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GENERATING FUNCTIONS FOR COLUORED 3D YOUNG DIAGRAMS AND THE DONALDSON-THOMAS INVARIANTS OF ORBIFOLDS BENJAMIN YOUNG, WITH AN APPENDIX BY JIM BRYAN ABSTRACT. We derive two multivariate generating functions for three- dimensional Young diagrams (also called plane partitions). The vari- ables correspond to a colouring of the boxes according to a finite abelian subgroup G of SO(3). These generating functions turn out to be orb- ifold Donaldson–Thomas partition functions for the orbifold [C3/G]. We need only the vertex operator methods of Okounkov–Reshetikhin– Vafa for the easy case G = Zn; to handle the considerably more difficult case G = Z2 × Z2, we will also use a refinement of the author’s recent q–enumeration of pyramid partitions. In the appendix, we relate the diagram generating functions to the Donaldson-Thomas partition functions of the orbifold [C3/G]. We find a relationship between the Donaldson-Thomas partition functions of the orbifold and its G-Hilbert scheme resolution. We formulate a crepant resolution conjecture for the Donaldson-Thomas theory of local orb- ifolds satisfying the Hard Lefschetz condition.
- INTRODUCTION A 3D Young diagram, or 3D diagram for short, is a stable pile of cubical boxes which sit in the corner of a large cubical room. More formally, a 3D Young diagram is a finite subset π of (Z≥0)3 such that if any of (i + 1, j, k), (i, j + 1, k), (i, j, k + 1) are in π, then (i, j, k) ∈π. The ordered triples are the “boxes”; the closure condition means that the boxes of a 3D partition are stacked stably in the positive octant, with gravity pulling them in the direction (−1, −1, −1). 3D Young diagrams are well–studied; they are also called plane parti- tions or 3D partitions elsewhere in the literature. The first result on 3D Young diagrams is due to Dr. Percy MacMahon [19]. MacMahon was the Date: May 31, 2018. 1 arXiv:0802.3948v2 [math.CO] 2 Jul 2008
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2 BENJAMIN YOUNG, WITH AN APPENDIX BY JIM BRYAN first to “q–count” (i.e. to give a generating function for) 3D Young diagrams by volume: (1) X π 3D diagram q|π| = Y n 1 1 −qn n , where |π| denotes the number of boxes in π. Generating functions of this form will appear frequently, so we adopt the following notation: Definition 1.1. Let M(x, q) = ∞ Y n=1 1 1 −xqn n f M(x, q) = M(x, q)M(x−1, q) We call M(x, q) and f M(x, q) the MacMahon and MacMahon tilde func- tions, respectively. Strictly speaking, f M(x, q) lies in the ring of formal power series Zx, x−1, q. However, in all of our applications, we will spe- cialize x and q in such a way that no negative powers of any variables appear in the formulae (see Theorems 1.4 and 1.5). Since MacMahon, there have been many proofs of (1), spanning many fields: combinatorics, statistical mechanics, representation theory, and oth- ers. Recently, there has been a thorough study of the various symmetry classes of 3D Young diagrams [5], and of many macroscopic properties of large random 3D Young diagrams [25]. There is also active research in al- gebraic geometry which relies upon enumerations of various types of 3D partitions [20]. We will derive two refinements of MacMahon’s generating function. Fix a set of colours C, and replace the variable q with a set of variables, Q = {qg | g ∈C}. We will need to assign a colour to each point of the first orthant. In par- ticular, we will usually have C = G, a finite Abelian group. In this case, addition in Z3 ≥0 must respect the group law of G. Definition 1.2. A colouring is a map K : (Z≥0)3 →C.
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COLOURED 3D YOUNG DIAGRAMS 3 If C = G is a finite Abelian group, then a G–colouring is a colouring which is also a homomorphism of additive monoids. Note that a G–colouring is uniquely determined by K(1, 0, 0), K(0, 1, 0) and K(0, 0, 1), and that K(0, 0, 0) is the identity element of G. There is a simple way of defining a G–colouring KG when G is a three– dimensional matrix group G. Decompose G as a direct sum of one–dimensional representations Rx, Ry, Rz. The set of irreducible representations of any Abelian G forms a group bG ≃G under tensor product, so let ψ be an iso- morphism ψ : bG −→G and define KG(i, j, k) = ψ(R⊗i x ⊗R⊗j y ⊗R⊗k z ). Both of the colourings used in this paper are of this form. We next define the multivariate generating function ZG = ZG(Q) which “Q–counts” diagrams (that is, ZG counts each diagram with the Q–weight of its boxes): Definition 1.3. For g ∈G, let |π|g be the number of g–coloured boxes in π, |π|g = |K−1 G (g) ∩π|. Define the G-coloured partition function ZG = X π3D partition Y g∈G q|π|g g . The question of determining ZG, though completely combinatorial, has its genesis in a field of enumerative algebraic geometry called Donaldson- Thomas theory. When G is a finite Abelian subgroup of SO(3) (which forces G = Zn or Z2 ×Z2), there is a colouring induced by the natural three dimensional representation for which the generating function ZG is, up to signs of the variables, the orbifold Donaldson–Thomas partition function for the quotient stack [C3/G] (see Appendix A). Although it is not yet clear why, these seem to be precisely the groups G for which ZG has a product formula.
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4 BENJAMIN YOUNG, WITH AN APPENDIX BY JIM BRYAN Theorem 1.4. Let G = Zn and let the colouring KZn be given by KZn(1, 0, 0) = 1 KZn(0, 1, 0) = −1 KZn(0, 0, 1) = 0. Let q = q0 · · · qn−1 and for a, b ∈[1, n −1], let q[a,b] = qaqa+1 · · · qb. Then ZZn = M(1, q)n Y 0<a≤b<n f M(q[a,b], q). The proof of Theorem 1.4 is straightforward; it is essentially a simple modification of the methods used in [27] (or, indeed, a special case of the extremely general methods of [25]). We include it for completeness and as an introduction to the vertex operator calculus used to prove Theorem 1.5. There are several other ways to prove Theorem 1.4, some of which have (at least implicitly) appeared in the literature. For example, [1, 14] both com- pute a generating function with variables xk(k ∈Z) which can be easily specialized to ZZn. The result [1] is particularly notable, as it is a direct computer algebra implementation of MacMahon’s techniques of combina- tory analysis. The following theorem, however, is new: Theorem 1.5. Let G = Z2 ×Z2 = {0, a, b, c} and let the colouring KZ2×Z2 be given by KZ2×Z2(1, 0, 0) = a KZ2×Z2(0, 1, 0) = b KZ2×Z2(0, 0, 1) = c. Let q = q0qaqbqc. Then ZZ2×Z2 = M(1, q)4 · f M(qaqb, q)f M(qaqc, q)f M(qbqc, q) f M(−qa, q)f M(−qb, q)f M(−qc, q)f M(−qaqbqc, q) . See Figure 1 for pictures of a partition coloured in the manner described by these theorems. As an application of these theorems, we will compute the Donaldson- Thomas invariants of the orbifolds [C3/Zn] and [C3/Z2 ×Z2]. The orbifold Donaldson-Thomas partition function of [C3/G] has variables labeled by
