# Conjectures on counting associative 3-folds in G 2 -manifolds

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1 in G 2 -manifolds Dominic Joyce, Oxford University Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics, First Annual Meeting, New York City, September Based on arxiv: These slides available at joyce/. 1 / 23 Dominic Joyce, Oxford University in G 2 -manifolds

2 Plan of talk: 1 G 2 -manifolds and associative 3-folds 2 3 Disclaimer Everything may be false. 2 / 23 Dominic Joyce, Oxford University in G 2 -manifolds

3 1. G 2 -manifolds and associative 3-folds Let (X, g) be a Riemannian manifold, and x X. The holonomy group Hol(g) is the group of isometries of T x X given by parallel transport using the Levi-Civita connection around loops in X based at x. They were classified by Berger: Theorem (Berger, 1955) Suppose X is simply-connected of dimension n and g is irreducible and nonsymmetric. Then either: (i) Hol(g) = SO(n) [generic]; (ii) n = 2m 4 and Hol(g) = U(m), [Kähler manifolds]; (iii) n = 2m 4 and Hol(g) = SU(m), [Calabi Yau m-folds]; (iv) n = 4m 8 and Hol(g) = Sp(m), [hyperkähler]; (v) n =4m 8 and Hol(g)=Sp(m) Sp(1), [quaternionic Kähler]; (vi) n = 7 and Hol(g) = G 2, [exceptional holonomy] or (vii) n = 8 and Hol(g) = Spin(7) [exceptional holonomy]. 3 / 23 Dominic Joyce, Oxford University in G 2 -manifolds

4 We are interested in 7-manifolds (X, g) with holonomy G 2. Then g is Ricci-flat. Any such X has a natural closed 3-form ϕ and closed 4-form ϕ, and I refer to (X, ϕ, ϕ) as a G 2 -manifold. Many examples of compact 7-manifolds (X, ϕ, ϕ) with holonomy G 2 were constructed by Joyce (1996), Kovalev (2003), and Corti Haskins Nordström Pacini (2015). They are important in String Theory and M-theory. The moduli space of holonomy G 2 -metrics on a compact X is smooth of dimension b 3 (X ). 4 / 23 Dominic Joyce, Oxford University in G 2 -manifolds

5 We are interested in 7-manifolds (X, g) with holonomy G 2. Then g is Ricci-flat. Any such X has a natural closed 3-form ϕ and closed 4-form ϕ, and I refer to (X, ϕ, ϕ) as a G 2 -manifold. Many examples of compact 7-manifolds (X, ϕ, ϕ) with holonomy G 2 were constructed by Joyce (1996), Kovalev (2003), and Corti Haskins Nordström Pacini (2015). They are important in String Theory and M-theory. The moduli space of holonomy G 2 -metrics on a compact X is smooth of dimension b 3 (X ). If (X, ϕ, ϕ) is a G 2 -manifold then ϕ, ϕ are calibrations on (X, g), in the sense of Harvey Lawson, so we have natural classes of calibrated submanifolds, called associative 3-folds, and coassociative 4-folds. They are minimal submanifolds in (X, g). The deformation theory of compact associatives and coassociatives was studied by McLean (1998). Both are controlled by an elliptic equation, and so are well behaved. Moduli spaces of associative 3-folds N may be obstructed, and have virtual dimension 0. Moduli spaces of coassociative 4-folds C are smooth of dimension b 2 +(C). 4 / 23 Dominic Joyce, Oxford University in G 2 -manifolds

6 An analogy between G 2 -manifolds and Calabi Yau 3-folds There is a strong analogy: G 2 -manifolds (X, ϕ, ϕ) Calabi Yau 3-folds (Y, J, h) G 2 3-form ϕ Kähler form ω of h G 2 4-form ϕ complex structure J Associative 3-folds N J-holomorphic curves Σ in Y Coassociative 4-folds C special Lagrangians L in Y. For instance, if Y is a Calabi Yau 3-fold then X = Y S 1 is a G 2 -manifold, and J-holomorphic curves Σ, special Lagrangians L map to associative 3-folds Σ S 1 and coassociative 4-folds L S 1 in Y S 1. Now lots of exciting mathematics is known about Calabi Yau 3-folds, so we can ask how much of this may extend to G 2 -manifolds. Today we ask whether theories on counting J-holomorphic curves in Calabi Yau 3-folds extend to theories on counting associative 3-folds in G 2 -manifolds. 5 / 23 Dominic Joyce, Oxford University in G 2 -manifolds

