Abstract
Highest-weight representations of infinite-dimensional Lie algebras and Hilbert schemes of points are considered, together with the applications of these concepts to partition functions, which are most useful in physics. Partition functions (elliptic genera) are conveniently transformed into product expressions, which may inherit the homology properties of appropriate (poly)graded Lie algebras. Specifically, the role of (Selberg-type) Ruelle spectral functions of hyperbolic geometry in the calculation of partition functions and associated q-series is discussed. Examples of these connections in quantum field theory are considered (in particular, within the AdS/CFT correspondence), as the AdS3 case where one has Ruelle/Selberg spectral functions, whereas on the CFT side, partition functions and modular forms arise. These objects are here shown to have a common background, expressible in terms of Euler-Poincaré and Macdonald identities, which, in turn, describe homological aspects of (finite or infinite) Lie algebra representations. Finally, some other applications of modular forms and spectral functions (mainly related with the congruence subgroup of \(SL(2, \mathbb{Z})\)) to partition functions, Hilbert schemes of points, and symmetric products are investigated by means of homological and K-theory methods.
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- 1.
Vanishing theorems for the type (0, q) cohomology of locally symmetric spaces can be found in [15]. Again, if χ is acyclic (H(X; χ) = 0), the Ray-Singer norm (3) is a topological invariant: it does not depend on the choice of the metric on X and χ used in the construction. If X is a complex manifold (smooth C ∞-manifold or topological space), then \(\mathbb{E} \rightarrow X\) is the induced complex (or smooth, or continuous) vector bundles. We write \({H}^{p,q}(X; \mathbb{E}) \simeq {H}^{0,q}(X{;\varLambda }^{p,0}X \otimes \mathbb{E})\) holonomic vector bundles Λ p, 0 X → X (see [15] for details).
- 2.
The modular forms in question are the forms for the congruence subgroup of \(SL(2, \mathbb{Z})\), which is viewed as the group that leaves fixed one of the three nontrivial spin structures on an elliptic curve.
- 3.
In the case when V has the one-dimensional odd degree part only (the bilinear form is (r, r) = 1 for a nonzero vector r ∈ V ) and the above condition is not satisfied, we can modify the definition of the corresponding super-Heisenberg algebra by changing the bilinear form on W as \((r \otimes {t}^{i},r \otimes {t}^{j}) =\delta _{i+j,0}\). The resulting algebra is termed an infinite-dimensional Clifford algebra. The above representation R is the fermionic Fock space in physics and it can be modified as follows: the representation of the even degree part was realized as the space of polynomials of infinitely many variables; the Clifford algebra is realized on the exterior algebra \(R {=\varLambda }^{{\ast}}(\bigoplus _{j}\mathbb{Q}dp_{j})\) of a vector space with a basis of infinitely many vectors. For j > 0 we define r ⊗ t −j as an exterior product of dp j and r ⊗ t j as an interior product of ∂∕∂ p j .
- 4.
A fundamental example of \(\tilde{H}\varGamma _{N}\)-vector superbundles over X N (X compact) is the following: given a Γ-vector bundle V over X, consider the vector superbundle V ⊕ V over X with the natural \(\mathbb{Z}_{2}\)-grading. One can endow the Nth outer tensor product (V ⊕ V ) ⊠ N with a natural \(\tilde{H}\varGamma _{N}\)-equivariant vector superbundle structure over X N.
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Acknowledgements
AAB would like to acknowledge the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq, Brazil) and Fundaçao Araucaria (Parana, Brazil) for the financial support. EE’s research has been partly supported by MICINN (Spain), contract PR2011-0128 and projects FIS2006-02842 and FIS2010-15640, and by the CPAN Consolider Ingenio Project and by AGAUR (Generalitat de Catalunya), contract 2009SGR-994. EE’s research was done in part while visiting the Department of Physics and Astronomy, Dartmouth College, NH, USA.
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Bytsenko, A.A., Elizalde, E. (2014). On Partition Functions of Hyperbolic Three-Geometry and Associated Hilbert Schemes. In: Milovanović, G., Rassias, M. (eds) Analytic Number Theory, Approximation Theory, and Special Functions. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0258-3_5
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