Holomorphic Anomaly Research Articles

Resurgent Structure of the Topological String and the First Painlevé Equation

We present an explicit formula for the Stokes automorphism acting on the topological string partition function. When written in terms of the dual partition function, our formula implies that flat coordinates in topological string theory transform as quantum periods, and according to the Delabaere-Dillinger-Pham formula. We first show how the formula follows from the non-linear Stokes phenomenon of the Painlevé I equation, together with the connection between its $\tau$-function and topological strings on elliptic curves. Then, we show that this formula is also a consequence of a recent conjecture on the resurgent structure of the topological string, based on the holomorphic anomaly equations, and it is in fact valid for arbitrary Calabi-Yau threefolds.

Non-perturbative topological string theory on compact Calabi-Yau 3-folds

We obtain analytic and numerical results for the non-perturbative amplitudes of topological string theory on arbitrary, compact Calabi-Yau manifolds. Our approach is based on the theory of resurgence and extends previous special results to the more general case. In particular, we obtain explicit trans-series solutions of the holomorphic anomaly equations. Our results predict the all orders, large genus asymptotics of the topological string free energies, which we test in detail against high genus perturbative series obtained recently in the compact case. We also provide additional evidence that the Stokes constants appearing in the resurgent structure are closely related to integer BPS invariants.

A note on rank 5/2 Liouville irregular block, Painlevé I and the H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{H} $$\\end{document}0 Argyres-Douglas theory

We study 4d type H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{H} $$\\end{document}0 Argyres-Douglas theory in Ω-background by constructing Liouville irregular state of rank 5/2. The results are compared with generalized Holomorphic anomaly approach, which provides order by order expansion in Ω-background parameters ϵ1,2. Another crucial test of our results provides comparison with respect to Painlevé I τ-function, which was expected to be hold in self-dual case ϵ1 = −ϵ2. We also discuss Nekrasov-Shatashvili limit ϵ1 = 0, accessible either by means of deformed Seiberg-Witten curve, or WKB methods.

On the resurgent structure of quantum periods

Quantum periods appear in many contexts, from quantum mechanics to local mirror symmetry. They can be described in terms of topological string free energies and Wilson loops, in the so-called Nekrasov-Shatashvili limit. We consider the trans-series extension of the holomorphic anomaly equations satisfied by these quantities, and we obtain exact multi-instanton solutions for these trans-series. Building on this result, we propose a unified perspective on the resurgent structure of quantum periods. We show for example that the Delabaere-Pham formula, which was originally obtained in quantum mechanical examples, is a generic feature of quantum periods. We illustrate our general results with explicit calculations for the double-well in quantum mechanics, and for the quantum mirror curve of local \mathbb{P}^2ℙ2.

Partition functions of non-Lagrangian theories from the holomorphic anomaly

The computation of the partition function in certain quantum field theories, such as those of the Argyres-Douglas or Minahan-Nemeschansky type, is problematic due to the lack of a Lagrangian description. In this paper, we use the holomorphic anomaly equation to derive the gravitational corrections to the prepotential of such theories at rank one by deforming them from the conformal point. In the conformal limit, we find a general formula for the partition function as a sum of hypergeometric functions. We show explicit results for the round sphere and the Nekrasov-Shatashvili phases of the Ω background. The first case is relevant for the derivation of extremal correlators in flat space, whereas the second one has interesting applications for the study of anharmonic oscillators.

Vertex algebras and quantum master equation

We contribute to study the two dimensional chiral conformal field theories that arise from chiral deformations of free theories such as the $\beta \gamma$‑system, bc‑system and chiral bosons. Our method is based on the effective Batalin–Vilkovisky quantization theory. We establish an exact correspondence between renormalized quantum master equations for chiral deformations and Maurer–Cartan equations for the Lie algebra of modes of chiral vertex algebras. The generating functions on the torus are proven to have modular property with mild holomorphic anomaly. As an application, we construct an exact solution of quantum B‑model (BCOV theory) in complex one dimension that solves the full higher genus mirror symmetry conjecture on elliptic curves.