7 Invariant theories in Symplectic Geometry Let (Y, ω) be a compact symplectic manifold (e.g. a Calabi Yau 3-fold) and J an almost complex structure on Y compatible with ω. Then symplectic geometers can define: The Gromov Witten invariants GW g,k (α) of (Y, ω) counting J-holomorphic curves Σ with genus g and k marked points in homology class α H 2 (Y ; Z). The Lagrangian Floer cohomology groups HF (L 1, L 2 ) of compact Lagrangians L 1, L 2 in Y. The Fukaya category D b F(Y ). All of these are defined by counting J-holomorphic curves Σ in Y satisfying some conditions, but have the magic property that they are independent of the choice of J. That is, we can deform J through a smooth family J t : t [0, 1], but the structures defined using J t do not change (up to canonical isomorphism). 6 / 23 Dominic Joyce, Oxford University in G 2 -manifolds

8 Invariant theories in G 2 geometry? So by analogy with Calabi Yau 3-folds, we can ask: Question 1 Given a compact G 2 -manifold (X, ϕ, ϕ), can we define interesting theories analogous to Gromov Witten invariants, etc., by counting associative 3-folds in X, so that the answer is unchanged under continuous deformations of ϕ? Actually, this is not yet a good question. In the symplectic case we fix ω and vary J. We have an analogy ϕ ω, ϕ J, so it would seem natural to consider varying the G 2 -structure so that ϕ is fixed and ϕ varies. But ϕ is determined by ϕ, so this makes no sense. Our solution is to enlarge the class of G 2 -manifolds we consider. Following Donaldson and Segal, we define Tamed Almost G 2 -manifolds, or TA G 2 -manifolds, (X, ϕ, ψ) to be a 7-manifold X with a closed G 2 3-form ϕ and a closed G 2 4-form ψ which satisfy a pointwise compatibility condition that is weaker than ψ = ϕ. 7 / 23 Dominic Joyce, Oxford University in G 2 -manifolds

9 We can add an extra line to our analogy: TA G 2 -manifolds (X, ϕ, ψ) symplectic manifold (Y, ω) with compatible almost structure J. Then associative 3-folds N in a TA G 2 -manifold (X, ϕ, ψ) depend only on ψ, but the natural notion of volume is vol(n) = N ϕ, just as J-holomorphic curves Σ depend only on J but have volume ω. We refine Question 1 to: Σ Question 2 Given a compact TA G 2 -manifold (X, ϕ, ψ), can we define interesting theories analogous to Gromov Witten invariants, etc., by counting associative 3-folds in X, so that the answer is unchanged under continuous deformations of (ϕ, ψ) fixing ϕ? This is related to understanding topological M Theory. Answering this depends on understanding the singular behaviour of associative 3-folds, as singularities can break deformation-invariance. 8 / 23 Dominic Joyce, Oxford University in G 2 -manifolds

10 2. Many important areas of geometry concern invariants. These have the following general structure: We begin with a primary geometric object X we want to study, e.g. a compact oriented 4-manifold, or a symplectic manifold. We choose some secondary geometric data G on X, e.g. a Riemannian metric g on the 4-manifold, or an almost complex structure J on the symplectic manifold (X, ω). Usually G lives in a large, connected, infinite-dimensional family. Using X, G we define a nonlinear elliptic equation for objects E X, e.g. instantons on oriented Riemannian 4-manifold (X, J), or J-holomorphic curves in (X, ω). These objects E form (compactified) moduli spaces M E. We count the moduli spaces M E to get invariants [M E ] virt. We prove the resulting numbers (or homology groups, etc.) are independent of the secondary data G, and so depend only on X. 9 / 23 Dominic Joyce, Oxford University in G 2 -manifolds

