Galilei”, Universit`a di PadovaVia Marzolo 8, 35131 Padova, Italy, and INFN sezione di Padova, Italy 4International School for Advanced Studies SISSA/ISASVia Beirut 2 - 4, 34014 Trieste,
Introduction
Instantons, moduli of punctured spheres and recursion relations
We establish a correspondence between the instanton moduli space of N=2 supersymmetric Yang–Mills theory with SU(2) gauge group and the moduli space of punctured Riemann spheres, presenting the reconstruction of the instanton moduli space in terms of punctured-sphere moduli as the central result of this work This punctured-sphere formulation exposes algebraic‑geometric features of the instanton moduli space that are less transparent in other descriptions In this framework, the natural Kähler form is the Weil–Petersson (WP) two-form on the moduli space of punctured spheres, which not only induces a natural metric on the instanton moduli space but also reveals the Wolpert restriction phenomenon The Wolpert restriction phenomenon guarantees the localization of integrals to the boundary of the moduli spaces for particular integrands, highlighting a deep link between boundary geometry and the integral structures encountered in the study.
More precisely, we will show some evidence that the moduli space M I n of the n- instanton is mapped to the moduli space of the sphere with 4n+ 2 punctures, namely
By reconstructing the instanton moduli space from the Seiberg–Witten solution in terms of the moduli space of punctured spheres M0,n, we establish a precise map to connect MIn → M0,4n+2 This program leverages the Liouville description of the M0,n spaces, which provides a natural framework for the Weil–Petersson (WP) volumes and their bilinear recursion, a feature arising from the Deligne–Knudsen–Mumford (DKM) compactification together with Wolpert’s restriction phenomenon A productive way to capture this structure is to describe the moduli of punctured spheres in Liouville theory, where the classical Liouville action serves as the Kähler potential for the WP metric It has been shown that the bilinear recursive structure of the integrals of WP forms on M0,n is preserved when the WP volume forms ωn are slightly deformed and the deformed volumes are used in place of the usual WP volume.
M 0,n ωn n − 3 , a deformed volume in which we replace the last insertion with an arbitrary closed two-form
Liouville F-models, defined as rational intersection theories on M0,n, constitute a universal class within string theory These deformations were originally proposed to describe the nonperturbative aspects of pure quantum Liouville gravity in the continuum formulation In this paper, we show that the SU(2) Seiberg–Witten (SW) solution provides another explicit example within the Liouville F-model framework.
In the Liouville background, the evaluation of the integral defines the expectation value of the two-form ωF This formulation guarantees that these expectation values obey a master equation, refining the original Liouville F-models described in [18] All the recursive structures of the integral, including its coefficients, are now captured by the differential operators Fn, which characterize the master equation.
Treating the coefficient of the n-th instanton amplitude as an integral over the moduli space of punctured spheres described by Liouville F-models implies that the instanton moduli space inherits the algebraic-geometric properties of these models, in particular their recursive structure tied to the DKM compactification On this basis, the construction indicates we should expect a bilinear recursion relation among instanton coefficients that shares the same features as the recursion for Weil-Petersson (WP) volumes Indeed, analysis from the Seiberg-Witten (SW) solution shows that the coefficients of the instantons satisfy the following bilinear relation.
An unexpected but productive link ties WP volumes to instantons: on the WP side the bilinear recursion for WP volumes yields a nonlinear ordinary differential equation that is the inverse of a linear differential equation satisfied by the generating function of those volumes By contrast, in N=2 gauge theory the periods obey a linear differential equation—the Picard-Fuchs equation—which, when inverted, gives a nonlinear ODE that encodes the recursion among instanton coefficients When we identify the effective gauge coupling constant τ(a) with the coupling constant of the WP-volume generating function, a concrete map emerges between N=2 supersymmetric Yang-Mills theory and WP volumes, with the latter described in terms of classical Liouville theory.
Explicit maps to Mg,n that simplify calculations are well known in the literature A striking example is the Hurwitz space, the moduli space of meromorphic functions that define ramified coverings, such as those of the sphere This space admits a compactification Hg,n consisting of stable meromorphic functions In particular, the projection Hg,n → Mg,n extends to
As we will see, explicit results are simply obtained thanks to such a map In particular,calculations simplify considerably in the genus zero case.
The Stringy point of View
By constructing instanton amplitudes from a bilinear recursion relation and Liouville F-models, we reconstruct the moduli space of instantons as punctured spheres and uncover meaningful connections to the stringy setup of N=2 SYM theory The work highlights the geometric engineering approach and the noncritical string perspective as central facets of these connections.
Let us return to the instanton amplitudes: direct instanton calculations have established that the integrands localize on the moduli space This localization has a natural counterpart in our construction of the instanton moduli space, where the underlying moduli-geometry governs the behavior of the amplitudes.
Within the Dijkgraaf–Vafa (DV) correspondence, there is a map between N=2 and N=1 supersymmetric Yang–Mills theories When described in terms of punctured spheres, the DKM boundary of the moduli space plays a significant role in evaluating the integral Moreover, the bilinear recursion relation suggests a dynamical selection of the boundary: the boundary that contributes to the amplitude is the divisor which separates the number of punctures by multiples of four (+2), aligning with the naive expectation that the boundary of the instanton moduli space is given by the collision of two or more instantons.
Remarkably, the puncture boundary, originally four units, can be reduced to a single unit, enabling the DKM compactification to operate and making the recursion relation resemble that of topological gravity This observation suggests interpreting the punctured sphere within a geometric engineering framework for N=2 SYM theory Traditionally, N=2 SYM can be engineered by the topological A-model on a noncompact Calabi–Yau manifold, where gauge instanton coefficients arise from worldsheet instantons wrapping cycles inside the threefold Here we propose an alternative worldsheet formulation in which the full topological A-model is treated as a perturbation around the theory obtained in the corresponding limit, offering a new perspective on the same physics.
The large-N limit corresponds to the semiclassical regime of the gauge theory, with a denoting the Higgs vacuum expectation value Perturbations are implemented by deforming the worldsheet CFT Since the gauge-theory prepotential in a flat background equals the tree-level (sphere) free energy of the A-model, the instanton coefficients arise as integrals over the moduli space of n-punctured spheres This construction extends naturally to a nontrivial gravitational background, the graviphoton, whose corrections to the prepotential have attracted considerable recent attention.
Building on the direct consequence of our recursion relation and the construction of instanton amplitudes in terms of the moduli space of punctured spheres, we predict the asymptotic form of the Gromov-Witten invariants for the local Hirzebruch surface, which in a certain limit yields N=2 SU(2) SYM Our bilinear recursion relation reproduces the rescaled version of the bilinear recursion for the asymptotic growth of the Gromov-Witten invariants Furthermore, by expressing the instanton amplitudes through the intersection theory on the moduli space of punctured spheres, we show that the asymptotic growth of the Gromov-Witten invariants is calculable via the rational intersection theory on M0,n.
Quantum Liouville theory, or the c = 0 noncritical string (see review [24]), is very akin to supersymmetric gauge theory in a precise sense Like supersymmetric theories, Liouville theory exhibits holomorphic dependence on its parameters, and this property significantly contributes to its solvability Moreover, the standard route to computing Liouville correlation functions follows from the approaches of Goulian and Li [25] and Dorn and Otto [26], highlighting the parallels between the two frameworks.
Zamolodchikov–Zamolodchikov remind us of the instanton calculation in the supersymmetric gauge theory—we compute the amplitude in the perturbative regime and then analytically continue to general cases using symmetry arguments—and in the world-sheet N=2 super Liouville theory there is a direct connection: the Liouville superpotential can be derived from U(1) vortex condensation, i.e., instanton effects in two dimensions from the parent N=2 U(1) gauged linear sigma model, so the dependence of the cosmological constant in correlation functions is essentially an instanton effect in this perspective Since it has been conjectured that the bosonic noncritical string theory is closely related to the topological twist of the N=2 super Liouville theory, the cosmological-constant dependence in bosonic Liouville theory may share the same origin, and this paper pushes that idea forward by exploiting a new bilinear recursion relation to establish a more direct link: the instanton contribution to the gauge theory prepotential can be rewritten as the genus expansion of a particular noncritical string theory, which we call the ‘instanton string theory.’ This theory bears a striking resemblance to the c=0 Liouville theory in its structure, placing them in the same Liouville F-model universality class, with the most intriguing feature that the amplitude arises solely from the boundary of the moduli space, much like in topological gravity, and the bilinear recursion relation functions as the string equation in this perspective.