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COLOURED 3D YOUNG DIAGRAMS 5 Figure 1: A partition coloured according to KZ2×Z2 and to KZ3 (a) KZ2×Z2 – weight q30 0 q29 a q31 b q28 c (b) KZ3 – weight q40 0 q38 1 q40 2 representations of G (see Appendix) and hence has the same variables as the G-coloured diagram partition function. In the Appendix, we prove that the diagram partition function and the Donaldson-Thomas partition function are related by simple sign changes on the variables: Theorem 1.6. The orbifold Donaldson-Thomas partition functions of the orbifolds [C3/Z2 × Z2] and [C3/Zn] are given by ZDT C3/Zn(q0, q1, …, qn−1) = ZZn(−q0, q1, …, qn−1) ZDT C3/Z2×Z2(q0, qa, qb, qc) = ZZ2×Z2(q0, −qa, −qb, −qc) where q and q[a,b] are defined as in Theorems 1.4 and 1.5. There is a striking similarity between the Donaldson-Thomas partition functions of the orbifold [C3/G] and the crepant resolution given by the G–Hilbert scheme. The following is proved in the Appendix: Theorem 1.7. Let YG −→C3/G be the crepant resolution of C3/G given by the G-Hilbert scheme. YG has a natural basis of curve classes indexed by non-trivial elements of G. The Donaldson-Thomas partition functions of
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6 BENJAMIN YOUNG, WITH AN APPENDIX BY JIM BRYAN YZn and YZ2×Z2 are given by ZDT YZn = M(1, −q)n Y 0<a≤b<n M(q[a,b], −q), ZDT YZ2×Z2 = M(1, −q)4 M(qaqb, −q)M(qbqc, −q)M(qaqc, −q) M(qa, −q)M(qb, −q)M(qc, −q)M(qaqbqc, −q), where {q1, …, qn−1} and {qa, qb, qc} are the variables corresponding to curve classes and q is the variable corresponding to Euler number. We see from these theorems that the reduced partition function of the orbifold [C3/G] is obtained from the reduced partition function of the res- olution by identifying the variables appropriately and then simply writing a tilde over every factor of M in the formula! A similar phenomenon was observed by Szendr˝oi for the partition function of the (non–commutative) conifold singularity and its crepant resolution [31]. It would be very desirable to have even a conjectural understanding of the relationship between the Donaldson–Thomas theory of an arbitrary Calabi– Yau orbifold and its crepant resolution(s). We formulate a conjecture for the case of a local orbifold satisfying the hard Lefschetz condition (see Conjec- ture A.6. Theorem 1.5 is not straightforward to prove. Essentially none of the stan- dard proofs of MacMahon’s colourless result can be modified to work in this situation. The generating function was first conjectured by Jim Bryan based on some related phenomena from Donaldson–Thomas theory; concurrently, Kenyon made an (unpublished) equivalent conjecture for Z2×Z2–weighted dimer models on the hexagon lattice, based on computational evidence. Having this conjectured formula was crucial for finding the proof of The- orem 1.5, which involves a somewhat bizarre detour: one must first Q– count pyramid partitions (see Figure 4). One then performs a computation with vertex operators to make ZZ2×Z2 emerge. We discovered this idea serendipitously while trying to generalize our earlier work on pyramid par- titions [32].
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COLOURED 3D YOUNG DIAGRAMS 7 2. REVIEW: THE INFINITE WEDGE SPACE Our general strategy will be to think of a 3D diagram π as a set of diag- onal slices, {πk | k ∈Z}, where πk is the set of all bricks which lie in the plane x −y = k. We will then analyze how one passes from one slice to the next. Since we will be summing over all 3D Young diagrams, it is very helpful to consider (possibly infinite) formal sums of the form X λ∈some set of partitions fλ(Q) · λ, where fλ(Q) is a power series in the elements of Q. A nice way of de- scribing the set of all such sums is the charge–zero subspace of the infinite wedge space, (Λ∞/2)0V where V is a vector space with a basis labeled by the elements of Z + 1 2. This setting allows one to define, quite naturally, several useful operators on partitions. The use of (Λ∞/2)0V , and its associated operators, was in part popular- ized by [24, Appendix A], and we shall adhere to the notation established there. In this section, we have collected the minimum number of formulae necessary for our purposes. We will use Dirac’s “bra–ket” notation ⟨λ |µ⟩ to denote the inner product under which the partitions are orthonormal. We will need need the bosonic creation and annihilation operators αn, defined in [24, Appendix A] in the section on Bosons and Vertex Operators. The operators αn satisfy the Heisenberg commutation relations, (2) [αn, α−m] = nδm,n. Concretely, α−n acts on a 2D Young diagram λ by adding a single border strip of length n onto λ in all possible ways, with sign (−1)h+1, where h is the height of the border strip (see Figure 2). The operator αn is adjoint to α−n, and acts by deleting border strips.
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8 BENJAMIN YOUNG, WITH AN APPENDIX BY JIM BRYAN Figure 2: Applying α−3 to a partition h = 3 h = 2 + − + h = 1 α−3 Let xj(j ≥1) be indeterminates; and define the homogeneous, elemen- tary, and power sum symmetric functions as usual: X i hi(x1, x2, …)ti = Y i 1 1 −xit X i ei(x1, x2, …)ti = Y i (1 + xit) pi(x1, x2, …) = X j≥1 xi j For a comprehensive reference on symmetric functions, see [30]. We next define the vertex operators Γ±: Definition 2.1. Γ±(x1, x2, …) = exp X k pk k α±k The matrix coefficients (with respect to the orthonormal basis formed by the 2D Young diagrams) of the Γ± operators turn out [24, A.15] to be the skew Schur functions, ⟨λ |Γ−(x1, x2, …)| µ⟩= ⟨µ |Γ+(x1, x2, …)| λ⟩= sλ/µ(xi). We will need the following well-known theorem from representation theory (see, for example, [13, Chapter 8]) to work with Γ± and other exponentiated operators.
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COLOURED 3D YOUNG DIAGRAMS 9 Theorem 2.2. (Campbell-Baker-Hausdorff) If A and B are operators, then log(exp(A) exp(B)) = A + B + 1 2[A, B] + · · · , where the higher–order terms are multiples of nested commutators of A and B. It is certainly possible to give more terms in the expansion, but we shall only need the following two corollaries. Corollary 2.3. If A and B are commuting operators, then exp(A) exp(B) = exp(A + B). Corollary 2.4. If A and B are operators such that [A, B] is a central ele- ment, then we have exp(A) exp(B) = exp([A, B]) exp(B) exp(A). 3. THE OPERATORS Γ(x), Γ′(x), AND Qg Our next goal is to define precisely what it means for two diagonal slices λ, µ to sit next to one another in a 3D Young diagram, and to define opera- tors for working with such slices. Definition 3.1. Let λ, µ be two 2D Young diagrams. We say that λ interlaces with µ, and write λ ≻µ, if µ ⊆λ and the skew diagram λ/µ contains no vertical domino. For example, (6, 3, 2) ≻(4, 2), because the skew diagram (6, 3, 2)/(4, 2) has no two boxes in the same column. The following lemma is easy to check: Lemma 3.2. The following are equivalent: (1) λ ≻µ. (2) The row lengths λi, µi satisfy λ1 ≥µ1 ≥λ2 ≥µ2 ≥· · · . (3) λi −µi = 0 or 1, for each pair of columns λi, µi. (4) λ and µ are two adjacent diagonal slices of some 3D Young dia- gram.
