Chern–Simons invariants on hyperbolic manifolds and topological quantum field theories
 354 Downloads
Abstract
We derive formulas for the classical Chern–Simons invariant of irreducible SU(n)flat connections on negatively curved locally symmetric threemanifolds. We determine the condition for which the theory remains consistent (with basic physical principles). We show that a connection between holomorphic values of Selbergtype functions at point zero, associated with Rtorsion of the flat bundle, and twisted Dirac operators acting on negatively curved manifolds, can be interpreted by means of the Chern–Simons invariant. On the basis of the Labastida–Mariño–Ooguri–Vafa conjecture we analyze a representation of the Chern–Simons quantum partition function (as a generating series of quantum group invariants) in the form of an infinite product weighted by Sfunctions and Selbergtype functions. We consider the case of links and a knot and use the Rogers approach to discover certain symmetry and modular form identities.
Keywords
Dirac Operator Chern Character Adiabatic Limit Simons Theory Flat Connection1 Introduction
The Chern–Simons theory is one of the two archetypal field theories in physics, together with Yang–Mills theory, that describe the interaction of gauge fields. In 3D, the number of dimensions we wish to consider in this paper, the dynamics is so constrained as to leave room only for nondynamical (topological) correlators. The corresponding topological quantum field theory was defined and developed by Witten [1] and Reshetikhin and Turaev [2], and it was applied to the mathematical theory of knots and links in threedimensional manifolds. From a field theory point of view, the observables of the Chern–Simons theory are correlators of Wilson lines (beside the partition function). In this paper we will analyze some properties of the latter.
A wellknown characteristic of the Chern–Simons path integral is that its welldefiniteness is connected to the properties of a topological invariant in fourdimensional manifolds. Such invariant is related to the Chern–Simons form via the transgression formula. The invariant of a fourmanifold in the topological field theory involves its signature and Euler characteristic (see for example [3]). The Chern character allows one to map the analytical Dirac index in terms of Ktheory classes into a topological index, which can be expressed in terms of cohomological characteristic classes. This results in a connection between the Chern–Simons action and the Atiyah–Singer index theorem. Such a connection will be used in this paper in order to determine the Chern–Simons invariant of irreducible SU(n)flat connections on negatively curved locally symmetric threemanifolds. Indeed a critical point of the Chern–Simons functional is just a flat connection; it corresponds to a representation of the fundamental group \(\pi _1(\textsf {X})\) associated to a threemanifold \(\textsf {X}\). The value of the Chern–Simons functional at a critical point can be regarded as a topological invariant of a pair \((\textsf {X}, \rho )\), where \(\rho \) is a representation of \(\pi _1(\textsf {X})\). Due to a wellknown adiabatic argument, knowing these invariants allows us to compute the partition function.
On the other hand, the Chern–Simons partition function is a generating series of quantum group invariants weighted by Sfunctions. Recall that the Chern–Simons theory has been conjectured to be equivalent to a topological string theory 1 / N expansion in physics. The Chern–Simons/topological string duality conjecture identifies the generating function of Gromov–Witten invariants as Chern–Simons knot invariants [4]. The existence of a sequence of integer invariants is conjectured [4, 5] in a similar spirit to the Gopakumar–Vafa setting [6]. This provides essential evidence of the duality between Chern–Simons theory and topological string theory. Such an integrality conjecture is called the LMOV conjecture. In the context of this conjecture we derive a new representation of the Chern–Simons quantum partition function in the form of an infinite product in terms of Selbergtype functions.
Deeply related with the content of this paper is the problem of anomalies. In field theory anomalies may prevent the path integral from being well defined. In the case of CS in threedimensional manifolds there are no local anomalies, but there may be global anomalies. To guarantee their absence one must restrict to integer values the (suitably normalized) coupling appearing in front of the action. For this reason it is of utmost importance to know the value of the Chern–Simons invariant in any given space–time. Strictly connected with this is the issue of existence of fermionic path integrals (fermion determinants) in threedimensional manifolds. There are also other indeterminacies in this theory when links and knots are involved, related to the evaluation of overlapping Wilson loops. The problem of such framing anomalies was pointed out and solved by Witten [7].

