Tau Functions from Joyce Structures

We argued in [Proc. Sympos. Pure Math., Vol. 103, American Mathematical Society, Providence, RI, 2021, 1-66, arXiv:1912.06504] that, when a certain sub-exponential growth property holds, the Donaldson-Thomas invariants of a 3-Calabi-Yau triangulated category should give rise to a geometric structure...

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Published in:	Symmetry, Integrability and Geometry: Methods and Applications
Date:	2024
Main Author:	Bridgeland, Tom
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Language:	English
Published:	Інститут математики НАН України 2024
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Cite this:	Tau Functions from Joyce Structures. Tom Bridgeland. SIGMA 20 (2024), 112, 26 pages

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author	Bridgeland, Tom
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citation_txt	Tau Functions from Joyce Structures. Tom Bridgeland. SIGMA 20 (2024), 112, 26 pages
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container_title	Symmetry, Integrability and Geometry: Methods and Applications
description	We argued in [Proc. Sympos. Pure Math., Vol. 103, American Mathematical Society, Providence, RI, 2021, 1-66, arXiv:1912.06504] that, when a certain sub-exponential growth property holds, the Donaldson-Thomas invariants of a 3-Calabi-Yau triangulated category should give rise to a geometric structure on its space of stability conditions called a Joyce structure. In this paper, we show how to use a Joyce structure to define a generating function which we call the τ-function. When applied to the derived category of the resolved conifold, this reproduces the non-perturbative topological string partition function of [J. Differential Geom. 115 (2020), 395-435, arXiv:1703.02776]. In the case of the derived category of the Ginzburg algebra of the A2 quiver, we obtain the Painlevé I τ-function.
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 20 (2024), 112, 26 pages Tau Functions from Joyce Structures Tom BRIDGELAND Department of Pure Mathematics, University of Sheffield, Sheffield, S3 7RH, UK E-mail: t.bridgeland@sheffield.ac.uk Received July 26, 2024, in final form December 12, 2024; Published online December 18, 2024 https://doi.org/10.3842/SIGMA.2024.112 Abstract. We argued in [Proc. Sympos. Pure Math., Vol. 103, American Mathematical Society, Providence, RI, 2021, 1–66, arXiv:1912.06504] that, when a certain sub-exponential growth property holds, the Donaldson–Thomas invariants of a 3-Calabi–Yau triangulated category should give rise to a geometric structure on its space of stability conditions called a Joyce structure. In this paper, we show how to use a Joyce structure to define a gener- ating function which we call the τ -function. When applied to the derived category of the resolved conifold, this reproduces the non-perturbative topological string partition function of [J. Differential Geom. 115 (2020), 395–435, arXiv:1703.02776]. In the case of the derived category of the Ginzburg algebra of the A2 quiver, we obtain the Painlevé I τ -function. Key words: Donaldson–Thomas invariants; topological string theory; hyperkähler geometry; twistor spaces; Painlevé equations 2020 Mathematics Subject Classification: 53C26; 53C28; 53D30; 34M55; 14N35 1 Introduction This paper is the continuation of a programme which attempts to encode the Donaldson–Thomas (DT) invariants of a CY3 triangulated category D in a geometric structure on the space of stability conditions M = Stab(D). The relevant geometry is a kind of non-linear Frobenius structure, and was christened a Joyce structure in [12] in honour of the paper [28] where the main ingredients were first discovered. In later work with Strachan [17], it was shown that a Joyce structure can be re-expressed in terms of a complex hyperkähler structure on the total space of the tangent bundle X = TM . An introduction to Joyce structures and their twistor spaces can be found in [14]. The procedure for producing Joyce structures from DT invariants is conjectural, and re- quires solving a family of non-linear Riemann–Hilbert (RH) problems [10] involving maps from the complex plane into a torus (C∗)n with prescribed jumps across a collection of rays. These problems are only defined if the DT invariants of the category satisfy a sub-exponential growth condition. So far there are no general results on existence or uniqueness of solutions. Nonethe- less, there are several situations where it is possible to find natural solutions to the RH problems and explicitly describe the resulting Joyce structures. It was discovered in [11] that when D is the derived category of coherent sheaves on the resolved conifold, the solutions to the associated RH problems can be repackaged in terms of a single function, which can moreover be viewed as a non-perturbative A-model topological string partition function, since its asymptotic expansion coincides with the generating function for the Gromov–Witten invariants. This function was introduced in a rather ad hoc way however, and it was unclear how to extend its definition to more general settings. The aim of this paper is to formulate a general definition of such generating functions and study their properties. We associate to a Joyce structure on a complex manifold M , equipped with choices of certain additional data, a locally-defined function τ : X = TM → C∗ which we call the τ -function. In the case of the derived category of the resolved conifold, and for appropriate mailto:t.bridgeland@sheffield.ac.uk https://doi.org/10.3842/SIGMA.2024.112 2 T. Bridgeland choices of the additional data, the restriction of this τ -function to a natural section M ⊂ TM coincides with the non-perturbative partition function obtained in [11]. The RH problems associated to the DT theory of the resolved conifold are rather spe- cial, in that the jumps across any two rays commute. A more representative class of ex- amples can be obtained from the DT theory of CY3 triangulated categories of class S[A1]. These categories D = D(g,m) are indexed by a genus g ≥ 0 and a collection of pole or- ders m = {m1, . . . ,ml}. When l > 0, they can be defined using quivers with potential asso- ciated to triangulations of marked bordered surfaces [31], or via Fukaya categories of certain non-compact Calabi–Yau threefolds [37]. The relevant threefolds Y (g,m) are fibred over a Rie- mann surface C, and are described locally by an equation of the form y2 + uv = Q(x). It was shown in [16] that when l > 0 the space of stability conditions on the category D(g,m), quotiented by the group of auto-equivalences, is the moduli space of pairs (C,Q) consisting of a Riemann surface C of genus g, equipped with a quadratic differential Q with poles of or- der {m1, . . . ,ml} and simple zeroes. The associated DT invariants were shown to be counts of finite-length horizontal trajectories, as predicted by earlier work in physics [25, 30]. These results have been extended by Haiden [26] to the case l = 0 involving holomorphic quadratic differentials. The general story described above then leads one to look for a natural Joyce struc- ture on this space. In the case of differentials without poles this was constructed in [13], and the generalisation to meromorphic differentials will appear in the forthcoming work [38]. The key ingredient in these constructions is the existence of isomonodromic families of bundles with connections. In several examples, a non-perturbative completion of the B-model topological string par- tition function of the threefold Y (g,m) is known to be related, via the Nekrasov partition function of the associated four-dimensional supersymmetric gauge theories of class S[A1], to an isomonodromic τ -function [7, 8, 9]. It therefore becomes natural to try to relate the τ -function associated to a Joyce structure of class S[A1] to an isomonodromic τ -function. That something along these lines should be true was suggested by the work of Teschner and collaborators [19, 20]. The definition of the Joyce structure τ -function was then reverse-engineered using the work of Bertola and Korotkin [5] on the moduli-dependence of isomonodromic τ -functions. We shall treat one example of class S[A1] in detail below. It corresponds to taking g = 0 and m = {7}. The resulting category D(g,m) is the derived category of the CY3 Ginzburg algebra associated to the A2 quiver. The corresponding Joyce structure was constructed in [15]. We show that with appropriate choices of the additional data the resulting Joyce structure τ -function coincides with the Painlevé I τ -function studied by Lisovyy and Roussillon [32]. Plan of the paper. We begin in Section 2 by reviewing material from [14, 17]. We introduce the notion of a pre-Joyce structure on a complex manifold M and the corresponding complex hyperkähler structure on the total space X = TM of the tangent bundle. A Joyce structure is defined as a pre-Joyce structure with certain extra symmetries which are controlled by an additional structure on M called a period structure. In Section 3, we define the twistor space p : Z → P1 associated to a Joyce structure and recall some of its basic properties. We also give a brief preview of the definition of the τ -function which is revisited in detail in Section 6. In Section 4, we give a conjectural description of a class of Joyce structures relating to theories of class S[A1]. The base M of these structures parameterises Riemann surfaces of genus g equipped with a meromorphic quadratic differential having poles of fixed orders {m1, . . . ,ml}. In the case l = 0 of holomorphic differentials, these are constructed rigorously by a different method in [13]. The general case will be treated in [38]. In Section 5, we consider certain additional structures on the twistor space of a Joyce structure which are present in many examples, and which are relevant for the definition of the τ -function. These are: (i) the structure of a cotangent bundle on the fibre Z0, (ii) the existence of collections Tau Functions from Joyce Structures 3 of preferred Darboux co-ordinates on the fibre Z1, and (iii) a preferred choice of a Lagrangian submanifold in Z∞. We recall from [14] that the combination of (i) and (iii) gives rise to an integrable Hamiltonian system. The definition of the τ -function associated to a Joyce structure appears in Section 6. It depends on the choice of certain additional data, namely symplectic potentials on the twistor fibres Z0, Z1 and Z∞. We explain how these choices relate to the additional properties of the twistor space discussed in the previous section. We then show that when restricted to various loci the τ -function produces generating functions for certain naturally associated symplectic maps. We also explain the relation with τ -functions in the usual sense of Hamiltonian systems. In Section 7, we consider the Joyce structure τ -functions associated to uncoupled BPS struc- tures. When restricted to a section of the projection π : X → M we show that our definition reproduces the τ -functions defined in [10]. In particular, this applies to the non-perturbative partition function of the resolved conifold computed in [11]. In Section 8, we consider the Joyce structure arising from the DT theory of the A2 quiver. This was constructed in [15] using the monodromy map for the deformed cubic oscillator. We show that the Joyce structure τ -function coincides with the Painlevé I τ -function extended as a function of monodromy exactly as described by Lisovyy and Roussillon [32]. Conventions. We work throughout in the category of complex manifolds and holomorphic maps. All symplectic forms, metrics, bundles, connections, sections etc. are holomorphic. The tangent bundle of a complex manifold M is denoted TM , and the derivative of a map of complex manifolds f : M → N is denoted f∗ : TM → f∗(TN ). The map f is called étale if f∗ is an isomorphism. We use the symbol L to denote the Lie derivative. 