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10 BENJAMIN YOUNG, WITH AN APPENDIX BY JIM BRYAN Note that we have used the convenient, but slightly nonstandard, notation λi to denote the columns of λ. Part (3) will become relevant in Section 5, when we will see that adjacent diagonal slices of a pyramid partition also interlace. We are mainly interested in two specializations of Γ±(x1, x2, …) which create interlacing partitions, and which depend only upon a single indeter- minate q. The first will be denoted Γ±(q), and is obtained by performing the specialization x1 7→q, xi 7→0 for i > 1. Its formula is (3) Γ±(q) = exp X k qk k α±k. Recall [30, Chapter 7] that if λ, µ are partitions, then we may define the skew Schur function sλ/µ(x1, x2, · · · ) by P T xT, where T runs over the set of semistandard tableaux of shape λ/µ. Following [27], we see that sλ/µ(q, 0, 0, …) = q|λ|−|µ| if λ ≻µ 0 if λ ̸≻µ. One can then show [26] that Γ−(q)µ = X λ≻µ q|λ|−|µ|λ Γ+(q)λ = X µ≺λ q|λ|−|µ|µ. (4) For the second specialization, recall that there is an involution ω on the algebra of symmetric functions [30, Chapter 7.6], given by any one of the following equivalent definitions: ek(xi) ←→hk(xi) pk(xi) ←→(−1)k−1pk(xi) sλ/µ(xi) ←→sλ′/µ′(xi) Here, λ′ is the transpose partition of λ. To obtain the second specialization, called Γ′ ±(q), we first perform the involution pk 7→ωpk and then specialize x1 7→q, xi 7→0 (i > 1) as before. We obtain the formula (5) Γ′ ±(q) = exp X k (−1)k−1qk k α±k
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COLOURED 3D YOUNG DIAGRAMS 11 with the property that Γ′ −(q)µ = X λ′≻µ′ q|λ|−|µ|λ Γ′ +(q)λ = X µ′≺λ′ q|λ|−|µ|µ. Lemma 3.3. If a and b are commuting variables, then we have the following multiplicative commutators in Ca, b: [Γ+(a), Γ′ −(b)] = 1 + ab [Γ′ +(a), Γ−(b)] = 1 + ab [Γ+(a), Γ−(b)] = 1 1 −ab [Γ′ +(a), Γ′ −(b)] = 1 1 −ab Proof. Let us compute the first of these commutators; the others are similar. Let us apply (2), and then use Corollary 2.4 to rephrase the answer as the exponential of a commutator. We have [Γ′ +(a), Γ−(b)] = exp X j,k (−1)j−1ajbk jk [αj, α−k] = exp
− X j (−ab)j j ! = exp(log(1 −(−ab))). □ We next define diagonal operators Qg for assigning weights to 2D parti- tions. Definition 3.4. For g ∈G, define the weight operator Qg by Qg |λ⟩= q|λ| g |λ⟩. The operator Qg can be commuted past any of the Γ± operators, at the expense of changing the argument of Γ±: Γ+(x)Qg = QgΓ+(xqg) QgΓ−(x) = Γ−(xqg)Qg Γ′ +(x)Qg = QgΓ′ +(xqg) QgΓ′ −(x) = Γ′ −(xqg)Qg.
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12 BENJAMIN YOUNG, WITH AN APPENDIX BY JIM BRYAN 4. COUNTING WITH Zn COLOURING As a motivating example, let us use (4) to write down a vertex operator expression which computes MacMahon’s generating function (1), using the variable q = q0. This formula appears in [27] with marginally different notation. Consider a 3D Young diagram π and its diagonal slices: φ ≺· · · ≺π−2 ≺π−1 ≺π0 ≻π1 ≻π2 ≻· · · ≻φ, where φ denotes the empty partition. Each such π contributes q|π| 0 = q P |πn| 0 to the generating function, so we have X π 3D diagram q|π| = * φ ∞ Y i=1 (Γ+(1)Q0) ∞ Y i=1 (Γ−(1)Q0) φ + . This works because the operators Γ−and Γ+ pass from one slice to the next larger (respectively smaller) slice in all possible ways, and the Q0 operators assign the proper weight to each slice. One then commutes all the Γ−oper- ators to the left and all the Γ+ operators to the right (following the method outlined in [27]) to compute the generating function. Let us now write down a vertex operator expression which computes ZZn. Here, Q = {q0, … , qn−1}, q = q0q1 · · · qn−1, and K = KZn. The compu- tation is straightforward (following precisely the method of [27]) but awk- ward, so it is helpful to organize the work by collecting together n vertex operators at a time. Note that the diagonal slices of π are all monochrome (see Figure 3), so we define A±(x) = Γ±(x)Q1Γ±(x)Q2 · · · Qn−1Γ±(x)Q0 Then, the following vertex operator product counts Zn–coloured 3D dia- grams: (6) ZZn =
φ · · · A+(1)A+(1)A+(1)A−(1)A−(1)A−(1) · · · φ
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COLOURED 3D YOUNG DIAGRAMS 13 Figure 3: Slicing a Z3–coloured 3D diagram Let q = q0q1 · · · qn−1, and let Q = Q0Q1 · · · Qn−1. We use the commutation relations of the previous section to compute A+(x) = Q · Γ+ (xq1q2q3 · · · qn−1q0) Γ+ (xq2q3 · · · qn−1q0) · · · Γ+ (xq0) A−(x) = Γ+(x)Γ+(xq1) · · · Γ+(xq1q2 · · · qn−1) · Q = Γ+ �xqq−1 1 q−1 2 · · · q−1 n−1q−1 0 Γ+ �xqq−1 2 q−1 3 · · · q−1 n−1q−1 0 · · · · · · Γ+ �xqq−1 0 · Q Next, set A+(x) = Q−1A+(x); A−(x) = A−(x)Q−1. From this expression, it is clear that A+(x)A−(y) = C(x, y) · A−(y)A+(x) where C(x, y) is the following product of the n2 commutators obtained by moving a Γ+ past a Γ−: C(x, y) = 1 1 −qxy n Y 0≤a≤b<n 1 1 −(qaqa+1 · · · qb)qxy · Y 0≤a≤b<n 1 1 −(qaqa+1 · · · qb)−1qxy
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14 BENJAMIN YOUNG, WITH AN APPENDIX BY JIM BRYAN We now follow the derivation of MacMahon’s formula in [27]. Starting with (6), we convert all of the A± into A± and move the resulting weight functions to the outside of the product (where they act trivially). This gives ZZn =
φ · · · A+(q2)A+(q)A+(1)A−(1)A−(q)A−(q2) · · · φ . We then commute all A+ operators to the right and all A−to the left: ZZn = ⟨φ| · · · A+(q2)A+(q) A+(1)A−(1) A−(q)A−(q2) · · · |φ⟩ = C(1, 1) φ · · · A+(q2) A+(q)A−(1)A+(1)A−(q) A−(q2) · · · φ = · · ·
∞ Y i,j=0 C(qi, qj) ·
φ A−(1)A−(q)A−(q2) · · · A+(q2)A+(q)A+(1) φ . The vertex operator product in the final line is now equal to 1, because ⟨φ| A−(x) = ⟨φ| and A+(x) |φ⟩= |φ⟩. Finally, we rewrite the remaining product with MacMahon functions: ZZn = ∞ Y i,j=0 C(qi, qj) = M(1, q)n Y 0≤a≤b<n M(qa · · · qb, q)M(q−1 a · · · q−1 b , q) = M(1, q)n Y 0≤a≤b<n f M(qa · · · qb, q), and Theorem 1.4 is proven. □ 5. PYRAMID PARTITIONS The methods of the Section 4 may also be used to Q-count a similar type of three-dimensional combinatorial object, called pyramid partitions. Essentially, we want to replace Z3 ≥0 with the upside–down pyramid shaped stack of bricks shown in Figure 4. Note that the bricks have ridges and grooves set into them; this helps to remind us how the bricks are meant to stack.
