Pentagrams, inscribed polygons, and Prym varieties

The pentagram map is a discrete integrable system on the moduli space of planar polygons. The corresponding first integrals are so-called monodromy invariants $E_1, O_1, E_2, O_2,\dots$ By analyzing the combinatorics of these invariants, R.Schwartz and S.Tabachnikov have recently proved that for polygons inscribed in a conic section one has $E_k = O_k$ for all $k$. In this paper we give a simple conceptual proof of the Schwartz-Tabachnikov theorem. Our main observation is that for inscribed polygons the corresponding monodromy satisfies a certain self-duality relation. From this we also deduce that the space of inscribed polygons with fixed values of the monodromy invariants is an open dense subset in the Prym variety (i.e., a half-dimensional torus in the Jacobian) of the spectral curve. As a byproduct, we also prove another conjecture of Schwartz and Tabachnikov on positivity of monodromy invariants for convex polygons.


Introduction and main results
The pentagram map was introduced by R. Schwartz [14] in 1992, and is now one of the most renowned discrete integrable systems which has deep connections with many different subjects such as projective geometry, integrable PDEs, cluster algebras, etc. The definition of the pentagram map is illustrated in Figure 1: the image of the polygon P under the pentagram map is the polygon P ′ whose vertices are the intersection points of consecutive "short" diagonals of P (i.e., diagonals connecting second-nearest vertices). P P ′ Figure 1: The pentagram map.
Since this construction is projectively invariant, one usually regards the pentagram map as a dynamical system on the space of polygons in P 2 modulo projective equivalence. (Here P 2 denotes the real or complex projective plane. More generally, one can consider polygons in the projective plane over any field.) The pentagram map also naturally extends to a bigger space of so-called twisted polygons. A bi-infinite sequence of points vi ∈ P 2 is called a twisted n-gon if vi+n = M (vi) for every i ∈ Z and a fixed projective transformation M , called the monodromy. The case M = Id corresponds to closed polygons. The pentagram map is welldefined on the space of projective equivalence classes of twisted polygons and preserves the conjugacy class of the monodromy.
From the beginning there was a strong indication that the pentagram map is integrable. In [15] Schwartz established a first result in this direction proving that the pentagram map is recurrent. Further, in [17] he constructed two sequences E k , O k of so-called monodromy invariants preserved by the pentagram map. The functions E k , O k are, roughly speaking, weighted homogeneous components of spectral invariants of the monodromy matrix. Remarkably, this construction, essentially based on the notion of a twisted polygon, provides invariants for the pentagram map in both twisted and closed cases.
V. Ovsienko, R. Schwartz, and S. Tabachnikov [12] proved that the pentagram map on twisted polygons has an invariant Poisson bracket, and that the monodromy invariants Poisson commute, thus establishing Arnold-Liouville integrability in the twisted case. F. Soloviev [21] showed that the pentagram map is algebraically integrable, both in twisted and closed cases. An alternative proof of integrability in the closed case can be found in [13].
We also mention, in random order, several works which generalize the pentagram map and explore its relations to other subjects. M. Glick [5] interpreted the pentagram map in terms of cluster algebras. M. Gekhtman, M. Shapiro, S. Tabachnikov, and A. Vainshtein [4] generalized Glick's work by including the pentagram map into a family of discrete integrable systems related to weighted directed networks. In this family one finds a discrete version of the relativistic Toda lattice, as well as certain multidimensional generalizations of the pentagram map defined on so-called corrugated polygons. Other integrable generalizations of the pentagram map were studied by B. Khesin and F. Soloviev [9,10], G. Marí Beffa [11], and R. Felipe with G. Marí Beffa [1]. Some of these maps have been recently put in the context of cluster algebra by M. Glick and P. Pylyavskyy [7]. We finally mention the work of V. Fock and A. Marshakov [2], which, in particular, relates the pentagram map to Poisson-Lie groups, and the paper [8] by R. Kedem and P. Vichitkunakorn, which interprets the pentagram map in terms of T-systems.