In Sect. 2.1 we derive the formula for the Chern–Simons invariant of irreducible SU(n)flat connections on a locally symmetric manifold of nonpositive sectional curvature. For this Chern–Simons invariant our result, Eq. (2.19), determines the condition for which the quantum field theory is consistent. The results of Sect. 2.1 are preparatory for the generalization of the Chern–Simons invariant to the case of other manifolds (\(X = S^3/\Gamma \), for example) and of nontrivial U(n)bundle over X (Sect. 2.4).

In Sect. 2.4 \(\textsf {X}= \Gamma \backslash \overline{\textsf {X}}\) with \(\overline{\textsf {X}}\) is a globally symmetric space of noncompact type and \(\Gamma \) a discrete, torsionfree, cocompact subgroup of orientationpreserving isometries. \(\textsf {X}\) inherits a locally symmetric Riemannian metric g of nonpositive sectional curvature. If \({{\mathfrak D}}: C^{\infty }(\textsf {X}, V)\rightarrow C^{\infty }(\textsf {X},V)\) is a differential operator acting on the sections of the vector bundle V, then \({\mathfrak D}\) can be extended canonically to a differential operator \({\mathfrak D}_{\varphi }: C^{\infty }(\textsf {X},V\otimes F)\rightarrow C^{\infty }(\textsf {X},V\otimes {F})\), uniquely characterized by the property that \({\mathfrak D}_{\varphi }\) is locally isomorphic to \({\mathfrak D}\otimes \cdots \otimes {\mathfrak D}\) (\(\mathrm{dim}\,{F}\) times) [8]. We show that a connection between holomorphic values of Selbergtype functions at point zero, associated with Rtorsion of the flat bundle, and twisted Dirac operators \({\mathfrak D}_\varphi \) on negatively curved locally symmetric spaces, can be interpreted by means of the Chern–Simons invariant. This leads to our main result, Eq. (2.18). We also briefly describe the possibility to derive the Chern–Simons invariant for locally symmetric spaces of higher rank in terms of the spectral function \({{\mathcal R}}(s; \varphi )\).

The quantum \(\mathfrak {sl}_N\) invariant in the case of links and a knot is analyzed in Sect. 3. On the basis of LMOV conjecture we derive a new representation of the Chern–Simons quantum partition function in the form of an infinite product in terms of Selbergtype functions. In addition, we discuss the symmetry and modular form properties of infiniteproduct formulas.
2 Chern–Simons invariants for negatively curved manifolds
2.1 Flat connections and gauge bundles
Flat connections on fibered hyperbolic manifolds The Chern character allows one to map the analytical Dirac index in terms of Ktheory classes into a topological index which can be expressed in terms of cohomological characteristic classes. This results in a connection between the Chern–Simons action and the celebrated Atiyah–Singer index theorem. The goal of this section is to use this fact in order to present explicit formulas for the Chern classes and gauge Chern–Simons invariant of an irreducible SU(n)flat connection on real compact hyperbolic threemanifolds.
Let \({P}=\textsf {X}\times {G}\) be a trivial principal bundle over \(\textsf {X}\) with the gauge group \({G}=SU(n)\) and let \(\Omega ^1(\textsf {X};{\mathfrak g})\) be the space of all connections on P; this space is an affine space of oneforms on \(\textsf {X}\) with values in the Lie algebra \({\mathfrak g}\) of G. Let \({\mathcal A}_{\textsf {\textsf {X}}}= \Omega ^1(\textsf {X};{\mathfrak g})\) be the space of connections and \({\mathcal A}_{\textsf {\textsf {X}}, F}=\{A\in {\mathcal A}_{\textsf {X}}F_A= \mathrm{d}A+A\wedge A=0\}\) be the space of flat connections on P. Then the gauge transformation group \({\mathcal G}_{\textsf {X}}\cong C^\infty (\textsf {X}, G)\) acts on \({\mathcal A}_{\textsf {X}}\) via pullback: \({\mathfrak g}^*A = {\mathfrak g}^{1}A{\mathfrak g} + {\mathfrak g}^{1}d{\mathfrak g}\), \({\mathfrak g}\in {\mathcal G}_{\textsf {X}}\), \(A\in {\mathcal A}_{\textsf {X}}\). This action preserves \({\mathcal A}_{\textsf {\textsf {X}},F}\).