2 Joyce structures In this section, we introduce the geometric structures that will appear throughout the rest of the paper. They can be described either in terms of flat pencils of symplectic non-linear connections [12], or via complex hyperkähler structures as in [17]. Most of this material is standard in the twistor-theory literature, see, for example, [18, 21], and goes back to the work of Plebański [36]. We base our treatment on [14] to which we refer for further details. 2.1 Pre-Joyce structures Let π : X → M be a holomorphic submersion of complex manifolds. There is a short exact sequence of vector bundles 0 −→ V (π) i−→ TX π∗−→ π∗(TM ) −→ 0, where V (π) = ker(π∗) is the sub-bundle of vertical tangent vectors. Recall that a non-linear (or Ehresmann) connection on π is a splitting of this sequence, given by a map of bundles h : π∗(TM )→ TX satisfying π∗ ◦ h = 1. Writing H = im(h) and V = V (π), the tangent bundle of X decomposes as a direct sum TX = H ⊕ V . We call tangent vectors and vector fields horizontal or vertical if they lie in H or V respectively. A vector field u ∈ H0(M,TM ) can be lifted to a horizontal vector field h(u) ∈ H0(X,TX) by composing the pullback π∗(u) ∈ H0(X,π∗(TM )) with the map h. The connection h is flat if the following equivalent conditions hold: (i) for every x ∈ X there are local co-ordinates (x1, . . . , xn) on X at x, and (y1, . . . , yd) on M at π(x), such that xi = π∗(yi) and h ( ∂ ∂yi ) = ∂ ∂xi for 1 ⩽ i ⩽ d, (ii) the sub-bundle H = im(h) ⊂ TX is involutive: [H,H] ⊂ H. 4 T. Bridgeland Consider the special case in which π : X = TM →M is the total space of the tangent bundle of M . There is then a canonical isomorphism ν : π∗(TM )→ V (π) identifying the vertical tangent vectors in the bundle with the bundle itself, and we set v = i ◦ ν 0 // V (π) i // TX π∗ // π∗(TM ) hϵ �� ν ee // 0. Suppose that M is equipped with a holomorphic symplectic form ω ∈ H0 ( M,∧2T ∗ M ) . Via the isomorphism ν we obtain a relative symplectic form Ωπ ∈ H0 ( X,∧2T ∗ X/M ) which restricts to a translation-invariant symplectic form ωm on each fibre Xm = TM,m. We say that the connection h on π is symplectic if for any path γ : [0, 1] → M the partially-defined parallel transport maps PTγ(t) : Xγ(0) → Xγ(t) take Ωγ(0) to Ωγ(t). Definition 2.1. A pre-Joyce structure (ω, h) on a complex manifold M consists of (i) a symplectic form ω on M , (ii) a non-linear connection h on the tangent bundle π : X = TM →M , such that for each ϵ ∈ C∗ the connection hϵ = h+ ϵ−1v is flat and symplectic. Given local co-ordinates (z1, . . . , zn) on M there are associated linear co-ordinates (θ1, . . . , θn) on the tangent spaces TM,p obtained by writing a tangent vector in the form ∑ i θi · ∂/∂zi. We thus get induced local co-ordinates (zi, θj) on the space X = TM . We always assume that the co-ordinates zi are Darboux, in the sense that ω = 1 2 ∑ p,q ωpq · dzp ∧ dzq, with (ωpq) n p,q=1 a constant skew-symmetric matrix. We denote by (ηpq) n p,q=1 the inverse matrix. We can express them maps v and h in the form vi = v ( ∂ ∂zi ) = ∂ ∂θi , hi = h ( ∂ ∂zi ) = ∂ ∂zi + ∑ p,q ηpq · ∂Wi ∂θp · ∂ ∂θq , (2.1) for locally-defined functions Wi on X. The connection hϵ is flat precisely if [ hi + ϵ−1vi, hj + ϵ−1vj ] = 0 for all i, j. A short calculation shows that this holds for all ϵ ∈ C∗ precisely if ∂ ∂θk ( ∂Wi ∂θj − ∂Wj ∂θi ) = 0, ∂ ∂θk ( ∂Wi ∂zj − ∂Wj ∂zi − ∑ p,q ηpq · ∂Wi ∂θp · ∂Wj ∂θq ) = 0. (2.2) 2.2 Joyce structures A Joyce structure is a pre-Joyce structure with certain extra symmetries. These symmetries are controlled by an additional structure called a period structure. Definition 2.2. A period structure on a complex manifold M is given by a collection of dis- tinguished charts ϕi : Ui → Cn whose transition functions are maps of the form zi 7→ ∑ j aijzj with (aij) n i,j=1 ∈ GLn(Z). The local co-ordinate systems (z1, . . . , zn) associated to distinguished charts will be called integral linear co-ordinates. Tau Functions from Joyce Structures 5 A Joyce structure on a complex manifold M consists of a period structure and a pre-Joyce structure satisfying several compatibility relations. For the precise definition, we refer the reader to [14, Section 3]. Here we will just give a description in terms of local co-ordinates. Henceforth, (z1, . . . , zn) will always denote a local system of co-ordinates on M which are integral linear for the period structure underlying our Joyce structure. We then consider the associated co- ordinates (zi, θj) on X = TM as above. There is a well-defined vector field on M Z = ∑ i zi · ∂ ∂zi . We also consider the corresponding lifted vector field on X given by E = ∑ i zi · ∂ ∂zi . (2.3) We refer to either of these vector fields as Euler vector fields. We say that the Joyce structure is homogeneous if they generate C∗-actions on M and X respectively. For simplicity, we will often assume that this is the case in what follows. It was shown in [14] that in the case of a Joyce structure one can rewrite the vector fields (2.1) in the form vi = v ( ∂ ∂zi ) = ∂ ∂θi , hi = h ( ∂ ∂zi ) = ∂ ∂zi + ∑ p,q ηpq · ∂2W ∂θi∂θp · ∂ ∂θq , where W is a single locally-defined function on X called the Plebański function. This can be normalised by requiring W (z1, . . . , zn, 0, . . . , 0) = 0 = ∂W ∂θi (z1, . . . , zn, 0, . . . , 0). The flatness conditions (2.2) then become Plebański’s second heavenly equations ∂2W ∂θi∂zj − ∂2W ∂θj∂zi = ∑ p,q ηpq · ∂2W ∂θi∂θp · ∂2W ∂θj∂θq . The first axiom of a Joyce structure is that the inverse of the symplectic form satisfies ηpq ∈ 2πiZ. The other axioms are equivalent to the following symmetry properties of the function W : ∂2W ∂θi∂θj (z1, . . . , zn, θ1 + 2πik1, . . . , θn + 2πikn) = ∂2W ∂θi∂θj (z1, . . . , zn, θ1, . . . , θn), (2.4) W (λz1, . . . , λzn, θ1, . . . , θn) = λ−1 ·W (z1, . . . , zn, θ1, . . . , θn), (2.5) W (z1, . . . , zn,−θ1, . . . ,−θn) = −W (z1, . . . , zn, θ1, . . . , θn), where (k1, . . . , kn) ∈ Zn and λ ∈ C∗. 2.3 Complex hyperkähler structure We recall the following definition from [14]. Definition 2.3. A complex hyperkähler structure (g, I, J,K) on a complex manifold X con- sists of a non-degenerate symmetric bilinear form g : TX ⊗ TX → OX , together with endomor- phisms I, J,K ∈ End(TX) satisfying the quaternion relations I2 = J2 = K2 = IJK = −1, which preserve the form g, and which are parallel with respect to the Levi-Civita connection. 6 T. Bridgeland A pre-Joyce structure on a complex manifold M induces a complex hyperkähler structure on the total space X = TM . The action of the quaternions is given by I(hi) = i · hi, I(vi) = −i · vi, J(hi) = −vi, J(vi) = hi, K(hi) = ivi, K(vi) = ihi, and the metric g is defined by g(hi, hj) = 0, g(hi, vj) = 1 2 ωij , g(vi, vj) = 0. There are associated closed 2-forms on X given by ΩI(w1, w2) = g(I(w1), w2), Ω±(w1, w2) = g((J ± iK)(w1), w2). Explicitly, we have Ω+ = 1 2 · ∑ p,q ωpq · dzp ∧ dzq, 2iΩI = − ∑ p,q ωpq · dzp ∧ dθq, Ω− = 1 2 · ∑ p,q ωpq · dθp ∧ dθq + ∑ p,q ∂2W ∂θp∂θq · dθp ∧ dzq + ∑ p,q ∂2W ∂zp∂θq · dzp ∧ dzq. (2.6) There are identities LE(Ω+) = 2Ω+, LE(ΩI) = ΩI , LE(Ω−) = 0, (2.7) which follow immediately from (2.3) and the homogeneity property (2.5). Remark 2.4. So as to be able to include certain interesting examples such as those discussed in Section 4 below, it is often useful to weaken the axioms of a Joyce structure to allow the connections hϵ : π ∗(TM )→ TX to have poles. When expressed in terms of local co-ordinates as above, this just means that the function W (z, θ) is meromorphic. We will refer to the resulting structures as meromorphic Joyce structures. 3 Twistor space The geometry of a Joyce structure is often clearer when viewed through the lens of the associated twistor space [4, 27]. In this section, we define the twistor space and give some basic properties. We also give a preview of the definition of the τ -function which is the main topic of this paper. 