In the present paper, we study the interaction of the pentagram map with polygons inscribed in conic sections. Schwartz and Tabachnikov [20] proved that the restrictions of the monodromy invariants to inscribed polygons satisfy the following identities. The proof of Schwartz and Tabachnikov is rather hard and is based on combinatorial analysis of explicit formulas for the monodromy invariants. Our first result is a new proof of Theorem 1.1. Our argument does not rely on explicit formulas but employs the definition of E k , O k in terms of the spectrum of the monodromy. Namely, we show that, up to conjugation, the monodromy matrix for inscribed polygons satisfies a self-duality relation where z is the spectral parameter. As a corollary, the spectral curve, defined, roughly speaking, as the zero locus of the characteristic polynomial det(M (z) − wId), is invariant under the involution σ : Remark 1.2. Note that Theorem 1.1 is, in general, not true for polygons inscribed in degenerate conics. Although any degenerate conic C can be approximated by nondegenerate conics Cε → C, a twisted polygon inscribed in C cannot be, in general, approximated by twisted polygons inscribed in Cε. For this reason, one cannot apply a limiting argument to conclude that E k = O k in the degenerate case, and, in fact, there are twisted polygons inscribed in degenerate conics with E k = O k . Nevertheless, for closed polygons, Theorem 1.1 is true in both nondegenerate and degenerate cases.
We also obtain a geometric characterization of inscribed polygons with fixed values of the monodromy invariants and describe the behavior of this set under the pentagram map. For simplicity, consider the case of n-gons with odd n. In this case, the variety of polygons with fixed generic monodromy invariants E k , O k is identified with an open dense subset in the Jacobian of the spectral curve, while the pentagram map is a shift relative to the group structure on the Jacobian [21]. Consider a level set of the monodromy invariants which contains at least one inscribed polygon. In this case, there is an involution σ : (z, w) ↔ (z −1 , w −1 ) on the spectral curve X, defining a double covering π : X → Y := X/σ. Now recall that, given a ramified (non-étale) double covering of curves π : X → Y , one can decompose the Jacobian of X in a sum of Abelian subvarieties of complementary dimensions: Jac(X) = Jac(Y ) + Prym(X|Y ).
Here Jac(Y ) is the Jacobian of the base curve Y embedded in Jac(X) by means of the pullback homomorphism π * , while Prym(X|Y ) is the Prym variety of X over Y , which can be defined as the kernel of the pushforward (or norm) homomorphism π * . The intersection Jac(Y ) ∩ Prym(X|Y ) is the finite set of order 2 points in Jac(Y ).
In the following theorem, we consider twisted polygons in the complex projective plane.   fixed points, therefore, by the Riemann-Hurwitz formula, the genus of Y is q. So, dim Jac(Y ) = q, and dim Prym(X|Y ) = dim Jac(X) − dim Jac(Y ) = q.
We conclude that the moduli space of inscribed twisted (2q + 1)-gons is a fibration over a (q + 1)dimensional base with q-dimensional fibers, and thus has total dimension 2q + 1. Note that the latter dimension can also be computed by identifying inscribed polygons with polygons in P 1 .
Remark 1.5. The reason why in part 2 of Theorem 1.3 one needs to relabel the vertices to identify the squared pentagram map with a translation along Jac(Y ) is the following. There are two equally natural ways (the so-called left and right labeling schemes) how to define the pentagram map on polygons with labeled vertices. Figure 3 depicts these two labeling schemes and the squares of the corresponding maps. Notice that in both cases, the squared map exhibits a shift of indices. At the same time, the translation along Jac(Y ) corresponds to the "canonical squared pentagram map" depicted in Figure 4 (the latter is, in fact, not a square of any map).
The proof of Theorem 1.3 is somewhat involved and is based on relation (1). Details of the proof will be published elsewhere.
The structure of the paper is as follows. In Section 2, we recall the derivation of the monodromy invariants. In Section 3, we show that the monodromy invariants arise as coefficients in the equation of the spectral curve. In Section 4, we derive a formula for the monodromy in terms of the corner invariants. Finally, in Section 5, we use these results to prove Theorem 1.1. Section 6 is devoted to open questions. Additionally, the paper contains an appendix where we apply our technique to prove another conjecture of Schwartz and Tabachnikov on positivity of monodromy invariants for convex polygons.