Outline of the Paper
Section 2 reviews the basic facts on the uniformization property and the moduli space of punctured spheres, establishing the mathematical foundation of the paper We demonstrate the crucial connection between Liouville theory and the Weil–Petersson (WP) volumes and introduce the DKM compactification of the moduli space of punctured spheres This framework, together with the Wolpert restriction phenomenon, yields a bilinear recursion relation for the WP volumes.
Section 3 presents Liouville F-models and the Liouville background, viewing Liouville F-models as a universality class in string theory that naturally carries a bilinear recursive structure We formulate a master equation that offers a general scheme to treat these bilinear structures within the theory, enabling a unified approach to their recursive dynamics and interactions under the Liouville framework.
In section 4, we discuss the relation between the moduli space of gauge theory in- stantons and that of the punctured spheres We propose from the algebraic-geometrical
In this setting, the N = 2 super Liouville theory emerges in the description of singular Calabi–Yau spaces, and the topological B-model, which can be formulated in terms of matrix models, is intimately related to these Calabi–Yau geometries; this suggests a perspective in which the Liouville description is captured on the matrix-model side We also explore stable compactifications of moduli spaces and present an explicit example of a map into the moduli space of punctured spheres, realized as the map from the space of meromorphic functions on Riemann surfaces, a construction that is closely tied to Hurwitz numbers.
Section 5 shows that the instanton coefficients are expressed as integrals over the moduli space of punctured spheres, linking nonperturbative data to geometric moduli We then introduce a particular Liouville F-model whose master equation predicts a bilinear recursion relation for these coefficients, a structure that will be demonstrated in Section 6 by deriving it from the Picard–Fuchs equations of Seiberg–Witten theory.
Section 6 presents the final Liouville F-model for SU(2) N=2 SYM and derives the bilinear recursion relation hidden in the Seiberg–Witten solution with explicit coefficients Although anticipated from our earlier discussion, this section provides a concrete form and precise coefficients for the bilinear recursion, tying its existence to the condition that the inverse of the PF potential be at most quadratic As a side remark, starting from the bilinear recursion ansatz with the one-instanton coefficient allows us to rederive the Seiberg–Witten solution itself.
In section 7, we discuss the physical interpretation of this bilinear relation from the geometric engineering point of view [33] as well as from the noncritical string theory per- spective In the former approach, by expressing the instanton amplitudes as integrals on the moduli space ofn-punctured spheres, we derive the perturbed CFT expression for the geometric engineering topological A-model In the latter approach, we show that the gauge coupling constant can be written as the second derivative of a certain noncritical string theory All these different approaches are based on the underlying Liouville theory.
In section 8 we propose some speculations and future directions We first discuss the possible dualities among various approach to the SW theory in our view point based on the Liouville geometry Then we show the extension to the graviphoton background and the relation of our bilinear relation to the underlying recursive structure of the Gromov-Witten invariants On the relation to the graviphoton background, we point out an intriguing analogy with the self-dual YM equations for the gravitational version of SU(2) Finally, we also speculate on the extension of our results to the higher rank gauge theories.
In section 9 we address some concluding remarks.
In Appendix we report the simple proof of Wolpert’s restriction phenomenon and the derivation of the Weil-Petersson divisor.
Classical Liouville theory and Weil-Petersson volumes
Liouville theory and uniformization of punctured spheres
Here we are primarily interested in the punctured Riemann spheres Σ^0,n = ℂ̂ \ {z1, , zn}, with ℂ̂ ≡ ℂ ∪ {∞} Since three punctures can be fixed by a PSL(2,ℂ) transformation, different complex structures may arise only for n ≥ 3 Let us introduce the moduli space of the punctured Riemann spheres, M_{0,n}, which parametrizes these structures up to projective equivalence.
M_{0,n} is the moduli space of n distinct labeled points on the Riemann sphere, represented by (z1, , zn) with zj ≠ zk for j ≠ k, modulo the joint action of Symm(n) (permuting the labels) and PSL(2,C) (acting by linear fractional transformations) The Symm(n) action accounts for relabeling, while PSL(2,C) captures Möbius equivalence among configurations Using these PSL(2,C) transformations, one can fix three points to canonical positions, for example setting z_{n-2} = 0, z_{n-1} = 1, and z_n = ∞, yielding a standard normalization of the configuration.
A fundamental object in the theory of Riemann surfaces is the uniformizing mapping
3 It turns out that the WP two-form is also in the same cohomological class of the Fenchel- Nielsen two-form (see Appendix).
It would be interesting to investigate whether this important property of the Liouville action—the ability to generate the metric of both spaces—holds for other theories as well Let H denote the upper-half plane, H = {w | Im w > 0}, equipped with the Poincaré metric ds^2 = |dw|^2 / (Im w)^2, the standard hyperbolic metric on H This perspective raises questions about the universality of Liouville action in producing the geometric data associated with different theories and whether similar metric-generation mechanisms persist beyond the original setting.
H, is the metric of constant scalar curvature −1 Since w = JH −1 (z), this induces on the Riemann surface the metric ds 2 =e ϕ |dz| 2 , where e ϕ = |JH − 1 ′ | 2
The fact that the metric has constant curvature −1 is the same of the statement that ϕ satisfies the Liouville equation
An important object is the Liouville stress tensor
2ϕ 2 z , (2.8) where {f(z), z} = f ′′′ /f ′ − 3 2 (f ′ /f ′′ ) 2 is the Schwarzian derivative In the case of the punctured Riemann spheres we have
, (2.9) where the accessory parameters c 1 , , c n−1 are functions on V (n) They satisfy the two conditions n − 1
Let us write down the Liouville action [35]
It turns out that the accessory parameters are strictly related to S (n) evaluated at the classical solution More precisely, we have the Polyakov conjecture (see also [36] for a recent discussion) ck =− 1
2π∂z kS cl (n) , (2.13) which has been proved in [35] Furthermore, it turns out that Scl is the K¨ahler potential for the Weil-Petersson two-form [35] ω (n) W P = i
Equation (2.14) shows that Liouville theory describes the geometry of the moduli space of Riemann surfaces, tying Liouville dynamics directly to the structure of moduli space in conformal field theory As a consequence, even in critical string theory, the classical Liouville action appears in the string measure on moduli space, underscoring its fundamental role in the integration over worldsheet geometries.
We now consider the DKM stable compactification V (n) of the moduli space V (n)
Consider a nontrivial cycle in Σ0,n; shrinking it changes the complex structure and traces a path in V(n) In the limit, when the cycle is fully contracted, the resulting degenerate surface leaves V(n) A similar phenomenon occurs when two punctures collide: in the stable compactification a long, thin neck forms, yielding two surfaces glued by a node (a double puncture), each with at least three punctures, so the involved Riemann surfaces remain negatively curved Removing the node yields two Riemann spheres with k+2 and n−k punctures, where k runs from 1 to n−3 and depends on how many punctures were encircled This exclusion principle—that punctures never collide—is essentially a consequence of Gauss–Bonnet The DKM boundary of V(n) consists of the moduli spaces corresponding to such degenerate configurations Since dim V(n) = n−3, the boundary has codimension one, and although two punctures are added overall, the total Euler characteristic remains unchanged.
A basic property of the DKM compactification is that it has a clear recursive structure
5 Note that each of the two punctures of the node belong to different surfaces and in thePoincar´e metric their distance is infinity.
This recursive structure already suggests that for some suitable forms one may get the localization property
Equation (2.15) leads to a bilinear recursion relation, which implies that the generating function for these integrals satisfies nonlinear differential equations From the boundary structure one can perform an exact resummation, yielding a nonperturbative result Remarkably, in several cases, including N=2 SYM and Weil–Petersson (WP) volumes, these equations are essentially the inverses of linear ones.