3.1 Definition of twistor space Let (ω, h) be a pre-Joyce structure on a complex manifold M . There is then a pencil of flat, non-linear connections hϵ = h + ϵ−1v on the tangent bundle π : X = TM → M . For a given point ϵ ∈ C∗ ∪ {∞}, we can consider the sub-bundle H(ϵ) := im(hϵ) ⊂ TX . Since hϵ is flat this sub-bundle is involutive: [H(ϵ), H(ϵ)] ⊂ H(ϵ). The twistor fibre Zϵ is defined to be the space of leaves of the associated foliation on X. We denote by qϵ : X → Zϵ the quotient map. More globally, we consider the product P1×X and the projection maps P1 π1←− P1×X π2−→ X. As ϵ ∈ P1 varies, the sub-bundles H(ϵ) ⊂ TM combine to give a sub-bundle H ⊂ π∗ 2(TX). We can then view this as a sub-bundle of TP1×X via the canonical decomposition TP1×X = π∗ 1(TP1) ⊕ π∗ 2(TX). This sub-bundle is involutive, and we define the twistor space Z to be the space of leaves of the associated foliation. We denote by q : P1 ×X → Z the quotient map. Tau Functions from Joyce Structures 7 There is an induced projection p : Z → P1 satisfying p ◦ q = π1. The fibre p−1(ϵ) over a point 0 ̸= ϵ ∈ P1 coincides with the space Zϵ defined before. The fibre Z0 = p−1(0) is the space of leaves of the distribution V = ker(π∗) ⊂ TX and is therefore identified with M . The situation is summarised in the following diagram: X � � // qϵ �� P1 ×X q �� π1 \|\| Zϵ � � // �� Z p �� {ϵ} � � // P1 in which the horizontal arrows are the obvious closed embeddings. Remark 3.1. Unfortunately, to obtain a well-behaved twistor space p : Z → P1 we cannot in general just take the space of leaves of the foliation on P1 ×X. Rather, we should consider the holonomy groupoid, which leads to the analytic analogue of a Deligne–Mumford stack [33]. We will completely ignore these subtleties here, and essentially pretend that Z is a complex manifold. In fact, for what we do here, nothing useful would be gained by a more abstract point of view, because we are only really using the twistor space as a useful and suggestive shorthand. All statements we make about the space Z can be easily translated into statements only involving objects on X. For example, a symplectic form on the twistor fibre Zϵ is nothing but a closed 2-form on X whose kernel coincides with the sub-bundle H(ϵ) ⊂ TX . Similarly, an étale map from the twistor fibre Zϵ to some complex manifold Y is just a holomorphic map f : X → Y such that ker(f∗) = H(ϵ). Recall the complex hyperkähler structure (g, I, J,K) on X and the associated closed 2- forms Ω± and ΩI . When ϵ ∈ C∗ ∪ {∞} a simple calculation shows that H(ϵ) = im ( h+ ϵ−1v ) = ker ( ϵ−2(J + iK) + 2iϵ−1I + (J − iK) ) ⊂ TX . It follows that there is a symplectic form Ωϵ on the twistor fibre Zϵ such that q∗ϵ (Ωϵ) = ϵ−2Ω+ + 2iϵ−1ΩI +Ω−. (3.1) More globally, the right-hand side of (3.1) defines a twisted relative symplectic form on the twistor space p : Z → P1. This is a section of the bundle p∗(O(2)) ⊗ ∧2T ∗ Z/P1 and restricts to give a 2-form Ωϵ on each twistor fibre Zϵ which is well defined up to multiplication by a nonzero constant. When ϵ ̸= 0, we can fix this scale by imposing (3.1). We specify the scale on the fibre Z0 = M by taking Ω0 = ω. Thus q∗0(Ω0) = Ω+, q ∗ ∞(Ω∞) = Ω−. In what follows, we will generally try to distinguish between objects living on a twistor fibre Zϵ and their pullbacks via the quotient map qϵ : X → Zϵ. When writing formulae, however, this distinction tends to be a distraction and we will sometimes drop it. 3.2 Twistor space of a Joyce structure Let us now consider a homogeneous Joyce structure on a complex manifoldM , and the associated twistor space p : Z → P1. In particular, there is a C∗-action onM generated by the vector field Z, and a C∗-action on X generated by the lifted vector field E. We then consider the diagonal C∗- action on P1×X, where the action on P1 is the standard one rescaling ϵ with weight 1. It follows from the conditions (2.7) that this action descends along the quotient map q : P1 ×X → Z to give a C∗-action on Z. 8 T. Bridgeland The map p : Z → P1 intertwines the C∗-actions, and it follows that there are essentially three distinct twistor fibres: Z0, Z1 and Z∞, of which Z0 = M is the base of the Joyce structure. The C∗-action on Z restricts to C∗-actions on the fibres Z0 and Z∞. The relations (2.7) show that the symplectic form Ω0 on Z0 is homogeneous of weight 2, whereas Ω∞ on Z∞ is C∗-invariant. We introduce 1-forms on X via the formulae α+ = iE(Ω+), αI = iE(ΩI), α− = iE(Ω−). The relations (2.7) together with the Cartan formula imply that dα+ = 2Ω+, dαI = ΩI , dα− = 0. (3.2) The closed form α− was discussed in detail in [14, Section 5.1]. The forms α+ and αI will play an important role below. In terms of local co-ordinates, we have α+ = ∑ p,q ωpq · zpdzq, 2iαI = − ∑ p,q ωpq · zpdθq. (3.3) The form α+ clearly descends to the twistor fibre Z0 = M . Thus we can write α+ = q∗0(α0), with α0 a form on Z0, and we see that the symplectic form Ω0 is exact, with canonical symplectic potential 1 2α0. Moreover, this potential is homogeneous of weight 2 for the C∗-action on Z0. The form αI similarly provides a canonical symplectic potential for the symplectic form ΩI on X, and is homogeneous of weight 1. 3.3 Preview of the τ -function We now give a preview of the definition of the τ -function associated to a Joyce structure. Consider the identity of closed 2-forms on X = TM q∗1(Ω1) = Ω+ + 2iΩI +Ω− (3.4) obtained by setting ϵ = 1 in (3.1). Let us choose locally-defined 1-forms ΘI and Θ± on X satisfying dΘI = ΩI , dΘ± = Ω±. It is natural to assume that the forms Θ+ and Θ− descend to the twistor fibres Z0 and Z∞, respectively, and to require the homogeneity properties LE(Θ+) = 2Θ+, LE(ΘI) = ΘI , LE(Θ−) = 0. Let us also choose a locally-defined 1-form Θ1 on the twistor fibre Z1 satisfying dΘ1 = Ω1. Then the corresponding τ -function is the locally-defined function on X uniquely specified up to multiplication by constants by the relation d log(τ) = Θ+ + 2iΘI +Θ− − q∗1(Θ1). (3.5) Note that (3.4) implies that the right-hand side of (3.5) is closed. The definition (3.5) is of course vacuous without some prescription for the symplectic po- tentials appearing. From the discussion of Section 3.2, there are canonical choices Θ+ = α+ and ΘI = αI , so what remains is to specify Θ1 and Θ∞. However we will see below that even for ΘI and Θ+ it may be useful to make different choices. In this paper, we will only make partial progress towards resolving these issues. Before revisiting them in more detail in Section 6, we will first introduce an important class of examples of Joyce structures, and use them to motivate some relevant geometric properties of the twistor space. Tau Functions from Joyce Structures 9 4 Joyce structures of class S[A1] As discussed in the introduction, there is a class of meromorphic Joyce structures which are related to supersymmetric theories of class S[A1]. The base M parameterizes pairs consisting of a Riemann surface equipped with a quadratic differential having simple zeroes and poles of fixed orders. In the case of holomorphic quadratic differentials, the Joyce structure was constructed in [13]. The construction in the meromorphic case will appear in [38]. In this section, we give a sketch of a ‘quick and dirty’ approach to these Joyce structures which will be enough to understand some of their general features, and which may be of independent interest. 4.1 Quadratic differentials We begin by fixing the data of a genus g ≥ 0 and a collection of integers m = {m1, . . . ,ml} with all mi ≥ 2. There is then a space Quad(g,m) parameterizing pairs (C,Q0), where C is a compact, connected Riemann surface of genus g, and Q0 is a meromorphic section of ω⊗2 C with simple zeroes, and l poles xi ∈ C with multiplicities mi. For the basic properties of these spaces, we refer the reader to [16]. We always assume that k := 6g − 6 + l∑ i=1 (mi + 1) > 0. The space Quad(g,m) is then a non-empty complex orbifold of dimension k. Given a point (C,Q0) ∈ Quad(g,m) there is a spectral curve p : Σ → C branched at the zeroes and odd order poles of Q0. Locally it is given by writing y2 = Q0(x). There is a covering involution σ : Σ → Σ and a canonical meromorphic 1-form λ = ydx satisfying λ⊗2 = p∗(Q0). We define Σ0 ⊂ Σ to be the complement of the poles of λ. We consider the homology group H1 ( Σ0,Z )− . The superscript signifies anti-invariance for the covering involution: we consider only classes satisfying σ∗(γ) = −γ. A calculation shows that H1 ( Σ0,Z )− ∼= Z⊕k is free of rank k. Given a basis (γ1, . . . , γk) ⊂ H1 ( Σ0,Z )− we define zi = ∫ γi λ ∈ C. (4.1) As the point (C,Q0) ∈ Quad(g,m) varies the homology groups H1 ( Σ0,Z )− fit together to form a local system over Quad(g,m). Transporting the basis elements γi to nearby points, the resulting functions (z1, . . . , zk) form local co-ordinates on Quad(g,m). In particular, the tangent space at a point of Quad(g,m) is naturally identified with the cohomology group H1(Σ0,C)−. The inclusion i : Σ0 ↪→ Σ induces a surjection i∗ : H1 ( Σ0,Z )− → H1(Σ,Z)−. The intersection form on H1(Σ,Z) pulls back to an integral skew-symmetric form ⟨−,−⟩ on H1 ( Σ0,Z )− . This induces a Poisson structure on Quad(g,m) satisfying {zi, zj} = 2πi⟨γi, γj⟩. The kernel of i∗ is spanned (at least rationally) by classes ±βi ∈ H1 ( Σ0,Z )− defined up to sign by the difference of small loops around the two inverse images of an even-order pole of Q0. The residue of Q0 at such a pole xi ∈ C is defined to be the period resxi(Q0) = ± ∫ βi λ ∈ C. (4.2) It is well defined up to sign and is a Casimir for the above Poisson structure. Fixing these residues locally cuts out a symplectic leaf in Quad(g,m) of dimension n = 2d, where d = 3g − 3 + l∑ i=1 ⌊ 1 2 (mi + 1) ⌋ . (4.3) 10 T. Bridgeland There are 10 choices of the data (g,m) with g = 0 for which n = 2, and these correspond to the Painlevé equations (see, for example, [9, Section 2.1]). We will consider the particular symplectic leaf M = M(g,m) ⊂ Quad(g,m) obtained by requiring all residues (4.2) at even order poles to be zero. Note that a pole of order 2 with zero residue is actually a pole of order 1. The tangent space to M at a point (C, q) is identified with the cohomology group H1(Σ,C)−. Choosing a basis (γ1, . . . , γn) ⊂ H1(Σ,Z)− gives local co-ordinates (z1, . . . , zn) defined by the formula (4.1). These preferred local co-ordinates define a period structure on M . The inverse to the Poisson structure (4.2) defines a symplectic form ω on M . The remaining data we need to define a (meromorphic) Joyce structure onM is a pencil of flat, symplectic, non-linear (meromorphic) connections hϵ on the tangent bundle π : X = TM → M . Recall that the period structure defines a subsheaf TZ M ⊂ TM and consider the quotient X# = T# M = TM/(2πi)TZ M . We aim to construct a non-linear connection on the projection π : X# → M . This then pulls back to give a connection on π : X →M which moreover automatically satisfies the periodicity condition (2.4). Note that the local co-ordinates on X# are (zi, ξj) where ξj = eθj . 