\(\eta \) is a spectral invariant which measures the symmetry of the spectrum of an operator \({\mathfrak D}\), and admits a meromorphic extension to the whole splane, with at most simple poles at \((\mathrm{dim} \textsf {X}k)/(\mathrm{ord}\, {\mathfrak D})\) (\(k=0,1,2,\ldots \)) and locally computable residues. For \(\textsf {X}\) a compact oriented (4n1)dimensional Riemannian manifold of constant negative curvature, a remarkable formula relating \(\eta \) to the closed geodesics on \(\textsf {X}\) has been proved in [13]. Citing [8], the appropriate class of Riemannian manifolds for which a result of this type can be expected is that of nonpositively curved locally symmetric manifolds, while the class of selfadjoint operators whose eta invariants are interesting to compute is that of Diractype operators, even with additional coefficients in locally flat bundles. It is one of the purpose of this paper to formulate and prove for Chern–Simons invariants (2.12) below such an extension as the one in [13] (see Sect. 2.4).
2.2 The Chern–Simons and the \(\eta \)invariants
2.3 Determinant line bundles
In the previous subsections we have studied the connection of the CS invariant with the \(\eta \)invariant. The latter, on the other hand, features also in the context of anomaly formulas, which are related to the geometry of the determinant line bundles. In this subsection we would like to recall such a connection. We have in mind in particular a threedimensional manifold \(\textsf {X}\), as above, but it is possible to stick to a more general treatment.
Fiber bundles Let us discuss some aspects of Dirac operators in the case of fiber bundles. Let \(\pi : {\textsf {W}}\rightarrow \textsf {Z}\) be a smooth fiber bundle with a Riemannian metric on the tangent bundle \(T(\textsf {W}/\textsf {Z})\), which is endowed with spin structure. Here and in the following \(\textsf {W}/\textsf {Z}\) denotes the fiber of \(\textsf {W}\rightarrow \textsf {Z}\). A spin structure on a manifold means a spin structure on its tangent bundle, in this case the tangent bundle \(T(\textsf {W}/\textsf {Z})\) along the fibers. Every point in \(\textsf {Z}\) determines a Dirac operator acting on the corresponding fiber. We will eventually identify the fiber \(\partial \textsf {W}/\textsf {Z}\) with a threedimensional manifold \(\textsf {X}\) without boundary, but for the time being we keep as general as possible.
Assume that the Riemannian metric on the fibers is a product near the boundary. The determinant line carries the Quillen metric and a canonical connection \(\nabla \) [15] and the exponentiated \(\xi \)invariant is a smooth section \(\tau _{\textsf {W}/\textsf {Z}}: \textsf {Z}\rightarrow \mathrm{det}_{{\partial \textsf {W}}/\textsf {Z}}^{1}\).
Let us summarize. The differential geometry of determinant line bundles has been developed in [19] in a special case and in [15, 16] in general. In [20, 21] the results on \(\xi \)invariants were used to redemonstrate the holonomy formula for determinant line bundles, known as Witten’s global anomaly formula [18]. For a family of Dirac operators the exponentiated \(\xi \)invariant is a section of the inverse determinant line bundle over the parameter space. In [20] the usual formula for the variation of the \(\xi \)invariant was been generalized to a formula for the covariant derivative. The variational formula relates the exponentiated \(\xi \)invariant to the natural connection on the (inverse) determinant line bundle [20]. One can use such a connection to compute the holonomy, or global anomaly. The latter can expressed as the adiabatic limit of the exponentiated \(\xi \)invariant.