To conclude this introduction, we also mention two works [19,18] both studying the pentagram map or its variations for polygons inscribed or circumscribed about a conic section. We find it an interesting problem to obtain an algebraic geometric explanation of these results in the spirit of the present paper. Richard Schwartz, and Sergei Tabachnikov for fruitful discussions.

Monodromy invariants
In this section, we briefly recall the derivation of first integrals for the pentagram map -the monodromy invariants E k , O k . Details can be found in almost any paper on the subject, for instance in [12,17]. Let Pn be the space of twisted n-gons modulo projective transformations. Recall that a twisted n-gon is a bi-infinite sequence of points vi ∈ P 2 such that vi+n = M (vi) for every i ∈ Z and a fixed projective transformation M , called the monodromy. For an equivalence class of polygons P ∈ Pn, the monodromy M ∈ PGL3 is well-defined up to conjugation (this means that, as a matrix, the monodromy is defined up to conjugation and scalar multiplication).
In what follows, we only consider polygons satisfying the following genericity assumption: any three consecutive vertices vi, vi+1, vi+2 are not collinear. Note that for inscribed polygons this always holds.
Any twisted n-gon is uniquely determined by its n consecutive vertices and its monodromy M ∈ PGL3. Therefore, the dimension of the space of twisted n-gons is 2n + 8, while the dimension of the moduli space Pn is 2n. As coordinates on the space Pn, one can take the so-called corner invariants. To every vertex vi of a twisted n-gon, one associates two cross-ratios xi, yi, as shown in Figure 5. We note that while there are several different ways how to define the cross-ratio, the one traditionally used in the definition of corner invariants is Clearly, the sequences (xi), (yi) are n-periodic and depend only on the projective equivalence class of the polygon (vi). Furthermore, {x1, y1, . . . , xn, yn} is a coordinate system on an open dense subset of Pn [17].
To define the pentagram map, we use the right labeling scheme (see Figure 3). The image of a polygon (vi) This map descends to a densely defined map T : Pn → Pn (which we also call the pentagram map). In terms of the corner invariants xi, yi, the map T is given by Now, we recall the construction of the monodromy invariants. For any equivalence class P ∈ Pn of twisted ngons, the corresponding monodromy M is a 3 × 3 matrix defined up to scalar multiplication and conjugation. Therefore, the quantities are well-defined 2 . Furthermore, since the pentagram map on polygons preserves the monodromy, the pentagram map T : Pn → Pn on equivalence classes preserves Ω1, Ω2. Further, let yi.
Then from formulas (2) it immediately follows that the functions On, En are invariant under the pentagram map. Therefore, the quantitiesΩ 1 := O 2 n EnΩ1,Ω2 := OnE 2 n Ω2 are also invariant. In [17] it is shown that theΩ1,Ω2 are polynomials in corner invariants xi, yi.
Therefore, all coefficients in z of the polynomials R * z (Ω1), R * z (Ω2), where R * z denotes the natural action of Rz on functions of corner invariants, are also first integrals. In [17] it is shown that where O k , E k are certain polynomials in corner invariants. The monodromy invariants are polynomials in xi, yi preserved by the pentagram map. In [12] they are used to prove that the pentagram map is a completely integrable system.

The spectral curve
In this section, we define the spectral curve and relate it to the monodromy invariants. The pentagram spectral curve first appeared in [21], where it was used to prove algebraic integrability.
Let P ∈ Pn be an equivalence class of twisted n-gons. Then the rescaling operation Rz : Pn → Pn defines a family Pz ∈ Pn, Pz := Rz(P ) parametrized by z ∈ C * . For each equivalence class Pz, we have the corresponding conjugacy class of monodromies. Take any representative of this conjugacy class and lift it to GL3. This gives a family M (z) of 3 × 3 matrices parametrized by z ∈ C * . Definition 3.1. We call M (z) a scaled monodromy matrix of the equivalence class P .