11 We can divide invariant problems into four types, in decreasing order of niceness: (A) Absolute invariants: the numbers (or homology classes, etc.) [M E ] virt are completely independent of secondary data G. (B) Invariants with cohomological wall crossing. The [M E ] virt do depend on G, but in a nice way: from G we can define some cohomology classes c(g), and the [M E ] virt change according to a rigid wall-crossing formula when c(g) crosses certain real hypersurfaces in its cohomology group. (C) No invariant numbers, but subtle homological information conserved. The [M E ] virt depend on G in a nasty way, the individual numbers change chaotically and unpredictably with G, but from the family of all [M E ] virt we can extract some nontrivial information such as a cohomology group which is independent of G. (D) No conserved information. Nothing works. It s not always easy to tell the difference between (C) and (D). 10 / 23 Dominic Joyce, Oxford University in G 2 -manifolds

12 Examples of invariant problems of each type (A) Absolute invariants. Closed String. Donaldson or Seiberg Witten invariants counting instantons on closed oriented 4-manifolds X with b 2 +(X ) > 1. Gromov Witten invariants counting J-holomorphic curves in symplectic manifolds/smooth schemes. (B) Invariants with wall-crossing. Counting branes/bps states. Donaldson/SW invariants counting instantons on closed oriented 4-manifolds X with b 2 +(X ) = 1. Donaldson Thomas invariants. (C) Invariant structures from homological algebra. Open string. Morse flow lines and Morse homology. Instanton/SW Floer homology. J-holomorphic curves with boundary in Lagrangians; Lagrangian Floer cohomology; Fukaya categories. (D) No conserved information. Number of squirrels in Central Park. 11 / 23 Dominic Joyce, Oxford University in G 2 -manifolds

13 Examples of invariant problems of each type (A) Absolute invariants. Closed String. Donaldson or Seiberg Witten invariants counting instantons on closed oriented 4-manifolds X with b 2 +(X ) > 1. Gromov Witten invariants counting J-holomorphic curves in symplectic manifolds/smooth schemes. (B) Invariants with wall-crossing. Counting branes/bps states. Donaldson/SW invariants counting instantons on closed oriented 4-manifolds X with b 2 +(X ) = 1. Donaldson Thomas invariants. (C) Invariant structures from homological algebra. Open string. Morse flow lines and Morse homology. Instanton/SW Floer homology. J-holomorphic curves with boundary in Lagrangians; Lagrangian Floer cohomology; Fukaya categories. Counting associative 3-folds? Counting G 2 -instantons (D S)? (D) No conserved information. Number of squirrels in Central Park. 11 / 23 Dominic Joyce, Oxford University in G 2 -manifolds

14 Examples of invariant problems of each type (A) Absolute invariants. Closed String. Donaldson or Seiberg Witten invariants counting instantons on closed oriented 4-manifolds X with b 2 +(X ) > 1. Gromov Witten invariants counting J-holomorphic curves in symplectic manifolds/smooth schemes. (B) Invariants with wall-crossing. Counting branes/bps states. Donaldson/SW invariants counting instantons on closed oriented 4-manifolds X with b 2 +(X ) = 1. Donaldson Thomas invariants. (C) Invariant structures from homological algebra. Open string. Morse flow lines and Morse homology. Instanton/SW Floer homology. J-holomorphic curves with boundary in Lagrangians; Lagrangian Floer cohomology; Fukaya categories. Counting associative 3-folds? Counting G 2 -instantons (D S)? (D) No conserved information. Number of squirrels in Central Park.... until Kontsevich s Homological Squirrel Conjecture? 11 / 23 Dominic Joyce, Oxford University in G 2 -manifolds

15 More about problems of type (C) A simple example: Morse homology and Morse flow lines Let X be a compact manifold, and f : X R a fixed Morse function. Pick some generic Riemannian metric g on X (the secondary data G), and for critical points p, q of f, consider the moduli spaces M(p, q) of gradient flow-lines γ : R X of f (i.e. d dt γ(t) = f γ(t)) with lim t γ(t) = p, lim t γ(t) = q. Then (simplifying a bit) M(p, q) is an oriented manifold with corners, of dimension µ(p) µ(q) 1, where µ(p) is the Morse index (number of negative e-values of Hess p (f )), with boundary M(p, r) = q Crit(f ) M(p, q) M(q, r). (1) This boundary behaviour comes from broken flow-lines a real codimension 1 singular behaviour of flow-lines. 12 / 23 Dominic Joyce, Oxford University in G 2 -manifolds

