We show that modularity and the gap condition make the holomorphic anomaly equation completely integrable for non-compact Calabi-Yau manifolds. This leads to a very efficient formalism to solve the topological string on these geometries in terms of almost holomorphic modular forms. The formalism provides in particular holomorphic expansions everywhere in moduli space including large radius points, the conifold loci, Seiberg-Witten points and the orbifold points. It can be also viewed as a very efficient method to solve higher genus closed string amplitudes in the $\frac{1}{N^2}$ expansion of matrix models with more then one cut.
We study the instanton partition functions of two well-known superconformal field theories with mass deformations. Two types of anomaly equations, namely, the modular anomaly and holomorphic anomaly, have been discovered in the literature. We provide a clean solution to the long standing puzzle about their precise relation, and obtain some universal formulas. We show that the partition function is invariant under the SL(2,Z) duality which exchanges theories at strong coupling with those of weak coupling.
Recently, Okuyama and Sakai proposed a novel holomorphic anomaly equation for the partition function of 2d Yang-Mills theory on a torus, based on an anholomorphic deformation of the propagator in the bosonic formulation. Using the boson-fermion correspondence, we derive the formula for the deformed partition function in fermionic description and give a proof of the holomorphic anomaly equation.
We show that the BRS operator of the topological string B model is not holomorphic in the complex structure of the target space. This implies that the so-called holomorphic anomaly of topological strings should not be interpreted as a BRS anomaly.
We show that, in analyzing differential equations obeyed by one-loop gauge theory amplitudes, one must take into account a certain holomorphic anomaly. When this is done, the results are consistent with the simplest twistor-space picture of the available one-loop amplitudes.
We show that a general solution to the extended holomorphic anomaly equations for the open topological string on D-branes in a Calabi-Yau manifold, recently written down by Walcher in arXiv:0705.4098, is obtained from the general solution to the holomorphic anomaly equations for the closed topological string on the same manifold, by shifting the closed string moduli by amounts proportional to the 't Hooft coupling.
In this paper, we study the open topological string amplitudes on Calabi-Yau threefolds by the extended holomorphic anomaly equation. The disk two-point function determined by the domainwall tension, together with the Yukawa couplings, solves the amplitudes with high genus and boundaries recursively. The BPS invariants encoded in the amplitudes are extracted by mirror symmetry.
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 deform…
The special geometry ($(t,{\bar t})$-equations) for twisted $N=2$ strings are derived as consistency conditions of a new contact term algebra. The dilaton appears in the contact terms of topological and antitopological operators. The holomorphic anomaly, which can be interpreted as measuring the background dependence, is obtained from the contact algebra relations.
In this short note, we report a curious appearance of the recently discovered 4d-5d connection of extremal blackholes in the topological string B-model. The holomorphic anomaly equations in the Schrodinger-Weil representation are written {\it formally} in terms of M2 charges. In the phase space the 4d-5d charges are related by a non-linear canonical transformation. The blackhole partition function factors into M2-anti-M2 contributions in leading approximation.
Recently, it has been shown that the holomorphic anomaly of unitarity cuts can be used as a tool in determining the one-loop amplitudes in N=4 super Yang-Mills theory. It is interesting to examine whether this method can be applied to more general cases. We present results for a non-MHV N=1 supersymmetric one-loop amplitude. We show that the holomorphic anomaly of each unitarity cut correctly reproduces the action on the amplitude's imaginary part of the differential operators corresponding to collinearity in twistor space. We find that the use of the holomorphic anomaly to evaluate the amplit…
In this paper we study the one-loop dilatation operator of the full scalar field sector of Leigh-Strassler deformed N=4 SYM theory. In particular we map it onto a spin chain and find the parameter values for which the Reshetikhin integrability criteria are fulfilled. Some years ago Roiban found an integrable subsector, consisting of two holomorphic scalar fields, corresponding to the XXZ model. He was pondering about the existence of a subsector which would form generalisation of that model to an integrable su_q(3) model. Later Berenstein and Cherkis added one more holomorphic field and showed…
We consider four-dimensional Omega-deformed N=2 supersymmetric SU(2) gauge theory on A1 space and its lift to five dimensions. We find that the partition functions can be reproduced via special geometry and the holomorphic anomaly equation. Schwinger type integral expressions for the boundary conditions at the monopole/dyon point in moduli space are inferred. The interpretation of the five-dimensional partition function as the partition function of a refined topological string on A1x(local P1xP1) is suggested.
