The Holomorphic Anomaly Of Topological Strings

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Topological strings on CYs In terms of complexity: Noncritical strings < Topological strings < Superstrings Still, highly nontrivial! For the A model on a CY, TS amplitudes encode an enormous amount of geometric information: Σ pw 1 1 e−dt d,w i pw h h Σ g,h F g,h(t,p 1, ,p h)= F Σ e−dt g(t)= d Σ g 2

E ective actions and topological strings

The anomaly equation ful lled by the modi ed elliptic genus is shown to be consistent with the holomorphic anomaly equations observed in the context of N= 4 topological Vafa-Witten theory on P2 and theories of E-strings obtained from wrapping M5-branes on del Pezzo surfaces. In selected examples it is argued how the holomorphic anomaly equation

Black hole attractors and the topological string

arXiv:hep-th/0405146v2 1 Jun 2004 hep-th/0405146 HUTP-04/A020, CALT-68-2501 Black Hole Attractors and the Topological String Hirosi Ooguri,a Andrew Strominger,b and Cumrun Vafab

Re ned topological strings on compact Calabi-Yau spaces

the topological vertex, [math/0408426 [math.AG]]. A long standing problem: How to solve topological strings on compact Calabi-Yau spaces? At higher genus, the most e ective approach is the mirror symmetry and use holomorphic anomaly equationM. Bershadsky, S. Cecotti, H. Ooguri, C. Vafa (BCOV), 1993. This was done byBCOV up to genus 2.

The topological open string wavefunction

alently, one can introduce a non-holomorphic dependence in the amplitudes which is governed by the holomorphic anomaly equations of [10]. As shown in [1], the transformation properties of the closed string amplitudes can be derived from the fact that the total closed string partition function (summed over all genera) is a wavefunction [44].

Non-Perturbative Quantum Mechanics from Non-Perturbative Strings

quantum systems is encoded in the so-called re ned holomorphic anomaly equations of topological string theory [4{6]. This makes it possible to calculate the quantum periods (also known as Voros multipliers [7]) to all orders, by using both modularity and the direct integration of the holomorphic anomaly equations [8,9].

Topological strings, black holes, and matrix models Minxin Huang

Topological strings: A N = (2,2) supersymmetric non-linear sigma model from world sheet Σ to target space X. Φi: Σ → X Topological string theory is the most interesting and free of world sheet anomaly, when the target space Xis a Calabi-Yau 3-fold. There are two types of topological twistings: A-model and B-model.

1 Preface -

An idiosyncratic view of the geometry underlying the holomorphic anomaly of topological strings. (Nothing to do with moduli spaces of Riemann surfaces.) Explain how the anomaly equation of BCOV can be rewritten precisely in the form of a heat equation, and how Walcher s extension is a heat equation with convection.

Towards Integrability of Topological Strings I

correlators in topological string theory (partition function) this wave function should be defined in the linear polarization of the symplectic manifold H3(M,R) associated with some reference complex structure on M. The dependence on the reference holomorphic structure is governed by the holomorphic anomaly equation [1],[2],[3]. However, even

Published for SISSA by Springer

version of special K ahler geometry which we characterise. The holomorphic anomaly equation arises in this framework from the integrability condition for the existence of a Hesse potential. Keywords: Di erential and Algebraic Geometry, Supergravity Models, Topological Strings ArXiv ePrint: 1511.06658 Open Access, c The Authors. Article funded

The holomorphic anomaly for open string moduli

Main result:Complete the holomorphic anomaly equations for topological strings with their dependence on open moduli. How: by standard path integral arguments generalizing the analysis of BCOV to strings with boundaries and open moduli. In particular: study anti-holomorphic dependence of string partition functions

International School of Advanced Studies (SISSA) and INFN

Topological adventures in the world of tadpoles Giulio Bonelli, Andrea Prudenziati, Alessandro Tanzini, and Jie Yang International School of Advanced Studies (SISSA) and INFN, Sezione di Trieste via Beirut 2-4, 34014 Trieste, Italy ABSTRACT In this paper we analyze the problem of tadpole cancellation in open topological strings.

