Quantum mirror curve of periodic chain geometry
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Springer
Received: October 28, 2018 Accepted: April 11, 2019 Published: April 24, 2019
Taro Kimuraa and Yuji Sugimotob a
Department of Physics, Keio University, Kanagawa 223-8521, Japan b Osaka City University Advanced Mathematical Institute (OCAMI), 3-3-138, Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan
E-mail: [email protected], [email protected] Abstract: The mirror curves enable us to study B-model topological strings on noncompact toric Calabi-Yau threefolds. One of the method to obtain the mirror curves is to calculate the partition function of the topological string with a single brane. In this paper, we discuss two types of geometries: one is the chain of N P1 ’s which we call “N chain geometry,” the other is the chain of N P1 ’s with a compactification which we call “periodic N -chain geometry.” We calculate the partition functions of the open topological strings on these geometries, and obtain the mirror curves and their quantization, which is characterized by (elliptic) hypergeometric difference operator. We also find a relation between the periodic chain and ∞-chain geometries, which implies a possible connection between 5d and 6d gauge theories in the larte N limit. Keywords: Topological Strings, String Duality, Supersymmetric Gauge Theory ArXiv ePrint: 1810.01885
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP04(2019)147
JHEP04(2019)147
Quantum mirror curve of periodic chain geometry
Contents 1 Introduction and summary
1
2 Chain geometries and its compactification 2.1 Chain geometries 2.2 Compactification
3 3 7 11 12 14 17
4 Discussion
20
A Definitions and notations A.1 Mathematical preliminaries A.2 Topological vertex A.3 Vertex operators
21 21 22 22
1
Introduction and summary
Topological string introduced in [1] is known as a toy model to understand string theory compactified on Calabi-Yau threefolds. Two formulations, the A-model topological string which depends on the K¨ ahler moduli, and the B-model topological string depending on the complex modulus, are related to each other via mirror symmetry [2]. For the noncompact toric Calabi-Yau threefolds, the gravity sector is decoupled, and it ends up with gauge theory with supersymmetry: the A-model topological string computes the partition function of 5d N = 1 supersymmetric gauge theories compactified on a circle with the self-dual omega background [3]. In the B-model topological strings, the partition function is closely related to the entropy of the supersymmetric black hole [4]. The B-model topological sring is encoded to the complex 1-dimensional manifold in the toric Calabi-Yau threefold Σ = {(x, p) ∈ C∗ × C∗ | H(x, p) = 0} where x, p ∈ C∗ = C\{0}, the so-called “mirror curve” [5]. Recently, it has been shown that quantization of the mirror curves introduced in [5, 6] provides the non-perturbative effects in the B-model topological strings on the toric CalabiYau threefolds [7]. This formalism is now known to be relevant to ABJM theory [8, 9], integrable system [10, 11], and ev
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