拓扑字符串与颤动：条带几何的深刻联系

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"这篇文章探讨了拓扑字符串理论与条带几何和颤动表示理论之间的紧密联系，特别是在一类没有紧凑四环的复曲面Calabi-Yau流形上。作者揭示了开放拓扑字符串理论的量，如分区函数、Gromov-Witten不变式和开放BPS不变式，可以通过相应颤动表示的模空间特征来表示。这一发现不仅证明了动机Donaldson-Thomas不变量的完整性，而且为任意带状几何体的经典开放式BPS不变量提供了明确的表达式，产生了丰富的数论完整性陈述。此外，特定情况下的开放拓扑字符串分区函数以广义q超几何函数的形式出现，给出了这些函数在量子对数和积分不变量方面的新表述。研究还涉及与颤动相关的量子曲线和A多项式，及其各种限制和条带几何的特殊化。复曲面与颤动的关联可视为更一般Calabi-Yau几何的结颤动对应关系的推广。" 这篇学术论文详细阐述了拓扑弦理论在特定数学结构中的应用，主要关注的是那些没有紧凑四环的复曲面Calabi-Yau流形，也被称为条带几何。拓扑弦理论是一种量子场论，它研究的是在多维空间中的闭合和开口弦的动力学，这些理论在理论物理和数学中都有重要应用。文章的核心发现是，开放拓扑字符串理论的性质可以与颤动表示理论相结合，颤动表示是代数几何中的一个重要概念，涉及到向量空间的线性组合。作者通过这种方式，建立了开放BPS不变量与动机Donaldson-Thomas不变量之间的联系，前者是研究弦理论中稳定对象的数量，后者是研究三维 Calabi-Yau 流形上某些类别的计数问题。这种联系的发现提供了一种直接证明动机Donaldson-Thomas不变量完整性的方法，即它们完全列举了所考虑的几何体上的稳定对象。此外，论文还讨论了带状几何体中的开放式BPS不变量，这些不变量是描述在有界区域内的弦的行为的关键。作者给出了这些不变量的显式表达式，这导致了一系列的数论性质。特别是，当开放拓扑字符串的分区函数在特定条件下时，它可以被表示为广义q超几何函数，这是一种复杂的数学函数，它允许对字符串理论的计算进行更深入的理解。进一步的研究集中在与颤动相关的量子曲线和A多项式上，这些都是理解弦理论动力学的重要工具。通过对这些对象的限制和特殊化，可以探索弦理论在不同几何背景下的行为。最后，作者指出，复曲面流形与颤动的关联是对更广泛的Calabi-Yau几何中结颤动对应原理的一种推广，这表明了拓扑弦理论的普适性和数学深度。这篇论文展示了拓扑弦理论如何与复几何、代数几何以及数论的多个分支交织在一起，为理论物理和数学领域的交叉研究提供了新的视角和工具。

JHEP01(2019)124

properties of BPS invariants, which follow from the relation to quiver representation the-

ory. Finally, in section 5 we consider several examples of strip geometries, and illustrate in

such examples various structures introduced earlier. In appendix A we discuss various con-

ventions related to the deﬁnition of motivic Donaldson-Thomas invariants and positivity

of these invariants.

2 Topological string theory, strip geometries, and brane amplitudes

Topological string amplitudes count, in appropriate sense, maps from Riemann surfaces into

a target space. Open topological string amplitudes count maps from Riemann surfaces with

boundaries, and the boundary conditions may be encoded by appropriately chosen branes.