Resurgent Transseries in Topological String Theory

How much nonperturbative information can we uncover for topological string theory assuming it has a resurgent structure? Starting point: Perturbative free energy, F(0) = P 1 g=0 g 2g s F (0) g;F g ˘c(2 g 1)A 2g+1. Plan: Analyze the large-order growth of F(0) g assuming resurgence. Extend holomorphic anomaly equations to compute higher

Asymptotic Freedom and the Spectral Index of String Vacua

holomorphic and anti-holomorphic dependence on the complex structure moduli, which is captured by the so-called holomorphic anomaly equation of [3]. In the B model, it has the following origin. Quantization of the topological string can be viewed as a quantization of the space H3(X,R) with the natural symplectic structure.

Holomorphic anomaly and matrix models

2. Holomorphic anomaly and topological strings 3 2.1 The holomorphic anomaly equations 3 2.2 The local Calabi-Yau case and the Dijkgraaf-Vafa conjecture 5 3. Review of matrix models 7 3.1 Formal matrix models and algebraic geometry 7 3.2 Variations of the matrix model free energies 9 3.2.1 Variations wrt filling fractions 11 4.

Topological M-theory as unification of form theories of gravity

that the holomorphic anomaly of topological strings [7] can be viewed as the statement that the partition function of topological string is a state in some 7-dimensional theory, with the Calabi Yau 3-fold realized as the boundary of space [8] (see also [9]). Parallel to the new discoveries about topological strings was the discovery

Santiago Codesido S anchez, Unige

Quantum mechanics and the holomorphic anomaly 1 Introduction QM and topological strings 2 The double-well oscillator WKB periods Holomorphic anomaly 3 Beyond perturbation An application 4 Conclusions and outlook

Preprint - UNIGE

holomorphic anomaly equations [8,9]. So far, the results in [3] are purely perturbative: the ~ expansion in the WKB method is treated as a genus expansion in topological string theory, and obtained, order by order, by using the recursive structure of the holomorphic anomaly. It is now natural to ask whether the

Blowup Equations for Refined Topological Strings

Blowup Equations for Re ned Topological Strings I The holomorphic anomaly equationBCOV 92 I W(t i; t i) = e P 1 g=2 g 2g s Fg( t i; i) is the wave function of the quantization of x i = t i;p i space. I BCOV as a quantization condition. I Wave function is invariant under symplectic transformations of phase space parameters (x i;p i) !Modular

Open string amplitudes and large order behavior in

in topological string theory Marcos Marin˜o1 Department of Physics, CERN Geneva 23, CH-1211 Switzerland [email protected] Abstract We propose a formalism inspired by matrix models to compute open and closed topo-logical string amplitudes in the B-model on toric Calabi Yau manifolds. We find closed


holomorphic anomaly equations, combined with modularity and appropriate boundary condi-tions, can be used to compute recursively (and e ciently) the topological string free energies at all genera [17,18]. Since topological strings are closely related to many other physical systems,

Quaternionic geometry, BPS black holes and topological strings

bundles over symmetric spaces, holomorphic modular forms. Quaternionic discrete series similarly realized in H1(Z;O(¡k)). A rep in the limit of discrete series is very closely related to holomorphic anomaly; this rep has also been constructed by geometric quantization, using hyperk˜ahler geometry of S. H1 related to indeflnite metric in c


topological N = 2 string amplitudes, with the replacement of holomorphic anomaly with harmonicity equation. Another aspect of N= 2 string, is the topological interpretation of what it is comput-ing. We show that quite generally N= 4 topological strings, are a slightly (but crucially) modified form of N= 2 topological string amplitudes.

TheHolomorphic Anomaly of Topological Strings

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. 1 Introduction andConclusions There exist two topological sigma models whose target space is a Calabi-Yau complex three-fold X

In Copyright - Non-Commercial Use Permitted Rights / License

In mathematical terms, the topological string describes the Gromov-Witten invariants of that Calabi-Yau manifold. The B-model exhibits a so-called holomorphic anomaly. This can be expressed in recursively defined differential equations which are satisfied by the amplitudes. The deviation of the amplitudes from being holomorphic, as

String Amplitudes, Topological Strings and the Omega-deformation

String Amplitudes, Topological Strings and the Omega-deformation Ahmad Zein Assi CERN Strings @ Princeton 26 - 06 - 2014 Based on work with I. Antoniadis I. Florakis S. Hohenegger K. S. Narain 1302.6993 [hep-th] 1309.6688 [hep-th] 1406.xxxx [hep-th]