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COLOURED 3D YOUNG DIAGRAMS 15 Figure 4: (a) The set B of bricks (b) A pyramid partition removed from B Szendr˝oi [31] introduced us to the ideas in this section, albeit in a dif- ferent context. He proves that counting pyramid partitions with a slightly simpler colour scheme (namely specializing q0 = qc, qb = qa) yields a cer- tain noncommutative Donaldson–Thomas partition function. We shall bor- row some of Szendr˝oi’s terminology, but not much of the machinery that he developed. We will start by giving a rather algebraic definition for the bricks in a pyramid partition. Consider the quiver (or directed graph) P shown in Fig- ure 5(a). The vertices of P are the elements of Z2 × Z2 = {0, a, b, c}. The edges are labelled {v1, w1, v2, w2}. Definition 5.1. A word in P is the concatenation of the edge labels of some directed path in P. We may optionally associate a base to a word; the base is the starting vertex of the path. Note that a word based at 0 may also be based at c, but not at b or a. Any path in P is uniquely determined by its base and its word.
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16 BENJAMIN YOUNG, WITH AN APPENDIX BY JIM BRYAN Figure 5: v2 w2 v1 w1 v1 w1 v2 w2 0 c a b (a) The quiver P (b) A pyramid partition π Definition 5.2. Form the path algebra CP spanned by all words in P, and define the noncommutative quotient ring A = CP/IW, where IW = ⟨v1wiv2 −v2wiv1, w1vjw2 −w2vjw1⟩, i, j ∈{1, 2}. If B is a word in CP, we write [B] for its residue class in CP/Iw. Definition 5.3. A brick is an element [B] of CP/IW, where B is a word based at the vertex 0. Let B be the set of all bricks. To understand how to draw Figure 4, we interpret the edge labels of P as vectors in Z3. Definition 5.4. Let v1 = (−1, 1, 0) v2 = (1, 1, 0) w1 = (0, 1, −1) w2 = (0, 1, 1). The position of a brick [B] is the sum of the vectors corresponding to the edge labels in [B]. The brick corresponding to the empty walk, [], is located at the origin. We next define a “colouring” on B. Definition 5.5. Define Kpyramid : B −→Z2 × Z2
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COLOURED 3D YOUNG DIAGRAMS 17 Figure 6: A view of the pyramid B from below. The bricks have been shrunk to points, and some checkerboard–coloured slices are shown. q0, qc qb, qa qa, qb by setting Kpyramid([B]) to be the final vertex of any path whose word is B. We call Kpyramid([B]) the colour of B. For an example of all of these concepts, define the brick [B] by the word B = v2w2v2w1. The brick [B] is based at the vertex 0, ending at the vertex b. The position of [B] is (2, 4, 0); [B] is the c-coloured brick in the top layer of Figure 5b. Note that the colour is completely determined by the x and y coordinates of [B]; Figure 6 shows the colouring as viewed from along the z axis. Definition 5.6. A pyramid partition π is a subset of B such that if [B] ∈π then every prefix of B also represents a brick in π. Note that pyramid partitions may also be defined algebraically, although it is unnecessary to do so for this paper. A pyramid partition corresponds to a framed cyclic CP/IW–module based at 0, much in the same way that a 3D Young diagram corresponds to a monomial ideal in C[x, y, z]. We refer the reader to [31] for further details of this approach.
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18 BENJAMIN YOUNG, WITH AN APPENDIX BY JIM BRYAN Our next goal is to show that the diagonal slices of a pyramid partition in- terlace with one another. This will allow us to reuse the strategy of Section 4 to obtain a nice generating function for pyramid partitions. Definition 5.7. Let π be a pyramid partition. Define the kth diagonal slice of π, written πk, to be the set of all bricks in π whose position (x, y, z) satisfies x −y = k. Lemma 5.8. Let k ≥0. Then π−2k = {[(v1w2)kW]} ∩π, π2k = {[(v2w1)kW]} ∩π, where W runs over all words in v1w1 and v2w2, and π−2k−1 = {[(v1w2)kv1W ′]} ∩π, π2k+1 = {[(v2w1)kv2W ′]} ∩π, where W ′ runs over all words in w1v1 and w2v2. Moreover, the bricks of πk form a 2D Young diagram; the slices are single–coloured, as follows: Colour of πk = 0 if k = 0 (mod 4) b if k = 1 (mod 4) c if k = 2 (mod 4) a if k = 3 (mod 4). Proof. Let us prove the first equation; the other three are similar. Note that the brick represented by the word (v1w2)k is in position (−k, 2k, k) and thus lies in the −2kth diagonal. Appending v1w1 or v2w2 to this word adds (−1, 2, −1) or (1, 2, 1) to the position, which does not alter x −y. To see that the bricks of π−2k form a 2D Young diagram, observe that w1v1 and w2v2 commute in CP/Iw. The suffix (w1v1)i(w2v2)j corresponds to the (i, j) box in the Young diagram. Again, the other cases are similar. The colours are easy to check directly. □ Figure 7 shows the central slice π0 of the pyramid partition of Figure 5b. Every brick has been replaced with a square tile to make the orientation of the 2D Young diagram clear.
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COLOURED 3D YOUNG DIAGRAMS 19 Figure 7: A diagonal slice of a pyramid partition, interpreted as a 2D Young diagram Lemma 5.9. For k ≥0, we have the following interlacing properties (where the prime denotes transposition of 2D Young diagrams): π2k ≻π2k+1 π′ −2k ≻π′ −2k−1 π′ 2k+1 ≻π′ 2k+2 π−2k−1 ≻π−2k−2 Proof. Let us handle the first case. Let Rj k be the set of bricks in the jth column of πk, and suppose that |Rj k| = ℓj k. Explicitly, Rj 2k+1 = {(v2w1)kv2(w1v1)j(w2v2)i | 0 ≤i < ℓj 2k+1} = {(v2w1)k(v1w1)j(v2w2)iv2 | 0 ≤i < ℓj 2k+1}, Rj 2k = {(v2w1)k(v1w1)j(v2w2)i | 0 ≤i < ℓj 2k}. In particular, each of the bricks in Rj 2j ∪Rj 2j+1 may be represented as some prefix of the word (v2w1)k(v1w1)j(v2w2)max{ℓj 2k,ℓj 2k+1}. Informally speaking, Rj 2k and Rj 2k+1 form a chain of bricks, each of which rests on the previous one (see Figure 8). It follows from Definition 5.6 that ℓj 2k −ℓj 2k+1 ∈{0, 1}; then part (3) of Lemma 3.2 says that π2k ≻π2k+1.