To formalize the description, one must count how many times the component V(k+2) × V(n−k) appears in the boundary of V(n); this number is determined by the different ways to encircle a fixed number of punctures To organize this count, we introduce divisors D1, , D[n/2]−1, which are codimension-one subvarieties with real dimension 2n−8 Each Dk encodes, via the combinatorics, a splitting surface that, after removing the node, separates into two Riemann spheres with k+2 and n−k punctures In particular, Dk consists of C(k) copies of V(k+2) × V(n−k).
, (2.16) k = 1, ,(n−3)/2, fornodd In the case of evennthe unique difference is fork =n/2−1, for which we have
It turns out that the image of the divisors D k ’s provides a basis in H 2n−8 (M0,n,R) The DKM boundary simply consists of the union of the divisors D k ’s, that is
For future purposes it is convenient to extend the range of the index of Dj by setting
D k =D n − k − 2 , k = 1, , n−3 (2.19) Let us consider the WP volume 6
As we will see, the volumes are rational numbers up to powers of π, so we set
Vn=π 2(3 − n) (n−3)!VolW P(V (n) ) , (2.21) and will consider the rescaled WP two-form ωn = ω W P (n) π 2 , (2.22) as it will lead to rational cohomology.
In a remarkable paper [38], Zograf calculated such volumes recursively This construc- tion is simple and elegant It is instructive to illustrate the main features leading to his recursion relation.
Because the divisors D_k form a basis for the cohomology group H^{2n−8}(M_{0,n}, R), the Weil–Petersson two-form ω_n has a Poincaré dual that can be expressed as a linear combination of the D_k Consequently, V_n reduces to an integral over the boundary ∂V(n).
Section (2) reveals the recursive structure of the DKM boundary: the organization of the Dk’s implies that Vn is expressed as a sum of integrals on V(k+2) × V(n−k) This establishes the recursive decomposition pattern of the boundary, with each term representing an integral over the product space that combines higher- and lower-dimensional factors, demonstrating how Vn decomposes via the V(k+2) × V(n−k) structure. -**Support Pollinations.AI:** -🌸 **Ad** 🌸Powered by Pollinations.AI free text APIs [Support our mission](https://pollinations.ai/redirect/kofi) to keep AI accessible for everyone.
V (k+2) ×V (n−k) ρ k , (2.23) where c k are some combinatorial factors.
(3) The last step is the observation that the form ρk is expressed just in terms of the WP two-forms of V (k+2) and V (n − k) This is due to the Wolpert restriction phenomenon
[39], according to which the restriction of the WP two-form ω n on each component
V (k+2) of the DKM boundary is in the same cohomological class of ω k+2 More precisely, we have 8
Theorem (Wolpert [39]) Let i denote the natural embedding i :V (m) →V (m) × ∗ →V (m) ×V (n−m+2) →∂V (n) →V (n) , (2.24) for n > m, where ∗ is an arbitrary point in V (n − m+2) Then
7 The derivation of the Poincar´e dual to ω W P is reported in Appendix.
8 For an excellent updating on the WP metric see [40].
We report in the Appendix a simple proof of this theorem This theorem implies that (2.23) becomes
The above is the essential account of Zograf’s recursion relation [38]
2.4 The Equation for the WP Volumes Generating Function
Zograf’s recursion relation originated a series of interesting results in the framework of quantum cohomology [41] The first observation is that a rescaling ofVksimplifies Zograf’s recursion relation considerably, that is by setting [18] a k = V k
From the recursion (2.29), one can introduce a generating function for the Weil–Petersson (WP) volumes, defined as g(x) = ∑_{k=3}^{∞} a_k x^{k−1}, which then satisfies the nonlinear differential equation x(x−g) g'' = x g'^2 + (x−g) g' This recursion has been crucial in formulating a nonperturbative model of Liouville quantum gravity as a Liouville F-model, viewed as a deformation of the WP volumes, and it supports the view that the relevant integrations over the moduli space of higher-genus Riemann surfaces can be reduced to integrations over the moduli space of punctured Riemann spheres.
It has been shown by Kaufmann, Manin and Zagier [43] that this nonlinear ODE is essentially the inverse of a linear one More precisely, defining g=x 2 ∂ x x −1 h, one has that (2.31) implies xh ′′ −h ′ = (xh ′ −h)h ′′ (2.32)
Differentiating (2.32) we get yy ′′ =xy 3 , (2.33) where y=h ′ Then, interchanging the rˆoles of x and y, one can transform (2.33) into the
One known solution of this equation is a modified Bessel function, which is exactly the inverse of the generating function of the WP volumes More precisely, we have [43] x(y) =−√ yJ 0 ′ (2√ y) , (2.35) where
The modified Bessel function is convergent over the entire complex plane, whereas its inverse is not The boundary of convergence for the inverse is determined by the first zero of the derivative, where x′(y) = J0(2√y) This zero marks the point at which the inverse ceases to converge Through this framework, one can calculate the asymptotic form of the WP volumes in this way.
There is a surprising similarity between the general structure involved in deriving WP volumes and the instanton sector of N=2 SU(2) Yang–Mills theory In Seiberg–Witten theory one starts from the linear differential equation satisfied by a(u) and, by inverting it, obtains a nonlinear differential equation for u, namely u = G(a) = ∑_{k=0}^∞ a^{2−4k} G_k This in turn implies a recursion relation for the instanton coefficients G_k, which are related to the prepotential coefficients F_k by G_k = 2π i k F_k In the WP-volume case, one begins from a similar starting point and, through the same linear-to-nonlinear inversion, arrives at a corresponding recursion for the WP-volume coefficients that govern the expansion.
A surprising similarity
There is a surprising parallel between the derivation of the WP volumes and the instanton structure in N=2 supersymmetric Yang–Mills theory with gauge group SU(2) In Seiberg–Witten theory one starts from the linear differential equation for a(u) and inverts it to obtain a nonlinear differential equation satisfied by u(a), written as u(a) = G(a) = ∑_{k=0}^∞ a^{2−4k} G_k This, in turn, yields a recursion for the instanton coefficients G_k, which are related to the prepotential via G_k = 2π i k F_k By contrast, for WP volumes one proceeds from the opposite direction: the recursion is computed directly using the DKM compactification and the Wolpert restriction phenomenon, leading to the nonlinear differential equation (2.31) that can be compared with the SW equation in [2].
What is crucial is that, like (2.38), also (2.31) is essentially the inverse of a linear ODE (2.34).
An especially tight analogy suggests that the N=2 SYM results could be reobtained starting from Zograf’s point of departure by directly evaluating the recursion relation with algebraic-geometrical techniques in instanton theory, and this would be feasible within the Liouville F-models framework introduced in [18] In this context it is notable that, beyond the recursive nature of the DKM compactification and the Wolpert restriction phenomenon, two steps underpin localization phenomena, among which one observation is that the original recursion relation collapses to the simple form (2.29) Furthermore, a key advance is that the recursive structure identified by Zograf admits a significant generalization; namely, in [18] it was shown that the recursive structure arising in the evaluation of the integrals R
M 0,nω n n−3 persists even if one considers the replacement ω n n−3 −→ω n n−4 ∧ω F , (2.39) for suitable closed two-forms ω F This led to the nonperturbative formulation of Liouville quantum gravity in the continuum [18].
The Liouville F-models and the master equation
The Liouville background
With the effective action S_F defining a closed form ω_F = e^{-S_F} ω_n up to exact terms, a notable feature arises: the amplitudes Z_n^F possess a recursive structure that distinguishes Liouville F-models The replacement (2.39) leads to new recursion relations As observed in [18], the recursion property of the WP volume form can be preserved not only for ω_n^{n−3} but also for other (2n−6)-forms To understand this, note that applying the replacement (2.39) makes the volumes V_n be replaced by ω_{n−4} ∧ ω_F.
The condition can be weakened for our purposes: the equality only needs to hold after integrating and wedging with ω^{n−4} over the moduli space of punctured spheres Here ∩ denotes the topological cup product and D F is the Poincaré dual to [ω F] Expanding D F in terms of the chosen homology basis yields equation (3.4).