Regularized Integrals on Elliptic Curves and Holomorphic Anomaly Equations

We derive residue formulas for the regularized integrals (introduced by Li and Zhou in Commun Math Phys 388:1403–1474, 2021) on configuration spaces of elliptic curves. Based on these formulas, we prove that the regularized integrals satisfy holomorphic anomaly equations, providing a mathematical formulation of the so-called contact term singularities. We also discuss residue formulas for the ordered A-cycle integrals and establish their relations with those for the regularized integrals.

Holomorphic anomalies, fourfolds and fluxes

We investigate holomorphic anomalies of partition functions underlying string compactifications on Calabi-Yau fourfolds with background fluxes. For elliptic fourfolds the partition functions have an alternative interpretation as elliptic genera of N = 1 supersymmetric string theories in four dimensions, or as generating functions for relative, genus zero Gromov-Witten invariants of fourfolds with fluxes. We derive the holomorphic anomaly equations by starting from the BCOV formalism of topological strings, and translating them into geometrical terms. The result can be recast into modular and elliptic anomaly equations. As a new feature, as compared to threefolds, we find an extra contribution which is given by a gravitational descendant invariant. This leads to linear terms in the anomaly equations, which support an algebra of derivatives mapping between partition functions of the various flux sectors. These geometric features are mirrored by certain properties of quasi-Jacobi forms. We also offer an interpretation of the physics from the viewpoint of the worldsheet theory, and comment on holomorphic anomalies at genus one.

Towards refining the topological strings on compact Calabi-Yau 3-folds

We make a proposal for calculating refined Gopakumar-Vafa numbers (GVN) on elliptically fibered Calabi-Yau 3-folds based on refined holomorphic anomaly equations. The key examples are smooth elliptic fibrations over (almost) Fano surfaces. We include a detailed review of existing mathematical methods towards defining and calculating the (unrefined) Gopakumar-Vafa invariants (GVI) and the GVNs on compact Calabi-Yau 3-folds using moduli of stable sheaves, in a language that should be accessible to physicists. In particular, we discuss the dependence of the GVNs on the complex structure moduli and on the choice of an orientation. We calculate the GVNs in many instances and compare the B-model predictions with the geometric calculations. We also derive the modular anomaly equations from the holomorphic anomaly equations by analyzing the quasi-modular properties of the propagators. We speculate about the physical relevance of the mathematical choices that can be made for the orientation.

Quasi-Jacobi forms, elliptic genera and strings in four dimensions

We investigate the interplay between the enumerative geometry of Calabi-Yau fourfolds with fluxes and the modularity of elliptic genera in four-dimensional string theories. We argue that certain contributions to the elliptic genus are given by derivatives of modular or quasi-modular forms, which may encode BPS invariants of Calabi-Yau or non-Calabi-Yau threefolds that are embedded in the given fourfold. As a result, the elliptic genus is only a quasi-Jacobi form, rather than a modular or quasi-modular one in the usual sense. This manifests itself as a holomorphic anomaly of the spectral flow symmetry, and in an elliptic holomorphic anomaly equation that maps between different flux sectors. We support our general considerations by a detailed study of examples, including non-critical strings in four dimensions.For the critical heterotic string, we explain how anomaly cancellation is restored due to the properties of the derivative sector. Essentially, while the modular sector of the elliptic genus takes care of anomaly cancellation involving the universal B-field, the quasi-Jacobi one accounts for additional B-fields that can be present.Thus once again, diverse mathematical ingredients, namely here the algebraic geometry of fourfolds, relative Gromow-Witten theory pertaining to flux backgrounds, and the modular properties of (quasi-)Jacobi forms, conspire in an intriguing manner precisely as required by stringy consistency.

Refinement and modularity of immortal dyons

Extending recent results in mathcal{N} = 2 string compactifications, we propose that the holomorphic anomaly equation satisfied by the modular completions of the generating functions of refined BPS indices has a universal structure independent of the number mathcal{N} of supersymmetries. We show that this equation allows to recover all known results about modularity (under SL(2, ℤ) duality group) of BPS states in mathcal{N} = 4 string theory. In particular, we reproduce the holomorphic anomaly characterizing the mock modular behavior of quarter-BPS dyons and generalize it to the case of non-trivial torsion invariant.