Motivated by the recently proposed connection between N=2 BPS black holes and topological strings, I study the attractor equations and their interplay with the holomorphic anomaly equation. The topological string partition function is interpreted as a wave-function obtained by quantizing the real cohomology of the Calabi-Yau. In this interpretation the apparent background dependence due to the holomorphic anomaly is caused by the choice of complex polarization. The black hole attractor equations express the moduli in terms of the electric and magnetic charges, and lead to a real polarization i…
Using wall-crossing formulae and the theory of mock modular forms we derive a holomorphic anomaly equation for the modified elliptic genus of two M5-branes wrapping a rigid divisor inside a Calabi-Yau manifold. The anomaly originates from restoring modularity of an indefinite theta-function capturing the wall-crossing of BPS invariants associated to D4-D2-D0 brane systems. We show the compatibility of this equation with anomaly equations previously observed in the context of N=4 topological Yang-Mills theory on P^2 and E-strings obtained from wrapping M5-branes on a del Pezzo surface. The non-…
Blowup equations and holomorphic anomaly equations are two universal yet completely different approaches to solve refined topological string theory on local Calabi-Yau threefolds corresponding to A- and B-model respectively. The former originated from comparing Nekrasov partition functions of 4d $\mathcal{N}=2$ gauge theories on $Ω$ defomed spacetime $\mathbb{C}^2_{ε_1,ε_2}$ and its one-point blown-up, while the latter takes root in the degeneration of wordsheet Riemann surfaces. The relation between the two approaches is an open question. In this short note, we find a novel recursive equation…
The partition function of an N=2 gauge theory in the Omega-background satisfies, for generic value of the parameter beta=-eps_1/eps_2, the, in general extended, but otherwise beta-independent, holomorphic anomaly equation of special geometry. Modularity together with the (beta-dependent) gap structure at the various singular loci in the moduli space completely fixes the holomorphic ambiguity, also when the extension is non-trivial. In some cases, the theory at the orbifold radius, corresponding to beta=2, can be identified with an "orientifold" of the theory at beta=1. The various connections …
In a previous paper we had proposed a specific route to relating the entropy of two charge black holes to the degeneracy of elementary string states in N=4 supersymmetric heterotic string theory in four dimensions. For toroidal compactification this proposal works correctly to all orders in a power series expansion in inverse charges provided we take into account the corrections to the black hole entropy formula due to holomorphic anomaly. In this paper we demonstrate that similar agreement holds also for other N=4 supersymmetric heterotic string compactifications.
We use the holomorphic anomaly equation to solve the gravitational corrections to Seiberg-Witten theory and a two-cut matrix model, which is related by the Dijkgraaf-Vafa conjecture to the topological B-model on a local Calabi-Yau manifold. In both cases we construct propagators that give a recursive solution in the genus modulo a holomorphic ambiguity. In the case of Seiberg-Witten theory the gravitational corrections can be expressed in closed form as quasimodular functions of Gamma(2). In the matrix model we fix the holomorphic ambiguity up to genus two. The latter result establishes the Di…
Studying the quadratic field theory on seven dimensional spacetime constructed by a direct product of Calabi-Yau three-fold by a real time axis, with phase space being the third cohomology of the Calabi-Yau three-fold, the generators of translation along moduli directions of Calabi-Yau three-fold are constructed. The algebra of these generators is derived which take a simple form in canonical coordinates. Applying the Dirac method of quantization of second class constraint systems, we show that the Schrödinger equations corresponding to these generators are equivalent to the holomorphic anomal…
We consider boundary conditions compatible with discrete holomorphicity for the dilute O(n) and C_2^(1) loop models. In each model, for a general set of boundary plaquettes, multiple types of loops can appear. A generalisation of Smirnov's parafermionic observable is therefore required in order to maintain the discrete holomorphicity property in the bulk. We show that there exist natural boundary conditions for this observable which are consistent with integrability, that is to say that, by imposing certain boundary conditions, we obtain a set of linear equations whose solutions also satisfy t…
We show how the holomorphic anomaly found in hep-th/0409245 can be used to efficiently compute certain classes of unitarity cuts of one-loop N=4 amplitudes of gluons. These classes include all cuts of n-gluon one-loop MHV amplitudes and of n-gluon next-to-MHV amplitudes with helicities (1+,2+,3+,4-,..., n-). As an application of this method, we present the explicit computation of the (1,2,3)-cut of the n-gluon one-loop N=4 leading-color amplitude A_{n;1}(1+,2+,3+,4-,..., n-). The answer is given in terms of scalar box functions and provides information about the corresponding amplitudes. A pos…
We consider a class of gauged linear sigma models (GLSMs) in two dimensions that flow to non-compact (2,2) superconformal field theories in the infra-red, a prototype of which is the SL(2,\IR)/U(1) (cigar) coset. We compute the elliptic genus of the GLSMs as a path-integral on the torus using supersymmetric localization. We find that the result is a Jacobi-like form that is non-holomorphic in the modular parameter $τ$ of the torus, with mock modular behavior. This agrees with a previously-computed expression in the cigar coset. We show that the lack of holomorphicity of the elliptic genus aris…
We study Gromov-Witten invariants of a rational elliptic surface using holomorphic anomaly equation in [HST1](hep-th/9901151). Formulating invariance under the affine $E_8$ Weyl group symmetry, we determine conjectured invariants, the number of BPS states, from Gromov-Witten invariants. We also connect our holomorphic anomaly equation to that found by Bershadsky,Cecotti,Ooguri and Vafa [BCOV1](hep-th/9302103).