4.2 Pencils of projective structures We will construct a pencil of non-linear (meromorphic) connections on the projection π :X#→M by associating to a generic point of the space X# a pencil of projective structures with apparent singularities. The required connections will then be obtained by considering isomonodromic deformations of these projective structures. We refer to [3] for a review of projective structures on Riemann surfaces. These objects can be equivalently described as PGL2(C)-opers. We will consider meromorphic projective structures on a compact Riemann surface C of the form f ′′(x) = Q(x, ϵ) · f(x), Q(x, ϵ) = ϵ−2 ·Q0(x) + ϵ−1 ·Q1(x) +Q2(x), (4.4) where ϵ ∈ C∗ and Qi(x) are meromorphic functions. More invariantly, Q0 = Q0(x)dx ⊗2 and Q1 = Q1(x)dx ⊗2 are meromorphic quadratic differerentials on C, and Q2(x) represents a meromorphic projective structure. We will require that the pair (C,Q0) defines a point of the space M . At a point xi ∈ C where Q0 has a pole of order mi we will insist that Q1 and Q2 have poles of order at most⌊ 1 2(mi + 1) ⌋ . We will allow Q1 and Q2 to have poles at exactly d other points qi ∈ C, with d given by (4.3). At these points Q1, Q2 will be required to have the leading-order behaviour Q1(x) = pi x− qi + ri +O(x− qi), Q2(x) = 3 4(x− qi)2 + si x− qi + ui +O(x− qi), with pi, ri, si, ui ∈ C. We will then insist that for all ϵ ∈ C∗ the equation (4.4) has apparent singularities at the points x = qi, i.e., that the monodromy of the associated linear system is trivial asd an element of PGL2(C). This is equivalent to the condition( ϵ−1pi + si )2 = ϵ−2Q0(qi) + ϵ−1ri + ui, for all ϵ ∈ C∗ (see, e.g., [15, Lemma 2.1]), and hence to the equations p2i = Q0(qi), ri = 2pisi, ui = s2i . (4.5) Tau Functions from Joyce Structures 11 The first of these relations shows that the pair (qi, pi) defines a point of the double cover Σ associated to the quadratic differential (C,Q0) ∈M . The condition on the pole orders of Q1 at the points xi, together with the equations (4.5), ensures that the anti-invariant differential Q1/ √ Q0 on the double cover Σ has simple poles at the points (qi,±pi) with residues ±1 and no other poles. It follows that the expressions ξi = exp ( − ∫ γi Q1 2 √ Q0 ) ∈ C∗ (4.6) are well defined. 4.3 Conjectural Joyce structure Let O = O(g,m) denote the space of data (C,Q0, Q1, Q2) satisfying the conditions described in the previous subsection. There is a diagram of maps O α // π �� X# π �� M, where π : O → M sends a point (C,Q0, Q1, Q2) ∈ O to the point (C,Q0) ∈ M , and α sends a point (C,Q0, Q1, Q2) ∈ O to the point of X# whose local co-ordinates (zi, ξj) are given by the formulae (4.1) and (4.6). Conjecture 4.1. The map α : O→ X# is generically étale. Sketch proof. We must show that at a generic point of O the functions (zi, ξj) give local co- ordinates. We know that the co-ordinates zi locally determine (C,Q0). The next step is to show that the numbers ξi ∈ C∗ locally determine Q1, at least generically. Consider the line bundle L = OΣ ( d∑ i=1 (qi, pi)− d∑ i=1 (qi,−pi) ) . Since the 1-form η = Q1/ √ Q0 has simple poles with residues ±1 at the points (qi,±pi), it follows that the expression ∂ = d−η defines an anti-invariant connection on L. Moreover, ξ2i ∈ C∗ is the holonomy of this connection around the cycle γi ∈ H1(Σ,Z)−. The abelian Riemann–Hilbert correspondence shows that these holonomies uniquely determine (L, ∂). But now, by a version of the Jacobi inversion theorem, generically the bundle L uniquely determines the points (qi, pi). The final step is to show that the meromorphic projective structure Q2 is uniquely determined by the other data. Any two choices differ by a quadratic differential having poles of order at most ⌊ 1 2(mi + 1) ⌋ at the points xi, and by (4.5), regular and vanishing at the points qi. Such an object is a section of the line bundle ω⊗2 C  l∑ i=1 ⌈mi 2 ⌉ · xi − k∑ j=1 (qi, pi)  , which has Euler characteristic d − d = 0. Thus for general positions of the points qi ∈ C, the projective structure Q2 is uniquely determined. ■ For each ϵ ∈ C∗, we can attempt to define a non-linear connection hϵ on the projection O→M by considering isomonodromic deformations. That is, as (C,Q0) varies, we specify the variation of (Q1, Q2) by insisting that the generalised monodromy of (4.4) is constant. Unfortunately, the existence of such an isomonodromy connection has not yet been established in the literature. 12 T. Bridgeland Conjecture 4.2. (i) For each ϵ ∈ C∗, there is a flat, meromorphic connection gϵ on the projection π : O → M whose leaves define deformations of the projective structure (4.4) with constant generalised monodromy. (ii) There exist meromorphic connections hϵ on the projection π : X# →M whose pullback via the map α : O→ X# are the connections gϵ of part (i). (iii) There is a meromorphic Joyce structure on M obtained by combining the period structure and symplectic form on M defined above with the connections hϵ of part (ii). Note that if Conjecture 4.1 holds, then there can be at most one meromorphic connection on π : X# → M which lifts to a given meromorphic connection on π : O → M . We note that Conjecture 4.2 is known to hold for some specific choices of the data (g,m). A different construction of the Joyce structures of Conjecture 4.2 will be given in [38] using bundles with connection rather than projective structures with apparent singularities. 5 Further geometry of twistor space In this section, we discuss further structures on the twistor spaces of a Joyce structure which relate to choices of symplectic potentials on the twistor fibres Z0, Z1 and Z∞, and are highly relevant to the definition of the τ -function in Section 6. We try to explain how these structures appear naturally in the conjectural examples of class S[A1] of Section 4, although many details are still missing. It would be interesting to study these structures in further examples, and examine the extent to which they can be defined in general. 5.1 Cotangent bundle structure on M = Z0 Let M be a complex manifold with a holomorphic symplectic form ω ∈ H0 ( M,∧2T ∗ M ) . By a cotangent bundle structure on M we mean the data of a complex manifold B and an open embedding M ⊂ T ∗ B, such that ω is the restriction of the canonical symplectic form on T ∗ B. We denote by ρ : M → B the induced projection map, and by λ ∈ H0(M,T ∗ M ) the restriction to M of the canonical Liouville 1-form on T ∗ B. Given local co-ordinates (t1, . . . , td) on B, there are induced linear co-ordinates (s1, . . . , sd) on the cotangent spaces T ∗ B,b obtained by writing a 1-form as ∑ i sidti. In the resulting co- ordinates (si, ti) on M , we have ω = ∑ i dti ∧ dsi, λ = ∑ i sidti. Note that d(−λ) = ω. When M = Z0 is the base of a homogeneous Joyce structure there is a C∗-action on M satisfying LE(ω) = 2ω. It is then natural to seek a cotangent bundle structure M ⊂ T ∗ B such that LE(λ) = 2λ. Note that this implies that the C∗-action on M preserves the distribution of vertical vector fields for ρ, since this coincides with the kernel of λ. It follows that there is a C∗-action on B, and that the C∗-action on M is given by the combining the induced action on T ∗ B with a rescaling of the fibres with weight 2. Consider the conjectural Joyce structures of class S[A1] of Section 4, and assume for simplicity that l = 0. Then M parameterises pairs (C,Q0) consisting of a Riemann surface of genus g equipped with a holomorphic quadratic differential Q0 ∈ H0 ( C,ω⊗2 C ) with simple zeroes. The base M then has a natural cotangent bundle structure, with B the moduli space of curves of genus g, and ρ : M → B the obvious projection. Indeed, the tangent space to the moduli space of curves is H1(C, TC), and Serre duality gives H0 ( C,ω⊗2 C ) = H1(C, TC) ∗. Thus M is an open subset of T ∗ B obtained by requiring that Q0 has simple zeroes. We note in passing that in the class S[A1] case the fibres of ρ have a highly non-trivial compat- ibility with the Joyce structure: in the language of [13, Section 8] they are good Lagrangians. Tau Functions from Joyce Structures 13 See also the closely related notion of a projectable hyper-Lagrangian foliation from [22]. In the meromorphic case l > 0, there is a similar picture, but the space B is a moduli space of wild curves: the map ρ remembers the curve C together with the most singular half of the meromorphic tail of the differential Q0 at each pole xi ∈ C. 5.2 Cluster-type structure on Z1 A crucial part of the definition of the τ -function in Section 6 will be the existence of collections of preferred co-ordinate systems (x1, . . . , xn) on the twistor fibre Z1. These should be Darboux in the sense that Ω1 = 1 2 ∑ i,j ωij · dxi ∧ dxj . Although it will not be needed below, the preferred co-ordinates (x1, . . . , xn) are expected to have the following asymptotic property. Consider an integral linear co-ordinate system on M and the associated co-ordinate system on X and take a point x ∈ X with co-ordinates (zi, θj). For ϵ ∈ C∗, consider the point ϵ−1 · x ∈ X with co-ordinates ( ϵ−1zi, θj ) obtained by acting by ϵ−1 ∈ C∗ on the point x ∈ X. The claim is that provided x ∈ X is suitably generic there should be a preferred system of co-ordinates (x1, . . . , xn) on Z1 such that xi ( ϵ−1 · x ) ∼ −ϵ−1zi + θi (5.1) as ϵ→ 0 in the half-plane Re(ϵ) > 0. When the Joyce structure is constructed via the DT RH problems of [12], the required preferred co-ordinates systems xi are given directly by the solutions to the RH problems, and the property (5.1) holds by definition. Thus for the Joyce structures of interest in DT theory there is no problem finding this additional data. It is still interesting however to ask whether such distinguished co-ordinate systems exist for general Joyce structures. A heuristic explanation for why this might be the case can be found in [14, Section 4.3]. Consider the conjectural Joyce structures of class S[A1] of Section 4. Associated to the data g ≥ 0 and m = {m1, . . . ,ml} is a topological object (S,M) called a marked bordered surface. It is a compact, connected, oriented surface with boundary S, together with a finite set of marked points M ⊂ ∂S. We specify the pair (S,M) up to diffeomorphism by requiring that S has genus g, and l boundary components, and that the numbers of marked points on the various boundary components are the integers mi − 2 ≥ 0. Let us fix ϵ ∈ C∗. Ignoring some subtleties [3], the generalised monodromy of the projective structure (4.4) then defines a PGL2(C) framed local system on (S,M) in the sense of Fock and Goncharov [23]. By the definition of the Joyce structure in Conjecture 4.2, this generalised monodromy is constant along the leaves of the foliation defined by hϵ. We therefore expect an étale map µϵ : Zϵ → X(g,m)/MCG(g,m), (5.2) where X(g,m) is the space of framed local systems on the surface (S,M), and MCG(g,m) denotes the mapping class group. When l > 0 the preferred co-ordinate systems are expected to be the logarithms of Fock– Goncharov co-ordinates [23, 25], and the asymptotic property (5.1) should be a consequence of exact WKB analysis. In the case g = 0 and m = {7}, this story is treated in detail in [15]. In the case l = 0 of holomorphic quadratic differentials, the expected picture is less well understood, but it has been suggested that the required co-ordinates on the character variety are the Bonahon– Thurston shear-bend co-ordinates [6]. 