We study (1 + 1)-dimensional SU (N ) spin systems in the presence of the global SU (N ) rotation and lattice translation symmetries. By matching the mixed anomaly of the P SU (N ) × Z symmetry in the continuum limit, we identify a topological index for spin model evaluated as the total number of Young-tableaux boxes of spins per unit cell modulo N , which characterizes the "ingappability" of the system. A nontrivial index implies either a ground-state degeneracy in a gapped phase, which can be regarded as a field-theory version of the Lieb-Schultz-Mattis theorem, or a restriction of the possible universality classes in a critical phase -the symmetry-protected critical phase, e.g. only a class of SU (N ) Wess-Zumino-Witten theories can be realized in the low-energy limit of the given lattice model in the presence of the symmetries. Similar constraints also apply when a higher global symmetry emerges in the model with a lower symmetry. Our prediction agrees with several examples known in previous studies of SU (N ) models.

For a projective algebraic surface X, with an ample line bundle H, let M X H (c) be the moduli space H-semistable sheaves E of class c in the Grothendieck group K(X). We write c = (r, c 1 , c 2 ), or c = (r, c 1 , χ) with r the rank, c 1 , c 2 , the Chern classes and χ the holomorphic Euler characteristic. We also write M Xis the determinant line bundle associated to a line bundle on X. More generally for suitable classes c * ∈ K(X) there is a determinant line bundle D c,c * on M X H (c). We first compute some generating functions for K-theoretic Donaldson invariants on P 2 and rational ruled surfaces, using the wallcrossing formula of [11].Then we show that Le Potier's strange duality conjecture relating H 0 (M X H (c), D c,c * ) and H 0 (M X H (c * ), D c * ,c ) holds for the cases c = (2, c 1 = 0, c 2 > 2) and c * = (0, L, χ = 0) with L = −K X on P 2 , and L = −K X or −K X + F on P 1 × P 1 and P 2 with F the fiber class of the ruling, and also the case c = (2, H, c 2 ) and c * = (0, 2H, χ = −1) on P 2 .

Abstract. Let M H X (u) be the moduli space of semi-stable pure sheaves of class u on a smooth complex projective surface X. We specify u = (0, L, χ(u) = 0), i.e. sheaves in u are of dimension 1. There is a natural morphism π from the moduli space M H X (u) to the linear system |L|. We study a series of determinant line bundles λ c r n on M H X (u) via π. Denote g L the arithmetic genus of curves in |L|. For any X and g L ≤ 0, we compute the generating function Z r (t) = n h 0 (M H X (u), λ c r n )t n . For X being P 2 or P(O P 1 ⊕ O P 1 (−e)) with e = 0, 1, we compute Z 1 (t) for g L > 0 and Z r (t) for all r and g L = 1, 2. Our results provide a numerical check to Strange Duality in these specified situations, together with Göttsche's computation. And in addition, we get an interesting corollary (Corollary 4.2.13) in the theory of compactified Jacobian of integral curves.

Let M(d, χ) be the moduli space of semistable sheaves of rank 0, Euler characteristic χ and first Chern class dH (d > 0), with H the hyperplane class in P 2 . We give a description of M(d, χ), viewing each sheaf as a class of matrices with entries in i≥0 H 0 (O P 2 (i)). We show that there is a big open subset of M(d, 1) isomorphic to a projective bundle over an open subset of a Hilbert scheme of points on P 2 . Finally we compute the classes of M(4, 1), M(5, 1) and M(5, 2) in the Grothendieck ring of varieties, especially we conclude that M(5, 1) and M(5, 2) are of the same class.Theorem 1.3 (Theorem 5.2). [M(4, 1)] = 17 i=0 b 2i L i and b 0 = b 34 = 1, b 2 = b 32 = 2, b 4 = b 30 = 6, b 6 = b 28 = 10, b 8 = b 26 = 14, b 10 = b 24 = 15, b 12 = b 14 = b 16 = b 18 = b 20 = b 22 = 16.In particular the Euler number e (M(4, 1)) of the moduli space is 192. Theorem 1.4 (Theorem 6.1). [M(5, 1)] = [M(5, 2)] = 26 i=0 b 2i L i and b 0 = b 52 = 1, b 2 = b 50 = 2, b 4 = b 48 = 6, b 6 = b 46 = 13, b 8 = b 44 = 26, b 10 = b 42 = 45, b 12 = b 40 = 68, b 14 = b 38 = 87, b 16 = b 36 = 100, b 18 = b 34 = 107, b 20 = b 32 = 111, b 22 = b 30 = 112, b 24 = b 26 = b 28 = 113.In particular the Euler number of both moduli spaces is 1695.We then have µ(E) + 1 = µ(F ) for E, F in the sequence (3.1).Definition 3.1. We say a pair (E, f ) is (semi)stable if for any subsheaf E ′ E and E ′ a direct sum of line bundles such that f −1 (E ′ ) ≃ E ′ ⊗ O P 2 (−1), we have µ(E ′ )(≤) < µ(E).Lemma 3.2. θ induces a bijection from semistable pairs to semistable sheaves.

We construct immersions of trivalent abstract tropical curves in the Euclidean plane and embeddings of all abstract tropical curves in higher dimensional Euclidean space. Since not all curves have an embedding in the plane, we define the tropical crossing number of an abstract tropical curve to be the minimum number of self-intersections, counted with multiplicity, over all its immersions in the plane. We show that the tropical crossing number is at most quadratic in the number of edges and this bound is sharp. For curves of genus up to two, we systematically compute the crossing number. Finally, we use our immersed tropical curves to construct totally faithful nodal algebraic curves via lifting results of Mikhalkin and Shustin.1991 Mathematics Subject Classification. 14T05.