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20 BENJAMIN YOUNG, WITH AN APPENDIX BY JIM BRYAN Figure 8: The jth columns of two adjacent slices π0, π1 Figure 9: Row– and column–interlacing behaviour for adjacent diagonal slices of a pyramid partition Next let us see that π′ 2k+1 ≻π′ 2k+2. Let Ri,k be the ith row of πk, with |Ri,k| = ℓi,k. We have Ri,2k+2 = {(v2w1)k(v2w2)i(v1w1)j | 0 ≤j < ℓi,2k+2} = {(v2w1)kv2(w2v2)i(w1v1)j | 0 ≤j < ℓi,2k+2}, Ri,2k+1 = {(v2w1)kv2(w2v2)i(w1v1)j | 0 ≤j < ℓi,2k+1}. from which it follows that ℓj,2k+1 −ℓj,2k+2 ∈{0, 1}. This means that π′ 2k+1 ≻π′ 2k+2. See Figure 9 for an illustration of the difference between the row–interlacing and column–interlacing behaviour. The remaining cases are similar. □
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COLOURED 3D YOUNG DIAGRAMS 21 6. A GENERATING FUNCTION FOR PYRAMID PARTITIONS We will now compute the following generating function for pyramid par- titions. Definition 6.1. Let π be a pyramid partition. For g ∈Z2 × Z2, let |π|g = |K−1 pyramid(g) ∩π| denote the number of boxes coloured g in π. Define Zpyramid = X π pyramid partition Y g∈Z2×Z2 q|π|g g . Theorem 6.2. Zpyramid = M(1, q)4f M(qbqc)f M(qaqc) f M(−qa, q)f M(−qb, q)f M(−qc, q)f M(−qaqbqc, q) , where q = q0qaqbqc. Theorem 6.2 may seem unrelated to the other theorems in this paper, but it will turn out that it is the key to computing ZZ2×Z2. The proof is much like that of Theorem 1.4. Proof. We define a vertex operator product which counts pyramid parti- tions. Let us first define an operator which sweeps out four slices of the pyramid partition at the same time. Let A ′ ±(x) = Γ±(x)QbΓ′ ±(x)QcΓ±(x)QaΓ′ ±(x)Q0, so that Zpyramid = D φ · · · A ′ +(1)A ′ +(1)A ′ −(1)A ′ −(1) · · · φ E . It is simple to check this product against Lemmas 5.9 and 5.8 to be sure that it describes the correct colouring and interlacing behaviour. Set A′ +(x) = Q−1 0 Q−1 b Q−1 c Q−1 a A ′ +(x) A′ −(x) = A ′ +(x)Q−1 0 Q−1 b Q−1 c Q−1 a . Commuting the weight operators past the vertex operators gives A′ +(x) = Γ+(xqbqcqaq0)Γ′ +(xqcqaq0)Γ+(xqaq0)Γ′ +(xq0)
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22 BENJAMIN YOUNG, WITH AN APPENDIX BY JIM BRYAN and A′ −(y) = Γ−(yqq−1 b q−1 c q−1 a q−1 0 )Γ′ −(yqq−1 c q−1 a q−1 0 ) ·Γ−(yqq−1 a q−1 0 )Γ′ −(yqq−1 0 ) so that (7) Zpyramid =
φ · · · A′ +(q2)A′ +(q)A′ +(1)A′ −(1)A′ −(q)A′ −(q2) · · · φ . The commutation relation for these A′ operators is A′ +(x)A′ −(y) = (1 + qbxyq)(1 + qbqcqaxyq)(1 + q−1 b xyq)(1 + qcxyq) (1 −xyq)(1 −qbqcxyq)(1 −xyq)(1 −qcqaxyq) ·(1 + q−1 c xyq)(1 + qaxyq)(1 + (qbqcqa)−1xyq)(1 −q−1 a xyq) (1 −(qbqc)−1xyq)(1 −xyq)(1 −(qcqa)−1xyq)(1 −xyq) ·A′ −(y)A′ +(x). Because of the mixed Γ and Γ′ operators, some of the commutation factors now appear in the numerator. We now move the A′
- operators in (7) to the left of the expression, while moving the A′ −operators to the right. As in the proof of Theorem 1.4, all of the A′ vanish, and we are left with the commutator Zpyramid = M(1, q)4f M(qbqc, q)f M(qaqc, q) f M(−qa, q)f M(−qb, q)f M(−qc, q)f M(−qaqbqc, q) , where q = q0qaqbqc. □ Note that this method gives an alternate proof of the result in [32], when we specialize q0 = qc, q1 = qa = qb.
- COUNTING Z2 × Z2–COLOURED 3D YOUNG DIAGRAMS We will now prove Theorem 1.5. Let us name the elements of Z2 × Z2 {0, a, b, c} as in the previous section, and recall the definition of KZ2×Z2 from Theorem 1.5. Our set of indeterminates is Q = {q0, qa, qb, qc}. Let q = q0qaqbqc. Before we proceed to compute this generating function, consider the kth diagonal slice x −y = k of the positive octant. Note that we have KZ2×Z2(x, x + k, z) = (x −z)c + kb.
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COLOURED 3D YOUNG DIAGRAMS 23 Figure 10: Slicing a Z2 × Z2–coloured 3D diagram In particular, the box (x, x + k, z) is coloured k · b if x ≡z (mod 2), and k·b+c otherwise. In other words, each diagonal slice of π is now coloured in a checkerboard fashion, whereas in the Zn case, they were single–coloured (see Figure 10). We need to introduce a two–coloured weight function if we are to use vertex operators to compute ZZ2×Z2. Definition 7.1. For g, h ∈Z2 × Z2, let Qghλ = q#{(i,j)∈λ | i≡j (mod 2)} g · q#{(i,j)∈λ | i̸≡j (mod 2)} h · λ. We may write down a vertex operator product which sweeps out a 3D diagram in diagonal slices, according to the Z2 × Z2 colouring. It is ZZ2×Z2 = ⟨φ| · · · QbaΓ+(1)Q0cΓ+(1)QbaΓ+(1)Q0c ·Γ−(1)QabΓ−(1)Q0cΓ−(1)Qab · · · |φ⟩ (8) Unfortunately, Γ± no longer commutes nicely with the Qgh operators, so our usual approach to computing with vertex operators fails here. The prob- lem is fundamental, and it does not appear that we can resolve it in a natural way. We need a new idea. However, there are two clues which tell us how to proceed. The first clue is that the desired formula for ZZ2×Z2 is very close to Zpyramid, so it would suffice to prove the following:
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24 BENJAMIN YOUNG, WITH AN APPENDIX BY JIM BRYAN Lemma 7.2. ZZ2×Z2 = f M(qaqb, q) · Zpyramid. The second clue is that if we attempt to compute Zpyramid by slicing along lines x + y = k, rather than x −y = k, then the slices of the pyramid partition are checkerboard–coloured as well! See Figure 6, which shows the colouring scheme from below. The heavy black lines represent two edges of the pyramid, corresponding to prefixes of the words (w1v1)k and (w2v2)k, so bricks which lie on these lines represent the corners of the slices. So, using our checkerboard coloured weight operators, we can write down a different vertex operator product which still counts pyramid par- titions. Lemma 7.3. Zpyramid = ⟨φ| · · · QbaΓ′ +(1)Q0cΓ+(1)QbaΓ′ +(1)Q0c ·Γ−(1)QabΓ′ −(1)Q0cΓ−(1)Qab · · · |φ⟩ Proof. One must check that the interlacing behaviour of the slices is correct, and that the correct weights are assigned to each slice. This is similar to the proofs of Lemmas 5.8 and 5.9. □ Observe that (8) is very similar to the product in Lemma 7.3, so we shall look for a way to transform Γ±(x) into Γ′ ±(x). Definition 7.4. Define E±(x) = exp X k≥1 x2k k α±2k. Lemma 7.5. The operators E± have the following properties: Γ±(x) = Γ′ ±(x)E±(x). [E±, Γ±] = 0. E+(x)Γ−(y) = 1 1 −(xy)2Γ−(y)E+(x). Γ+(x)E−(y) = 1 1 −(xy)2E−(y)Γ+(x). Γ′ +(x)E−(y) = (1 −(xy)2)E−(y)Γ′ +(x).