14 T. Bridgeland 5.3 Lagrangian submanifolds of Z∞ At various points in what follows it will be convenient to choose a C∗-invariant Lagrangian submanifold R ⊂ Z∞ in the twistor fibre at infinity. For example, in the next section we will see that the choice of such a Lagrangian, together with a cotangent bundle structure on Z0, leads to a time-dependent Hamiltonian system. Unfortunately, for the Joyce structures arising from the DT RH problems of [12], the geometry of the twistor fibre Z∞ is currently quite mysterious, since it relates to the behaviour of solutions to the RH problems at ϵ =∞. So it is not yet clear whether such a Lagrangian submanifold can be expected to exist in general. When discussing the twistor fibre Z∞ for Joyce structures of class S[A1], it is important to distinguish the cases l = 0 and l > 0. When l = 0 the argument leading to the étale map (5.2) applies also when ϵ = ∞, because the monodromy of the projective structure (4.4) lives in the same space for all ϵ−1 ∈ C. In contrast, in the case l > 0, the projective structure (4.4) has poles of order mi at the points xi ∈ C for all ϵ ∈ C∗, but when ϵ = ∞ these poles are of order ⌊ 1 2(mi + 1) ⌋ . This means that the direct analogue of the map (5.2) for ϵ =∞ takes values in a space of framed local systems of a strictly lower dimension, and the fibres of this map are not well understood at present. A closely related problem is to understand the behaviour of the Fock–Goncharov co-ordinates of the projective structure (4.4) in the limit as ϵ→∞. Another relevant difference between the holomorphic and meromorphic cases can be seen in the action of C∗ on Z∞. The action of C∗ on X corresponds to fixing the curve C and rescaling the differentials (Q0, Q1, Q2) with weights (2, 1, 0) respectively. In the holomorphic case l = 0 the twistor fibre Z∞ is locally parameterising the pair (C,Q2) and the C∗-action is therefore trivial. As in the previous paragraph, in the meromorphic case l > 0 the twistor fibre Z∞ is not so easily described, and the induced action on Z∞ is in general non-trivial, as can be seen explicitly in the example of Section 8. 5.4 Hamiltonian systems A time-dependent Hamiltonian system consists of the following data: (i) a submersion f : Y → B with a relative symplectic form Ω ∈ H0 ( Y,∧2T ∗ Y/B ) , (ii) a flat, symplectic connection k on f , (iii) a section ϖ ∈ H0(Y, f∗(T ∗ B)). For each vector field u ∈ H0(B, TB) there is an associated function Hu = (f∗(u), ϖ) : Y → C. There is then a pencil kϵ of symplectic connections on f defined by kϵ(u) = k(u) + ϵ−1 · Ω♯(dHu). The system is called strongly-integrable if these connections are all flat. These definitions become more familiar when expressed in local co-ordinates. If we take co- ordinates ti on the baseB, which we can think of as times, and k-flat Darboux co-ordinates (qi, pi) on the fibres of f , we can write ϖ = ∑ iHidti and view the functions Hi : Y → C as time- dependent Hamiltonians. The connection kϵ is then given by the flows kϵ ( ∂ ∂ti ) = ∂ ∂ti + 1 ϵ · ∑ j ( ∂Hi ∂pj ∂ ∂qj − ∂Hi ∂qj ∂ ∂pj ) . (5.3) The condition that the system is strongly-integrable is that∑ r,s ( ∂Hi ∂qr · ∂Hj ∂ps − ∂Hi ∂qs · ∂Hj ∂pr ) = 0, ∂Hi ∂tj = ∂Hj ∂ti . (5.4) Tau Functions from Joyce Structures 15 Let L ⊂ Y denote a leaf of the foliation k1. The restriction ϖ\|L is then closed, so we can write ϖ\|L = d log(τL) for some locally-defined function τL : L → C∗. In terms of co-ordinates, since the projection π : L→ B is a local isomorphism, we can lift ti to co-ordinates on L, whence we have ∂ ∂ti log(τL) = Hi\|L. (5.5) A τ -function in this context is a locally-defined function τ : Y → C∗ whose restriction τ \|L to each leaf L ⊂ Y satisfies (5.5). Note that this definition only specifies τ up to multiplication by the pullback of an arbitrary function on the space of leaves. 5.5 Hamiltonian systems from Joyce structures Let M be a complex manifold equipped with a Joyce structure. Suppose also given (i) a cotangent bundle structure M ⊂ T ∗ B, (ii) Lagrangian submanifold R ⊂ Z∞. For the notion of a cotangent bundle structure, see Section 5.1. We denote by ρ : M → B the induced projection, and by β ∈ H0(M,ρ∗(T ∗ B)) the tautological section. Note that β is almost the same as the Liouville form λ ∈ H0(M,T ∗ M ) appearing in Section 5.1: they correspond under the inclusion ρ∗(T ∗ B) ↪→ T ∗ M induced by ρ. Set Y = q−1 ∞ (R) ⊂ X, and denote by i : Y ↪→ X the inclusion. There are maps Y � � i // X π //M ρ // B. Define p : Y → M and f : Y → B as the composites p = π ◦ i and f = ρ ◦ π ◦ i. We make the following transversality assumption: (⋆) For each b ∈ B the restriction of q1 : X → Z1 to the fibre f−1(b) ⊂ Y ⊂ X is étale. The following result was proved in [14]. Theorem 5.1. Given the above data there is a strongly-integrable time-dependent Hamiltonian system on the map f : Y → B uniquely specified by the following conditions: (i) the relative symplectic form Ω is induced by the closed 2-form i∗(2iΩI) on Y ; (ii) for each ϵ ∈ C∗ the connection kϵ on f : Y → B satisfies im(kϵ) = TY ∩ im(hϵ) ⊂ TX ; (iii) the Hamiltonian form is ϖ = p∗(β) ∈ H0(Y, f∗(T ∗ B)). To make condition (iii) more explicit, take local co-ordinates (t1, . . . , td) on B, and ex- tend them to co-ordinates (si, tj) on M as in Section 5.1. We can also extend to local co- ordinates (ti, qj , pk) on Y as in Section 5.4. Then β = ∑ i si · ρ∗(dti) and the connection kϵ is given by the flows (5.3) with Hamiltonians Hi = p∗(si). The main non-trivial claim is that the conditions (5.4) hold for these Hamiltonians. 6 The Joyce structure τ -function In this section, we define the τ -function associated to a Joyce structure on a complex manifoldM , and discuss some of its basic properties. It is most naturally viewed as the unique up-to-scale local flat section of a flat line bundle on X = TM . Given a choice of section of this line bundle it becomes a locally-defined function on X. 16 T. Bridgeland 6.1 Definition of the τ -function Let M be a complex manifold equipped with a Joyce structure as above, and let p : Z → P1 be the associated twistor space. We set ΘI = iE(ΩI), so that as in Section 3.2 we have dΘI = ΩI . Recall the identity of closed 2-forms on X q∗1(Ω1) = q∗0(Ω0) + 2iΩI + q∗∞(Ω∞). (6.1) We start by giving the definition of the τ -function in explicit local form. The geometrically- minded reader is encouraged to read the next section first. Definition 6.1. Choose locally-defined symplectic potentials: (i) Θ0 on Z0 satisfying dΘ0 = Ω0 and LE(Θ0) = 2Θ0, (ii) Θ1 on Z1 satisfying dΘ1 = Ω1, (iii) Θ∞ on Z∞ satisfying dΘ∞ = Ω∞ and LE(Θ∞) = 0. Then the corresponding τ -function is the locally-defined function on X uniquely specified up to multiplication by constants by the relation d log(τ) = q∗0(Θ0) + 2iΘI + q∗∞(Θ∞)− q∗1(Θ1). (6.2) In the case of a homogeneous Joyce structure, we can pull back τ via the action map m : C∗× X → X so that it becomes a function also of ϵ ∈ C∗. Restricted to the slice {ϵ} × X it then satisfies d log(τ) = ϵ−2q∗0(Θ0) + 2iϵ−1ΘI + q∗∞(Θ∞)− q∗ϵ (Θϵ), (6.3) where the symplectic potential Θϵ is defined by the relation q∗ϵ (Θϵ) = m∗ ϵ−1(q ∗ 1(Θ1)), and we used the relation LE(ΘI) = ΘI . Although the extra parameter ϵ is redundant, it is frequently useful to introduce it, for example, so as to expand τ as an asymptotic series. To obtain a well-defined τ -function, we of course need to give some prescription for choosing the symplectic potentials Θ0, Θ1 and Θ∞. We make some general remarks about this in Sec- tion 6.3, although some aspects are still unclear. In particular, the choice of Θ∞ remains quite mysterious. In later sections, we will show how to make appropriate choices in specific examples. 6.2 Global description Let us assume that the de Rham cohomology class of the form 1 2πi · Ω1 on the twistor fibre Z1 is integral. As explained in Appendix A, this implies that there is a line bundle with connec- tion (L1,∇1) on Z1 with curvature form Ω1. Note that the forms Ω0 and 2iΩI are exact by (3.2), so the relation (3.1) shows that the same integrality condition holds for all the twistor fibres Zϵ. In particular, there are line bundles with connection (L0,∇0) and (L∞,∇∞) on the twistor fibres Z0 and Z∞, with curvature forms Ω0 and Ω∞, respectively. The identity (6.1) shows that the connection( q∗0(∇0)⊗ 1⊗ 1 ) + ( 1⊗ q∗∞(∇∞)⊗ 1 ) − ( 1⊗ 1⊗ q∗1(∇1) ) + 2iΘI on the line bundle q∗0(L0)⊗ q∗∞(L∞)⊗ q∗1(L1) −1 is flat. Locally on X there is therefore a unique flat section up to scale, which we call the τ -section. Given a section s0 ∈ H0(U,L0) over an open subset U ⊂ Z0 we can write ∇0(s0) = Θ0 · s0 for a 1-form Θ0 on U . Then dΘ0 = Ω0\|U , so that Θ0 is a symplectic potential for Ω0 on this open subset. This construction defines a bijection between local sections of L0 up to scale, and Tau Functions from Joyce Structures 17 local symplectic potentials for Ω0. Let us take local sections s0, s1 and s∞ of the bundles L0, L1 and L∞ respectively, and let Θ0, Θ1 and Θ∞ be the corresponding symplectic potentials. Then we can write the τ -section in the form τ · ( s0 ⊗ s∞ ⊗ s−1 1 ) , and the resulting locally-defined function τ on X will satisfy the equation (6.2). We can constrain the possible choices of local sections s0 and s∞ using the C∗-actions on the fibres Z0 and Z∞. The relations LE(Ω0) = 2Ω0 and LE(Ω∞) = 0 show that the symplectic forms Ω0 and Ω∞ are homogeneous for this action. It follows that the action lifts to the pre- quantum line bundles L0 and L∞. We can then insist that the local sections s0 and s∞ are C∗-equivariant, which in terms of the corresponding symplectic potentials translates into the conditions LE(Θ0) = 2Θ0 and LE(Θ∞) = 0 appearing in Definition 6.1. Remark 6.2. For Joyce structures arising in DT theory the integrality property of the form 1 2πi ·Ω1 amounts to an extension of the usual wall-crossing formula. In brief, the twistor fibre Z1 is covered by preferred Darboux co-ordinate charts whose transition functions are symplectic maps given by time 1 Hamiltonian flows of logarithms of products of quantum dilogarithms. The cocycle condition is the wall-crossing formula in DT theory. The generating functions for these symplectic maps are given by formulae involving the Rogers dilogarithm, and the cocycle condition for the pre-quantum line bundle (L1,∇1) is then an extension of the wall-crossing formula involving these generating functions. This picture was explained by Alexandrov, Persson and Pioline [1, 2], and by Neitzke [35], and in the case of theories of class S[A1] plays an important role in the work of Teschner et al. [19, 20]. The Rogers dilogarithm identities were described in the context of cluster theory by Fock and Goncharov [24, Section 6], and by Kashaev and Nakanishi [29, 34]. 