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Relations in the tautological ring R. Pandharipande and A. Pixton 1 1 October 2010 0 2 n a J The notes below cover our series of three lectures at Humboldt Uni- 1 1 versity inBerlinfortheOctoberconference Intersection theory on mod- ] uli space (organized by G. Farkas). The topic concerns relations among G the κ classes in the tautological ring of the moduli space of curves A . After a discussion of classical constructions ending in Theorem 1, h. Mg we derive an explicit set of relations from the moduli space of stable t a quotients. In a series of steps, the stable quotient relations are trans- m formed to simpler and simpler forms. The first step, Theorem 3, comes [ almost immediately from the virtual geometry of the moduli space of 1 v stable quotients. After a certain amount analysis, the simpler form 6 3 of Proposition 3 is found. Our final result, Theorem 5, establishes a 2 previously conjectural set of tautological relations proposed a decade 2 . ago by Faber-Zagier. A detailed presentation of the proof will appear 1 0 in [7]. 1 1 : A. Chern vanishing relations v i X Faber’s original relations in Conjectural description of the tautolog- r a ical ring [1] are obtained from a very simple geometric construction. Let π : g C → M be the universal curve over the moduli space, and let πd : d g C → M be the map associated to the dth fiber product of the universal curve. For every point [C,p ,...,p ] d, we have the restriction map 1 d ∈ C (1) H0(C,ω ) H0(C,ω ) . C → C|p1+...+pd 1 2 Sincethecanonical bundleω hasdegree2g 2, themap(1)isinjective C − if d > 2g 2. Over the moduli space d, we obtain the exact sequence − C 0 E Ω Q 0 d → → → → where E is the rank g Hodge bundle, Ω is the rank d > 2g 2 bundle d − with fiber H0(C,ω ), and Q is the quotient bundle of rank C|p1+...+pd d g. Hence, − c (Q) = 0 Ak( d) for k > d g . k ∈ C − After cutting such vanishing c (Q) down with cotangent line and di- k agonal classes in d and pushing-forward via πd to , we arrive at C ∗ Mg Faber’s relations in R∗( ). g M From our point of view, at the center of Faber’s relations in Conjec- tural description of the tautological ring [1] is the function ∞ d ( 1)dxd Θ(t,x) = (1+it) − . d! td d=0 i=1 XY The differential equation d t(x+1) Θ+(t+1)Θ = 0 dx is easily found. Hence, we obtain the following result. Lemma 1. Θ = (1+x)−t+t1 . We introduce a variable set z indexed by pairs of integers z = z i 1, j i 1 . i,j { | ≥ ≥ − } For monomials zσ = zσi,j, i,j i,j Y we define ℓ(σ) = iσ , σ = jσ . i,j i,j | | i,j i,j X X Of course Aut(σ) = σ ! . | | i,j i,j The variables z are used to define a differential operator Q i d = z tj x . i,j D dx i,j (cid:18) (cid:19) X 3 After applying exp( ) to Θ, we obtain D ΘD = exp( ) Θ D ∞ d ( 1)dxd dℓ(σ)t|σ|zσ = (1+it) − d! td Aut(σ) σ d=0 i=1 | | XXY where σ runs over all monomials in the variables z. Define constants Cr(σ) by the formula d ∞ ∞ xd log(ΘD) = Cr(σ) tr zσ . d d! σ d=1 r=−1 XX X By an elementary application of Wick, the t dependence of log(ΘD) has at most simple poles. Finally, we consider the following function, B ∞ ∞ xd γ = 2i κ t2i−1 + Cr(σ) κ tr zσ . 2i(2i 1) 2i−1 d r d! i≥1 − σ d=1r=−1 X XX X Denote the trxdzσ coefficient of exp( γ) by − exp( γ) Q[κ ,κ ,κ ,κ ,...] . − trxdzσ ∈ −1 0 1 2 Our form of Fa(cid:2)ber’s equa(cid:3)tions is the following result. Theorem 1. In Rr( ), the relation g M exp( γ) = 0 − trxdzσ holds when r > g + σ (cid:2)and d > 2(cid:3)g 2. − | | − In the tautological ring R∗( ), the conventions g M κ = 0, κ = 2g 2 −1 0 − willalwaysbefollowed. Forfixedg andr, Theorem1provides infinitely many relations by increasing d. While the proof of Theorem 1 is appealingly simple, the relations do not seem to fit the other forms we will see later. The variables z i,j efficiently encode both the cotangent and diagonal operations studied in Conjectural description of the tautological ring [1]. In particular, the relations of Theorem 1 are equivalent to the mixing of all cotangent and diagonal operations studied there. 