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COLOURED 3D YOUNG DIAGRAMS 25 Proof. These are all simple applications of Corollaries 2.4 and 2.3 as well as (3) and (5). □ In fact, unlike Γ±(x), E±(x) also commutes nicely with the checkerboard weight operators Qgh. Lemma 7.6. E−(x)Qgh = QghE−(x√qgqh); QghE+(x) = E+(x√qgqh)Qgh. Proof. The operator α2n has the effect of adding all possible border strips R of length 2n to the boundary of a 2D Young diagram. Since the length of the strips R is even, any such R has the same Qgh–weight. Indeed, Qgh · R = (qgqh)n · R = (√qgqh)|R| · R, It follows that X n x2n n α−2n ! Qgh · λ = Qgh X n (q√qgqh)2n n α−2n ! · λ and thus E−(x)Qgh = QghE−(x√qgqh). The case of E+ is similar. □ Finally, we have the following property of E±, inherited from the corre- sponding property of α±n: ⟨φ| E−(x) = ⟨φ| E+(x) |φ⟩= |φ⟩
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26 BENJAMIN YOUNG, WITH AN APPENDIX BY JIM BRYAN Proof of Lemma 7.2. Let us alter the first line of (8) by transforming half of the Γ+(1) operators into Γ′ +(1) operators, * φ ∞ Y i=1 Γ+(1)QbaΓ+(1)Q0c
φ ∞ Y i=1 Γ+(1)QbaΓ′ +(1)E+(1)Q0c
φ ∞ Y i=1 Γ+(1)QbaΓ′ +(1)Q0c · ∞ Y i=0 E+(qi)E+(Qi√qbqa) . Now, continue to move the E+ term to the right through the second line of (8). We have ∞ Y i=0 E+(qi)E+(Qi√qbqa) · ∞ Y i=1 Γ−(1)QabΓ−(1)Q0c φ + = C · ∞ Y i=1 Γ−(1)QabΓ−(1)Q0c · ∞ Y i=0 E+(qi)E+(Qi√qbqa) φ + = C · ∞ Y i=1 Γ−(1)QabΓ−(1)Q0c φ + , where C = M(1, q)M(q−1 b q−1 c , q) is the product of the commutators gen- erated by Lemma 7.5. Next, change half of the Γ−to Γ′ −in the above expression, ∞ Y i=1 Γ−(1)QabΓ′ −(1)E−(1)Q0c φ + and commute them out to the left. This time, we pick up the multiplicative factor M(qaqb, q) M(1, q) , and the factors M(1, q) cancel. We have shown that ZZ2×Z2 = Zpyramid · f M(qaqb). so Lemma 7.2 and Theorem 1.5 are now proven. □
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COLOURED 3D YOUNG DIAGRAMS 27 8. FUTURE WORK It would obviously be wonderful to have a combinatorial proof of these identities; such a proof might be analagous to the “n–quotient” on 2D Young diagrams, which decomposes a Young diagram into n smaller Young diagrams and an n–core. The authors suspect, however, that such a proof would be rather difficult to find. One indication of this is that there are for- mulae for 3D Young tableaux which fit inside an A × B × C box, but com- putational evidence suggests that there is no such nice formula for Z2 ×Z2– coloured partitions. One could attempt to compute the Donaldson–Thomas partition func- tions of arbitrary toric Calabi-Yau orbifolds. To this aim, it should be pos- sible to develop an orbifold version of the topological vertex formalism following [20]; this is a work in progress. It would also be interesting to try to extend Szendr˝oi’s work [31] in noncommutative Donaldson–Thomas theory using the results of this paper. One box counting problem which is of great interest is to take G = Z3 and the colouring K given by K(1, 0, 0) = K(0, 1, 0) = K(0, 0, 1) = 1. However, this problem appears to be rather difficult. The group representa- tion does not naturally embed into SO(3) or SU(2), so Donaldson–Thomas theory does not generate any conjectures as to what the answer might look like. Indeed, Kenyon [16] conjectures that there is no nice product formula in this example. One unifying theme between 3D diagrams and pyramid partitions is quiv- ers: both objects arise from a quiver path algebra modulo an ideal generated by a superpotential [31]. Perhaps one can extend the methods to other quiv- ers and superpotentials. However, the most intriguing direction for future work is simply to try to understand these proofs more fully. The reader may perhaps have no- ticed that the appearance of pyramid partitions seems somewhat unmoti- vated. Undoubtedly, there is some underlying geometric or representation- theoretic reason why these product formulae exist.
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28 BENJAMIN YOUNG, WITH AN APPENDIX BY JIM BRYAN APPENDIX A. DONALDSON-THOMAS THEORY OF C3/G AND ITS CREPANT RESOLUTION (BY DR. JIM BRYAN) A.1. Review of Donaldson-Thomas theory. Donaldson-Thomas theory, in its incarnation due Maulik, Okounkov, Nek- rasov, and Pandharipande, constructs subtle integer valued deformation in- variants of a threefold X out of the Hilbert scheme of subschemes of X. If X is a Calabi-Yau threefold, i.e., KX is trivial, then this invariant has a simple formulation due to Behrend. It is given by the weighted topological Euler characteristic of the Hilbert scheme where the weighting is by ν, an integer valued constructible function which is canonically associated to any scheme [2]. Let X be a (not necessarily compact) threefold with trivial canonical class. Let In(X, β) be the Hilbert scheme of subschemes Z ⊂X hav- ing proper support of dimension less than or equal to one and with [Z] = β ∈H2(X) and n = χ(OZ). We define the Donaldson-Thomas invariant N n β (X) to be N n β (X) = e(In(X, β), ν)
X k∈Z ke �ν−1(k) where e(·) denotes topological Euler characteristic and ν is Behrend’s con- structible function . The invariants are assembled into the partition function ZDT X as follows. Let C1, … , Cl be a basis for H2(X, Z) such that any effective curve class β is given by d1C1 + · · · + dlCl with di ≥0. Let v1, … , vl be correspond- ing variables and let vβ = vd1 1 · · · vdl l . The Donaldson-Thomas partition function of X is defined by ZDT X (v, q) = X β∈H2(X,Z) X n∈Z N n β (X)vβqn.
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COLOURED 3D YOUNG DIAGRAMS 29 Define the reduced partition function by ZDT X (v, q)′ = ZDT X (v, q) ZDT X (0, q) = M(1, q)−e(X)ZDT X (v, q) where the second equality is a theorem proved by [3, 17, 18]. Maulik, Nekrasov, Okounkov, and Pandharipande conjecture that Don- aldson-Thomas theory is equivalent to Gromov-Witten theory. We assemble GW g β(X), the genus g Gromov-Witten invariants of non-zero degree β into the reduced Gromov-Witten partition function as follows: ZGW X (v, λ)′ = exp X β̸=0 ∞ X g=0 GW g β(X)vβλ2g−2 ! . Conjecture A.1. [21] Under the change of variables q = −eiλ the reduced partition functions of Donaldson-Thomas and Gromov-Witten theory are equal: ZDT X (v, q)′ = ZGW X (v, λ)′. This conjecture has been proven in the case where X is a toric local surface [21] and when X is a local curve [10, 23]. A.2. Orbifold Donaldson-Thomas theory of [C3/G]. Extending Donaldson-Thomas theory to the case of three dimensional orbifolds is expected to be routine since the Hilbert scheme of substacks of a Deligne-Mumford stack has been constructed by Olsson and Starr [28], although it isn’t clear how best to choose the discrete data in general. The orbifolds that we consider are simple enough that we can identify the Hilbert scheme directly. Let G be a finite subgroup of SU(3). A sub- stack of [C3/G] may be regarded as a G-invariant subscheme of C3, and consequently we can regard the Hilbert scheme of [C3/G] as a subset of the Hilbert scheme of C3. Since we require our substacks to have proper sup- port, we need only consider zero dimensional subschemes of C3. For any G- representation R of dimension d we identify HilbR([C3/G]) ⊂Hilbd(C3) as follows: HilbR([C3/G]) = Z ⊂C3 : Z is G-invariant with H0(OZ) = R
.