The scaled monodromy matrix of a given equivalence class is determined up to multiplication by a scalar function of z and conjugation by a z-dependent matrix. Now, given an equivalence class P ∈ Pn, we take its scaled monodromy M (z) and set where λ(z) := (z n det M (z)) 1/3 , and I is the identity matrix. The function R(z, w) is the characteristic polynomial of M (z) normalized in such a way that it does not change if M (z) is multiplied by a scalar function of z. Furthermore, R(z, w) depends only on the conjugacy class of M (z), thus being a well-defined function of z, w, and the equivalence class P . Explicitly, we have where and E k , O k are the monodromy invariants of the equivalence class P .
Definition 3.2. The zero locus of R(z, w) in the complex torus (C * ) 2 is called the spectral curve. 2 There is a typo in the corresponding formula (2.8) of [12]. To be consistent with subsequent definitions, Ω 1 should be defined using M −1 , and Ω 2 using M , not the other way around. Figure 6: The Newton polygon of the spectral curve.
The spectral curve encodes all the monodromy invariants E k , O k . In particular, we have the following immediate corollary of (4).  Remark 3.5. Note that in [21] the spectral curve is defined using the matrix M (z −1 ) −1 . Although this definition is not completely equivalent to ours, the corresponding curves are related by a change of variables.

The monodromy matrix via corner invariants
In this section, we express the monodromy in terms of the corner invariants. Our approach is similar to that of [21], but we do not assume that the number of vertices is not divisible by 3. To get rid of this assumption, we use the same idea as in Remark 4.4 of [12].
Lemma 4.1. Let (vi) be a twisted n-gon in P 2 with monodromy M ∈ PGL3 and corner invariants (xi, yi). Then M is conjugate to any of the matrices 3 Mi := LiLi+1 · · · Li+n−1, where i ∈ Z is arbitrary, and Proof. Using that vi, vi+1, vi+2 are not collinear, we lift the points vi to vectors Vi ∈ C 3 in such a way that (Here and in what follows, we regard Vi's as column vectors.) From (6) it follows that for some sequences ai, bi ∈ C. In matrix form, this can be written as It follows that Wi+n = Wi(NiNi+1 · · · Ni+n−1).
Further, take an arbitrary liftM ∈ GL3 of the monodromy M . Note that since M (vi) = vi+n, we havẽ M Vi = tiVi+n for some sequence ti ∈ C. This can be rewritten as where Di is a diagonal matrix Di := diag(ti, ti+1, ti+2). Comparing (10) and (11), we conclude thatM (and thus M ) is conjugate toM i := (NiNi+1 · · · Ni+n−1)Di, (12) for any i ∈ Z. Now, we use the result of Lemma 4.5 of [12] which says that given any lift (Vi) of a twisted polygon satisfying (7), the corner invariants are given by Using these formulas, one easily verifies that the matrix Lj given by (5) is related to Nj given by (9) by means of a gauge type transformation: Therefore, for the product Mi = LiLi+1 · · · Li+n−1, we have whereMj is given by (12), and ci ∈ C. Further, multiplying (8) by the monodromy matrixM from the left and using (11), we get Wi+n+1Di+1 = Wi+nDiNi.
Comparing the latter equation with (8), we see that Spelling out this equation, we get the relations ti+3 = ti, ai+n = ai ti+2 ti , bi+n = bi ti+1 ti (cf. Remark 4.4 of [12]). This implies the following quasiperiodicity condition for Λi: Using the latter equation, (13) can be rewritten as SinceMi−2 is conjugate to the monodromy matrix M , this proves the lemma.

Monodromy invariants for inscribed polygons
In this section, we show that for inscribed polygons the scaled monodromy satisfies a certain self-duality relation and then use this to prove Theorem 1.1.