Equation (3.5) uses the Wolpert restriction phenomenon, as described in the Appendix The unique contribution comes from the binomial expansion terms that have the correct dimension with respect to V(k+2) and V(n−k), so the final result is governed by these dimensionally consistent terms.
Thus, for suitable ω F ’s one has recursive relations.
The theories we consider are the ones with an effective actionS F such that
This notation synthesizes some of the peculiar properties enjoyed by the moduli space of punctured spheres and of its Liouville geometry To explain this, first recall that since ω n = i
Equation (3.9) shows 2π^2 ∂∂S cl(n), and the notation in (3.8) defines how a given two-form is evaluated in the Liouville background We have demonstrated that for suitably chosen ωF, the substitution (2.39) preserves the recursive structure of the integrals In this sense, the notation bundles all these pieces: during the integration one uses the DKM compactification and the Wolpert restriction phenomenon, while ωn provides the Liouville background.
To define the divisor D F, we first observe that expressing the theory through integrals on V(n) relies on its recursive structure rather than the specific numerical values of the parameters By organizing the construction around the hierarchy embedded in V(n), D F emerges from a sequence of integral relations that reflect the geometry at each level This recursive framework provides a stable, parameter-robust definition of the divisor and clarifies how it interacts with the ambient space across scales In practice, focusing on the recursive structure enables concise integral formulas on V(n) to characterize D F, paving the way for efficient computation and deeper understanding of the divisor in relation to the theory.
WP volumes Therefore we introduce the normalized divisors
, (3.10) where [σk] is the Poincar´e dual to Dk, so that 10 hàkin = 1 , (3.11) with [à k ] the dual of Dk.
Intersection theory and the bootstrap
From the preceding discussion, this formulation can be viewed as a deformation of the WP volumes In particular, each divisor D_k is built from copies of V_{k+2} × V_{n−k}, labeled by two distinct subscripts, a detail that will become clearer in what follows A general deformation is conveniently expressed by introducing two parameters, s and t Consequently, we define the two-form η(s,t), whose cohomology class is specified by its Poincaré dual.
Understanding the class [η(s, t)] rests on the fact that the theories under consideration share the recursive geometry of the moduli space M0,n, so the divisor Dη encodes the physical data carried by the correlators on the products M0,k+2 × M0,n−k Once the theory is defined, we evaluate the rational intersection of the divisor associated to ωF with the DKM boundary Even without an explicit expression for ωF, the iterative structure of M0,n allows us to define ωF on M0,n in terms of the ωF’s defined on the factors M0,k+2 × M0,n−k Consequently, only the first correlator ⟨ωF⟩3 is needed as the initial condition to generate the entire tower of ωF’s This means that we compute ωF on M0,n by induction from the initial correlator ⟨ωF⟩3.
Starting from the three-point function hωF_i3, we bootstrap the full family of n-point functions hωF_n, so that M0,n is determined by this initial data This bootstrap follows from the recursive structure of the DKM compactification, which expresses higher-point correlators in terms of lower-point inputs The Wolpert restriction phenomenon enforces compatibility constraints that propagate through the recursion, ensuring each hωF_n is fixed by the starting three-point data In short, the n-point function hωF_n is entirely encoded by the initial three-point function via the DKM-driven recursion together with the Wolpert restrictions.
The master equation and bilinear relations
As we have seen so far, the bilinear recursion relation for the WP volume (2.29) has a natural generalization:
Due to the DKM boundary structure, the coefficients Fn(k+2, n−k) that define the general bilinear recursion relation—obtained by evaluating suitable integrations on M0,n—can be chosen to be symmetric under the exchange k+2 ↔ n−k, ensuring that, in general, the recursion respects this natural symmetry This symmetry reduces redundancy, aligns the recursion with the intrinsic k–n symmetry of the problem, and simplifies the computation of Fn for all admissible (k, n) pairs.
Because many coefficients F_n are redundant, we focus on the equivalence class of recurrence relations that differ only by trivial rescalings To select a canonical representative from each class, we can normalize by setting h_j = 0 for every negative index j < 0 and requiring h_0 ≠ 0, as in (3.15).
11 To be precise, this does not uniquely fix h 0 However, this is irrelevant for our purpose.
The general Liouville F-models are defined by the generating function for the Z n F where [ω F ] is given by the master equation
Equation (3.16) states that [ωF] = Fn(∂s, ∂t)[η0], with η0 ≡ η(0,0) This equation fixes the recursion relations once the initial condition, namely the three-point function ⟨ωF⟩3 (written here as hωF i3), is specified In terms of the Poincaré dual, the master equation reads.
D F =Fn(∂s, ∂t)Dη 0 (3.17) Evaluating the master equation (3.16) on the Liouville background hω F in = F n (∂ s , ∂ t )hη 0 in , (3.18) and by (3.11) hω F in n − 3
Equation (3.19) concisely encapsulates the recursive properties of M0,n, showing how higher-order structures arise from simpler components In this view, the recursion acts as a generalization of the Riemann bilinear relations for the M0,n setting, highlighting a deep connection between the two formalisms Indeed, both Eq (3.19) and the classical Riemann bilinear relations express the volume form as a linear combination of products of two lower-dimensional integrals, underscoring a shared structural pattern across dimensions.
Building on the previous analysis, we used the recursive properties of the DKM compactification to define F-models in terms of a suitable set of cohomological classes [ωF], whose evaluation on the Liouville background retains these recursive properties Beyond the DKM compactification, it is the Liouville background itself—via the Wolpert restriction phenomenon—that generates the recursion relations essential to the framework.
Pure Liouville quantum gravity
The generating functions for the Liouville F-models are [18]
Equation (3.20) describes the family X∞_{n=3} x_n h_ω F_in, which are classified by α, F_n and h_ω F_i^3 The Liouville F-models include pure quantum Liouville gravity, which can be formulated in the continuum as deformations of Weil–Petersson volumes, with the geometry described by the classical Liouville theory [18].
One of the distinguishing features of pure quantum Liouville gravity is that, like in the case of WP volumes, we havehj = 0,∀j 6= 0, and the master equation takes the simple form
Evaluating this equation on the Liouville background gives hω F in =h 0 hη 0 in =h 0 n−3X k=1 hω F ik+2hω F in−k , (3.22) n≥4, where in the case of pure Liouville gravity [18] h 0 = 3
By setting Z(t) = Z_F^{12,5}(t_5) and applying the master equation, we find that the specific heat of pure Liouville quantum gravity is governed by the series formed from the two-form ω_F evaluated on the Liouville background.
X∞ n=3 t 5n hω F in , (3.24) which in fact satisfies the Painlev´e I
3Z ′′ =t , (3.25) with initial conditionsZ(0) =Z ′ (0) = 0 Recalling that ω n = i
2π 2 ∂∂S cl (n) , (3.26) we see that the formulation in the continuum of pure Liouville quantum gravity is expressed in terms of the Liouville action evaluated at the classical solution.
Since this solution is non-perturbative rather than standard perturbative, we summarize its qualitative features before concluding the subsection The model’s physical consequences have been explored in [42], showing that it corresponds to quantum Liouville theory with the Einstein–Hilbert action acquiring an imaginary part of π/2 In other words, Eq (3.24) implements a Θ-vacuum structure in the genus expansion, as discussed in [42].
2 , (3.27) where the specific heat is defined asZ(t) =−F ′′ (t) The effect of the Θ-term is to convert the expansion into a series of alternating signs which is Borel summable.
An important observation is that the model’s specific heat displays a physically meaningful behavior Under the standard thermodynamic definition, where the specific heat is taken as the second derivative of the free energy (as in [45]), it should be negative However, [42] shows that the specific heat remains negative for all t > 0, while the conventional boundary-condition choice used in the asymptotic expansion (as in [45]) leads to a positive specific heat only for sufficiently large values of l.
The apparent effort to avoid the unphysical behavior of alternating signs in the asymptotic series arises from a misinterpretation: alternating signs in perturbation theory do not have a nonperturbative quantum-field-theoretical meaning, and this so-called unphysical behavior is only an artifact of the perturbation expansion What matters are the nonperturbative results, which are fully consistent with basic physical principles Hence the model's results agree with standard thermodynamics, and the theory is Borel summable.
