双曲几何谱函数与量子同源不变性：Weyl与正交群的应用

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本文深入探讨了S函数、双曲几何的谱函数以及顶点算子在量子同源不变性领域的应用，特别关注的是slN链同源性的Poincaré多项式。量子同调学是现代数学物理的一个重要分支，它结合了代数拓扑和量子场论，用于研究拓扑空间的量子性质。在计算过程中，当特定拓扑空间的同调维度明确时，Kovanov-Rozansky型同源性的计算可以通过Euler-Poincaré公式得到有效简化，这极大地提高了效率。 S函数和谱函数在双曲几何中扮演着关键角色，它们反映了这些几何结构的内在特性。作者利用这些函数来表达经典群，如Weyl群和正交群的不可约张量表示的特征，这在理论物理中具有重要意义，尤其是在理解弦理论和M理论中的对称性方面。具体来说，Labastida-Mariño-Ooguri-Vafa猜想指出，Chern-Simons分区函数可以以Ruelle谱函数的形式表示，这种函数不仅适用于无结情况，也包括了结和链结的复杂情形，其形式呈现为无限积。对于正交的Chern-Simons理论，文中提出了一个关于该理论的无限积公式，这不仅展示了理论的精确结构，而且揭示了其内在的奇异性和对称性。这个公式可能是通过分析Chern-Simons理论在不同边界条件下的行为得出的，它可能与拓扑量子场论中的某些关键物理现象直接相关。整个研究工作是由A.A.Bytsenko和M.Chaichian两位学者完成的，他们分别来自巴西利亚州立大学和赫尔辛基大学，他们的合作反映了国际科研合作在解决复杂物理问题上的价值。值得一提的是，该论文于2016年3月发表在《核物理学B》杂志上，编辑为Hubert Saleur，还特别献给了已故的朋友和同事Petr P. Kulish。这篇文章不仅深化了我们对量子同源不变性及经典群表示的理解，而且为Chern-Simons理论的谱函数形式提供了新的洞察，对于推动理论物理和数学交叉领域的研究具有重要意义。

A.A. Bytsenko, M. Chaichian / Nuclear Physics B 907 (2016) 258–285 263

,...,R



(q, t) :=



k,j∈Z

dim H

,...,R



k,j

(L). (2.4)

By deﬁnition, it is a polynomial in q

±1

and t

±1

with integer non-negative coefﬁcients. In addi-

tion, evaluating (2.4) at t =−1gives (2.3). In the case R

= 2 for all a = 1, ..., , the homology

;2,...,2

k,j

(L) is known as the Khovanov–Rozansky homology,

(KR)

k,j

(L). The further phys-

ical interpretation of homological link invariants via Hilbert spaces of BPS states leads to certain

predictions regarding the behavior of link homologies with rank N (for more discussion see

[35,37]).

In the case when dimensions of homologies of appropriate topological spaces are precisely

known the procedure of the calculation can be appreciably simpliﬁed. Such a situation hap-

pends, for example, if asymptotic behavior the total dimension of H

,··· ,R



∗,∗

(L) grows as

[34]: dimH

,··· ,R



∗,∗

(L)|

N→∞

→ N

, d =





j=1

dim R

3. Hopf algebraic approach and group theory

Usually the Euler–Poincaré formula applies to chain complexes of ﬁnite dimensional Lie

algebras. In the inﬁnite dimensional case matters can be ﬁxed up by considering polygraded

Lie algebras. Thus the partition functions can indeed be converted into product expressions. The

expression on the right-hand side of (2.1) looks like counting the states in the Hilbert space of a

second quantized theory

. Certain formulas for the partition functions or Poincaré polynomials



,...,n



1 − q

···q



dim g

,...,n



,...,n



1 − q

···q



rank g

,...,n

(3.1)

are associated with dimensions of homologies of appropriate topological spaces and linked to

generating functions and elliptic genera. Note that this conclusively explains the sequence of

dimensions (distinguished powers) of the simple Lie algebras.

Before e

xamine a ring of formal power series (of type (3.1)) we provide a short discussion

on the theory of a higher-weight modules over Lie algebra. We start with very well known Lie

algebra gl(n, k).

Results for gl(n, k) survive the passage to the limit n →∞, if one assumes that

gl(∞, k) is the Lie algebra of inﬁnite ﬁnitary matrices, which means



gl(n, k). In this remark

we deal with the Lie algebra gl



(k) of generalized Jacobian matrices.

The algebra gl



(k) can

be considered as a nontrivial one-dimensional central extension of the Lie algebra gl

(k) (for

details, see [38]). It is obvious that gl

(k) ⊃ gl

(∞, k). The importance of this Lie algebra

stems from the following facts:

• Man

y of the classical constructions of the theory of representations of the Lie algebra gl

(k)

can be also applied to the algebra gl



(k). This creates a sizable supply of gl



(k)-modules.

• Important inﬁnite-dimensional Lie algebras can be embedded in gl



(k). Thus, the already

mentioned representations of gl



(k) become representations of these algebras.

The symbol k denotes the ﬁeld of real numbers R or the ﬁeld C of complex numbers. In particular, gl(n, C) is the

Lie algebra of all complex n × n matrices with the operation A, B → [A, B] = AB − BA.

The bilateral matrix a



i,j∈Z

is called a generalized Jacobian matrix if it has a ﬁnite number of nonzero diagonals

(that is, if there exists a positive N such that a

= 0for |j − i| >N). It is clear that the set of generalized Jacobian

matrices constitutes a Lie algebra, with respect to the usual commutation rule.

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