Returning to Chern–Simons invariants, we have seen that the latter are strictly connected to the \(\eta \) invariants and to the global anomaly formula. In the original papers, Atiyah, Patodi, and Singer discuss the relationship of \(\eta \)invariants (and so exponentiated \(\xi \)invariants) to classical Chern–Simons invariants for closed manifolds. It has been shown [10, 11, 12] that certain ratios of exponentiated \(\xi \)invariants are topological invariants which live in \(K^{1}\)theory with \({\mathbb R}/{\mathbb Z}\) coefficients. The exponentiated \(\xi \)invariant is local and therefore it can serve as an action for a field theory, the same one can say for the Chern–Simons invariant. But there is also a crucial difference: the Chern–Simons invariant is multiplicative in coverings, whereas the exponentiated \(\xi \)invariant is not (nevertheless the gluing law does exhibit some local properties of the \(\eta \)invariant).
The last considerations lead us to the physical interpretation of the material collected so far.
The same anomaly (the socalled parity anomaly) also originates from the presence of massless Majorana fermions on the threedimensional space \(\textsf {X}\). They give rise to a determinant line bundle which leads precisely to the calculation outlined above. The result shows up in the form of the \(\eta \) invariant (or, better, the \(\tau \) invariant), but the analysis is parallel to the previous one and leads to the same conclusions.
In [23] the previous results are used to analyze the 3+1 dimensional theories that describe the socalled topological insulators and topological superconductors. These theories are defined on a \(3+1\) manifold with a boundary and it is usually necessary to enforce complementarity between the fermions in the bulk and those in the boundary in order to cancel the global anomalies and satisfy the condition analogous to (2.25).
2.4 Adiabatic limit and twisted spectral functions
Suppose that \(\textsf {X}= \Gamma \backslash \overline{\textsf {X}}\) with \(\overline{\textsf {X}}\) a globally symmetric space of noncompact type and \(\Gamma \) a discrete, torsionfree, cocompact subgroup of orientationpreserving isometries. Thus \(\textsf {X}\) inherits a locally symmetric Riemannian metric g of nonpositive sectional curvature. In addition the connected components of the periodic set of the geodesic flow \(\Phi \), acting on the unit tangent bundle \(T\textsf {X}\), are parametrized by the nontrivial conjugacy classes \([\gamma ]\) in \(\Gamma = \pi _1(\textsf {X})\). Therefore each connected component \(\textsf {X}_\gamma \) is itself a closed locally symmetric manifold of nonpositive sectional curvature.
Suppose that \(\varphi : \Gamma \rightarrow U({F})\) be a unitary representation of \(\Gamma \) on F. The Hermitian vector bundle \({E}= \textsf {X}\times _{\Gamma }{F}\) over \(\textsf {X}\) inherits a flat connection from the trivial connection on \(\overline{\textsf {X}}\times {F}\). For any vector bundle E over \(\textsf {X}\) let \(\overline{E}\) denote the pullback to \(\overline{\textsf {X}}\). If \({{\mathfrak D}}: C^{\infty }(\textsf {X}, V)\rightarrow C^{\infty }(\textsf {X},V)\) is a differential operator acting on the sections of the vector bundle V, then \({\mathfrak D}\) extends canonically to a differential operator \({\mathfrak D}_{\varphi }: C^{\infty }(\textsf {X},V\otimes F)\rightarrow C^{\infty }(\textsf {X},V\otimes {F})\), uniquely characterized by the property that \({\mathfrak D}_{\varphi }\) is locally isomorphic to \({\mathfrak D}\otimes \cdots \otimes {\mathfrak D}\) (\(\mathrm{dim}\,{F}\) times).
Example 2.1
In connection with a real compact hyperbolic manifold \(\textsf {X}\) consider a locally homogeneous Dirac bundle E over \(\textsf {X}\) and the corresponding Dirac operator \({\mathfrak D}: C^\infty (\textsf {X}, E)\rightarrow C^\infty (\textsf {X}, E)\). As before, assume that \(\textsf {X}= \partial \textsf {M}\), that E extends to a Clifford bundle on \(\textsf {M}\).^{3} and that \(\varphi : \pi _1(\textsf {X})\rightarrow U(F)\) extends to a representation of \(\pi _1(\textsf {M})\). Let \(\underline{A}_\varphi \) be an extension of a flat connection \(A_\varphi \) corresponding to \(\varphi \).