6.3 Choice of symplectic potentials It was explained in Section 3.2 that setting Θ0 = 1 2 iE(Ω0) = 1 2 ∑ i,j ωijzidzj , provides a canonical and global choice for Θ0. On the other hand, the key to defining Θ1 is to assume the existence of a distinguished Darboux co-ordinate system (x1, . . . , xn) on the twistor fibre Z1 as in Section 5.2. We can then take Θ1 = 1 2 ∑ i,j ωijxidxj . (6.4) The choice of Θ∞ remains quite mysterious in general. One way to side-step this problem is to choose a C∗-invariant Lagrangian R ⊂ Z∞ as in Section 5.3, since if we restrict τ to the inverse image Y = q−1 ∞ (R) we can then drop the term q∗∞(Θ∞) from the definition of the τ -function. It is not quite clear whether this procedure is any more than a convenient trick. In other words, it is not clear whether τ should be viewed as a function on X, depending on a choice of symplectic potential Θ+, or whether the natural objects are the Lagrangian R ⊂ Z∞, and a function τ defined on the corresponding submanifold Y ⊂ X. In practice, in examples, the above choices of symplectic potentials Θ0 and Θ1 do not always give the nicest results. We discuss several possible modifications here. Of course, from the global point-of-view, these variations correspond to expressing the same τ -section in terms of different local sections of the line bundles L0 and L1. 18 T. Bridgeland 6.3.1 Cotangent bundle Suppose we are given a cotangent bundle structure on M as discussed in Section 5.1. As explained there, it is natural to assume that the associated Liouville form λ satisfies LE(λ) = 2λ. We can then consider the following three choices of symplectic potential ΘL 0 = −λ, Θ0 = 1 2 iE(Ω0), ΘH 0 = iE(Ω0) + λ. The justification for the strange-looking primitive ΘH 0 will be explained in Section 6.4.4: it is the correct choice to produce τ -functions for the Hamiltonian systems of Section 5.5. Applying the Cartan formula gives ΘH 0 −Θ0 = Θ0 −ΘL = 1 2 iE(Ω0) + λ = −1 2 iE(dλ) + 1 2 LE(λ) = 1 2 diE(λ). Thus the resulting τ functions differ by the addition of the global function 1 2q ∗ 0(iE(λ)). 6.3.2 Polarisation Suppose the distinguished Darboux co-ordinate system (x1, . . . , xn) on Z1 is polarised, in the sense that the coefficients ωij appearing in (6.4) satisfy ωij = 0 unless \|j− i\| = d, where n = 2d. We can then take as symplectic potential on Z1 ΘP 1 = d∑ i=1 ωi,i+d · xidxi+d, This resulting τ -function will be modified by 1 2 ∑ i ωi,i+d · xixi+d. 6.3.3 Flipping ΘI Given local co-ordinates on X as in Section 2.1 there are in fact two obvious choices of symplectic potential for 2iΩI , namely 2iΘI = − ∑ p,q ωpqzpdθq, 2iΘ′ I = ∑ p,q ωpqθpdzq. In Section 6.4.2, it will be convenient to replace 2iΘI in the definition of the τ -function with 2iΘ′ I . This will change the τ -function by K = ∑ p,q ωpq · zpθq. Note that K : X → C is a globally- defined function, since it is the 1-form iE(ω) on M considered as a function on X = TM . It does not however descend to the quotient X# = TM/(2πi)TZ M . 6.4 Interpretations of the τ -function By restricting to various submanifolds of X, the τ -function can be viewed as a generating function in a confusing number of ways. 6.4.1 Restriction to the zero-section Let j : M ↪→ X = TM be the inclusion of the zero-section, defined by setting all co-ordinates θi = 0. Since this is the fibre q−1 ∞ (0) over the distinguished point 0 ∈ Z∞ of [14, Section 5.2], we have j∗q∗∞(Θ∞) = 0. The formula (3.3) for αI = ΘI shows that also j∗(ΘI) = 0. The defining relation of the τ -function then implies that d log(τ \|M ) = Θ0 − j∗q∗1(Θ1), so that log(τ \|M ) is the generating function for the symplectic map q1 ◦ j : M → Z1 with respect to the symplectic potentials Θ0 and Θ1. Tau Functions from Joyce Structures 19 6.4.2 Restriction to a fibre of q0 Let j : F = TM,p ↪→ X be the inclusion of a fibre of the projection π : X = TM →M . Restriction to this locus corresponds to fixing the co-ordinates zi. By (2.6), the restriction ΩF = j∗q∗∞(Ω∞) = 1 2 ∑ p,q ωpq · dθp ∧ dθq is the linear symplectic form on TM,p defined by the symplectic form ω. It follows that ΘF = j∗q∗∞(Θ∞) is a symplectic potential for this form. Let us take the flipped form 2iΘ′ I in the definition of the τ -function as in Section 6.3.3. The forms Θ0 and 2iΘ′ I vanish when restricted to F , so d log(τ \|F ) = j∗q∗∞(Θ∞)− j∗q∗1(Θ1), and log(τ \|F ) is the generating function for the symplectic map q1 ◦ j : F → Z1 with respect to the symplectic potentials ΘF and Θ1. 6.4.3 Restriction to a fibre of f Let us consider the setting of Section 5.5. Thus we have chosen a cotangent bundle struc- ture M ⊂ T ∗ B, with associated projection ρ : M → B, and a C∗-invariant Lagrangian submani- fold R ⊂ Z∞, and set Y = q−1 ∞ (R). Let j : F ↪→ Y be the inclusion of a fibre F = f−1(b) of the map f : Y → B. By Theorem 5.1 (i), the closed 2-form 2iΩI restricts to a symplectic form on F . As explained above, we can drop the term q∗∞(Θ∞) from the definition of the τ -function after restricting to Y . Let us take the symplectic potential on Z0 to be ΘL 0 = −dλ as in Section 6.3.1. Then since π(F ) ⊂ ρ−1(b) we also have j∗q∗0 ( ΘL 0 ) = 0. Thus d log(τ \|F ) = j∗(2iΘI)− j∗q∗1(Θ1), and log(τ \|F ) is the generating function for the symplectic map q1 ◦ j : F → Z1 with respect to the symplectic potentials j∗(2iΘI) and Θ1. 6.4.4 Hamiltonian system τ -function Consider again the setting of Section 5.5. Let us further assume that the Joyce structure is homogenous and that the Lagrangian R ⊂ Z∞ is preserved by the C∗-action. As before, after restriction to Y ⊂ X we can drop the term q∗∞(Θ∞) from the definition of the τ -function. Let us take the symplectic potential on Z0 to be ΘH 0 as defined in Section 6.3.1. Then we have d log(τ \|Y ) = i∗q∗0(λ) + i∗q∗0(iE(Ω0)) + i∗(iE(2iΩI))− i∗q∗1(Θ1) = p∗(λ) + i∗(iE(q ∗ 1(Ω1)))− i∗q∗1(Θ1). Here we used the identity (6.1), together with the assumption that the Lagrangian R ⊂ Z∞ is C∗-invariant, which ensures that iE(Ω∞)\|R = 0. Let L ⊂ X be a leaf of the connection k1 on f : Y → B. By construction of k1, this is the intersection of Y with a leaf of the con- nection h1 on π : X → M . Note that if u is a horizontal vector field for the connection h1 then iuiE(q ∗ 1(Ω1)) = −iEiu(q∗1(Ω1)) = 0. Thus d log(τ \|L) = p∗(λ)\|L. Applying the definition of Section 5.4, it follows that τ \|Y is a τ -function for the strongly- integrable time-dependent Hamiltonian system of Theorem 5.1. 20 T. Bridgeland 7 Example: uncoupled BPS structures In the paper [10], it was found that in certain special cases, solutions to DT RH problems could be encoded by a single generating function, which was denoted τ . The most basic case is the one arising from the DT theory of the doubled A1 quiver, where the resulting τ -function is a modified Barnes G-function [10, Section 5]. In the case of the DT theory of the resolved conifold τ was shown to be a variant of the Barnes triple sine function [11], and interpreted as a non-perturbative topological string partition function. In this section, we show that these τ -functions can be viewed as special cases of the more general definition given above. 7.1 Uncoupled BPS structures and associated τ -function Consider as in [10, Section 5.4] a framed, miniversal family of finite, integral BPS structures over a complex manifold M . At each point p ∈ M there is a BPS structure consisting of a fixed lattice Γ ∼= Z⊕n, with a skew-symmetric form ⟨−,−⟩, a central charge Zp : Γ → C, and a collection of BPS invariants Ωp(γ) ∈ Q for γ ∈ Γ. The miniversal assumption is that the central charges zi = Z(γi) of a collection of basis vectors γi ∈ Γ define local co-ordinates on M . The finiteness assumption ensures that only finitely many Ωp(γ) are nonzero for any given point p ∈M , and the integrality condition is that Ωp(γ) ∈ Z for a generic point p ∈M . Let us also assume that all the BPS structures parameterised by M are uncoupled, which means that Ωp(γi) ̸= 0 for i = 1, 2 implies ⟨γ1, γ2⟩ = 0. This is a very special assumption, which implies [10, Remark A.4] that the BPS invariants Ωp(γ) = Ω(γ) are independent of p ∈ M . We can then take a basis (γ1, . . . , γ2d), where n = 2d as before, such that ⟨γi, γj⟩ = 0 unless \|j − i\| = d, and such that Ω(γ) ̸= 0 implies that γ ∈ ⊕d i=1 Zγi. We set ηij = 2πi · ⟨γi, γj⟩ and take ωij to be the inverse matrix. Note that ωi,i+d · ηi,i+d = −1 for all 1 ⩽ i ⩽ d. At each point p ∈ M there is a DT RH problem depending on the BPS structure, and also on a twisted character ξ : Γ → C∗. For 1 ⩽ i ⩽ d the expressions exp ( −ϵ−1zi ) · ξ(γi) are solutions to this problem. We assume that ξ(γi) = 1 for all 1 ⩽ i ⩽ d which then implies that Ω(γ) ̸= 0 =⇒ ξ(γ) = 1. This amounts to fixing a Lagrangian R ⊂ Z∞. Then by [10, Theorem 5.3], the DT RH problem has a unique solution whose components Xi = exp(xi) can be written in the form xi = −ϵ−1zi + yi, yi = ∑ γ∈Γ Ω(γ) · ⟨γ, γi⟩ · log Λ ( Z(γ) 2πiϵ ) , where Λ(w) is the modified gamma function Λ(w) = ew · Γ(w) √ 2π · ww− 1 2 . Note that for 1 ⩽ j ⩽ d we have yi = 0, and 2πiωi,i+d · yi+d = − ∑ k1,...,kd∈Z Ω  d∑ p=1 kpγp  · ki · log Λ (2πiϵ)−1 d∑ p=1 kpzp  . This implies the relations ωi,i+d · ∂yi+d ∂zj = ωj,j+d · ∂yj+d ∂zi , d∑ j=1 zj · ∂yi+d ∂zj + ϵ · ∂yi+d ∂ϵ = 0. (7.1) The τ -function of [10] was then defined as a locally-defined function τ : M → C∗ satisfying ∂ ∂zi log(τ) = −ωi,i+d · ∂yi+d ∂ϵ , ∂ ∂zi+d log(τ) = 0 (7.2) for 1 ⩽ i ⩽ d, and homogeneous under simultaneous rescaling of all zi and ϵ. Tau Functions from Joyce Structures 21 7.2 Comparison of τ -functions We can give M a cotangent bundle structure in which the map ρ : M → B just projects to the co-ordinates (z1, . . . , zd). We then have λ = d∑ i=1 ωi,i+d · zi+ddzi, Ω0 = d∑ i=1 ωi,i+d · dzi ∧ dzi+d. Let us take the Hamiltonian system choice Θ0 = ΘH 0 from Section 6.3.1, and the polarised choice Θ1 = ΘP 1 from Section 6.3.2. Then Θ0 = d∑ i=1 ωi,i+d · zidzi+d, Θ1 = d∑ i=1 ωi,i+d · xidxi+d. Let us also restrict to a sectionM ⊂ X = TM by fixing the constant term ξ. Since the variables θi are then constant, the restriction of the form ΘI is zero. The definition (6.3) of the τ -function becomes d log(τ \|M ) = ϵ−1 · d∑ i=1 ωi,i+d · zidyi+d. Expressing τ as a function of the co-ordinates zi, we find that for 1 ⩽ i ⩽ d ∂ ∂zi log(τ \|M ) = ϵ−1 · d∑ j=1 ωj,j+d · zj ∂yj+d ∂zi , ∂ ∂zi+d log(τ \|M ) = 0. Using the relations (7.1), this gives ∂ ∂zi log(τ \|M ) = ϵ−1 · d∑ j=1 ωi,i+d · zj ∂yi+d ∂zj = −ωi,i+d · ∂yi+d ∂ϵ , which coincides with (7.2). Thus we see that the τ -functions of [10] are particular examples of the τ -functions introduced here. The non-perturbative topological string partition function for the resolved conifold ob- tained in [11] also fits into this framework. Although the relevant BPS structures are not finite, they are uncoupled, and the above analysis goes through unchanged. 