Holomorphic anomaly equation for and the Nekrasov-Shatashvili limit of local

AbstractWe prove a higher genus version of the genus$0$local-relative correspondence of van Garrel-Graber-Ruddat: for$(X,D)$a pair withXa smooth projective variety andDa nef smooth divisor, maximal contact Gromov-Witten theory of$(X,D)$with$\lambda _g$-insertion is related to Gromov-Witten theory of the total space of${\mathcal O}_X(-D)$and local Gromov-Witten theory ofD.Specializing to$(X,D)=(S,E)$forSa del Pezzo surface or a rational elliptic surface andEa smooth anticanonical divisor, we show that maximal contact Gromov-Witten theory of$(S,E)$is determined by the Gromov-Witten theory of the Calabi-Yau 3-fold${\mathcal O}_S(-E)$and the stationary Gromov-Witten theory of the elliptic curveE.Specializing further to$S={\mathbb P}^2$, we prove that higher genus generating series of maximal contact Gromov-Witten invariants of$({\mathbb P}^2,E)$are quasimodular and satisfy a holomorphic anomaly equation. The proof combines the quasimodularity results and the holomorphic anomaly equations previously known for local${\mathbb P}^2$and the elliptic curve.Furthermore, using the connection between maximal contact Gromov-Witten invariants of$({\mathbb P}^2,E)$and Betti numbers of moduli spaces of semistable one-dimensional sheaves on${\mathbb P}^2$, we obtain a proof of the quasimodularity and holomorphic anomaly equation predicted in the physics literature for the refined topological string free energy of local${\mathbb P}^2$in the Nekrasov-Shatashvili limit.

Duality and mock modularity

We derive a holomorphic anomaly equation for the Vafa-Witten partition function for twisted four-dimensional \mathcal{N} =4𝒩=4 super Yang-Mills theory on \mathbb{CP}^{2}ℂℙ2 for the gauge group SO(3)SO(3) from the path integral of the effective theory on the Coulomb branch. The holomorphic kernel of this equation, which receives contributions only from the instantons, is not modular but ‘mock modular’. The partition function has correct modular properties expected from SS-duality only after including the anomalous nonholomorphic boundary contributions from anti-instantons. Using M-theory duality, we relate this phenomenon to the holomorphic anomaly of the elliptic genus of a two-dimensional noncompact sigma model and compute it independently in two dimensions. The anomaly both in four and in two dimensions can be traced to a topological term in the effective action of six-dimensional (2,0)(2,0) theory on the tensor branch. We consider generalizations to other manifolds and other gauge groups to show that mock modularity is generic and essential for exhibiting duality when the relevant field space is noncompact.

Renormalization for Holomorphic Field Theories

We introduce the concept of a holomorphic field theory on any complex manifold in the language of the Batalin–Vilkovisky formalism. When the complex dimension is one, this setting agrees with that of chiral conformal field theory. Our main result concerns the behavior of holomorphic theories under renormalization group flow. Namely, we show that holomorphic theories are one-loop finite. We use this to completely characterize holomorphic anomalies in any dimension. Throughout, we compare our approach to holomorphic field theories to more familiar approaches including that of supersymmetric field theories.

Holomorphic anomaly of 2d Yang-Mills theory on a torus revisited

We study the large N ’t Hooft expansion of the chiral partition function of 2d U(N) Yang-Mills theory on a torus. There is a long-standing puzzle that no explicit holomorphic anomaly equation is known for the partition function, although it admits a topological string interpretation. Based on the chiral boson interpretation we clarify how holomorphic anomaly arises and propose a natural anti-holomorphic deformation of the partition function. Our deformed partition function obeys a fairly traditional holomorphic anomaly equation. Moreover, we find a closed analytic expression for the deformed partition function. We also study the behavior of the deformed partition function both in the strong coupling/large area limit and in the weak coupling/small area limit. In particular, we observe that drastic simplification occurs in the weak coupling/small area limit, giving another nontrivial support for our anti-holomorphic deformation.