3 Infinite products for the quantum \(\mathfrak {sl}_N\) invariant
In this section we consider more general correlators in a CS theory, and view them as generating series of quantum group invariants weighted by Sfunctions. The quantum group invariants can be defined over any semisimple Lie algebra \(\mathfrak g\). In the SU(N) Chern–Simons gauge theory we study the quantum \({\mathfrak s}{\mathfrak l}_N\) invariants, which can be identified as the socalled colored HOMFLY polynomials^{4}
One important corollary of the LMOV conjecture is the possibility to express a Chern–Simons partition function as an infinite product. In this article we derive such a product. During the calculations we use the characters of the symplectic groups. The latter were found by Weyl [30] using a transcendental method (based on integration over the group manifold). However, the appropriate characters may also be obtained by algebraic methods [31]. Following [32] we have used algebraic methods. This allows us to exploit the Hopf algebra methods to determine (sub)group branching rules and the decomposition of tensor products.
The motivation for studying an infiniteproduct formula, associated to topological string partition functions, based on a guess on the modular property of partition function, stimulated by properties of Sfunctions.
Given \(X= \{x_i\}_{i\ge 1}\), \(Y=\{y_j\}_{j\ge 1}\), define \( X*Y = \{x_i\cdot y_j\}_{i\ge 1, j\ge 1}. \) We also define \(X^d = \{ x_i^d\}_{i\ge 1}\). The dth Adams operation of a Schur function is given by \(s_A(X^d)\). (An Adams operation is type of algebraic construction; the basic idea of this operation is to implement some fundamental identities in Sfunctions. In particular, \(s_A(X^d)\) means operation of a power sum on a polynomial.)

\({\mathcal L}\) will denote a link and L the number of components in \({\mathcal L}\).

The irreducible \(U_q(\mathfrak {sl}_N)\) module associated to \(\mathcal L\) will be labeled by their highest weights, thus by Young diagrams. We usually denote it by a vector form \(\overrightarrow{A}=(A^1,\ldots ,A^L)\).
 Let \(\overrightarrow{X} =(x_1,\ldots ,x_L)\) be a set of L variables, each of which is associated to a component of \(\mathcal L\) and \(\overrightarrow{\mu } = (\mu ^1,\ldots ,\mu ^L)\in {\mathcal Y}^L\) be a tuple of L partitions. We write$$\begin{aligned}&[\overrightarrow{\mu }] = \prod _{\alpha =1}^L [\mu ^\alpha ], \quad \mathfrak {z}_{\overrightarrow{\mu }} = \prod _{\alpha =1}^L \mathfrak {z}_{\mu ^\alpha },\\&\quad \chi _{\overrightarrow{A}}(C_{\overrightarrow{\mu }}) = \prod _{\alpha =1}^L \chi _{A^\alpha }(C_{\mu ^\alpha }), \end{aligned}$$$$\begin{aligned}&s_{\overrightarrow{A}}(\overrightarrow{X})= \prod _{\alpha =1}^L s_{A^\alpha }(x_\alpha ), \quad p_{\mu }(X)=\overset{\ell (\mu )}{\prod _{i=1}}p_{\mu _{i}}(X),\\&\quad p_{\overrightarrow{\mu }}(\overrightarrow{X}) = \prod _{\alpha =1}^L p_{\mu ^\alpha }(x_\alpha ). \end{aligned}$$
Calculations in the case of Kauffman polynomials, relative to the orthogonal group, can be found in Ref. [37].
Footnotes
 1.