8 Example: the A2 quiver, cubic oscillators and Painlevé I This example arises from the DT theory of the A2 quiver, and is the particular example of the construction of Section 4 corresponding to g = 0 and m = {7}. It was studied in detail in [15] and [14, Section 9] to which we refer the reader for further details. We show that with the natural choices of symplectic potentials Θ0, Θ1, Θ∞, the resulting τ -function coincides with the Painlevé I τ -function considered by Lisovyy and Roussillon [32]. 8.1 Joyce structure The base of the Joyce structure is M = { (a, b) ∈ C2 \| 4a3 + 27b2 ̸= 0 } . Associated to a pair (a, b) ∈M is a quadratic differential Q0(x)dx ⊗2 = ( x3 + ax+ b ) dx⊗2 (8.1) 22 T. Bridgeland on P1 which has a single pole of order 7 at x = ∞ and simple zeroes. The associated double cover is an affine elliptic curve Σ0 = Σ0(a, b) = { (x, y) ∈ C2 \| y2 = x3 + ax+ b } . We consider the pencil of projective structures f ′′(x) = Q(x) · f(x), Q(x) = ϵ−2 ·Q0(x) + ϵ−1 ·Q1(x) +Q2(x), (8.2) where the terms in the potential are Q1(x) = p x− q + r, Q2(x) = 3 4(x− q)2 + r 2p(x− q) + r2 4p2 , and we impose the relation p2 = q3+aq+b, which ensures that (8.1) has an apparent singularity at x = q for all ϵ ∈ C∗. Thus the pencil of projective structures (8.2) is parameterised by a point (a, b) ∈M , a point (q, p) ∈ Σ0(a, b), and a point r ∈ C. Locally, we can take alternative co-ordinates (z1, z2, θ1, θ2) defined by the relations zi = ∫ γi ydx, θi = − ∫ γi ( p x− q + r ) dx 2y , where (γ1, γ2) ⊂ H1(Σ,Z) is a basis of cycles with intersection γ1 · γ2 = 1. The Joyce structure on X = TM is obtained by taking the isomonodromy connection hϵ for the above family of projective structures. Explicitly, it is given by hϵ ( ∂ ∂a ) = −2p ϵ ∂ ∂q − q ϵ ∂ ∂r + ( ∂ ∂a − r p ∂ ∂q − r2 ( 3q2 + a ) − qpr 2p3 ∂ ∂r ) , hϵ ( ∂ ∂b ) = −1 ϵ ∂ ∂r + ( ∂ ∂b + r 2p2 ∂ ∂r ) . To obtain the Plebański function W , one rewrites these flows in the co-ordinates (zi, θj). The explicit formula can be found in [14, Section 9]. 8.2 Euler vector fields and symplectic forms The Euler vector field on M is Z = z1 ∂ ∂z1 + z2 ∂ ∂z2 = 4a 5 ∂ ∂a + 6b 5 ∂ ∂b , and the lift to X is E = z1 ∂ ∂z1 + z2 ∂ ∂z2 = 4a 5 ∂ ∂a + 6b 5 ∂ ∂b + 2q 5 ∂ ∂q + r 5 ∂ ∂r . The following facts were obtained in [14, Section 9]. There are identities Ω0 = − 1 2πi dz1 ∧ dz2 = da ∧ db, 2iΩI = 1 2πi (dθ1 ∧ dz2 − dθ2 ∧ dz1) = dq ∧ dp+ da ∧ dr, (8.3) which then immediately give iE(Ω0) = − 6b 5 da+ 4a 5 db, iE(2iΩI) = 2q 5 dp− 3p 5 dq + 4a 5 dr − r 5 da. Moreover, the functions ϕ1 = q + ar p , ϕ2 = r 2p , descend to local co-ordinates on the twistor fibre Z∞, and satisfy the relation Ω∞ = dϕ1 ∧ dϕ2, which can be used to give an explicit expression for Ω∞. Tau Functions from Joyce Structures 23 8.3 Joyce structure τ -function Let us now consider the additional choices needed to define the Joyce structure τ -function. Firstly, there is a natural cotangent bundle structure onM for which ρ : M → B is the projection to the co-ordinate a, which is the Painlevé time. The associated Liouville form is λ = bda. We shall restrict to the Lagrangian R ⊂ Z∞ defined by the equation ϕ2 = 0. The inverse image Y = q−1 ∞ (L) ⊂ X is then the 3-dimensional locus r = 0. We can take (a, q, p) as co- ordinates on the space Y . The map f : Y → B of Section 5.5 is then the projection (a, q, p) 7→ a. By (8.3), the form 2iΩI induces the relative symplectic form dq ∧ dp. The horizontal leaves of the connection k∞ are obtained by varying a while keeping (q, p) fixed. The Hamiltonian form is ϖ = bda. The twistor fibre Z1 is the space of framed local systems, and is covered by birational Fock– Goncharov co-ordinate charts (exp(x1), exp(x2)). We take the polarised choice for the symplectic potential Θ1, and the Hamiltonian system choice for Θ0. Thus Θ0 = iE(Ω0) + λ, Θ1 = −(2πi)−1x1dx2. Omitting the pullbacks q∗ from the notation, the definition of the τ -function reads d log(τ) = ϵ−2 ·Θ0 + ϵ−1 · 2iΘI +Θ∞ −Θ1. With the above choices this gives d log(τ \|Y ) = ϵ−2 ( −6b 5 da+ 4a 5 db+ bda ) + ϵ−1 ( 2q 5 dp− 3p 5 dq ) + 1 2πi x1dx2. (8.4) We now compare with the Painlevé τ -function computed in [32] and use notation as there. Consider the form ω defined in [32, equation (3.4)]. It involves local co-ordinates t, ma, mb with ma, mb constant under Painlevé flow. Let us re-express ω in terms of the local co-ordinates t, q, p. Using the relations qt = p, pt = 6q2 + t and Ht = −q appearing on [32, page 1], we get ω 2 = Hdt+ 1 5 ( 4tdH + 3qtdq − 2qdqt − ( 4tHt + 3q2t − 2qqtt ) dt ) = −H 5 dt+ 4t 5 dH + 3p 5 dq − 2q 5 dp. It is shown in [32] that Ω = dω = 4πidν1 ∧ dν2 and then the τ -function τLR is defined by d log(τLR) = 1 2 ω − 2πiν1dν2. (8.5) To compare with the τ -function τTB of the previous section, set pLR = −2pTB, qLR = qTB, t = 2a, H = 2b. Here the subscript TB means as in this paper, whereas LR means as appearing in [32]. Also we should set r = 0 and ϵ = 1 2 . The projective structure (8.2) then becomes gauge equivalent to the system (2.1a) of [32]. To match the monodromy data, we set x1 = 2πiν1 and x2 = −2πiν2. Then in terms of the notation of this paper, the definition (8.5) becomes d log τLR = −4b 5 da+ 16a 5 db− 6p 5 dq + 4q 5 dp+ 1 2πi x1dx2, which coincides with (8.4). It follows that the two τ -functions, which are both well defined up to multiplication by a nonzero constant, coincide. 24 T. Bridgeland A Pre-quantum line bundles in the holomorphic setting The following result on pre-quantum line bundles in the holomorphic setting is standard and well known, but is so relevant to the definition of the τ -function that it seems worth briefly recalling the proof. As in the body of the paper, all line bundles, connections, symplectic forms etc., will be holomorphic, but to make clear the distinction from the more familiar geometric quantization story we will re-emphasize this at several places. Theorem A.1. Let M be a complex manifold equipped with a holomorphic symplectic form Ω. Assume that the de Rham cohomology class [Ω] satisfies the integrality condition 1 2πi · [Ω] ∈ H2(M,Z) ⊂ H2(M,C). Then there is a holomorphic line bundle with holomorphic connection whose curvature is Ω. Proof. Consider the following diagram of sheaves of abelian groups on M , in which dO is the sheaf of closed holomorphic 1-forms, d is the de Rham differential, and the unlabelled arrows are the obvious inclusions, Z �� ·2πi // C �� exp // C∗ �� Z ·2πi // O exp // d �� O∗ d log �� dO // dO. The sheaf of 1-forms satisfying dΘ = Ω is a torsor for the sheaf dO, and hence defines an element η ∈ H1(M,dO). The image of η via the boundary map in the central column is the class [Ω] ∈ H2(M,C). By the integrality assumption, the image of η via the boundary map in the right-hand column is 1 ∈ H2(M,C∗). Thus, by the long exact sequence in cohomology for the right-hand column there exist elements ϕ ∈ H1(M,O∗) satisfying d log(ϕ) = η. Such a class ϕ defines a line bundle L on M , and one can then see that L has a holomorphic connection with curvature Ω. Translating the above discussion into Čech cohomology gives the following. Take a covering of X by open subsets Ui such that all intersections Ui1,...,in = Ui1 ∩ · · · ∩ Uin are contractible. Choose 1-forms Θi on Ui satisfying dΘi = Ω\|Ui , and set Θij = Θi\|Uij − Θj \|Uij . Then dΘij = 0 and the collection {Θij} defines a Čech 1-cocycle for the sheaf dO. On Uij we can now write d log ϕij = Θij = Θi\|Uij −Θj \|Uij (A.1) for functions ϕij : Uij → C∗. The integrality assumption implies that, after replacing ϕij by rij · ϕij for constants rij ∈ C∗, we can assume that ϕij \|Uijk · ϕjk\|Uijk · ϕki\|Uijk = 1. We then define the line bundle L by gluing the trivial line bundles Li over Ui using multiplication by ϕij . The relations (A.1) show that the connections ∇i = d+Θi on Li glue to a connection ∇ on L. Since ∇i has curvature dΘi = Ω\|Ui , the glued connection ∇ has curvature Ω. ■ Remarks A.2. (i) The relation (A.1) can be phrased as the statement that the gluing map ϕij for the line bundle L is the exponential generating function relating the symplectic potentials Θi\|Uij and Θj \|Uij on Uij . Tau Functions from Joyce Structures 25 (ii) Suppose U ⊂ M is a contractible open subset. Then sections s ∈ H0(U,L) up to scale are in bijection with symplectic potentials Θ on U . Given s we can write ∇(s) = Θ · s with dΘ = Ω. Conversely, given another symplectic potential Θ′ on U we can write Θ′ − Θ = d log(f) and hence define a section s′ = f · s satisfying ∇(s′) = Θ′ · s′. Acknowledgements The ideas presented here have evolved from discussions with many people over a long period of time. I would particularly like to thank Sergei Alexandrov, Andy Neitzke, Boris Pioline and Jörg Teschner for sharing their insights, and for patiently explaining many basic things to me. I am also very grateful for discussions and correspondence with Murad Alim, Fabrizio Del Monte, Maciej Dunajski, Lotte Hollands, Kohei Iwaki, Omar Kidwai, Dima Korotkin, Oleg Lisovyy, Lionel Mason, Ian Strachan and Menelaos Zikidis. Finally, I thank the anonymous referees for their careful reading and useful suggestions for improvements. References [1] Alexandrov S., Manschot J., Persson D., Pioline B., Quantum hypermultiplet moduli spaces in N = 2 string vacua: a review, in String-Math 2012, Proc. Sympos. Pure Math., Vol. 90, American Mathematical Society, Providence, RI, 2015, 181–211, arXiv:1304.0766. [2] Alexandrov S., Persson D., Pioline B., Wall-crossing, Rogers dilogarithm, and the QK/HK correspondence, J. High Energy Phys. 2011 (2011), no. 027, 64 pages, arXiv:1110.0466. [3] Allegretti D.G.L., Bridgeland T., The monodromy of meromorphic projective structures, Trans. Amer. Math. Soc. 373 (2020), 6321–6367, arXiv:1802.02505. [4] Bailey T.N., Eastwood M.G., Complex paraconformal manifolds – their differential geometry and twistor theory, Forum Math. 3 (1991), 61–103. [5] Bertola M., Korotkin D., Tau-functions and monodromy symplectomorphisms, Comm. Math. Phys. 388 (2021), 245–290, arXiv:1910.03370. [6] Bonahon F., Shearing hyperbolic surfaces, bending pleated surfaces and Thurston’s symplectic form, Ann. Fac. Sci. Toulouse Math. 5 (1996), 233–297. [7] Bonelli G., Grassi A., Tanzini A., Seiberg–Witten theory as a Fermi gas, Lett. Math. Phys. 107 (2017), 1–30, arXiv:1603.01174. [8] Bonelli G., Grassi A., Tanzini A., New results in N = 2 theories from non-perturbative string, Ann. Henri Poincaré 19 (2018), 743–774, arXiv:1704.01517. [9] Bonelli G., Lisovyy O., Maruyoshi K., Sciarappa A., Tanzini A., On Painlevé/gauge theory correspondence, Lett. Math. Phys. 107 (2017), 2359–2413, arXiv:1612.06235. [10] Bridgeland T., Riemann–Hilbert problems from Donaldson–Thomas