Topological Strings and (Almost) Modular Forms

By Mina Aganagic, V. Bouchard, A. Klemm. 2006

281 citations recorded by the source corpus.

The B-model topological string theory on a Calabi-Yau threefold X has a symmetry group Γ, generated by monodromies of the periods of X. This acts on the topological string wave function in a natural way, governed by the quantum mechanics of the phase space H3(X). We show that, depending on the choice of polarization, the genus g topological string amplitude is either a holomorphic quasi-modular form or an almost holomorphic modular form of weight 0 under Γ. Moreover, at each genus, certain combinations of genus g amplitudes are both modular and holomorphic. We illustrate this for the local Calabi-Yau manifolds giving rise to Seiberg-Witten gauge theories in four dimensions and local IP2 and IP1 × IP1. As a byproduct, we also obtain a simple way of relating the topological string amplitudes near different points in the moduli space, which we use to give predictions for Gromov-Witten invariants of the orbifold $${{\mathbb {C}^3} / {\mathbb {Z}_3}}$$.

Subfields: Geometry and Topology, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/5f5f7c5179ce7e47c194df6ade8e23d7c62f0713

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Citations in the corpus

Localization techniques in quantum field theories (2016, methodology): “If the map π factors through a multiple covering S2 → S2 of degree m, then the homology class of the image takes the form η = mφ but, as argued in [82] in dimension three and [83] in arbitrary dimension (see also [84]), the count of maps is the same.”
ABJM theory as a Fermi gas (2011, background): “This property was spelled out in detail and exploited in [32].”
Topological Strings from Quantum Mechanics (2014, background): “48) The advantage of this formulation is that all the ingredients have now an analytic continuation to the orbifold point z =∞, and one finds (see for example [64, 65]):”
The Ω Deformed B-model for Rigid N = 2 Theories (2011, methodology): “In particular, the arguments of [42] that the change from real to holomorphic polarisation of the wave function corresponds to the change from quasimodular to almost holomorphic functions extend to the refined case, consistent with the modular invariant form of the F (n,g) in terms of Ê2 in holomorp”
Direct integration of the topological string (2007, methodology): “This has been also studied in [1], see also [27].”
The Crepant Resolution Conjecture (2006, unknown): “A G ] 7 J an 2”
Invariants of algebraic curves and topological expansion (2007, background): “(4-18) Remark 4.1 The special functions, except F (1) and F (0), are obtained by changing B and dS by B and dS in the diagrammatic rules defined in section 4.5.”
Open string amplitudes and large order behavior in topological string theory (2006, methodology): “Following the philosophy of [23, 40, 1], one can obtain a modular invariant object by adding an antiholomorphic piece in the closed string moduli,”
Asymptotics for rank and crank moments (2009, background): “These types of q-series arise quite frequently in mathematical physics in the study of topological Yangs-Mills from string theory [2, 23].”
Holomorphic anomaly in gauge theories and matrix models (2006, background): “Using this picture the failure of holomorphicity can be traded against a failure of modularity with a similar iteration [50], which makes the connection more naturally.”
Fivebranes and 3-manifold homology (2016, methodology): “The relation between asymptotic expansions in gs then should be given by a Fourier-like transform [101] (which follows from [102, 103]):”
The space of stability conditions on the local projective plane (2009, background): “[AGM94, DG00]; our presentation follows [ABK08] and [Asp05] most closely.”
Exact results in ABJM theory from topological strings (2009, methodology): “There, the interpolation from weak to strong coupling involves a different choice of flat coordinates and a different choice of duality frame (see for example [32] and specially the analysis of Wilson loop amplitudes in [18]).”
From Weak to Strong Coupling in ABJM Theory (2010, methodology): “As noted in [41], we can use the relation between the local F0 theory and Seiberg– Witten theory to write all the quantities in the model in terms of modular forms.”
Non-perturbative effects and the refined topological string (2013, methodology): “The grand potential J(μ) is related to this partition function by an inverse Legendre transform, and by using the results of [34] it is easy to see [8] that it is essentially given by the topological string free energy in the large radius frame.”
Nonperturbative aspects of ABJM theory (2011, background): “The different F (f) g are related by symplectic transformations and they transform as quasi-modular forms [28].”
Landau–Ginzburg/Calabi–Yau correspondence for quintic three-folds via symplectic transformations (2008, methodology): “y by A. Zinger after a great deal of hard work [Zi08]. Computations in higher genera are out of mathematicians’ reach for the moment. There is a physical method put forward by Aganagic–Bouchard–Klemm [ABK08]; their best record is g<53. In this current unsatisfactory state of aﬀairs, a natural ide”
Remodeling the B-Model (2007, methodology): “More precisely, the resulting amplitudes admit a non-holomorphic extension fixed by modular invariance (as in [1]) which satisfies the holomorphic anomaly equations of [12] in the local case.”