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30 BENJAMIN YOUNG, WITH AN APPENDIX BY JIM BRYAN This Hilbert scheme has a symmetric perfect obstruction theory induced by the G-invariant part of the of the perfect obstruction theory on Hilbd(C3) = Id(C3, 0). However, we do not need this construction since we can define the Donaldson-Thomas invariants directly using Behrend’s constructible function. Definition A.2. The Donaldson-Thomas invariants of [C3/G] are indexed by representations of G and are given by the Euler characteristics of the Hilbert schemes, weighted by Behrend’s ν function: N R(C3/G) = e(HilbR([C3/G]), ν). Let q0, … , qr be variables corresponding to R0, … , Rr, the irreducible representations of G. For a representation R = d0R0 + · · · + drRr, let qR denote qd0 0 · · · qdr r . We define the orbifold Donaldson-Thomas partition function by ZDT C3/G(q0, … , ql) = X R N R(C3/G)qR where R runs over all representations of G. We now restrict our attention to groups G which are subgroups of SO(3) ⊂ SU(3) and are Abelian. Finite subgroups of SO(3) admit an ADE classi- fication. They are the cyclic groups, the dihedral groups, and the platonic groups. The only Abelian groups from this list are the cyclic groups Zn and the Klein 4-group Z2 × Z2. The action of k ∈Zn on C3 is given by k(x, y, z) = (ωkx, ω−ky, z) where ω = exp � 2πi n . The action of Z2 × Z2 = {0, a, b, c} on C3 is given by a(x, y, z) = (x, −y, −z), b(x, y, z) = (−x, y, −z), c(x, y, z) = (−x, −y, z). As in the introduction, we choose an isomorphism ψ of the group of representations bG with G. Explicitly, we identify 1 ∈Zn with L, the rep- resentation of Zn where 1 ∈Zn acts by multiplication by exp � 2πi n . For
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COLOURED 3D YOUNG DIAGRAMS 31 Z2 × Z2 = {0, a, b, c} we identify a, b, and c with the representations α, β, and γ given by the action on the x, y, and z coordinates of C3 respectively. Theorem A.3. Let qk be the variable corresponding to the group element k ∈Zn and the character Lk. Then ZDT C3/Zn(q0, … , qn−1) = ZZn(−q0, q1, … , qn−1) where ZZn is the Zn-coloured 3D diagram partition function introduced and computed in the main body of the paper (Theorem 1.4). Let {q0, qa, qb, qc} be variables corresponding to {0, a, b, c}, the group elements of Z2 × Z2, and {1, α, β, γ}, the characters of Z2 × Z2. Then ZDT C3/Z2×Z2(q0, qa, qb, qc) = ZZ2×Z2(q0, −qa, −qb, −qc) where ZZ2×Z2 is the Z2 × Z2-coloured 3D diagram partition function intro- duced and computed in the main body of the paper (Theorem 1.5). PROOF: Let G be Zn or Z2 × Z2 and let T ⊂(C×)3 be the subtorus with t1t2t3 = 1. The action of T on C3 commutes with the action of G and hence defines a T-action on [C3/G] and on HilbR(C3/G). The fixed points of T in HilbR(C/G) ⊂Hilbdim R(C3) are isolated, even infinitesimally [3, Lemma 4.1], and they correspond to monomial ideals in C[x, y, z]. The monomial ideals in turn correspond to 3D Young diagrams π where if Z denotes the T-fixed subscheme of C3, then H0(OZ) = X (i,j,k)∈π ti 1tj 2tk 3 as a T-representation viewed as a polynomial in t1, t2, t3 modulo the rela- tion t1t2t3 = 1. Following [21], we adopt the notation Qπ = X (i,j,k)∈π ti 1tj 2tk 3. By [3, Prop. 3.3], the ν-weighted Euler characteristic of HilbR(C3/G) is given by a sum over the T-fixed points, counted with sign given by the parity of the dimension of the Zariski tangent space of HilbR(C3/G) at a fixed point corresponding to 3D diagram π. Hence both the Donaldson- Thomas and the diagram partition functions are given by a sum over 3D
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32 BENJAMIN YOUNG, WITH AN APPENDIX BY JIM BRYAN diagrams, weighted, up to a sign, by the same variables. Thus our main task is to determine the sign. Let π be a 3D diagram having N = |π| boxes and having |π|g boxes of colour g ∈G. Let Tπ denote the Zariski tangent space of HilbN(C3) at the subscheme corresponding to π. Let (Tπ)0 ⊂Tπ be the Zariski tangent space of HilbR([C3/G]) ⊂HilbN(C3) at the same point. Tπ can be regarded as both a T-representation and a G-representa- tion. (Tπ)0 is given by the G-invariant subspace of Tπ. The difference of Tπ and its dual T ∨ π , regarded as a virtual (C×)3-repre- sentation, is computed in [21, equation (13)] and given by Tπ −T ∨ π = Qπ − Qπ t1t2t3
- QπQπ (1 −t1)(1 −t2)(1 −t3) t1t2t3 where Qπ(t1, t2, t3) = Qπ(t−1 1 , t−1 2 , t−1 3 ). Using the relation t1t2t3 = 1 to eliminate t3 from the above expression, we can regard Tπ −T ∨ π as an element in R(T) ∼= Z[t1, t2, t−1 1 , t−1 2 ], the virtual representation ring of T. Following [21], we let Vπ = Qπ + QπQπ(1 −t1)(1 −t2)t−1 1 t−1 2 which satisfies the easily verified equation (9) Tπ −T ∨ π = Vπ −V ∨ π in R(T), and also has the crucial property that the constant term of Vπ is even [21, Lemma 10]. These facts allow us to use Vπ as a surrogate for Tπ when computing the parity of the dimension: Lemma A.4. Let (Tπ)0 and (Vπ)0 denote the G-invariant part of Tπ and Vπ respectively, then dim (Tπ)0 ≡vdim (Vπ)0 mod 2.