theory, Invent. Math. 216 (2019), 69– 124, arXiv:1611.03697. [11] Bridgeland T., Riemann–Hilbert problems for the resolved conifold, J. Differential Geom. 115 (2020), 395– 435, arXiv:1703.02776. [12] Bridgeland T., Geometry from Donaldson–Thomas invariants, in Integrability, Quantization, and Geome- try II. Quantum Theories and Algebraic Geometry, Proc. Sympos. Pure Math., Vol. 103, American Mathe- matical Society, Providence, RI, 2021, 1–66, arXiv:1912.06504. [13] Bridgeland T., Joyce structures on spaces of quadratic differentials, Geom. Topol., to appear, arXiv:2203.17148. [14] Bridgeland T., Joyce structures and their twistor spaces, arXiv:2407.18229. [15] Bridgeland T., Masoero D., On the monodromy of the deformed cubic oscillator, Math. Ann. 385 (2023), 193–258, arXiv:2006.10648. [16] Bridgeland T., Smith I., Quadratic differentials as stability conditions, Publ. Math. Inst. Hautes Études Sci. 121 (2015), 155–278, arXiv:1302.7030. [17] Bridgeland T., Strachan I.A.B., Complex hyperkähler structures defined by Donaldson–Thomas invariants, Lett. Math. Phys. 111 (2021), 54, 24 pages, arXiv:2006.13059. https://doi.org/10.1090/pspum/090/01523 https://arxiv.org/abs/1304.0766 https://doi.org/10.1007/JHEP12(2011)027 https://arxiv.org/abs/1110.0466 https://doi.org/10.1090/tran/8093 https://doi.org/10.1090/tran/8093 https://arxiv.org/abs/1802.02505 https://doi.org/10.1515/form.1991.3.61 https://doi.org/10.1007/s00220-021-04224-6 https://arxiv.org/abs/1910.03370 https://doi.org/10.1007/s11005-016-0893-z https://arxiv.org/abs/1603.01174 https://doi.org/10.1007/s00023-017-0643-5 https://doi.org/10.1007/s00023-017-0643-5 https://arxiv.org/abs/1704.01517 https://doi.org/10.1007/s11005-017-0983-6 https://arxiv.org/abs/1612.06235 https://doi.org/10.1007/s00222-018-0843-8 https://arxiv.org/abs/1611.03697 https://doi.org/10.4310/jdg/1594260015 https://arxiv.org/abs/1703.02776 https://doi.org/10.1090/pspum/103.2/01851 https://doi.org/10.1090/pspum/103.2/01851 https://arxiv.org/abs/1912.06504 https://arxiv.org/abs/2203.17148 https://arxiv.org/abs/2407.18229 https://doi.org/10.1007/s00208-021-02337-w https://arxiv.org/abs/2006.10648 https://doi.org/10.1007/s10240-014-0066-5 https://arxiv.org/abs/1302.7030 https://doi.org/10.1007/s11005-021-01388-z https://arxiv.org/abs/2006.13059 26 T. Bridgeland [18] Chakravarty S., Mason L., Newman E.T., Canonical structures on anti-self-dual four-manifolds and the diffeomorphism group, J. Math. Phys. 32 (1991), 1458–1464. [19] Coman I., Pomoni E., Teschner J., From quantum curves to topological string partition functions, Comm. Math. Phys. 399 (2023), 1501–1548, arXiv:1811.01978. [20] Coman I., Pomoni E., Teschner J., From quantum curves to topological string partition functions II, arXiv:2004.04585. [21] Dunajski M., Mason L.J., Hyper-Kähler hierarchies and their twistor theory, Comm. Math. Phys. 213 (2000), 641–672, arXiv:math.DG/0001008. [22] Dunajski M., Moy T., Heavenly metrics, hyper-Lagrangians and Joyce structures, J. Lond. Math. Soc. 110 (2024), e13009, 42 pages, arXiv:2402.14352. [23] Fock V., Goncharov A., Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci. 103 (2006), 1–211, arXiv:math.AG/0311149. [24] Fock V., Goncharov A., Cluster ensembles, quantization and the dilogarithm, Ann. Sci. Éc. Norm. Supér. 42 (2009), 865–930, arXiv:math.AG/0311245. [25] Gaiotto D., Moore G.W., Neitzke A., Wall-crossing, Hitchin systems, and the WKB approximation, Adv. Math. 234 (2013), 239–403, arXiv:0907.3987. [26] Haiden F., 3-D Calabi–Yau categories for Teichmüller theory, Duke Math. J. 173 (2024), 277–346, arXiv:2104.06018. [27] Hitchin N.J., Karlhede A., Lindström U., Roček M., Hyper-Kähler metrics and supersymmetry, Comm. Math. Phys. 108 (1987), 535–589. [28] Joyce D., Holomorphic generating functions for invariants counting coherent sheaves on Calabi–Yau 3-folds, Geom. Topol. 11 (2007), 667–725, arXiv:hep-th/0607039. [29] Kashaev R.M., Nakanishi T., Classical and quantum dilogarithm identities, SIGMA 7 (2011), 102, 29 pages, arXiv:1104.4630. [30] Klemm A., Lerche W., Mayr P., Vafa C., Warner N., Self-dual strings and N = 2 supersymmetric field theory, Nuclear Phys. B 477 (1996), 746–764, arXiv:hep-th/9604034. [31] Labardini-Fragoso D., Quivers with potentials associated to triangulated surfaces, Proc. Lond. Math. Soc. 98 (2009), 797–839, arXiv:0803.1328. [32] Lisovyy O., Roussillon J., On the connection problem for Painlevé I, J. Phys. A 50 (2017), 255202, 15 pages, arXiv:1612.08382. [33] Moerdijk I., Mrčun J., Introduction to foliations and Lie groupoids, Cambridge Stud. Adv. Math., Vol. 91, Cambridge University Press, Cambridge, 2003. [34] Nakanishi T., Periodicities in cluster algebras and dilogarithm identities, in Representations of Algebras and Related Topics, EMS Ser. Congr. Rep., European Mathematical Society, Zürich, 2011, 407–443, arXiv:1006.0632. [35] Neitzke A., On a hyperholomorphic line bundle over the Coulomb branch, arXiv:1110.1619. [36] Plebański J., Some solutions of complex Einstein equations, J. Math. Phys. 16 (1975), 2395–2402. [37] Smith I., Quiver algebras as Fukaya categories, Geom. Topol. 19 (2015), 2557–2617, arXiv:1309.0452. [38] Zikidis M., Joyce structures from meromorphic quadratic differentials, in preparation. https://doi.org/10.1063/1.529513 https://doi.org/10.1007/s00220-022-04579-4 https://doi.org/10.1007/s00220-022-04579-4 https://arxiv.org/abs/1811.01978 https://arxiv.org/abs/2004.04585 https://doi.org/10.1007/PL00005532 https://arxiv.org/abs/math.DG/0001008 https://doi.org/10.1112/jlms.13009 https://arxiv.org/abs/2402.14352 https://doi.org/10.1007/s10240-006-0039-4 https://doi.org/10.1007/s10240-006-0039-4 https://arxiv.org/abs/math.AG/0311149 https://doi.org/10.24033/asens.2112 https://arxiv.org/abs/math.AG/0311245 https://doi.org/10.1016/j.aim.2012.09.027 https://doi.org/10.1016/j.aim.2012.09.027 https://arxiv.org/abs/0907.3987 https://doi.org/10.1215/00127094-2023-0016 https://arxiv.org/abs/2104.06018 https://doi.org/10.1007/BF01214418 https://doi.org/10.1007/BF01214418 https://doi.org/10.2140/gt.2007.11.667 https://arxiv.org/abs/hep-th/0607039 https://doi.org/10.3842/SIGMA.2011.102 https://arxiv.org/abs/1104.4630 https://doi.org/10.1016/0550-3213(96)00353-7 https://arxiv.org/abs/hep-th/9604034 https://doi.org/10.1112/plms/pdn051 https://arxiv.org/abs/0803.1328 https://doi.org/10.1088/1751-8121/aa6e12 https://arxiv.org/abs/1612.08382 https://doi.org/10.1017/CBO9780511615450 https://doi.org/10.4171/101-1/9 https://arxiv.org/abs/1006.0632 https://arxiv.org/abs/1110.1619 https://doi.org/10.1063/1.522505 https://doi.org/10.2140/gt.2015.19.2557 https://arxiv.org/abs/1309.0452 1 Introduction 2 Joyce structures 2.1 Pre-Joyce structures 2.2 Joyce structures 2.3 Complex hyperkähler structure 3 Twistor space 3.1 Definition of twistor space 3.2 Twistor space of a Joyce structure 3.3 Preview of the tau-function 4 Joyce structures of class S[A_1] 4.1 Quadratic differentials 4.2 Pencils of projective structures 4.3 Conjectural Joyce structure 5 Further geometry of twistor space 5.1 Cotangent bundle structure on M=Z_0 5.2 Cluster-type structure on Z_1 5.3 Lagrangian submanifolds of Z_infty 5.4 Hamiltonian systems 5.5 Hamiltonian systems from Joyce structures 6 The Joyce structure tau-function 6.1 Definition of the tau-function 6.2 Global description 6.3 Choice of symplectic potentials 6.3.1 Cotangent bundle 6.3.2 Polarisation 6.3.3 Flipping Theta_I 6.4 Interpretations of the tau-function 6.4.1 Restriction to the zero-section 6.4.2 Restriction to a fibre of q_0 6.4.3 Restriction to a fibre of f 6.4.4 Hamiltonian system tau-function 7 Example: uncoupled BPS structures 7.1 Uncoupled BPS structures and associated tau-function 7.2 Comparison of tau-functions 8 Example: the A_2 quiver, cubic oscillators and Painlevé I 8.1 Joyce structure 8.2 Euler vector fields and symplectic forms 8.3 Joyce structure tau-function A Pre-quantum line bundles in the holomorphic setting References
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institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
issn	1815-0659
language	English
last_indexed	2026-03-21T18:28:50Z
publishDate	2024
publisher	Інститут математики НАН України
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spelling	Bridgeland, Tom 2026-02-11T10:37:31Z 2024 Tau Functions from Joyce Structures. Tom Bridgeland. SIGMA 20 (2024), 112, 26 pages 1815-0659 2020 Mathematics Subject Classification: 53C26; 53C28; 53D30; 34M55; 14N35 arXiv:2303.07061 https://nasplib.isofts.kiev.ua/handle/123456789/212780 https://doi.org/10.3842/SIGMA.2024.112 We argued in [Proc. Sympos. Pure Math., Vol. 103, American Mathematical Society, Providence, RI, 2021, 1-66, arXiv:1912.06504] that, when a certain sub-exponential growth property holds, the Donaldson-Thomas invariants of a 3-Calabi-Yau triangulated category should give rise to a geometric structure on its space of stability conditions called a Joyce structure. In this paper, we show how to use a Joyce structure to define a generating function which we call the τ-function. When applied to the derived category of the resolved conifold, this reproduces the non-perturbative topological string partition function of [J. Differential Geom. 115 (2020), 395-435, arXiv:1703.02776]. In the case of the derived category of the Ginzburg algebra of the A2 quiver, we obtain the Painlevé I τ-function. The ideas presented here have evolved from discussions with many people over a long period of time. I would particularly like to thank Sergei Alexandrov, Andy Neitzke, Boris Pioline, and Jörg Teschner for sharing their insights and for patiently explaining many basic things to me. I am also very grateful for discussions and correspondence with Murad Alim, Fabrizio Del Monte, Maciej Dunajski, Lotte Hollands, Kohei Iwaki, Omar Kidwai, Dima Korotkin, Oleg Lisovyy, Lionel Mason, Ian Strachan, and Menelaos Zikidis. Finally, I thank the anonymous referees for their careful reading and useful suggestions for improvements. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Tau Functions from Joyce Structures Article published earlier
spellingShingle	Tau Functions from Joyce Structures Bridgeland, Tom
title	Tau Functions from Joyce Structures
title_full	Tau Functions from Joyce Structures
title_fullStr	Tau Functions from Joyce Structures
title_full_unstemmed	Tau Functions from Joyce Structures
title_short	Tau Functions from Joyce Structures
title_sort	tau functions from joyce structures
url	https://nasplib.isofts.kiev.ua/handle/123456789/212780
work_keys_str_mv	AT bridgelandtom taufunctionsfromjoycestructures

Tau Functions from Joyce Structures

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