Lemma 5.1. Consider an equivalence class P ∈ Pn of polygons inscribed in a nondegenerate conic. Then the corresponding scaled monodromy matrix can be chosen to satisfy the self-duality relation Remark 5.2. We call (14) self-duality because the matrix (M (z −1 ) −1 ) t represents the scaled monodromy of the dual polygon. Remark 5.3. From (14) it follows, in particular, that the matrix M (1), i.e., the actual non-scaled monodromy of the polygon, is orthogonal. This has a geometric explanation: if a polygon is inscribed into a conic C, then the corresponding monodromy should preserve C. But the subgroup of projective transformations preserving a non-singular conic is conjugate (over C) to SO3 ⊂ PGL3.
Proof of Lemma 5.1. Take any polygon (vi) in the equivalence class P , and let C be the conic circumscribed about (vi). Using the isomorphism C ≃ P 1 , we get a well-defined notion of cross-ratio of four points on C. Referring to this cross-ratio, we set The sequence (pi) is n-periodic and depends only on the projective equivalence class of the polygon (vi). By Lemma 3.1 of [20], we have the following relation between pi's and the corner invariants: Using these formulas and Corollary 4.2, we express the scaled monodromy in terms of pi's: Further, we notice that the matrix Lj (z) is related to the matrix by a gauge type transformation: Therefore, the matrix Mi(z) is conjugate to a scalar multiple of M ′ i (z) := L ′ i (z)L ′ i+1 (z) · · · L ′ i+n−1 (z). Further, observe that the matrices L ′ j (z) satisfy the relation Therefore, the same relation is satisfied by their product M ′ i (z): Further, write the matrix S(z) as and set M (z) := T (z) −1 M ′ 1 (z)T (z). Note that since M (z) is conjugate to M ′ 1 (z), it is also conjugate to a scalar multiple of M1(z), and thus can be taken as a scaled monodromy matrix for P . Furthermore, from (15) and (16) it follows that M (z) satisfies (14), as desired.
Proof of Theorem 1.1. Consider a polygon (vi) inscribed in a nondegenerate conic. By Lemma 5.1, the corresponding equivalence class P ∈ Pn admits a scaled monodromy matrix with self-duality relation (14). From self-duality, it immediately follows that the corresponding normalized characteristic polynomial (4) satisfies which means that the spectral curve R(z, w) = 0 is invariant under the involution (z, w) ↔ (z −1 , w −1 ). In view of Corollary 3.3, this shows that E k = O k for all k, proving the theorem.

Discussion and open questions
Circumscribed polygons. By duality, Theorem 1.1 is also true for circumscribed polygons. Furthermore, there is a "circumscribed" analog of Theorem 1.3. Namely, circumscribed polygons fill another torus parallel to the Prym variety (see Figure 7). Note that, in general, the orbit of an inscribed polygon under the pentagram map does not have to intersect the circumscribed locus. However, there are some special cases when a certain iteration of an inscribed polygon under the pentagram map is circumscribed [19]. It would be interesting to find an algebraic geometric explanation of this phenomenon.
Continuous limit. In the continuous limit, when the number of vertices of a polygon is large, while the vertices are close to each other, the pentagram map becomes the Boussinesq flow on curves [12]. In the language of differential operators, the Boussinesq equation reads where L := ∂ 3 x + u(x)∂x + v(x), and (L 2/3 )+ denotes the differential part of the pseudodifferential operator L 2/3 . The continuous limit of both inscribed and circumscribed polygons are parametrized conics, corresponding to skew-adjoint operators L. This, in particular, means that in the limit the inscribed and circumscribed tori in Figure 7  One possible approach to this problem is to use the fact that matrices satisfying self-duality relation (14) form a subgroup in the loop group of PGL3, somewhat similar to a twisted loop group. Does this subgroup admit a Poisson-Lie structure? If the answer to the latter question is positive, one should be able to use this structure to construct a Poisson bracket on inscribed polygons.