4 B. Stable quotient relations I. The function Φ. The relations in the tautological ring R∗( ) obtained from Moduli g M of stable quotients [4] are based on the function ∞ d 1 ( 1)dxd Φ(t,x) = − . 1 it d! td d=0 i=1 − XY Define the coefficients Cr by the logarithm, d ∞ ∞ xd log(Φ) = Crtr . d d! d=1 r=−1 X X By an elementary application of Wick, the t dependence has at most a simple pole. Let B ∞ ∞ xd γ = 2i κ t2i−1 + Crκ tr . 2i(2i 1) 2i−1 d r d! i≥1 − d=1 r=−1 X X X Denote the trxd coefficient of exp( γ) by − exp( γ) Q[κ ,κ ,κ ,κ ,...] . − trxd ∈ −1 0 1 2 (cid:2) (cid:3) In fact, [exp( γ)] is homogeneous of degree r in the κ classes. The trxd − first tautological relations of Moduli space of stable quotients [4] are given by the following result. Theorem 2. In Rr( ), the relation g M exp( γ) = 0 − trxd (cid:2) (cid:3) holds when g 2d 1 < r and g r +1mod 2. − − ≡ For fixed r and d, if Theorem 2 applies in genus g, then Theorem 2 applies in genera h = g 2δ for all natural numbers δ N. The − ∈ genus shifting mod 2 property will also be present in the Faber-Zagier conjecture discussed later. 5 II. Partitions, differential operators, and logs. We will write partitions σ as (1a12a23a3 ...) with ℓ(σ) = a and σ = ia . i i | | i i X X The empty partition corresponding to (102030...) is permitted. In ∅ all cases, we have Aut(σ) = a !a !a ! . 1 2 3 | | ··· Consider the infinite set of variables p ,p ,p ,... . Monomials in the 1 2 3 p correspond to partitions i pa1pa2pa3 ... (1a12a23a3 ...) . 1 2 3 ↔ Given a partition σ, let pσ denote the corresponding monomial. Let ∞ d 1 ( 1)dxd dℓ(σ)t|σ|pσ Φp(t,x) = − 1 it d! td Aut(σ) σ d=0 i=1 − | | XXY where the first sum is over all partitions σ. The summand corresponding to the empty partition equals Φ(t,x). The function Φp is easily obtained from Φ, ∞ d Φp(t,x) = exp p tix Φ(t,x) . i dx ! i=1 X Let D denote the differential operator ∞ d D = p tix . i dx i=1 X Expanding the exponential of D, we obtain 1 1 (2) Φp = Φ+DΦ+ D2Φ+ D3Φ+... 2 6 DΦ 1D2Φ 1D3Φ = Φ 1+ + + +... . Φ 2 Φ 6 Φ (cid:18) (cid:19) Let γ∗ = log(Φ) be the logarithm, DΦ Dγ∗ = . Φ 6 After applying the logarithm to (2), we see 1 log(Φp) = γ∗ +log 1+Dγ∗ + (D2γ∗ +(Dγ∗)2)+ ... 2 (cid:18) (cid:19) 1 = γ∗ +Dγ∗ + D2γ∗ +... 2 where the dots stand for a universal expression in the Dkγ∗. In fact, a remarkable simplification occurs, ∞ d log(Φp) = exp p tix γ∗ . i dx ! i=1 X The result follows from a general identity. Proposition 1. If f is a function of x, then d d log exp λx f = exp λx log(f) . dx dx (cid:18) (cid:18) (cid:19) (cid:19) (cid:18) (cid:19) Proof. A simple computation for monomials in x shows d exp λx xk = (eλx)k . dx (cid:18) (cid:19) Hence, since the differential operator is additive, d exp λx f(x) = f(eλx) . dx (cid:18) (cid:19) (cid:3) The Proposition follows immediately. The coefficients of the logarithm may be written as ∞ ∞ xd log(Φp) = Cr(p) tr d d! d=1 r=−1 X X ∞ ∞ xd ∞ = Cr tr exp dp ti d d! i ! d=1 r=−1 i=1 X X X ∞ ∞ xd dℓ(σ)t|σ|pσ = Cr tr . d d! Aut(σ) σ d=1r=−1 | | XX X 7 III. Full system of tautological relations. Following Proposition 5 of Moduli of stable quotients [4], we can obtain a much larger set of relations in the tautological ring of by g M including several factors of π (saiωbi) in the integrand instead of just a ∗ single factor. We study the associated relations where the a are always i 1. The b then form the parts of a partition σ. i To state the relations we obtain, we start by enriching the function γ from Section B.I, B γp = 2i κ t2i−1 2i−1 2i(2i 1) i≥1 − X ∞ ∞ xd dℓ(σ)t|σ|pσ + Crκ tr . d r+|σ| d! Aut(σ) σ d=1 r=−1 | | XX X Let γp be defined