量子弦理论与双曲几何的谱函数研究

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"这篇文章是关于双曲几何在量子弦理论中的应用，特别是在拓扑顶点、弦幅值和谱函数方面。作者M.E.X.Guimarães, R.M.Luna和T.O.Rosa深入探讨了量子串生成函数与形式幂级数之间的同态关系，这些幂级数与李代数的链的尺寸和同调性相关。文章指出，这种分析方法可以直接应用于拉格朗日谱的划分函数、精制顶点和开放字符串划分函数，其中涉及到SL(2, Z)同余子群中的模形式和谱函数。此外，还提到了与q系列相关的双曲三几何的Selberg型谱函数在弦理论计算中的重要性。文章在2014年被Eur.Phys.J.C接收，并以Open Access的形式在Springerlink.com发布。" 在本文中，作者首先介绍了量子弦理论的基本概念，这是一个研究基本粒子和它们相互作用的理论，它结合了量子力学和广义相对论的元素。量子弦生成函数是该理论中的核心工具，它描述了弦的不同振动模式及其对应的能量状态。这些生成函数可以通过形式幂级数来表示，这些幂级数与链的维度和李代数的同调性有直接联系。接着，文章聚焦于拓扑顶点的概念，这是弦理论中的一个重要构造，它允许计算多弦散射过程的幅度。拓扑顶点提供了一种简化复杂计算的方法，尤其在考虑超对称或精制的设置时。作者使用了模形式和SL(2, Z)的同余子群来处理这些计算，这使得结果具有模性质，即在特定变换下保持不变。弦幅值是弦理论中的另一个关键量，它表示弦之间相互作用的强度。在双曲几何背景下，弦幅值与双曲三几何的Selberg型谱函数紧密相关。Selberg谱函数在数学上是一个重要的工具，特别是在组合优化和复分析领域，而在物理中，它与双曲空间的几何特性及量子弦的振动态有关。此外，开放字符串的划分函数也在这篇文章中被讨论。这些函数描述了开放弦在特定边界条件下的所有可能状态。通过形式幂级数，作者展示了如何编码李代数的性质来理解和计算这些函数。这篇论文通过深入分析双曲几何的数学结构，为理解和计算量子弦理论中的各种物理量提供了新的视角和方法。这种跨学科的研究方式，结合了数学的精深理论和物理学的实际问题，展现了理论物理学研究的深度和广度。

Eur. Phys. J. C (2014) 74:2880

DOI 10.1140/epjc/s10052-014-2880-9

Regular Article - Theoretical Physics

Topological vertex, string amplitudes and spectral functions

of hyperbolic geometry

M. E. X. Guimarães

1,a

,R.M.Luna

2,b

,T.O.Rosa

1,c

Instituto de Física, Universidade Federal Fluminense, Av. Gal. Milton Tavares de Souza, s/n, CEP 24210-346, Niterói, RJ, Brazil

Departamento de Física, Universidade Estadual de Londrina, Caixa Postal 6001, Londrina, Paraná, Brazil

Received: 28 March 2014 / Accepted: 28 April 2014 / Published online: 20 May 2014

Abstract We discuss the homological aspects of the con-

nection between quantum string generating function and the

formal power series associated to the dimensions of chains

and homologies of suitable Lie algebras. Our analysis can

be considered as a new straightforward application of the

machinery of modular forms and spectral functions (with

values in the congruence subgroup of SL(2, Z)) to the parti-

tion functions of Lagrangian branes, r eﬁned vertex and open

string partition functions, represented by means of formal

power series that encode Lie algebra properties. The com-

mon feature in our examples lies in the modular properties

of the characters of certain representations of the pertinent

afﬁne Lie algebras and in the role of Selberg-type spectral

functions of a hyperbolic three-geometry associated with q-

series in the computation of the string amplitudes.

1 Introduction

In this work, we deal with some applications of the modular

forms ( and spectral functions related to the congruence sub-

group of SL(2, Z)) to topological vertex and string generat-

ing functions. For mathematicians topological vertices (and

their respective string generating functions) may be associ-

ated to new mathematical invariants for spaces, while, for

physicists, they are related to quantum string partition func-

tions.

Having made these general remarks, let us now explain the

connection between the speciﬁc contents of the various sec-

tions in more detail. In Sect. 2 we will explore a remarkable

connection between Poincaré polynomials (generating func-

tions) and formal power s eries associated with dimensions of

chains and homologies of Lie algebras (Euler–Poincaré for-

e-mail: emilia@if.uff.br

e-mail: luna@uel.br

e-mail: t18rosa@if.uff.br

mula). From a concrete point of view this paper consists of

applications of spectral functions to quantum string partition

functions connected to suitable Lie algebras.

We show that the Poincaré polynomials (Sect. 3) and the

associated topological vertex and string amplitude can be

converted into product expressions which inherit cohomo-

logical properties (in the sense of characteristic classes of

foliations [1]) of appropriate polygraded Lie algebras.

The ﬁnal result for a single Lagrangian brane, stack of

branes (Sect. 4), reﬁned vertex, and the partition functions for

the case of Calabi–Yau threefolds O(−1) ⊕ O(−1) → P

and O(0) ⊕ O(−2) → P

(Sect. 5) is written in terms of

spectral functions of the hyperbolic three-geometry associ-

ated with q-series.

2 Graded algebras and spectral functions of hyperbolic

geometry

Before considering topological vertex and string amplitudes

we would like to spend some time on the relation between

formal power q-series and homologies of Lie algebras. We

would like to show how combinatorial identities could be

derived from the initial complex of (graded) Lie algebras.

Our interest is the Euler–Poincaré formula associated with a

complex consisting of linear spaces. The relations between

Lie algebras and combinatorial identities was ﬁrst discov-

ered by Macdonald; the Euler–Poincaré formula is useful for

combinatorial identities known as Macdonald identities. The

Macdonald identities are related to Lie algebras in one way

or another and can be associated with generating functions

in quantum theory.

Let g be a Lie algebra, and assume that it has a grad-

ing, i.e. g is a direct sum of homogeneous components g

(λ)

where the λ’s are elements of an abelian group, [g

(λ)

, g

(μ)

]⊂

(λ+μ)

. Let us consider a module k over g,org-module,

which is a left module over the universal enveloping algebra

123

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