Cf. Eq. (2.13): the \(\eta \)invariant of locally symmetric manifolds of nonpositive curvature can be expressed as spectral values of zetafunctions constructed out of the periodic geodesics [8, 14]. In this case a necessary regularization can be given by the geodesic spectrum. For the time being we assumed the other regularization for the \(\eta \)invariant, which involves the “dual” data, namely the spectrum of the Laplace operator associated to the metric [14].
 2.
The result is obtained by considering a theory of a massive fermion and taking the mass to 0.
 3.
A Clifford module bundle is called a Dirac bundle if it has a connection \(\nabla \) satisfying the compatibility condition \(\nabla _z(v\cdot s) = (\nabla _z^Rv) \cdot s + v\cdot (\nabla _z s)\). Here s is a local section of E, v is a local section of \(C\ell (\textsf {M})\), z a vector field and \((\cdot )\) denote the module multiplication. On a Dirac bundle one then has a Dirac operator defined by \({\mathfrak D}s = \sum _je_j\cdot (\nabla _{e_j}s)\), where \(\{e_j\}\) is any local orthonormal frame for \(\textsf {M}\).
 4.
The framed HOMFLY polynomial of links (an invariant of framed oriented links), is denoted by \(\mathcal {H}({\mathcal L})\), and can be normalized as follows: \(\mathcal {H}(\bigcirc ) = (t^{\frac{1}{2}}t^{\frac{1}{2}})/(q^{\frac{1}{2}}q^{\frac{1}{2}})\). (These invariants can be recursively computed through the HOMFLY skein.) The colored HOMFLY polynomials are defined through the satellite knot. A satellite of knot \({\mathcal K}\) is determined by choosing a diagram Q in the annulus. Draw Q on the annular neighborhood of \({\mathcal K}\) determined by the framing, to give a satellite knot \({\mathcal K}\star Q\). One can refer to this construction as decorating \({\mathcal K}\) with the pattern Q. The HOMFLY polynomial \(\mathcal {H}({\mathcal K}\star Q)\) of the satellite depends on Q only as an element of the skein \({\mathcal C}\) of the annulus. \(\{Q_\lambda \}_{\lambda \in {\mathcal Y}}\) form a basis of \({\mathcal C}\). \({\mathcal C}\) can be regarded as the parameter space for these invariants of \({\mathcal K}\), and can be called the HOMFLY satellite invariants of \({\mathcal K}\).
 5.
We also mention on this topic the deep results of [42, 43], where a number of identities in the representation theory of Kač–Moody Lie algebras has been obtained. A family of modular functions satisfying Rogers–Ramanujantype identities for arbitrary affine root systems has been obtained in [44]. Extensive work in the theory of partition identities shows that basic hypergeometric series provide the generating functions for numerous families of partition identities.
Notes
Acknowledgements
A. A. Bonora would like to acknowledge the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq, Brazil) and Coordenação de Aperfeiçamento de Pessoal de Nível Superior (CAPES, Brazil) for financial support. L.B. would like to thank the Departamento de Fisica da Universitade Estadual de Londrina for its kind hospitality.