by a similar formula, B b γp = 2i κ ( t)2i−1 2i−1 2i(2i 1) − i≥1 − X b ∞ ∞ xd dℓ(σ)t|σ|pσ + Crκ ( t)r . d r+|σ| − d! Aut(σ) σ d=1r=−1 | | XX X The sign of t in t|σ| does not change in γp. The κ terms which appear −1 will later be set to 0. The full system of relations are obbtain from the coefficients of the functions ∞ exp( γp), exp( κ trp ) exp( γp) r r+1 − − · − r=0 X b 8 Theorem 3. In Rr( ), the relation g M ∞ exp( γp) = ( 1)g exp( κ trp ) exp( γp) r r+1 − trxdpσ − − · − trxdpσ h i h Xr=0 i holds when g 2d 1+ σ < r. b − − | | Again, we see thegenus shifting mod2 property. If the relationholds in genus g, then the same relation holds in genera h = g 2δ for all − natural numbers δ N. ∈ In case σ = , Theorem 3 specializes to the relation ∅ exp( γ(t,x)) = ( 1)g exp( γ( t,x)) − trxd − − − trxd h i h i = ( 1)g+r exp( γ(t,x)) , − − trxd h i nontrivial only if g r +1 mod 2. If the mod 2 condition holds, then ≡ we obtain the relations of Theorem 2. Consider the case σ = (1). The left side of the relation is then ∞ ∞ dxd exp( γ(t,x)) Cs κ ts+1 . − · − d s+1 d! ! trxd h Xd=1 sX=−1 i The right side is ∞ ∞ dxd ( 1)g exp( γ( t,x)) κ t0 + Cs κ ( t)s+1 . − − − · − 0 d s+1 − d! ! trxd h Xd=1 sX=−1 i If g r +1 mod 2, then the large terms cancel and we obtain ≡ κ exp( γ(t,x)) = 0 . 0 − · − trxd h i Since κ = 2g 2 and 0 − (g 2d 1+1 < r) = (g 2d 1 < r), − − ⇒ − − we recover most (but not all) of the σ = equations. ∅ If g r mod 2, then the resulting equation is ≡ ∞ ∞ dxd exp( γ(t,x)) κ 2 Cs κ ts+1 = 0 − · 0 − d s+1 d! ! trxd h Xd=1 sX=−1 i when g 2d < r. − 9 IV. Expanded form. Let σ = (1a12a23a3 ...) be a partition of length ℓ(σ) and size σ . We | | can directly write the corresponding relation in R∗( ) obtained from g M Theorem 3. A subpartition σ′ σ is obtained by selecting a nontrivial subset of ⊂ the parts of σ. A division of σ is a disjoint union (3) σ = σ(1) σ(2) σ(3)... ∪ ∪ of subpartitions which exhausts σ. The subpartitions in (3) are un- ordered. Let (σ) be the set of divisions of σ. For example, S (1121) = (1121), (11) (21) , S { ∪ } (13) = (13), (12) (11) . S { ∪ } We will use the notation σ• to denote a division of σ with subpartitions σ(i). Let 1 Aut(σ) m(σ•) = | | . Aut(σ•) ℓ(σ•) Aut(σ(i)) | | i=1 | | Here, Aut(σ•) is the group permutinQg equal subpartitions. The factor m(σ•) may be interpreted as counting the number of different ways the disjoint union can be made. To write explicitly the pσ coefficient of exp(γp), we introduce the functions ∞ ∞ dnxd F (t,x) = Cs κ ts+m n,m − d s+m d! d=1s=−1 X X for n,m 1. Then, ≥ Aut(σ) exp( γp) = | |· − trxdpσ h i ℓ(σ•) exp( γ(t,x)) m(σ•) F . − · ℓ(σ(i)),|σ(i)| trxd h σ•X∈S(σ) Yi=1 i   The length ℓ(σ∗,•) is the number of unmarked subpartitions. Let σ∗,• be a division of σ with a marked subpartition, (4) σ = σ∗ σ(1) σ(2) σ(3)..., ∪ ∪ ∪ 10 labelled by the superscript . The marked subpartition is permitted to ∗ be empty. Let ∗(σ) denote the set of marked divisions of σ. Let S 1 Aut(σ) m(σ∗,•) = | | . Aut(σ•) Aut(σ∗) ℓ(σ∗,•) Aut(σ(i)) | || | i=1 | | Q Then, Aut(σ) times the right side of Theorem 3 may be written as | | ( 1)g+|σ| Aut(σ) exp( γ( t,x)) − | |· − − · h ℓ(σ∗) ℓ(σ∗,•)  m(σ∗,•) κσj∗−1(−t)σj∗−1 Fℓ(σ(i)),|σ(i)|(−t,x) trxd σ∗,•X∈S∗(σ) Yj=1 Yi=1 i   To write Theorem 3 in the simplest form, the following definition with the Kronecker δ is useful, m±(σ∗,•) = (1 δ ) m(σ∗,•). 0,|σ∗| ± · There are two cases. If g r+ σ mod 2, then Theorem 3 is equivalent ≡ | | to the vanishing of ℓ(σ∗) ℓ(σ∗,•) exp(−γ)· m−(σ∗,•) κσj∗−1tσj∗−1 Fℓ(σ(i)),|σ(i)| trxd. h σ∗,•X∈S∗(σ) Yj=1 Yi=1 i   If g r+ σ +1 mod 2, then Theorem 3 is equivalent to the vanishing ≡ | | of ℓ(σ∗) ℓ(σ∗,•) exp(−γ)· m+(σ∗,•) κσj∗−1tσj∗−1 Fℓ(σ(i)),|σ(i)| trxd. h σ∗,•X∈S∗(σ) Yj=1 Yi=1 i   In either case, the relations are valid in the ring R∗( ) only if the g M condition g 2d 1+ σ < r holds. − − | |