Gromov–Witten theory of elliptic fibrations : Jacobi forms and holomorphic anomaly equations

We conjecture that the relative Gromov-Witten potentials of elliptic fibrations are (cycle-valued) lattice quasi-Jacobi forms and satisfy a holomorphic anomaly equation. We prove the conjecture for the rational elliptic surface in all genera and curve classes numerically. The generating series are quasi-Jacobi forms for the lattice $E_8$. We also show the compatibility of the conjecture with the degeneration formula. As Corollary we deduce that the Gromov-Witten potentials of the Schoen Calabi-Yau threefold (relative to $\mathbb{P}^1$) are $E_8 \times E_8$ quasi-bi-Jacobi forms and satisfy a holomorphic anomaly equation. This yields a partial verification of the BCOV holomorphic anomaly equation for Calabi-Yau threefolds. For abelian surfaces the holomorphic anomaly equation is proven numerically in primitive classes. The theory of lattice quasi-Jacobi forms is reviewed. In the Appendix the conjectural holomorphic anomaly equation is expressed as a matrix action on the space of (generalized) cohomological field theories. The compatibility of the matrix action with the Jacobi Lie algebra is proven. Holomorphic anomaly equations for K3 fibrations are discussed in an example.

Crepant resolution and the holomorphic anomaly equation for [C3/Z3

We study the orbifold Gromov–Witten theory of the quotient [ C 3 / Z 3 ] in all genera. Our first result is a proof of the holomorphic anomaly equations in the precise form predicted by B-model physics. Our second result is an exact crepant resolution correspondence relating the Gromov–Witten theories of [ C 3 / Z 3 ] and local P 2 . The proof of the correspondence requires an identity proven in the Appendix by T. Coates and H. Iritani.

Vafa–Witten Theory and Iterated Integrals of Modular Forms

Vafa-Witten (VW) theory is a topologically twisted version of N=4 supersymmetric Yang-Mills theory. S-duality suggests that the partition function of VW theory with gauge group SU(N) transforms as a modular form under duality transformations. Interestingly, Vafa and Witten demonstrated the presence of a modular anomaly, when the theory has gauge group SU(2) and is considered on the complex projective plane P2. This modular anomaly could be expressed as an integral of a modular form, and also be traded for a holomorphic anomaly. We demonstrate that the modular anomaly for gauge group SU(3) involves an iterated integral of modular forms. Moreover, the modular anomaly for SU(3) can be traded for a holomorphic anomaly, which is shown to factor into a product of the partition functions for lower rank gauge groups. The SU(3) partition function is mathematically an example of a mock modular form of depth two.

Holomorphic Anomaly Equations for the Formal Quintic

We define a formal Gromov–Witten theory of the quintic threefold via localization on $${\mathbb {P}}^4$$ . Our main result is a direct geometric proof of holomorphic anomaly equations for the formal quintic in precisely the same form as predicted by B-model physics for the true Gromov–Witten theory of the quintic threefold. The results suggest that the formal quintic and the true quintic theories should be related by transformations which respect the holomorphic anomaly equations. Such a relationship has been recently found by Q. Chen, S. Guo, F. Janda, and Y. Ruan via the geometry of new moduli spaces.

Amplitudes from anomalous superconformal symmetry

We initiate a systematic study of the consequences of (super)conformal symmetry of massless scattering amplitudes. The classical symmetry is potentially broken at the quantum level by infrared and ultraviolet effects. We study its manifestations on the finite hard part of the scattering process. The conformal Ward identities in momentum space are second-order differential equations, difficult to analyze. We prefer to study superconformal symmetry whose generators are first-order in the momenta. Working in a massless mathcal{N} = 1 supersymmetric Wess-Zumino model, we derive on-shell superconformal Ward identities. They contain an anomaly due to collinear regions of loop momenta. It is given by an integral with one loop less than the original graph, with an extra integral over a collinear splitting parameter. We discuss the relation to the holomorphic anomaly that was previously studied in tree-level amplitudes and at the level of unitarity cuts. We derive and solve Ward identities for various scattering processes in the model. We classify the on-shell superamplitudes according to their Grassmann degree, in close analogy with the helicity classification of gluon amplitudes. We focus on MHV-like and NMHV-like amplitudes with up to six external particles, at one and two loops. Interestingly, the superconformal generator acting on the bosonic part of the amplitudes is Witten’s twistor collinearity operator. We find that the first-order differential equations, together with physically motivated boundary conditions, uniquely fix the answer. All the cases considered give rise to uniform weight functions. Our most interesting example is a five-point non-planar hexa-box integral with an off-shell leg. It gives first indications on the function space needed for Higgs plus two jets production at next-to-next-to leading order.