−j)}c_{4(g−j)−2}][b_{4(g−j)−3}c_{4(g−j)−1}][b_{4(g−j)−2}c_{4(g−j)}][b_{4(g−j)−1}c_{4(g−j)−3}])

is 0. Therefore, the coefficient of this cycle in S + W′ is z (and its proper rotations do not appear).

The non-degeneracy of (Q(τ), T (τ)+W) implies the non-degeneracy of (Q(σ), S +W′). Furthermore, it is easy to see that if we set I = {2g + 1, 2g + 2, …, 6g − 1, 6g}, then (Q(σ), S + W′) and I satisfy the hypotheses of [3, Proposition 2.4], from which we deduce that z ≠ 0. This finishes the proof of the proposition. □

Example 4.5. Figure 5 sketches the flip sequence in the proof of Proposition 4.4 in the case of a twice-punctured torus. Note that the first and last triangulations have the same shape.

\begin{matrix}q \\ q \\ q \\ q \\ q \\ q \\ q \\ q \\ q \\ p \\ p \\ p \\ p \\ p \\ p \\ p \\ p \\ p \\ p \\ p \\ p \\ p \\ p \\ p \\ p \\ q \\ q \\ q \\ q \\ q \\ q \\ q \\ q \\ q \\ p \\ p \\ p \\ p \\ p \\ p \\ p \\ p \\ p \\ p \\ p \\ p \\ p \\ p \\ p \\ p \\ q \\ q \\ q \\ q \\ q \\ q \\ q \\ q \\ q \\ p \\ p \\ p \\ p \\ p \\ p \\ p \\ p \\ p \\ p \\ p \\ p \\ p \\ p \\ p \\ p \end{matrix}

Figure 5. Proving Proposition 4.4 for the twice-punctured torus.

Proof of Theorem 4.1. Let (Σ, 𝕄) a be twice-punctured closed surface of positive genus, and let τ be a triangulation of (Σ, 𝕄) satisfying (2.1) and (2.2). By Lemma 2.4, every non-degenerate potential on Q(τ) is right-equivalent to a potential of the form T (τ) + U for some U which is rotationally disjoint from T (τ). By Proposition 2.6, T (τ)+U is rightequivalent to T (τ) + W for some potential that involves only positive powers of g-cycles. Theorem 4.1 now follows from Proposition 4.4, Lemma 4.3 and [3, Lemma 8.5]. □

Acknowledgements

We thank Christof Geiss and Jan Schröer for many helpful discussions.

The three authors were supported by the second author’s grant PAPIIT-IA102215. The first two authors were supported by the second author’s grant CONACyT-238754 as well. DLF received support from a Cátedra Marcos Moshinsky and the grant PAPIIT-IN112519.

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Jan Geuenich, Fakultät für Mathematik, Universität Bielefeld, Germany

Email address: jgeuenich@math.uni-bielefeld.de

Daniel Labardini-Fragoso, Instituto de Matemáticas, Universidad Nacional Autónoma de México, Mexico

Email address: labardini@im.unam.mx

José Luis Miranda-Olvera, Department of Mathematical Sciences, Carnegie Mellon University, USA

Email address: joseluismiranda@cmu.edu