Resurgent Transseries and the Holomorphic Anomaly

holomorphic anomaly equations and sets the ground to start addressing large-order analysis and resurgent nonperturbative completions within closed topological string theory. Contents 1. Introduction and Summary 332 2. The Holomorphic Anomaly Equations 337 2.1. Reviewing the Background Calabi Yau Geometry 337

Orbifold Gromov-Witten Invariants and Topological Strings

special geometry and the holomorphic anomaly equations of [7] are available. Our goal is to pursue this line of thought in the context of enumerative geom-etry of orbifolds. Since direct evaluation of the Hodge integrals entering into the definition of orbifold Gromov-Witten invariants is rather complex (see for instance

Topological String Theory

The holomorphic anomaly equations Interptetation: (1) On each tangent space, there is a Hilbert space. (2) The holomorphic anomaly equation is describing the paralell transport between tangent spaces at different points. More on (1): More on (2): has a symplectic structure. Topological string uses a holomorphic polarization. The polarization

Topological Strings on elliptic Calabi-Yau three-folds Minxin

Topological strings:AN =(2,2) supersymmetric non-linear sigma model from world sheet⌃to target space X. i:⌃! X Topological string theory is the most interesting and free of world sheet anomaly, when the target space X is a Calabi-Yau 3-fold. There are two types of topological twisting: A-model and B-model.

Topological Strings, Resurgence and Quantum Mechanics

oscillators can be addressed in the context of topological string theory. For example, Voros symbols can be calculated with the holomorphic anomaly equation. Conclusions The intermediate stage between topological strings and ordinary quantum mechanics is the quantum SW curve, which

Topological Strings and Their Diverse Applications

-Define topological strings worldsheet vs. target closed vs. open A vs. B regular vs. refined -Computational techniques: holomorphy and holomorphic anomaly large N-dualities: Chern-Simons Topological Vertex Matrix Models Topological recursion -BPS content of topological strings D=5 spinning black holes


Chapter 31. Topological Strings 585 31.1. Quantum Field Theory of Topological Strings 585 31.2. Holomorphic Anomaly 593 Chapter 32. Topological Strings and Target Space Physics 599 32.1. Aspects of Target Space Physics 599 32.2. Target Space Interpretation of Topological String Amplitudes601 32.3. Counting of D-branes and Topological String

Resurgent Transseries and the Holomorphic Anomaly: CP

Abstract: The holomorphic anomaly equations describe B-model closed topological strings in Calabi{Yau geometries. Having been used to construct perturbative expansions, it was re-

Holomorphic anomaly and quantum mechanics

string free energies at all genera [18]. Since topological strings are closely related to 17, many other physical systems, the holomorphic anomaly equations have provided very power - ful tools to study a variety of problems. For example, the large N expansion of matrix models is also governed by the holomorphic anomaly [1719, 20].

All loop N = 2 string amplitudes The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Ooguri, Hiros

The holomorphic anomaly for open string moduli

4. The open moduli holomorphic anomaly 7 5. Closed moduli in presence of Wilson lines 11 6. Open issues 13 1. Introduction The holomorphic anomaly equations [5] are a most powerful tool which potentially allows for the complete solution of topological string theories [30], once complemented with suit-able methods to fix the holomorphic

Nonperturbative Topological Strings

of the holomorphic anomaly of the B model: quantization of H3(X,R). [Witten] Passing to Ω P,Q gets rid of the holomorphic anomaly the number of black holes with charge P +Q is a canonically defined object which is independent of the choice of basis. ( Wigner function )

Holomorphic Anomaly Equations in Topological String Theory

we provide a derivation of the holomorphic anomaly equation for open strings and study aspects of the Ooguri, Strominger, and Vafa conjecture. Topological string theory is a computable theory. The amplitudes of the closed topological string satisfy a holomorphic anomaly equation, which is a recursive dif-ferential equation.

The Holomorphic Anomaly of Topological Strings

The Holomorphic Anomaly of Topological Strings Carlo Becchi, Stefano Giusto 1, Camillo Imbimbo 2 1 Dipartimento di Fisica dell Universit a di Genova Via Dodecaneso 33, I-16146 Genova 2 I.N.F.N. Sezione di Genova Via Dodecaneso 33, I-16146 Genova Abstract: We show that the BRS operator of the topological string B model is not holomorphic in the