Self-dual polygons. Another class of polygons satisfying E k = O k are self-dual polygons. Recall that, for a polygon (vi) ∈ P 2 , the vertices of the dual polygon are sides [vi, vi+1] of the initial one, regarded as points in the dual projective plane. A polygon (vi) ∈ P 2 is called self-dual if there exists some m ∈ Z and a projective map P 2 → (P 2 ) * taking the vertex vi to the vertex [vi+m, vi+m+1] of the dual polygon. Furthermore, for polygons with odd number of vertices, there exists a canonical notion of self-duality. We say that a (2q + 1)-gon (vi) ∈ P 2 is canonically self-dual if there exists a projective map P 2 → (P 2 ) * taking the vertex vi to the vertex [vi+q, vi+q+1] of the dual polygon (i.e., to the opposite side). Canonically self-dual polygons form a maximal dimensional stratum in the set of self-dual polygons [3]. Furthermore, one can show that canonically self-dual polygons correspond to a third torus in Figure 7 parallel to the inscribed and circumscribed tori.
This description suggests that there should exist a (possibly birational) isomorphism between the varieties of canonically self-dual and inscribed polygons. For closed 7-gons and 9-gons, such an isomorphism is constructed in [19]. In particular, for 7-gons it is given by the pentagram map. Are the varieties of canonically self-dual and inscribed polygons isomorphic in the general case?
Poncelet polygons. For certain singular spectral curves, the inscribed and circumscribed tori in Figure 7 coincide, giving rise to polygons which are simultaneously inscribed and circumscribed. Such polygons are known as Poncelet polygons, since they are closely related to the famous Poncelet porism. In [16] it is proved that Poncelet polygons are fixed points for the squared pentagram map. Is there an algebraic geometric interpretation of this statement? Does the squared pentagram map have other fixed points? Are the fixed points given by Poncelet polygons Lyapunov stable?
Degenerate conics. How do the results of the present paper generalize to polygons inscribed in degenerate conics? Consider, in particular, a 2q-gon with the following property: all its odd vertices lie on a straight line l1, while all even vertices lie a on a line l2. Such polygons exhibit an interesting behavior under the (inverse) pentagram map, known as the Devron property [6]. Is there an algebraic geometric explanation for the Devron property? Note that the corresponding spectral curves are highly singular and have the form for certain a, b ∈ C. We believe that it should it be possible to relate the Devron property with algebraic geometry of such singular curves.

Appendix: Positivity of the monodromy invariants for convex polygons
Following [20], we define the following "signed" versions of the monodromy invariants: and also set O * n := On, E * n := En. The aim of this appendix is to prove the following result conjectured in [20].  1. The signed monodromy invariants E * k can be written as polynomials with positive coefficients in terms of yi, 1 − xiyi.
2. The signed monodromy invariants O * k can be written as polynomials with positive coefficients in terms of xi, 1 − xiyi.
Proof. We begin with the first statement. Since the result obviously holds for E * n = En, we only consider the case k ≤ [n/2]. Using the matrix Mi(z) from Corollary 4.2 and the definition of the monodromy invariants, we get 1 + Therefore, to prove that E * k is a polynomial with positive coefficients in terms of yi, 1 − xiyi, it suffices to show that trace Mi(z) is a polynomial with positive coefficients in yi, 1 − xiyi, and −z −1 . To that end, notice that the matrix Lj(z) from Corollary 4.2 can be written as Therefore, the product Mi(z) = Li(z) · · · Li+n−1(z) satisfies the same relation Mi(z) = UMi(z)U −1 ,Mi(z) :=Li(z) · · ·Li+n−1(z).
To complete the proof, notice that since all non-zero entries of the matrixLj (z) are polynomials with positive coefficients in terms of yi, 1 − xiyi, and −z −1 , the same is true for the entries of the matrixMi(z), and thus traceMi(z) = trace Mi(z), as desired. Now, we prove the second statement. Let and then apply the same argument as in the proof of the first statement. Thus the lemma is proved. Remark 7.3. As was observed by R. Schwartz, Lemma 7.2 can also be proved using the concepts of right and left modifications introduced in Section 2.2 of [17].
Proof of Theorem 7.1. It is easy to see that for convex closed polygons the corner invariants satisfy the inequalities 0 < xi, yi < 1. So, for such polygons one has xi, yi, 1 − xiyi > 0, and the result of the theorem follows from Lemma 7.2.