As noted, Theta-vacua have a suggestive role in string theory, linking the moduli-space structure to unitarity considerations Understanding this link requires treating degenerate surfaces as corresponding to Feynman diagrams, so the Theta-vacua should emerge from a Feynman-diagram analysis of the string path integral at the boundary of moduli spaces Additionally, the presence of Theta-vacua is expected to improve the convergence of perturbation theory for critical strings, implying that string perturbation theory with Theta-vacua converges.
We now turn to the Liouville geometry of the N = 2 instantons and the moduli space of punctured spheres As briefly noted in the introduction, the derivation of the Seiberg–Witten prepotential from direct instanton calculations has progressed steadily (see [4] for a review) Although Seiberg–Witten provided an exact solution, performing direct instanton calculations via the ADHM construction [5] remained technically challenging at first Nevertheless, it has gradually become clear [6][7][8] that instanton amplitudes are topological objects to which localization can be applied To fully exploit localization, some desingularization of the instanton moduli space is necessary, and Hollowood’s work [8] achieved this by introducing noncommutative geometry (for a more mathematical treatment, see [46] and references therein) The all-instanton solution from direct instanton calculation based on localization is then presented by Nekrasov [9][10], where the Ω-background renders the enumerative evaluation of the localized integral possible.
12 Note that series with coefficients having alternating signs can be obtained just by changing the point of the expansion.
13 Recently this method has been extended to other gauge groups in [47][48].
Instanton moduli space and M 0,n
Stable compactification and the bubble tree
The program of using the stable compactification, and therefore quantum cohomology,for the instantons has been already considered in the literature (see for example [52]) The important quantity here is the moduli space of stable maps The original proposal of a similar compactification for moduli spaces of instantons has been the one by Parker and Wolfson [53] Such a compactification is referred to as the bubble tree compactification. Another feature indicating the existence of a stable compactification for the instan- ton moduli space is the fact that in this approach ‘punctures never collide’ Therefore, punctures can be considered like fermions, so that there is a sort of underlying exclusion principle This similarity can be explicitly formulated in the geometrical formulation of quantum Liouville theory [54][55] and in the framework of anyon theories [56] The lat- ter corresponds to a problem for particles with configuration space M0,n whose dynamics is described by a quantum Hamiltonian which is naturally given by the Laplacian with respect to the WP metric (thus giving a self-adjoint operator) In particular, both in Liou- ville and anyon theory one can associate a conformal weight to elliptic points [55][56] that in the limit of infinite ramification, corresponding to a puncture, just gives the value 1/2, which is the weight of a fermion Therefore, in some respect we can consider punctures behaving as noncommutative vertices
This would suggest a possible bridge between the stable compactification and the one considered by Hollowood with the noncommutative U(1) [8] We also note that in the ADHM construction it should be possible, in principle, to find a suitable embedding of the matrices in M0,n such that the degenerated configurations be naturally compactified `a la DKM.
The emergence of a fermionic degree of freedom in this framework reveals a link between Nekrasov's solution and the geometric engineering of N = 2 supersymmetric Yang–Mills theory In particular, the τ-conjecture proposes that the full-genus (graviphoton-corrected) partition function is governed by the quantum dynamics of chiral fermions on the Seiberg–Witten curve, while the mirror B-model formulation of the topological vertex—needed to compute Seiberg–Witten amplitudes—naturally introduces a chiral fermion that describes noncompact B-branes.
The Hurwitz moduli space
Beyond instantons, there is a parallel theory that maps to the moduli space Mg,n of Riemann surfaces, including punctured spheres This theory studies the space Hg,n of meromorphic functions of degree n on genus g Riemann surfaces that define degree-n ramified coverings of the sphere When the poles are simple and the critical values of the function sum to zero, Hg,n is a smooth complex orbifold fibered over Mg,n, with the fiber at each surface consisting of the associated meromorphic functions on that surface This space admits a natural compactification Hg,n consisting of stable meromorphic functions The key fact is that the projection Hg,n → Mg,n extends to the compactified moduli spaces, providing a complete fibered picture of these branched coverings within the moduli-theoretic framework.
Because the projection does not define a vector bundle—the fiber dimension varies with the base point—we study the fiberwise projectivization PHg,n This space carries a natural two-form, given by the first Chern class of the tautological sheaf: ψg,n = c1(O(1)) ∈ H^2(PHg,n), as in equation (4.5).
A consequence, which is of interest for our purpose, is that the space PH0,n turns out to be fibred on M0,n, whose fiber is the projective space P E where
E =⊕ n k=1L ∨ k , (4.6) is the Whitney sum of the tangent lines to the curve at the punctures This means that the cohomological algebra is generated by ψ≡ψ 0,n subject to the relation [58] ψ n +
By using the natural projection π: PH0,n → M0,n, any α ∈ H^{2d}(PH0,n) can be written as α = π^*(η_d) + π^*(η_{d−1}) ψ + π^*(η_{d−2}) ψ^2 + , with η_k ∈ H^*(M0,n) Since π^*(ψ^s) = c_{s−n+1}(−E), the degree of α can be evaluated through integrals on M0,n, enabling the computation deg α in terms of integrals over M0,n (see [59]). -**Support Pollinations.AI:** -🌸 **Ad** 🌸Powered by Pollinations.AI free text APIs [Support our mission](https://pollinations.ai/redirect/kofi) to keep AI accessible for everyone.
Evaluating c(E) yields a distinctly simpler form, showing that integrals on complex moduli spaces can collapse under a natural map to M0,n This explicit example supports the program of expressing instanton contributions as integrals over M0,n Importantly, the simplification arises not only from dimensional reduction of the moduli but also from possible higher-dimensional parametrizations that lead to the same tractable structure In this sense, M0,n serves as a fundamental space where difficult integrals become much more manageable, highlighting its role as a basic building block for simplifying the geometry behind instanton calculations.
4.3 The Geometry of WP Recursion Relation
To formulate N = 2 instanton contributions as integrals over the moduli space in a way that preserves the bilinear recursive structure of the DKM compactification, one must first understand how the DKM geometry feeds into the expected bilinear recursion relation In testing the existence of such a bilinear recursion, a characteristic feature of classical Liouville theory is that the recursion for Weil–Petersson (WP) volumes has particular properties Let us write Eq (2.29) again with a_3 = 1/2 and a_n = 1.
The bilinear recursion relation (4.11) arises because the DKM boundary is the union of products of two lower-order moduli spaces, a structure that is difficult to replicate in N = 2 SYM theories An analogy proposed in [50] is to construct a moduli space for instantons whose boundary contains components that include a product of three subspaces To see this point, recall that (4.11) is obtained as the DKM boundary contribution to an integral, and the dimension identity dim(M0,n) = dim(M0,k+2) + dim(M0,n−k) + 1 for k = 1,…,n−3 holds because dim(M0,n) = n−3 This suggests that a Liouville F-model formulation, such as expressing instanton contributions as integrals on the moduli space of punctured spheres, would require the resulting recursion to reflect the key properties of the WP volumes Therefore, besides being bilinear, the recursion should exhibit features following from (4.12): its range is k = 1,…,n−3, and the instanton contributions involved should be labeled by n−k and k+2 There is a subtlety, though, in that the data can shift by a simple global constant in the indices and in n If one defines a_k = b_k+m and rewrites Eq (4.11) for n+m, one obtains b_{3+m} = 1/2 and b_n = 1.
Equation (4.13), valid for n ≥ m+4, can be written as [(n−m)(n−m−2)] / [(n−m−1)(n−m−3)] × ∑_{k=1}^{n−m−3} b_{k+m+2} b_{n−k}, and this form contains the same content as equation (4.11) despite its different appearance This equivalence highlights that the bilinear structure is preserved across representations and motivates the general bilinear recursion relation d_n = ∑_{k=1}^p c_k d_{n+q−k}, valid for n ≥ r, as shown in equation (4.14).
The geometry of Weil-Petersson recursion relations
By (4.11), in the case of the WP volumes we find
N = 2 gauge theory as Liouville F-models
A master equation in N = 2 SYM?
We have seen that there is a lot of evidence for the existence of a formulation of instanton contributions in terms of integrals on the moduli space of punctured spheres.