We derive the framing anomaly of four-dimensional holomorphic-topological Chern-Simons theory formulated on the product of a topological surface and the complex plane. We show that the presence of this anomaly allows one to couple four-dimensional Chern-Simons theory to holomorphic field theories with Kac-Moody symmetry, where the Kac-Moody level $k$ is critical $k=-h^{\vee}$. Applying this result to a holomorphic sigma model into a complex coadjoint orbit, we derive that four-dimensional Chern-Simons theory admits holomorphic monodromy defects.
In arXiv:0905.3629 we described a new class of N=2 topological amplitudes that depends both on vector and hypermultiplet moduli. Here we find that this class is actually a particular case of much more general topological amplitudes which appear at higher loops in heterotic string theory compactified on K3 x T^2. We analyze their effective field theory interpretation and derive particular (first order) differential equations as a consequence of supersymmetry Ward identities and the 1/2-BPS nature of the corresponding effective action terms. In string theory the latter get modified due to anomal…
We show that the 4-dimensional N=1/2 supersymmetry algebra admits central extension. The central charges are supported by domain wall and the central charges are computed. We also determine the Konishi anomaly for N=1/2 supersymmetric gauge theory. Due to the new couplings in the Lagrangian, many terms appears. We show that these terms sum up to give the expected form for the holomorphic part of the Konishi anomaly. For the anti-holomorphic part, we give a simple argument that the naive generalization has to be modified. We suggest that the anti-holomorphic Konishi anomaly is given by a gauge …
We propose a systematic way to carry out the method introduced in hep-th/0410077 for computing certain unitarity cuts of one-loop N=4 amplitudes of gluons. We observe that the class of cuts for which the method works involves all next-to-MHV n-gluon one-loop amplitudes of any helicity configurations. As an application of our systematic procedure, we obtain the complete seven-gluon one-loop leading-color amplitude A_{7;1}(1-,2-,3-,4+,5+,6+,7+).
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 h…
Open topological string amplitudes on compact Calabi-Yau threefolds are shown to satisfy an extension of the holomorphic anomaly equation of Bershadsky, Cecotti, Ooguri and Vafa. The total topological charge of the D-brane configuration must vanish in order to satisfy tadpole cancellation. The boundary state of such D-branes is holomorphically captured by a Hodge theoretic normal function. Its Griffiths' infinitesimal invariant is the analogue of the closed string Yukawa coupling and plays the role of the terminator in a Feynman diagram expansion for the topological string with D-branes. The h…
We complete the holomorphic anomaly equations for topological strings with their dependence on open moduli. We obtain the complete system by standard path integral arguments generalizing the analysis of BCOV (Commun. Math. Phys. 165 (1994) 311) to strings with boundaries. We study both the anti-holomorphic dependence on open moduli and on closed moduli in presence of Wilson lines. By providing the compactification a' la Deligne-Mumford of the moduli space of Riemann surfaces with boundaries, we show that the open holomorphic anomaly equations are structured on the (real codimension one) bounda…
We present a geometrical framework which incorporates higher derivative corrections to the action of N = 2 vector multiplets in terms of an enlarged scalar manifold which includes a complex deformation parameter. This enlarged space carries a deformed version of special Kahler geometry which we characterise. The holomorphic anomaly equation arises in this framework from the integrability condition for the existence of a Hesse potential.