16 We define Morse homology MH (X ) as the homology of the complex (MC (X ), ), where MC k (X ) has basis p Crit(f ) with µ(p) = k, and p = q Crit(f ):µ(q)=k 1 #M(p, q) q. Here M(p, q) is a compact oriented 0-manifold, i.e. a finite set with signs, and we count with signs to get #M(p, q). Note that these numbers #M(p, q) are not invariant under deformations of g. However, the homology MH (X ) of the complex (MC (X ), ) is canonically isomorphic to H (X ; R), and so is independent of g. The deformation-invariant information is contained in the collection of all numbers #M(p, q), not in individual numbers. As we deform g through a family g t : t [0, 1], in real codimension 1 in t, a flow line γ : q q can appear with µ(q) = µ(q ), so that vdim(m(q, q )) = 1. Then the numbers change by: #M(p, q ) #M(p, q ) ± #M(p, q), µ(p) = µ(q) + 1, #M(q, r) #M(q, r) #M(q, r), µ(r) = µ(q) / 23 Dominic Joyce, Oxford University in G 2 -manifolds

17 3. Let (X, ϕ, ψ) be a TA G 2 -manifold, and α H 3 (X ; Z) a homology class. Write M α (ψ) for the moduli space of compact associative 3-folds N in X with [N] = α in H 3 (X ; Z). McLean proved the deformation theory of M α (ψ) is elliptic, possibly obstructed, with virtual dimension 0. We expect that if ψ is generic, then there are no obstructions: Conjecture 1 Let (X, ϕ, ψ) be a compact TA-G 2 -manifold with ψ generic in the infinite-dimensional family of closed 4-forms on X. Then moduli spaces M α (ψ) of associative 3-folds N with [N] = α H 3 (X ; Z) are smooth compact 0-manifolds (i.e. finite sets). 14 / 23 Dominic Joyce, Oxford University in G 2 -manifolds

18 Counting invariants: elementary considerations Assume Conjecture 1, and consider how we might define invariants counting associative 3-folds. Two big issues are: Orientations of moduli spaces. We want a way to define orientations on the moduli spaces M α (ψ). For ψ generic, the moduli spaces are compact 0-manifolds, and orientations are maps Or : M α (ψ) {±1}. Thus we have counts #M α (ψ) = Or([N]). [N] M α (ψ) Dependence on ψ. Let (ϕ 0, ψ 0 ), (ϕ 1, ψ 1 ) be given with ψ 0, ψ 1 generic, and (ϕ t, ψ t ) : t [0, 1] be a generic smooth path of TA-G 2 -structures between them. Then we have families M α (ψ t ) for t [0, 1]. We expect that at finite sets of values 0 < t 1 < < t k < 1 of t, the moduli spaces can change, becoming singular/obstructed. Then #M α (ψ t ) may change discontinuously at t = t i, and so not be invariants. 15 / 23 Dominic Joyce, Oxford University in G 2 -manifolds

19 Real codimension 1 singular behaviour So, the really important issue to understand is: how can moduli spaces M α (ψ t ) change in generic 1-parameter families (ϕ t, ψ t ) : t [0, 1]? That is, what is the possible singular behaviour of associative 3-folds which happens in codimension 1 of possible G 2 4-forms ψ? Although general singularities of associatives may be too horrible to feasibly understand, it seems plausible that there may be only finitely many kinds of singularity that occur generically in codimension 1 the most common kinds of singularity. We could hope to understand all these types of codimension 1 singular behaviour, at least conjecturally. Then we would understand the only ways moduli spaces M α (ψ t ) can change in generic families (ϕ t, ψ t ) : t [0, 1], and we could try to build invariants, cohomology groups, etc., unaffected by these changes. This is the approach of my paper (also compare Donaldson Segal). 16 / 23 Dominic Joyce, Oxford University in G 2 -manifolds