One expects that the instanton moduli space can be mapped to M0,n along the lines of the Hurwitz space Since the parameter count for a single instanton does not match the one complex coordinate of a puncture, the natural map should land in a product of moduli spaces of punctured spheres of a particular kind On dimensional grounds, one should associate four punctures to each instanton, so we would initially consider the space M0,4n whose dimension is 4n−3 However, this identification leads to problems with dimensional matching needed to obtain the bilinear recursion relation Actually, this identification will give issues with the dimensional structure required for the recursion.
In this arrangement, the left-hand side counts punctures in multiples of four, while the right-hand side never contains pairs of moduli spaces where both have a puncture count that is a multiple of four As a result, identifying the instanton contribution Fn to the Seiberg–Witten prepotential with an integral over M0,4n while also satisfying a bilinear recursion relation becomes cumbersome By contrast, for the case M0,4n+2 a consistent pattern emerges.
M0,4n+2 −→ M0,4n − k+2× M0,k+2 , (5.2) so that when k is a multiple of 4 we may identify Fn with an integral on M0,4n+2 and consistently having the bilinear recursion relation.
D 4k+2 σ k 4k − 1 , (5.3) whereσ k is some two-form onD4k+2 to be determined In order to understand the nature of such a form we consider the natural embedding i :V (4k+2) →V (4k+2) × ∗ →V (4k+2) ×V (4n − 4k+2) →∂V (4n+2) →V (4n+2) , n > k ,
(5.4) where ∗ is an arbitrary point in V (4n − 4k+2) On the other hand according to Wolpert theorem we have
[ω 4k+2 ] =i ∗ [ω 4n+2 ], n > k (5.5) Using (5.3)(5.4) and (5.5) , we can express Fn in the form
V (4n+2) ω 4n+2 4n−2 ∧ω F , (5.6) where [ω F ] is the dual of a linear combination of the divisors D4k+2 The above investiga- tion suggests that
The instanton contributions can be expressed as integrals on the moduli space of punctured Riemann spheres leading to a bilinear recursion relation.
This framework enables expressing Fn through the bilinear recursion intrinsic to the F-models In deriving the master equation, we note that there is a canonical method to select a bilinear recursion relation from those that differ only by a trivial rescaling of terms Consequently, we use a rescaled form ¯Fn instead of Fn, while recognizing that this global rescaling does not alter the fundamental structure of the recursion relations.
We find that the divisor DF receives contributions exclusively from the moduli spaces of Riemann spheres with 4k+2 punctures; in particular, for N = 4n+2 the quantity F_N(k+2, N−k) vanishes unless k is a multiple of 4, and equivalently one can consider the expansion of η with respect to the divisor D.
D η n−1X k=1 e (4k+2)s e (N −4k)t hω F i4k+2hω F iN −4kD4k (5.7) From the master equation, in the case of N = 2 SYM we should then have
F¯n =hω F iN = FN(∂s, ∂t)hη0iN , (5.8) which leads to hω F iN n−1X k=1
F N (4k+ 2,(N −4k))hω F i4k+2hω F iN −4k , (5.9) n ≥ 2 In the next section we will show that this relation holds and we will fix both
Relation to ADHM construction
Here we briefly relate this proposal to the standard treatment of the integral over instanton moduli spaces, which is based on the ADHM construction Explicit instanton calculations reveal that integrals over the full moduli space simplify considerably, suggesting that the original parametrization may not be the most convenient in certain cases The n-th instanton contribution to the SU(2) prepotential can be obtained by integrating over the ADHM instanton moduli space M_I^n, whose metric and hence its volume form d^8_n are given by (4.2).
Effective action S_eff is obtained by integrating over the fermionic coordinates In this calculation, four of the total 8n real moduli coordinates are identified with the centers of the instantons Dividing by the spacetime volume regularizes the infrared divergence associated with the center integration, leaving an integral over the remaining 8n−4 coordinates Interestingly, this same integral can be rewritten as an integral of the (4n−3)th power of a two-form [7].
The moduli space cMIn is obtained from the usual MIn by first dividing by the instanton center (as in (5.10)) and then integrating over the complex scale of the ADHM moduli The closed two-form dρ can be formally interpreted as the Euler class of the original ADHM moduli space MIn viewed as a U(1) bundle (see [7] for details) Dimensional analysis shows that both (5.10) and (5.11) are performed after fixing the instanton center, but the latter also removes an additional complex coordinate related to the rescaling of the ADHM moduli It is useful to introduce noncommutative U(1) instantons to regularize divergences.
As shown in [8], the method smooths out the singularities of the moduli space, and it is convenient to preserve the explicit four-dimensional integration over the centers of the instantons.
Naturally, one can consider a moduli space with real dimension 8n−2 obtained by rescaling the moduli, which is precisely the dimension of M4n+2 we chose earlier The DKM boundary of the moduli space of punctured spheres is deeply related to the negatively curved nature of the punctured spheres (with more than 2 punctures), and stability rules out spheres with two punctures In the hyperbolic metric, the distance between two punctures is infinite; in particular, after removing a node the two punctures remain infinitely separated in the Poincaré metric but have zero distance in the Euclidean metric This is clearer in the upper-half plane, where the hyperbolic distance to the boundary blows up like 1/y while the Euclidean distance is simply y, and the product of the two distances is constant From the Euclidean viewpoint, approaching a boundary puncture is an infrared problem, whereas in the hyperbolic metric the geometry is always long-distance (infrared), a dual UV/IR picture This duality suggests a physical interpretation of the DKM compactification and hints at a direct connection with Hollowood’s U(1) noncommutative compactification [8] It is also noted that infrared and ultraviolet regularization properties of negatively curved manifolds have been observed [60], and that this UV/IR regularization is closely related to the distortion theorems for univalent functions (see the second reference in [2]).
The bilinear relation
Inverting differential equations
Because the trilinear recursion relation (6.4) does not preserve the bilinear recursive structure intrinsic to the DKM compactification—which is built into our master equation—it cannot meet our needs Consequently, we ask when a bilinear recursion relation can be derived starting from the general potential V(x) in the PF equation In this subsection and the next, we outline the necessary conditions under which a bilinear recursion relation arises in this framework.
Since the following construction is general, we will use ψ(x) rather thana(u) Let us consider the second-order differential equation
∂ x 2 +V(x) ψ(x) = 0 , (6.5) and set x=G(ψ) Since ∂ x =G ′− 1 ∂ ψ and∂ x 2 =−G ′− 3 G ′′ ∂ ψ +G ′− 2 ∂ ψ 2 , where here ′ ≡∂ ψ , it follows that equation for G(ψ) satisfies the differential equation
Consider expanding G(ψ) as a power series and deriving a recursion for its expansion coefficients from (6.6) The resulting relation would be at least trilinear due to the third power of G′ Fortunately, for a certain class of functions V(x) there is a neat way to simplify the nonlinear equation to a bilinear form This simplification, however, holds only in the case where
V − 1 (x) is at most quadratic in its argument.
From trilinear to bilinear
Introduce a function H(ψ) such that H ′ = G(ψ) and define the auxiliary function f through the following relation
V − 1 +fH ′′ = 0 , (6.7) so that a factor H ′′ drops and (6.6) becomes fH ′′′ +ψH ′′ 2 = 0 (6.8)
On the other hand, since V is a function ofG =H ′ , we have∂ ψ V −1 =H ′′ ∂ H ′ V −1 , so that (6.9) becomes f ′ −ψH ′′ +∂ H ′ V −1 = 0 , (6.10) which we can easily integrate to obtain the following expression for the auxiliary function f =ψH ′ − H −
Plugging this solution into the defining equation (6.7) for f, we obtain our final equation ψH ′ − H −
Proposition By (6.12) it follows that if V − 1 (x) is a polynomial at most quadratic in x;then the recursion relation for the coefficients of the power expansion of x = G(ψ), with ψ(x) solution of (6.5), is bilinear.