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COLOURED 3D YOUNG DIAGRAMS 33 PROOF: From equation (9) we see that Tπ −Vπ is self-dual. Thus all non- constant monomials occur in pairs of the form aij(ti 1tj 2 +t−i 1 t−j 2 ). Moreover, the constant term of Vπ is even [21, Lemma 10] and the constant term of Tπ is zero [3, Lemma 4.1]. Thus we have vdim (Tπ −Vπ) ≡0 mod 2. Indeed, the above argument shows that if we restrict Tπ−Vπ to any self-dual collection of weights, the virtual dimension will be even. In particular, the G-invariant part of Tπ −Vπ has even virtual dimension, which proves the lemma. □ To compute the parity of the G-invariant part of Vπ, we work in the rep- resentation ring of G with mod 2 coefficients. The restriction map R(T) ∼= Z[t1, t2, t−1 1 , t−1 2 ] →RZ2(G) is explicitly given by (t1, t2) 7→(L, L−1) in the case where G = Zn, and by (t1, t2) 7→(α, β) in the case where G = Z2 × Z2. For any W ∈RZ2(G) and any irreducible representation ζ, let [W]ζ ∈Z2 denote the coefficient of ζ in W. We compute [Vπ]1 in RZ2(Zn) = Z2[L]/(Ln −1) as follows. [Vπ]1 = Qπ + QπQπ(1 −L)(1 −L−1) 1 = [Qπ]1 + QπQπ(L + L−1) 1 = [Qπ]1 + QπQπ L−1 + QπQπ L = [Qπ]1 = |π|0 mod 2
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34 BENJAMIN YOUNG, WITH AN APPENDIX BY JIM BRYAN Since [Vπ]1 is equal to the dimension of the Zn-invariant part of Tπ modulo 2, the 3D diagram π is counted with sign (−1)|π|0 in the Donaldson-Thomas partition function of C3/Zn. This proves first part of Theorem A.3. We now compute [Vπ]1 in RZ2(Z2 × Z2) = Z2[α, β]/(α2 −1, β2 −1). We use the fact that in this ring, the square of an arbitrary element is equal to the sum of its coefficients: (n1 + n2α + n3β + n4αβ)2 = n1 + n2 + n3 + n4, and we compute as follows. [Vπ]1 = Qπ + QπQπ(1 −α)(1 −β)αβ 1 = [Qπ]1 + Q2 π(1 + α + β + αβ) 1 = [Qπ]1 + [|π|(1 + α + β + αβ)]1 = |π|0 + |π| mod 2 = |π|a + |π|b + |π|c mod 2. Since [Vπ]1 is equal to the dimension of the Z2 × Z2-invariant part of Tπ modulo 2, the 3D diagram π is counted with sign (−1)|π|a+|π|b+|π|c in the Donaldson-Thomas partition function of C3/Z2 × Z2. This proves the re- maining part of Theorem A.3 and so the proof of Theorem is complete. □ Remark A.5. For any finite Abelian subgroup G ⊂SU(3), the Donaldson- Thomas invariants of C3/G are given by a signed count of boxes coloured by G. However, it is not always true that this sign is obtained by simply changing the signs of some of the variables. For example, consider the case of G = Z3 acting on C3 with equal weights on all three factors. The sign associated to a 3D partition π can be computed by the methods of this appendix and is given by (−1)σ, where σ = |π|1 + |π|2 + |π|0|π|1 + |π|0|π|2 + |π|1|π|2. Thus the coloured 3D diagram partition function and the Donaldson-Thom- as partition function are not related in an obvious way.
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COLOURED 3D YOUNG DIAGRAMS 35 A.3. The Donaldson-Thomas crepant resolution conjecture. A well known principle in physics asserts that string theory on a Calabi- Yau orbifold X is equivalent to string theory on any crepant resolution Y → X. Consequently, it is expected that mathematical counterparts of string theory, such as Gromov-Witten theory or Donaldson-Thomas theory, should be equivalent on X and Y . Precise formulations of these equivalences are known as crepant resolution conjectures. The crepant resolution conjecture in Gromov-Witten theory goes back to Ruan, and has recently undergone successive refinements [29, 8, 11, 12]. In this section we formulate a crepant resolution conjecture for Don- aldson-Thomas theory. Our conjecture has somewhat limited scope: we stick to the “local case” where X is of the form [C3/G], and (for reasons explained below) we impose the hard Lefschetz condition [8, Defn 1.1], which implies [7] that G is a finite subgroup of either SU(2) ⊂SU(3) or SO(3) ⊂SU(3). The most straightforward formulation of the crepant resolution conjec- ture in Donaldson-Thomas theory posits that the partition functions of the orbifold and its resolution are equal after some natural change of vari- ables. For the orbifold [C3/G], we saw in the previous section that the partition function has variables naturally indexed by irreducible G-repre- sentations. By the classical McKay correspondence, the crepant resolution YG →C3/G given by the G-Hilbert scheme has a basis of H∗(YG) also labelled by irreducible G-representations [6, 22]. However, the variables of the Donaldson-Thomas partition function of YG correspond to a basis of H0(YG) ⊕H2(YG). So in order to get the number of variables of ZDT YG and ZDT C3/G to match, we need H∗(YG) = H0(YG) ⊕H2(YG). This occurs if and only if YG →C3/G is a semi-small resolution. This con- dition is equivalent to the orbifold satisfying the hard Lefschetz condition. Conjecture A.6. Let X be a local, 3 dimensional, Calabi-Yau orbifold sat- isfying the hard Lefschetz condition, namely, X = XG = [C3/G] where G is a finite subgroup of either SU(2) ⊂SU(3) or SO(3) ⊂SU(3).
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36 BENJAMIN YOUNG, WITH AN APPENDIX BY JIM BRYAN Let q0, q1, … , ql be variables corresponding to the irreducible G-repre- sentations R0, R1, … , Rl where R0 is the trivial representation. Let YG → XG be the crepant resolution given by the G-Hilbert scheme and let v1, … , vl be the variables corresponding to the basis of curve classes in YG labelled by the non-trivial G-representations R1, … , Rl. Then the Donaldson-Thomas partition functions of YG and XG are re- lated by the formula ZDT XG (q0, … , ql) = M(1, q)−e(YG)ZDT YG (q, v1, … , vl)ZDT YG (q, v−1 1 , … , v−1 l ) under the identification of the variables vi = qi for i = 1, … , l, q = qRreg = qdim R0 0 · · · qdim Rl l . Proposition A.7. Conjecture A.6 holds for G Abelian, namely for G = Zn or G = Z2 × Z2. Remark A.8. Szendr˝oi proved [31] that a similar relationship holds be- tween the Donaldson-Thomas partition functions of the non-commutative conifold singularity and its small resolution. Remark A.9. The Gromov-Witten partition function of YG has been com- puted for all G in SU(2) or SO(3) in [7] (see also Remark A.10). This provides, via the MNOP conjecture, a prediction for ZDT YG and hence our conjecture A.6 gives a prediction for ZDT C3/G which can be tested term by term. Verification of this prediction for terms of low order has been ob- tained by D. Steinberg in the case where G is the quaternion 8 group. In light of Theorems 1.4, 1.5, and A.3, Proposition A.7 is equivalent to Theorem 1.7 which we prove here. A.3.1. Proof of Proposition A.7 / Theorem 1.7: Since G is Abelian, YG is toric and so via [21, Theorems 2 and 3], the reduced Donaldson-Thomas partition function of YG is equal to the reduced Gromov-Witten partition function of YG after the change of variables q = −eiλ. Thus it suffices to
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COLOURED 3D YOUNG DIAGRAMS 37 compute the Gromov-Witten partition function of YG
- The pithiest way to encode the Gromov-Witten invariants is in terms of Gopakumar-Vafa invariants, or so called BPS state counts. It is well know that each genus zero BPS state count n0 β contributes a factor of M(vβ, −eiλ)−n0 β to the Gro- mov-Witten partition function (see for example the proof of Theorem 3.1 in [4]). Thus the content of Theorem 1.7 is that YZn has genus 0 Gopakumar- Vafa invariants occurring in the classes Ca + · · · + Cb for 0 < a ≤b < n with value -1, and that YZ2×Z2 has genus 0 Gopakumar-Vafa invariants occurring in the classes Ca, Cb, Cc, and Ca + Cb + Cc with value 1 and in the classes Ca + Cb, Ca + Cc, and Cb + Cc with value -1. Moreover, all other Gopakumar-Vafa invariants are zero. These assertions are proved in [15]: the case of YZ2×Z2 is Corollary 16 and Proposition 19 and the case of YZn is Proposition 12. □ Remark A.10. The Gromov-Witten and Donaldson-Thomas theories of YG are equivariant theories and so in general depend on the choice of the torus action. In this paper, we have assumed that the torus is chosen to act triv- ially on the canonical class. This choice is required to apply the topolog- ical vertex formalism as we have done in the above proof. We warn the reader that the computation of the Gromov-Witten invariants of YG for gen- eral G ⊂SO(3) done in [7] is done using the C× action induced from the diagonal action on C3/G. This does not change which classes carry Gopakumar-Vafa invariants, but it can change the values of the invariants in those curve classes that admit deformations to infinity. REFERENCES
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