20 U(1)-invariant associative 3-folds in R 7 Consider associative 3-folds N in R 7 invariant under the U(1)-action e iθ : (x 1,..., x 7 ) (x 1, x 2, x 3, cos θ x 4 sin θ x 5, sin θ x 4 + cos θ x 5, cos θ x 6 + sin θ x 7, sin θ x 6 + cos θ x 7 ). Define U(1)-invariant quadratic polynomials y 1, y 2, y 3 on R 7 by y 1 (x 1,..., x 7 ) = x x 2 5 x 2 6 x 2 7, y 2 (x 1,..., x 7 ) = 2(x 4 x 7 + x 5 x 6 ), y 3 (x 1,..., x 7 ) = 2(x 4 x 6 x 5 x 7 ). Then y y y 2 3 = (x x x x 2 7 )2. Consider the map Π = (x 1, x 2, x 3, y 1, y 2, y 3 ) : R 7 R 6 = C 3. This induces a homeomorphism Π : R 7 /U(1) R 6. The U(1)-fixed set R 3 R 7 maps to L = R 3 = { (x 1, x 2, x 3, 0, 0, 0) : x j R } in R 6 = C / 23 Dominic Joyce, Oxford University in G 2 -manifolds

21 Proposition G 2 -manifolds and associative 3-folds There is a 1-1 correspondence between U(1)-invariant associative 3-folds N in R 7 and J-holomorphic curves Σ in R 6 with boundary Σ L = R 3 R 6 by Σ = Π(N/U(1)), where J is an almost complex structure on R 6 with singularities on L = R 3 R 6. Explicitly, writing u : R 6 [0, ), u(x 1, x 2, x 3, y 1, y 2, y 3 ) = (y1 2 + y y 3 2)1/2, then in block diagonal form on R 6 = R 3 R( 3, we have 0 1 ) J = 2 u 1/2 id R 3 2u 1/2, id R 3 0 so that J is singular when u = 0. But I do not expect the singularities of J to affect the heuristic behaviour of curves Σ. Conclusion Counting closed associative 3-folds should be similar to counting J-holomorphic curves with boundary. So it may be of type (C), and we should not look for invariants, even with wall-crossing, but for Floer-type groups and subtle homological algebra information. 18 / 23 Dominic Joyce, Oxford University in G 2 -manifolds

22 Invariant information from counting associative 3-folds In my paper I make a proposal for extracting invariant information from numbers of associatives #M α (ψ). There are three parts to it: (a) Canonical flags of associative 3-folds, and defining natural orientations for associative moduli spaces M α (ψ). (b) Conjectural description of codimension 1 singular behaviour of associative 3-folds six different kinds of singularity of moduli spaces. (c) Define a superpotential Φ ψ : U Λ, where Λ is a Novikov ring of formal power series, and U is an open set in H 3 (X ; Λ). This Φ ψ is very roughly a generating function for #M α (ψ) for associative Q-homology spheres N, plus higher contributions involving linking numbers and self-linking numbers of such N. Conjecture: Φ ψ is not invariant, but changes under deformations by Φ ψ Φ ψ Υ for Υ : U U a diffeomorphism of U as a rigid analytic space. Given a choice of critical point of Φ ψ, we define G 2 -quantum cohomology, an associative, supercommutative Λ-algebra. I expect it to be deformation-invariant up to isomorphism. 19 / 23 Dominic Joyce, Oxford University in G 2 -manifolds

23 (a) Canonical flags and orientations Let X be an oriented 7-manifold, and N X a compact, connected, oriented 3-submanifold. A flag [s] on N is (roughly) an isotopy class of nonvanishing sections s of the normal bundle ν N of N in X. It is like a framing of a knot in R 3. The set Flag(N) of flags on N is a Z-torsor. 20 / 23 Dominic Joyce, Oxford University in G 2 -manifolds

24 (a) Canonical flags and orientations Let X be an oriented 7-manifold, and N X a compact, connected, oriented 3-submanifold. A flag [s] on N is (roughly) an isotopy class of nonvanishing sections s of the normal bundle ν N of N in X. It is like a framing of a knot in R 3. The set Flag(N) of flags on N is a Z-torsor. I define a flag structure F on 7-manifolds X, a new algebro-topological structure, giving a sign F (N, [s]) = ±1 to each immersed flagged 3-submanifold (N, [s]), satisfying some rules. 20 / 23 Dominic Joyce, Oxford University in G 2 -manifolds