The N = 2 bilinear relation
We now readily apply this procedure to the PF equation (6.2) By introducingH ′ (a) G(a) and following the above steps we find the following nonlinear equation
Note the while (6.3) was trilinear, this last (6.13) is just bilinear Plugging the asymptotic expansion
3−4ka 3 − 4k , (6.14) into (6.13) we eventually find the bilinear recursion relation for ˆGn≡ Gn/(4n−3)
This shows in N = 2 SYM we have the master equation (5.8), where we identify ¯Fn = ˆGn and
By equation (6.1) we obtain the relation between the coefficients of the prepotential
F(a) = P ka 2 − 4k Fk and Gk, namely Gk = 2πikFk, k ≥ 1 Then we obtain the bilinear recursion relation for the instanton expansion of the prepotential as
Equation (6.19), (4k−3)[4(n−k)−3] g_{k,n}, completes the construction of the instanton moduli space in terms of the moduli space of punctured spheres and leads to the master equation for N = 2 SYM theory proposed in section 5 It also shows that the same invariants introduced in section 4.3 appear here, highlighting the consistency of the framework across the punctured-sphere formulation and the N = 2 SYM dynamics.
As a final remark, we note that the auxiliary function H ′ = G can be actually inte- grated by the PF equation (6.2) to
Observe that H has the same monodromy of the perioda This suggests the possibility to introduce its dual
9(u 2 −1)a ′ D , (6.23) satisfying the equation ∂ a D HD = u which is the dual of H ′ = u Also note that by
Because H naturally appears in the derivation of the bilinear relation, it warrants further examination within the Seiberg–Witten (SW) theory framework Its physical interpretation, which may emerge in the strong-coupling regime as discussed in [63], still needs to be clarified.
Before adopting a stringy interpretation of our result, an intriguing possibility emerges: the bilinear recursion relation, derived from the known Seiberg–Witten solution, can be turned around to reconstruct the SW solution even if its coefficients are left unspecified In other words, by assuming only the existence of a bilinear recursion relation without fixing its coefficients, one can deduce the SW solution Put simply, if the instanton amplitude is known to arise from a particular Liouville F-model (even if that model is initially unknown), this knowledge suffices to reproduce the SW solution and thereby identify the underlying Liouville F-model Our assumption is that the instanton amplitude corresponds to such a Liouville F-model, enabling this reconstruction.
The instanton contributions can be expressed as integrals on the moduli space of punctured Riemann spheres leading to a bilinear recursion relation.
We will also make use of the one-instanton contribution, which is G1 = 1/2 2 , and of the relation between the u-modulus and the prepotential [2] u =πi
This relation can be derived in two ways: via instanton analysis, which directly proves G_k = 2π i k F_k without the need to explicitly calculate F_k or G_k, or by employing the superconformal anomaly.
From the relation (6.25), we know thata(u) satisfies the PF equation with an unknown potential V(u) However, according to the Proposition, V −1 should have the form
Recalling that u=G(a) =H ′ (a), we have by (6.12)
H ′′ [aH ′ −(1 + 2A)H −Ba] +AH ′ 2 +BH ′ +C = 0 (6.27) Putting the asymptotic expansion
3−4ka 3−4k , (6.28) with G0 = 1/2, which follows by the asymptotic expansion of F(a) and Eq.(6.25), we obtain
Gj(4j−1)a 2−4j +C = 0 (6.29) Now observe that each term a 2 − 4k is multiplied only by a singular Gk, so that
The Seiberg-Witten solution
The coefficient C is determined by requiring thatG1 = 1/2 2 , which gives
Therefore we have obtained the PF equation for N = 2 SYM for SU(2) gauge group
We have shown that the conjectured relation between N = 2 SYM and the Liouville theory points to the existence of a bilinear recursion relation for N = 2, and this assumption, together with the initial condition—given in the N = 2 case by the values of G0 and G1—is sufficient to completely fix the entire solution.
Geometric engineering and noncritical strings
Geometric engineering
Instanton contributions can be mapped to integrals on the moduli space of punctured spheres, offering a geometric engineering interpretation of the problem The bilinear recursion relation (6.18) has the same structural form as that seen in Liouville F-models A key consequence of the dimension matching in (4.12) is that this matching holds for a generic decomposition of the moduli space of an (r+2)-punctured sphere, a feature tied to its DKM compactification origin Consequently, the n-instanton coefficient can be identified with an integral over the moduli space of an ann-punctured sphere If we focus on Liouville F-models and set Z_n^F = F_n − 2, the recursion relation for Z_n^F (3.19) is translated accordingly.
From equation (7.2), 2πi F_{n+2}(k+2, n+2−k) = 4n−3 n e_{k,n}, we obtain the desired recursion relation (6.18) The divisor introduced here encompasses all possible numbers of punctures, unlike the previous section where the divisor or its coefficients F(n, k) vanish unless k is a multiple of 4 In this reduction we no longer need to restrict our boundaries to multiples of 4, which makes the formulation more compact Though we lose the perspective of integrating over the instanton moduli space, we gain a new interpretation: the integral over the moduli space of the n-punctured spheres corresponds to the perturbative calculation of the free-energy in topological string theory Furthermore it yields a clue to extending the formulation to the case where the gauge theory is coupled to a gravitational background (see [65][66] for a comparison of the geometric engineering expression for the graviphoton–corrected prepotential with Nekrasov’s formula) The argument goes as follows.
Geometric engineering provides the Seiberg–Witten prepotential by analyzing a topological A-model on a local Calabi–Yau threefold, where the free energy of the A-model encodes the full prepotential of the four-dimensional N=2 SU(2) gauge theory In the SU(2) case, one uses the local Hirzebruch surface (a P1 bundle over P1) and, by taking its canonical line bundle, constructs a noncompact Calabi–Yau threefold The gauge-theory and geometric parameters are properly identified via e^{-T_B} β Λ^2, which links the geometric moduli to the gauge theory scale.
Let TB and TF denote the base P1 volume and the fiber volume, related by TB = 2aβ and TF = β~, with β introduced to realize the four-dimensional field theory limit β → 0 The same setup can be obtained from M-theory on the Calabi–Yau threefold times a circle of radius β/2π, yielding a five-dimensional gauge theory with eight supercharges In the β → 0 limit the circle shrinks to zero size, producing the four-dimensional theory, and the A-model string coupling is identified with the four-dimensional self-dual graviphoton field strength F+, so that after the field theory limit, the higher-genus topological string amplitudes correspond to gravitational corrections to the N = 2 four-dimensional prepotential F(a, β~) Therefore, for the usual gauge theory on flat space, what we need is the genus-zero free energy of the topological CFT, and the n-th gauge theory instanton contribution comes from worldsheet instantons wrapping the base P1 n times.
14 In order to obtain the canonical form introduced in section 3, we need to rescale F n and use
15 The difference among the various kind of Hirzebruch surfaces is irrelevant in this limit.
From a world-sheet perspective, we connect the prepotential to the A-model by expressing it as an integral over the moduli space of n-punctured spheres in the A-model setup Let a be the gauge-theory period and take the limit a → ∞, which corresponds to the semiclassical regime of the field theory, producing an A-model conformal field theory with action S∞ At finite a, the world-sheet action becomes a perturbed CFT about this reference theory, i.e., it can be written as a perturbation of the original S∞.
Let us consider a conformal field theory defined by the insertion Z_d^2 z O(z) for a certain operator O(z) (7.4) The 1/a^4 term in the expansion of the Seiberg–Witten prepotential F(a) is provided by the free energy of this CFT, which can be evaluated as a perturbative series in the perturbation Usually, the A-model free energy is obtained from a worldsheet instanton sum via Gromov–Witten invariants, but in this discussion the A-model is treated as an abstract topological CFT rather than through its geometric realization With that perspective, to determine the free energy of this CFT we analyze the perturbative insertion of the vertex operator a^{1/4}.
Introducing the R d^2 z_O(z) term adds an extra modulus corresponding to a puncture and requires integrating over its position, thereby reproducing the prepotential as the integral over the moduli space of n-punctured spheres On this basis, and in light of our Liouville F-model analysis, we expect that the same moduli-space integration framework will consistently encode the prepotential for the n-punctured sphere setup.