References
 1.E. Witten, Topological quantum field theory. Commun. Math. Phys 117, 353–386 (1988)ADSMathSciNetCrossRefMATHGoogle Scholar
 2.N. Yu, Reshetikhin, V.G. Turaev, Invariants of 3manifolds via link polynomials and quantum groups. Invent. Math. 103, 547–597 (1991)Google Scholar
 3.L. Crane, D. Yetter, A categorical construction of 4D topological quantum field theories, Quantum topology. In: Ser. Knots Everything, vol. 3, pp. 120–130 (World Sci. Publ, River Edge, 1993)Google Scholar
 4.H. Ooguri, C. Vafa, Knot invariants and topological strings. Nucl. Phys. B 577, 419–438 (2000)ADSMathSciNetCrossRefMATHGoogle Scholar
 5.J. M. F. Labastida, M. Mariño, C. Vafa, Knots, links and branes at large N. JHEP 11, 007 (2000)Google Scholar
 6.R. Gopakumar, C. Vafa, On the gauge theory/geometry correspondence. Adv. Theor. Math. Phys. 3, 1415–1443 (1999)MathSciNetCrossRefMATHGoogle Scholar
 7.E. Witten, Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121, 351–399 (1989)ADSMathSciNetCrossRefMATHGoogle Scholar
 8.H. Moscovici, R. Stanton, Eta invariants of Dirac operators on locally symmetric manifolds. Invent. Math. 95, 629–666 (1989)ADSMathSciNetCrossRefMATHGoogle Scholar
 9.H. Nishi, \(SU(n)\)Chern–Simons invariants of Seifert fibered 3manifolds. Int. J. Math. 9, 295–330 (1998)MathSciNetCrossRefMATHGoogle Scholar
 10.M.F. Atiyah, V.K. Patodi, I.M. Singer, Spectral asymmetry and riemannian geometry. I. Math. Proc. Camb. Phil. Soc. 77, 43–69 (1975)MathSciNetCrossRefMATHGoogle Scholar
 11.M.F. Atiyah, V.K. Patodi, I.M. Singer, Spectral asymmetry and riemannian geometry. II. Math. Proc. Camb. Phil. Soc. 78, 405–432 (1975)MathSciNetCrossRefMATHGoogle Scholar
 12.M.F. Atiyah, V.K. Patodi, I.M. Singer, Spectral asymmetry and riemannian geometry. III. Math. Proc. Camb. Phil. Soc. 79, 71–99 (1976)MathSciNetCrossRefMATHGoogle Scholar
 13.J.J. Milson, Closed geodesics and the etainvariant. Ann. Math. 108, 1–39 (1978)MathSciNetCrossRefGoogle Scholar
 14.H. Moscovici, R. Stanton, Rtorsion and zeta functions for locally symmetric manifolds. Invent. Math. 105, 185–216 (1991)ADSMathSciNetCrossRefMATHGoogle Scholar
 15.J.M. Bismut, D. S. Freed, The analysis of elliptic families I. Metrics and connections on determinant bundles. Commun. Math. Phys. 106, 159–176 (1986)Google Scholar
 16.J.M. Bismut, D. S. Freed, The analysis of elliptic families II. Dirac operators, eta invariants, and the holonomy theorem of Witten. Commun. Math. Phys. 107, 103–163 (1986)Google Scholar
 17.D.S. Freed, E. Witten, Anomalies in string theory with Dbranes. Asian J. Math. 3, 819–851 (1999)MathSciNetCrossRefMATHGoogle Scholar
 18.E. Witten, Global gravitational anomalies. Commun. Math. Phys. 100, 197–229 (1985)ADSMathSciNetCrossRefMATHGoogle Scholar
 19.D. Quillen, Determinants of Cauchy–Riemann operators over a Riemann surface. Funk. Anal. i Prilozen. 19, 37–41 (1985)MathSciNetMATHGoogle Scholar
 20.X. Dai, D. S. Freed, Eta invariants and determinant lines. J. Math. Phys. 35, 5155–5194 (1994). (Erratum: ibid 42, 2343–2344 (2001)) Google Scholar
 21.D. S. Freed, Determinant lines bundles revisted. In: Lecture Notes in Pure and Applied Mathematics, vol. 184, pp. 187–195 (Marcel Dekker, New York, 1997)Google Scholar
 22.L. Bonora, M. Cvitan, P. Dominis Prester, B. Lima de Souza, I. Smoli, Massive fermion model in 3d and higher spin currents. JHEP 1605, 072 (2016)Google Scholar
 23.E. Witten, Anomalies revisited. Lecture at strings 2015, ICTSTITR, Bangalore, June 22, 2015. In: Witten, E. (ed.) Fermion Path Integrals And Topological Phases. arXiv:1508.04715 [condmat.meshall]