In this paper we consider A-model (holomorphic) amplitudes for target spaces which are

toric Calabi-Yau threefolds that do not contain compact four-cycles; such manifolds are

referred to as strip geometries or generalized conifolds. In this section we ﬁrst brieﬂy

summarize the general structure of A-model amplitudes, as well as the topological vertex

formalism and its simpliﬁcations that arise for strip geometries. We then compute the

open partition function for the Aganagic-Vafa lagrangian brane in arbitrary framing and in

arbitrary strip geometry. This result is given in (2.23) and it will be of our main interest in

what follows. We also determine the quantum mirror curve operator (2.27) that annihilates

this brane partition function, identify the mirror curve that arises in the classical limit of

this operator (2.29), and ﬁnd the diﬀerential operator (2.32) that arises in the modiﬁed

classical limit. We also discuss the relation of partition functions (2.23) and the equations

they satisfy to generalized hypergeometric functions and hypergeometric equations.

2.1 Topological string amplitudes and BPS invariants

A-model topological string amplitudes depend on K¨ahler parameters Q = {Q

} of a given

target Calabi-Yau manifold M, and open moduli x = {x

} that characterize branes. They

are deﬁned in terms of the genus expansion in the topological string coupling ~, and var-

ious terms in such expansion encode closed or open Gromov-Witten invariants. The full

topological string amplitudes factorize into closed string contributions and — in presence

of branes — open contributions, that involve both open and closed moduli

Z = Z

closed

(Q) · ψ

open

(Q, x). (2.1)

From the spacetime interpretation of topological strings [36, 37] it follows that topological

string amplitudes can be expressed in a product form that represents counting of BPS

states, in terms of the variable q = e

. In particular closed string contributions take form

closed

(Q) =

β∈H

(M)

∞

l=1

(1 − Q

l+j

)

β,j

, (2.2)

where N

are conjecturally integer Gopakumar-Vafa invariants that count BPS states of

closed M2-branes. Note that for ﬁxed β and m, the contribution from the product over

l is a generalization of the MacMahon function M(q) =

∞

l=1

(1 − q

)

that counts plane

partitions.

– 7 –

JHEP01(2019)124

It is known that in certain systems open partition functions satisfy Schr¨odinger-like

equations, hence they are also referred to as wave-functions, and for this reason we denote

them by the symbol ψ

open

(Q, x). Spacetime interpretation of BPS counting implies that

in presence of branes open topological string amplitudes also have product decomposition.

First, as argued in [5–7], the open partition function can be written in the form

open

(Q, x) =

open

X = exp

∞

n=1

, q

)Tr

, (2.3)

where we encoded brane moduli x = {x

} in a matrix X = diag(x

, x

, . . .), Young diagrams

P under summations represent brane boundary conditions, and

(Q, q) =

β,j

P,β,j

1/2

− q

−1/2

(2.4)

are functions that encode integer multiplicities N

P,β,j

of open M2-branes in a relative class

β, with spacetime spin j, and labeled by P . Multiplicities N

P,β,j

are referred to as Ooguri-

Vafa invariants and they provide an interesting reformulation of open Gromov-Witten

invariants. In the context of knots N

P,β,j

are also referred to as Labastida-Mari˜no-Ooguri-

Vafa (LMOV) invariants, and ψ

open

are related to colored HOMFLY-PT polynomials [5–7].

Taking advantage of the relation

, (2.5)

where k

= {k

} and m

are respectively weights of the representation P and their

multiplicities, the open partition function (2.3) can be written in the product form

open

(Q, x) =

P,j,β,k

∞

l=1

(1 − x

l+j−1/2

)

P,β,j

. (2.6)

Note that for ﬁxed P, j, β, k

, the product over l represents the quantum dilogarithm (with

appropriate arguments), which can also be written as a special case (with n = ∞) of the

q-Pochhammer symbol

(Q; q)

n−1

i=0

(1 − Qq

). (2.7)

More precisely, a single trace Tr

X in (2.3) represents one stack of branes; for multiple

stacks the open amplitude would in general take form

open

(Q, x) =

}

open

}

. (2.8)

It is also convenient to write the total amplitude (2.1), including both closed and open

contributions, in the form

Z =

}

, Z

}

= Z

closed

· ψ

open

}

(2.9)

– 8 –

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