The merit of this interpretation is that it provides a concrete way to incorporate the graviphoton correction in the A-model side, essentially captured by the higher-genus free energy as described in (7.3) [68][69] This insight suggests a direct route to extend the formulation to a gravitational background: by using the same conformal field theory on higher-genus Riemann surfaces, the framework generalizes naturally to include gravity.
It would be very interesting to complete this program and we leave it as a future study.
It should be noted, however, that though we expect that the graviphoton corrected pre- potential is given by the integration over the moduli space of the punctured higher genus Riemann surfaces, the integrand needs not a priori coincide with (7.5).
16 This is only true if 1/a 4 is a special coordinate of the topological CFT Though it is a difficult problem to verify this, our formulation suggests that this is indeed true We also expect that O(z) is formally BRST exact so that it contributes to the amplitude only through the contact terms, as we can see from our recursion relation The whole construction seems more transparent if we use the mirror symmetry and move to the B-model, but in the abstract CFT language used here, there is no essential difference.
17 We postpone the connection with the Gromov-Witten invariants to the last section.
Noncritical string
Finally, we present a noncritical string interpretation of the bilinear recursion relation.
Noncritical string theory and the four-dimensional N=2 supersymmetric Yang–Mills (SYM) theory share many features, and the relation via the worldsheet U(1) instanton is a central thread When we shift to the space-time theory, the instanton part of the d=4, N=2 SYM coupling constant τ exhibits a striking structure: τ, the instanton contribution, obeys, in line with equation (6.18), a bilinear recursion relation of the form τ_n = ∑_{k=1}^{n−1} e′_{k,n} τ_k τ_{n−k}, with e′_{k,n} = (4n−3)(4n−2)(4n−1) n [4(n−k)−1][4(n−k)−2](4k−1)(4k−2) e_{k,n} The rationale for focusing on τ rather than F becomes clear through a strong analogy to c=0 Liouville theory, where the specific heat is the focal quantity and τ is dimensionless Our aim is to understand this relation within the framework of the string genus expansion.
It was proposed in [42] that the recursion relation can be recast in the perturbative noncritical string form, a construction that had been carried out for Painlevé I in the c = 0 noncritical string theory This reformulation helps explain how the path integral over the Liouville field reduces to the known cohomological objects of the moduli space of Riemann surfaces, and it is complemented by Zamolodchikov's recent argument [70] proposing a possible mechanism for this reduction.
Rewriting the SYM recursion relation in the language of string perturbation theory reveals several intriguing directions: it hints at a confining string description of SYM, which we term instanton string theory, and offers a route to understanding the negative expansion of the cosmological constant within Liouville theory It is also notable that Polyakov anticipated a deep link between noncritical string theory and four-dimensional (supersymmetric) gauge theories long before AdS/CFT, treating a particular noncritical string theory as a dual description of the gauge dynamics.
Yang–Mills theory faces the typical obstacle of a c = 1 barrier in noncritical string theory, but in higher dimensions the Liouville dimension can be encoded as warped geometry, offering a coherent alternative This proposal finds a concrete realization in N = 2 super Liouville theory, where the recently discovered duality in N = 2 Liouville theory underpins the warped-geometry construction and connects it to noncritical strings.
Understanding this cannot be achieved within the worldsheet instanton framework alone; the negative dependence of a cannot be explained by the usual instanton expansion, and the same limitation holds in the Goulian-Li approach to Liouville theory at higher genus It was conjectured and later proven that the Liouville potential has a dual realization as warped geometry, providing a dual description of Liouville dynamics in terms of warped spacetime The duality between the B-model on the conifold and the c=1 noncritical string theory exemplifies this phenomenon, with the conifold partition function encoding the universal nonperturbative effects of N=1 supersymmetric gauge theories and offering a link to Seiberg-Witten theory.
In N = 2 SYM theory, the genus expansion on the gauge side is governed by Λ^{4g} In the noncritical string picture, the corresponding expansion involves the string coupling g_s, with g_s^{2g} ∼ e^{−r^{2g}} where r is the renormalized cosmological constant Thus the two descriptions are related by Λ^4 = Λ_0^4 e^{−4/(8π)}.
Equation (7.8) with b = 2N_c − N_f = 4 in the SU(2) case highlights two key points First, the θ dependence on the SYM side resembles the stringy Θ parameter introduced in Bonelli–Marchetti–Matone’s framework (see (3.27) and [18]), signaling a parallel structure between the gauge theory and the stringy parameter Second, the mapping between the parameters a and ar appears to stem from a geometric origin, hinting at an intrinsic link between the field-theory variables and the underlying geometry.
In the N=2 Liouville-CY correspondence, the parameter r acts as the deformation of the complex moduli, linking the geometry of the Calabi–Yau setup to the four-dimensional gauge theory This r deformation encodes the variation of the complex structure and should be identified with the moduli of the 4d theory, specifically the N=2 SYM moduli a Consequently, shifts in r on the Liouville-CY side map directly to changes in the a-moduli of the 4D theory, establishing a concrete bridge between geometric moduli and the physical moduli of N=2 supersymmetric Yang–Mills.
A deep link emerges between the KPZ scaling in noncritical string theory and the (fractional) instanton contributions that shape physical observables in supersymmetric gauge theories In the KPZ framework, integrating over the Liouville zero mode yields the genus expansion of the cosmological constant, which can be identified with the moduli of N=2 supersymmetric Yang–Mills (SYM) theory In contrast, in N=2 pure SYM, nonperturbative corrections to the prepotential arise solely from instantons, with no contribution from fractional instantons, unlike N=1 gauge theories These gauge-theory structures, like the Liouville theory, are constrained by symmetry arguments The resulting coincidence likely shares its origin with the emergence of N=2 Liouville theory as the worldsheet description of the corresponding gauge theories.
From this perspective, amplitudes involving fractional instantons—common in N = 1 gauge theories—require analytic continuation This same mechanism mirrors the appearance of a fractional power of the cosmological constant in Liouville theory, where evaluating correlators also necessitates analytic continuation Together, these observations suggest a shared mathematical structure governing nonperturbative effects across these theories and explain why analytic continuation is essential for meaningful amplitude and correlator calculations.
To obtain the noncritical string expression for instanton string theory, we adapt the argument developed in [42] to our current context We study the moduli spaces of higher-genus punctured Riemann surfaces and define the corresponding expectation value within this framework.
19 One minor difference, however, is that the warped coordinate is only the Liouville direction. the corresponding Liouville background as 20 hσig,n≡ 1
Adopting the notations from section 3, the Liouville background framework is applicable here, with S_cl playing the role of the potential for the Weil–Petersson metric even on punctured higher-genus Riemann surfaces (see [35]) The genus expansion of the N=2 effective coupling constant is defined by τ_g, as expressed in equation (7.10).
Here we introduce the divisor basis consisting of D0 and Dk (k = 1, , g−1): D0 = M_g − 1,4 and Dk = M_k, 2 × M_g − k, 2 We also rescale these divisors by the normalized Weil–Petersson (WP) volumes as in Section 3, so that hωii,j ≡ hω i,j ii,j.
We now define a divisor DI, which we call ‘N = 2 divisor’, as the (6g−4)-cycle
X k=1 c (g) k hω I ik,2hω I ig − k,2Dk , (7.12) where the coefficients c (g) k will be given later We identify [ω I ] as the Poincar´e dual to
For each index I, D_I is defined so that the corresponding Kähler class satisfies [ω_I] = c1([D_I]), where [D] denotes the line bundle attached to a divisor D and c1 is the first Chern class The rescaling of the divisor D_I is carried out in the same spirit as in Section 3.
We fix the c^{(g)}_k by requiring that the τ^{(g)}_k defined in (7.10) satisfy the recursion relation (7.6) Two facts are crucial for obtaining this recursion: first, when evaluating the relevant integrals, only boundary components ∂M_g,2 of the form M_g−k,i × M_k,j with i = j = 2 and M_g−1,4 appear; second, ω^{g,2} satisfies the restriction phenomenon mentioned above In particular, we consider the natural embedding i: M_k,2 → M_k,2×* → M_k,2× M_g−k,2 → ∂M_g,2 → M_g,2, as in (7.13).
20 The omitted suffix to σ is the same of the one appearing as suffix of the bracket We will use this notation throughout this subsection.