 24.P.B. Kronheimer, H. Nakajima, Young–Mills instantons on ALE gravitational instantons. Math. Ann. 288, 263–307 (1990)MathSciNetCrossRefMATHGoogle Scholar
 25.A. Cayley, About the algebraic structure of the orthogonal group and the other classical groups in a field of characteristic zero or a prime characteristic. J. Reine Angew. Math. 32, 119–123 (1846)MathSciNetCrossRefGoogle Scholar
 26.D. Fried, Lefschetz Formulas for Flows, in the Lefschetz Centennial Conference, part III. Contempor. Math. 58, 19–69 (1987)CrossRefGoogle Scholar
 27.D. Fried, Counting circles. Dynamical systems. In: Lecture Notes in Mathematics, vol. 1348, 196–215 (Springer, Berlin, 2006)Google Scholar
 28.D.B. Ray, I. Singer, Rtorsion and the Laplacian on Riemannian manifolds. Adv. Math. 7, 145–210 (1971)MathSciNetCrossRefMATHGoogle Scholar
 29.D. Fried, Analytic torsion and closed geodesic on hyperbolic manifolds. Invent. Math. 84, 523–540 (1986)ADSMathSciNetCrossRefMATHGoogle Scholar
 30.H. Weyl, The Classical Groups Their Invariants and Representations, 2nd edn. (Princeton University Press, Princeton, 1946)MATHGoogle Scholar
 31.D.E. Littlewood, Invariant theory, tensors and group characters. Philos. Trans. R. Soc. A 239, 305–365 (1944)ADSMathSciNetCrossRefMATHGoogle Scholar
 32.B. Fauser, P. D. Jarvis, R. C. King, Plethysms, replicated Schur functions and series, with applications to vertex operators. J. Phys. A Math. Theor. 43, 405202 (2010)Google Scholar
 33.S. Zhu, Colored HOMFLY polynomial via Skein theory. JHEP 229, 1310 (2013)Google Scholar
 34.K. Liu, P. Peng, Proof of the Labastida–Mariño–Ooguri–Vafa conjecture. J. Differ. Geom. 85, 479–525 (2010)MathSciNetMATHGoogle Scholar
 35.K. Liu, P. Peng, New structure of knot invariants. Commun. Number Theory Phys. 5, 601–615 (2011)Google Scholar
 36.L. Bonora, A.A. Bytsenko, M. Chaichian, Elliptic genera and characteristic \(q\)series of superconformal field theory. Nucl. Phys. B 895, 192–205 (2015)Google Scholar
 37.A.A. Bytsenko, M. Chaichian, \(S\)functions, spectral functions of hyperbolic geometry, and vertex operators with applications to structure for weyl and orthogonal group invariants. Nucl. Phys. B 907, 258–285 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar
 38.L. Bonora, A.A. Bytsenko, Partition functions for quantum gravity, black holes, elliptic genera and Lie algebra homologies. Nucl. Phys. B 852, 508–537 (2011)ADSMathSciNetCrossRefMATHGoogle Scholar
 39.A.A. Bytsenko, M. Chaichian, R.J. Szabo, A. Tureanu, Quantum black holes. Elliptic genera and spectral partition functions. IJGMMP 11, 1450048 (2014)MathSciNetMATHGoogle Scholar
 40.E. Hecke, Über den Zusammenhang zwischen elliptischen Modulfunktionen und indefiniten quadratischen Formen. Math. Werke, Vandenhoeck and Ruprecht, pp. 418–427, Göttingen (1959)Google Scholar
 41.L.J. Rogers, On the expansion of some infinite products. Proc. Lond. Math. Soc. 24, 337–352 (1893)MathSciNetMATHGoogle Scholar
 42.V.G. Kač, D.H. Peterson, Affine Lie algebras and Hecke modular forms. Bull. Am. Math. Soc. 3, 1057–1061 (1980)MathSciNetCrossRefMATHGoogle Scholar
 43.V.G. Kač, D.H. Peterson, Infinitedimensional Lie algebras. Theta functions and modular forms. Adv. Math. 53, 125–264 (1984)MathSciNetCrossRefMATHGoogle Scholar
 44.I. Cherednik, B. Feigin, Rogers–Ramanujan type identities and NilDAHA. Adv. Math. 248, 1059–1088 (2013)MathSciNetCrossRefMATHGoogle Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Funded by SCOAP^{3}