The genus g free energies of matrix models can be promoted to modular invariant, non-holomorphic amplitudes which only depend on the geometry of the classical spectral curve. We show that these non-holomorphic amplitudes satisfy the holomorphic anomaly equations of Bershadsky, Cecotti, Ooguri and Vafa. We derive as well holomorphic anomaly equations for the open string sector. These results provide evidence at all genera for the Dijkgraaf--Vafa conjecture relating matrix models to type B topological strings on certain local Calabi--Yau threefolds.
Let $S$ be a K3 surface and let $E$ be an elliptic curve. We solve the reduced Gromov-Witten theory of the Calabi-Yau threefold $S \times E$ for all curve classes which are primitive in the K3 factor. In particular, we deduce the Igusa cusp form conjecture. The proof relies on new results in the Gromov-Witten theory of elliptic curves and K3 surfaces. We show the generating series of Gromov-Witten classes of an elliptic curve are cycle-valued quasimodular forms and satisfy a holomorphic anomaly equation. The quasimodularity generalizes a result by Okounkov and Pandharipande, and the holomorphi…
In this paper we develop a theory of Fourier-like transforms on the space of stable graphs. In particular, we introduce a duality theory of stable graphs. As an application, we derive the holomorphic anomaly equations for general propagators in the work of Aganagic, Bouchard and Klemm.
Developing on the ideas of R. Stora and coworkers, a formulation of two dimensional field theory endowed with extended conformal symmetry is given, which is based on deformation theory of holomorphic and Hermitian spaces. The geometric background consists of a vector bundle $E$ over a closed surface $Σ$ endowed with a holomorphic structure and a Hermitian structure subordinated to it. The symmetry group is the semidirect product of the automorphism group ${\rm Aut}(E)$ of $E$ and the extended Weyl group ${\rm Weyl}(E)$ of $E$ and acts on the holomorphic and Hermitian structures. The extended W…
We consider the generating function (prepotential) for Gromov-Witten invariants of rational elliptic surface. We apply the local mirror principle to calculate the prepotential and prove a certain recursion relation, holomorphic anomaly equation, for genus 0 and 1. We propose the holomorphic anomaly equation for all genera and apply it to determine higher genus Gromov-Witten invariants and also the BPS states on the surface. Generalizing Göttsche's formula for the Hilbert scheme of $g$ points on a surface, we find precise agreement of our results with the proposal recently made by Gopakumar and…
We study N=(0,2) deformed (2,2) two-dimensional sigma models. Such heterotic models were discovered previously on the world sheet of non-Abelian strings supported by certain four-dimensional N=1 theories. We study geometric aspects and holomorphic properties of these models, and derive a number of exact expressions for the beta functions in terms of the anomalous dimensions analogous to the NSVZ beta function in four-dimensional Yang-Mills. Instanton calculus provides a straightforward method for the derivation. The anomalous dimensions are calculated up to two loops implying that one of the b…
We derive global consistency condition for strongly coupled heterotic string in the presence of M5-branes. Its elliptic genus is interpretable as generating functional of anomaly polynomials and so, on anomaly-free vacua, the genus is both holomorphic and modular invariant. In holomorphic basis, we identify the modular properties by calculating the phase. By interpreting the refinement parameters as background curvature of tangent and vector bundles, we identify the extended Bianchi identity for Kalb--Ramond field of heterotic M-theory in the presence of arbitrary numbers of M5-branes.
The holomorphic prepotential of ultraviolet finite N=2 supersymmetric gauge theories is obtained by a partial twisting of N=1 gauge theory in six dimensions, compactified on $\IR^4\timesT^2$. We show that Ward identities for the conserved chiral $\R$-symmetry in these theories generate a set of constraints on the correlation functions of chiral ring operators. These correlators depend only on the coordinates of the $T^2$, and the constraints are analogs of the Knihnik-Zamolodchikov-Bernard equations at the critical level.
The Abelian Higgs model on a compact Riemann surface Σof genus g is considered. We show that for g > 1 the Bogomolny equations for multi-vortices at critical coupling can be obtained as compatibility conditions of two linear equations (Lax pair) which are written down explicitly. These vortices correspond precisely to SO(3)-symmetric Yang-Mills instantons on the (conformal) gravitational instanton Σ\times S^2 with a scalar-flat Kahler metric. Thus, the standard methods of constructing solutions and studying their properties by using Lax pairs (twistor approach, dressing method etc.) can be app…
We study certain differential rings over the moduli space of Calabi-Yau manifolds. In the case of an elliptic curve, we observe a close relation to the differential ring of quasi-modular forms due to Kaneko and Zagier.