25 (a) Canonical flags and orientations Let X be an oriented 7-manifold, and N X a compact, connected, oriented 3-submanifold. A flag [s] on N is (roughly) an isotopy class of nonvanishing sections s of the normal bundle ν N of N in X. It is like a framing of a knot in R 3. The set Flag(N) of flags on N is a Z-torsor. I define a flag structure F on 7-manifolds X, a new algebro-topological structure, giving a sign F (N, [s]) = ±1 to each immersed flagged 3-submanifold (N, [s]), satisfying some rules. Let (X, ϕ, ψ) be a TA-G 2 -manifold, and N X be a compact associative 3-fold, with unobstructed deformation theory. McLean s deformation theory for N gives a twisted Dirac operator D N : Γ (ν) Γ (ν), self-adjoint as dψ = 0. By comparing D N with d + d : Γ (Λ 0 T N Λ 2 T N) Γ (Λ 0 T N Λ 2 T N) and using spectral flow, I define a canonical flag [s N ] for N. 20 / 23 Dominic Joyce, Oxford University in G 2 -manifolds

26 (a) Canonical flags and orientations Let X be an oriented 7-manifold, and N X a compact, connected, oriented 3-submanifold. A flag [s] on N is (roughly) an isotopy class of nonvanishing sections s of the normal bundle ν N of N in X. It is like a framing of a knot in R 3. The set Flag(N) of flags on N is a Z-torsor. I define a flag structure F on 7-manifolds X, a new algebro-topological structure, giving a sign F (N, [s]) = ±1 to each immersed flagged 3-submanifold (N, [s]), satisfying some rules. Let (X, ϕ, ψ) be a TA-G 2 -manifold, and N X be a compact associative 3-fold, with unobstructed deformation theory. McLean s deformation theory for N gives a twisted Dirac operator D N : Γ (ν) Γ (ν), self-adjoint as dψ = 0. By comparing D N with d + d : Γ (Λ 0 T N Λ 2 T N) Γ (Λ 0 T N Λ 2 T N) and using spectral flow, I define a canonical flag [s N ] for N. Given a choice of flag structure F on X, we define orientations Or : M α (ψ) {±1} on associative moduli spaces by Or(N) = F (N, [s N ]). I claim this is a natural thing to do. 20 / 23 Dominic Joyce, Oxford University in G 2 -manifolds

27 (b) Conjectural codim 1 singular behaviour of associatives In my paper I describe 6 kinds of codim 1 singularities of associatives. Here I will explain one. Suppose we have a generic 1-parameter family of TA-G 2 -manifolds (X, ϕ t, ψ t ), t [0, 1], and compact, unobstructed associatives Nt 1, Nt 2 in (X, ϕ t, ψ t ) with [Nt] i = α i in H 3 (X ; Z). For generic t [0, 1] we expect Nt 1 Nt 2 =, but at t 0 (0, 1) we may have Nt 1 Nt 2 = {p}. Then (Nordström) we can create a new family of associatives Nt 3 for t (t 0, 1] with topology the connect sum Nt 3 = Nt 1 #Nt 2 and homology class [Nt 3 ] = α 1 + α 2, such that for t close to t 0, Nt 3 near p resembles a small Lawlor neck S 2 R in T p X = C 3 R. Thus #M α1 +α 2 (ψ t ) changes by ±1 as t crosses t 0, so it is not invariant. 21 / 23 Dominic Joyce, Oxford University in G 2 -manifolds