The gauge theoretic large N expansion yields an asymptotic series which requires a nonperturbative completion in order to be well defined. Recently, within the context of random matrix models, it was shown how to build resurgent transseries solutions encoding the full nonperturbative information beyond the 't Hooft genus expansion. On the other hand, via large N duality, random matrix models may be holographically described by B-model closed topological strings in local Calabi-Yau geometries. This raises the question of constructing the corresponding holographically dual resurgent transseries,…
We show that the all-orders WKB periods of one-dimensional quantum mechanical oscillators are governed by the refined holomorphic anomaly equations of topological string theory. We analyze in detail the double-well potential and the cubic and quartic oscillators, and we calculate the WKB expansion of their quantum free energies by using the direct integration of the anomaly equations. We reproduce in this way all known results about the quantum periods of these models, which we express in terms of modular forms on the WKB curve. As an application of our results, we study the large order behavi…
We prove a higher genus version of the genus $0$ local-relative correspondence of van Garrel-Graber-Ruddat: for $(X,D)$ a pair with $X$ a smooth projective variety and $D$ a nef smooth divisor, maximal contact Gromov-Witten theory of $(X,D)$ with $λ_g$-insertion is related to Gromov-Witten theory of the total space of $\mathcal{O}_X(-D)$ and local Gromov-Witten theory of $D$. Specializing to $(X,D)=(S,E)$ for $S$ a del Pezzo surface or a rational elliptic surface and $E$ a smooth anticanonical divisor, we show that maximal contact Gromov-Witten theory of $(S,E)$ is determined by the Gromov-Wit…
We discuss a link between the topological recursion relations derived algebraically by Witten and the holomorphic anomaly equation of Bershadsky, Cecotti, Ooguri and Vafa. This is obtained through the definition of an operator ${\cal{W}}_s$ that reproduces the recursion relations for topological string theory coupled to worldsheet gravity a la BCOV. This operator is contained inside an algebra that generalizes the tt* equations and whose direct consequence is the holomorphic anomaly equation itself.
Using the generalized Konishi anomaly (GKA) equations, we derive the effective superpotential of four-dimensional N=1 supersymmetric SU(n) gauge theory with n+2 fundamental flavors. We find, however, that the GKA equations are only integrable in the Seiberg dual description of the theory, but not in the direct description of the theory. The failure of integrability in the direct, strongly coupled, description suggests the existence of non-perturbative corrections to the GKA equations.
The holomorphic anomaly equations describe B-model closed topological strings in Calabi-Yau geometries. Having been used to construct perturbative expansions, it was recently shown that they can also be extended past perturbation theory by making use of resurgent transseries. These yield formal nonperturbative solutions, showing integrability of the holomorphic anomaly equations at the nonperturbative level. This paper takes such constructions one step further by working out in great detail the specific example of topological strings in the mirror of the local CP2 toric Calabi-Yau background, …
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 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 anomal…
We study a generalization of the Langevin equation, that describes fluctuations, of commuting degrees of freedom, for scalar field theories with worldvolumes of arbitrary dimension, following Parisi and Sourlas and correspondingly generalizes the Nicolai map. Supersymmetry appears inevitably, as defining the consistent closure of system+fluctuations and it can be probed by the identities satisfied by the correlation functions of the noise fields, sampled by the action of the commuting fields. This can be done effectively, through numerical simulations. We focus on the case where the target spa…
We calculate the gravitational corrections to the effective action of N=2 SU(2) Seiberg-Witten theory with matter using modularity, the holomorphic anomaly equation and expected behavior at the boundaries of the moduli space. As in pure gauge theory we show that the gap condition at the dyon singularities completely fixes the gravitational corrections. We discuss the behavior of the gravitational corrections at the conformal points. We compare our results with the recursive solution of the loop equation in the matrix model approach, which provides in addition open amplitudes.
We consider the QCD scattering amplitudes at high energies sqrt{s} and fixed momentum transfers sqrt{-t} in the leading logarithmic approximation at a non-zero temperature T in the t-channel. It is shown that the BFKL Hamiltonian has the property of holomorphic separability. The Pomeron wave function for arbitrary T is calculated using an integral of motion. In multi-colour QCD, the holomorphic Hamiltonian for n-reggeized gluons at temperature T is shown to coincide with the local Hamiltonian of an integrable Heisenberg model and can be obtained from the T=0 Hamiltonian by an unitary transform…