28 (b) Conjectural codim 1 singular behaviour of associatives In my paper I describe 6 kinds of codim 1 singularities of associatives. Here I will explain one. Suppose we have a generic 1-parameter family of TA-G 2 -manifolds (X, ϕ t, ψ t ), t [0, 1], and compact, unobstructed associatives Nt 1, Nt 2 in (X, ϕ t, ψ t ) with [Nt] i = α i in H 3 (X ; Z). For generic t [0, 1] we expect Nt 1 Nt 2 =, but at t 0 (0, 1) we may have Nt 1 Nt 2 = {p}. Then (Nordström) we can create a new family of associatives Nt 3 for t (t 0, 1] with topology the connect sum Nt 3 = Nt 1 #Nt 2 and homology class [Nt 3 ] = α 1 + α 2, such that for t close to t 0, Nt 3 near p resembles a small Lawlor neck S 2 R in T p X = C 3 R. Thus #M α1 +α 2 (ψ t ) changes by ±1 as t crosses t 0, so it is not invariant. We might hope to compensate for this by also counting pairs (Nt 1, Nt 2 ) weighted by a linking number l(nt 1, Nt 2 ), such that l(nt 1, Nt 2 ) changes by 1 when Nt 1, Nt 2 cross, to cancel the change from Nt 3. However, no such l(nt 1, Nt 2 ) exists. On the face of it, this looks fatal for invariants counting associatives! 21 / 23 Dominic Joyce, Oxford University in G 2 -manifolds

29 (c) A superpotential counting associatives Consider TA-G 2 -structures (ϕ, ψ) on X with [ϕ] = γ H 3 (X ; R) fixed. Let F be the field Q, R or C. Write Λ for the Novikov ring Λ = { i=1 c iq α i : c i F, α i R, α i as i }, with q a formal variable. Write Λ >0 Λ for the ideal with all α i > 0. Write U for the set of group homomorphisms U = Hom ( H 3 (X ; Z), 1 + Λ >0 ). Then U is a smooth rigid analytic space, roughly an open set in H 3 (X ; Λ) in the analytic topology. It has U = Λ b 3(X ) >0. We define a superpotential Φ ψ : U Λ >0, of the form Φ ψ = associatives N: Or(N) H 1 (N; Z) q [ϕ] [N] + higher order terms. H 1 (N; Z) finite These higher order terms do involve a kind of linking number l(n 1, N 2 ), but it needs arbitrary choices to define. They also involve canonical flags [s N ], interpreted as a self-linking number. I conjecture that different arbitrary choices, and deformations, change Φ ψ by reparametrizations Φ ψ Φ ψ Υ, Υ an isomorphism. 22 / 23 Dominic Joyce, Oxford University in G 2 -manifolds

30 G 2 quantum cohomology Motivated by an analogy with bounding cochains in Fukaya Oh Ohta Ono s Lagrangian Floer theory, I believe a natural thing to do is to choose a critical point θ of the superpotential Φ ψ. This is, in a sense, a deformation-invariant thing to do, as if we change Φ ψ Φ ψ = Φ ψ Υ, then we also change θ θ = Υ 1 (θ). Such critical points θ need not exist, and we call (X, ϕ, ψ) unobstructed if Crit(Φ ψ ). We can then define the G 2 quantum cohomology QHθ (X ; Λ), which is a supercommutative algebra over the Novikov ring Λ, a natural deformation of H (X ; Λ), which is similar to Quantum Cohomology in Symplectic Geometry, and should be unchanged under deformations of (X, ϕ, ψ) fixing [ϕ] in H 3 (X ; R). 23 / 23 Dominic Joyce, Oxford University in G 2 -manifolds

31 G 2 quantum cohomology Motivated by an analogy with bounding cochains in Fukaya Oh Ohta Ono s Lagrangian Floer theory, I believe a natural thing to do is to choose a critical point θ of the superpotential Φ ψ. This is, in a sense, a deformation-invariant thing to do, as if we change Φ ψ Φ ψ = Φ ψ Υ, then we also change θ θ = Υ 1 (θ). Such critical points θ need not exist, and we call (X, ϕ, ψ) unobstructed if Crit(Φ ψ ). We can then define the G 2 quantum cohomology QHθ (X ; Λ), which is a supercommutative algebra over the Novikov ring Λ, a natural deformation of H (X ; Λ), which is similar to Quantum Cohomology in Symplectic Geometry, and should be unchanged under deformations of (X, ϕ, ψ) fixing [ϕ] in H 3 (X ; R). In my paper I also point out some possible problems in Donaldson Segal s programme for defining invariants counting G 2 -instantons, and propose to fix these problems by including a choice of critical point θ of Φ ψ in the picture. 23 / 23 Dominic Joyce, Oxford University in G 2 -manifolds

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