| 1. |
- Berg, Marcus, 1973-, et al.
(författare)
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Massive deformations of Maass forms and Jacobi forms
- 2021
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Ingår i: Communications in Number Theory and Physics. - : INTERNATIONAL PRESS. - 1931-4523 .- 1931-4531. ; 15:3, s. 575-603
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Tidskriftsartikel (refereegranskat)abstract
- We define one-parameter "massive" deformations of Maass forms and Jacobi forms. This is inspired by descriptions of plane gravitational waves in string theory. Examples include massive Green's functions (that we write in terms of Kronecker-Eisenstein series) and massive modular graph functions.
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| 2. |
- D'Hoker, Eric, et al.
(författare)
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Identities among higher genus modular graph tensors
- 2022
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Ingår i: Communications in Number Theory and Physics. - : INT PRESS BOSTON, INC. - 1931-4523 .- 1931-4531. ; 16:1, s. 35-74
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Tidskriftsartikel (refereegranskat)abstract
- Higher genus modular graph tensors map Feynman graphs to functions on the Torelli space of genus-h compact Riemann surfaces which transform as tensors under the modular group Sp(2h, Z), thereby generalizing a construction of Kawazumi. An infinite family of algebraic identities between one-loop and tree-level modular graph tensors are proven for arbitrary genus and arbitrary tensorial rank. We also derive a family of identities that apply to modular graph tensors of higher loop order.
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| 3. |
- Dimofte, T., et al.
(författare)
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Exact results for perturbative Chern-Simons theory with complex gauge group
- 2009
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Ingår i: Communications in Number Theory and Physics. - : International Press of Boston. - 1931-4523 .- 1931-4531. ; 3:2, s. 363-443
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Tidskriftsartikel (refereegranskat)abstract
- We develop several methods that allow us to compute all-loop partition functions in perturbative Chern-Simons theory with complex gauge group GC, sometimes in multiple ways. In the background of a non-abelian irreducible flat connection, perturbative GC invariants turn out to be interesting topological invariants, which are very different from the finite-type (Vassiliev) invariants usually studied in a theory with compact gauge group G and a trivial flat connection. We explore various aspects of these invariants and present an example where we compute them explicitly to high loop order. We also introduce a notion of "arithmetic topological quantum field theory" and conjecture (with supporting numerical evidence) that SL(2, C) Chern-Simons theory is an example of such a theory.
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| 4. |
- Dunin-Barkowski, Petr, et al.
(författare)
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Combinatorial structure of colored HOMFLY-PT polynomials for torus knots
- 2019
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Ingår i: Communications in Number Theory and Physics. - : INT PRESS BOSTON, INC. - 1931-4523 .- 1931-4531. ; 13:4, s. 763-826
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Tidskriftsartikel (refereegranskat)abstract
- We rewrite the (extended) Ooguri-Vafa partition function for colored HOMFLY-PT polynomials for torus knots in terms of the free-fermion (semi-infinite wedge) formalism, making it very similar to the generating function for double Hurwitz numbers. This allows us to conjecture the combinatorial meaning of full expansion of the correlation differentials obtained via the topological recursion on the Brini-Eynard-Marino spectral curve for the colored HOMFLY-PT polynomials of torus knots. This correspondence suggests a structural combinatorial result for the extended Ooguri-Vafa partition function. Namely, its coefficients should have a quasi-polynomial behavior, where non-polynomial factors are given by the Jacobi polynomials (treated as functions of their parameters in which they are indeed non-polynomial). We prove this quasi-polynomiality in a purely combinatorial way. In addition to that, we show that the (0,1)- and (0,2)-functions on the corresponding spectral curve are in agreement with the extension of the colored HOMFLY-PT polynomials data, and we prove the quantum spectral curve equation for a natural wave function obtained from the extended Ooguri-Vafa partition function.
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| 5. |
- Fedosova, Ksenia, et al.
(författare)
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Fourier expansions of vector-valued automorphic functions with non-unitary twists
- 2023
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Ingår i: Communications in Number Theory and Physics. - 1931-4531 .- 1931-4523. ; 17:1, s. 173-248
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Tidskriftsartikel (refereegranskat)abstract
- We provide Fourier expansions of vector-valued eigenfunctions of the hyperbolic Laplacian that are twist-periodic in a horocycle direction. The twist may be given by any endomorphism of a finite-dimensional vector space; no assumptions on invertibility or unitarity are made. Examples of such eigenfunctions include vector-valued twisted automorphic forms of Fuchsian groups. We further provide a detailed description of the Fourier coefficients and explicitly identify each of their constituents, which intimately depend on the eigenvalues of the twisting endomorphism and the size of its Jordan blocks. In addition, we determine the growth properties of the Fourier coefficients.
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| 6. |
- Fleig, Philipp, et al.
(författare)
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Fourier expansions of Kac-Moody Eisenstein series and degenerate Whittaker vectors
- 2014
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Ingår i: Communications in Number Theory and Physics. - 1931-4531 .- 1931-4523. ; 8:1, s. 41-100
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Tidskriftsartikel (refereegranskat)abstract
- Motivated by string theory scattering amplitudes that are invariant under a discrete U-duality, we study Fourier coefficients of Eisenstein series on Kac-Moody groups. In particular, we analyse the Eisenstein series on E_9(R), E_10(R) and E_11(R) corresponding to certain degenerate principal series at the values s=3/2 and s=5/2 that were studied in 1204.3043. We show that these Eisenstein series have very simple Fourier coefficients as expected for their role as supersymmetric contributions to the higher derivative couplings R^4 and \partial^{4} R^4 coming from 1/2-BPS and 1/4-BPS instantons, respectively. This suggests that there exist minimal and next-to-minimal unipotent automorphic representations of the associated Kac-Moody groups to which these special Eisenstein series are attached. We also provide complete explicit expressions for degenerate Whittaker vectors of minimal Eisenstein series on E_6(R), E_7(R) and E_8(R) that have not appeared in the literature before.
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| 7. |
- Freiberg, Tristan, et al.
(författare)
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Poisson distribution for gaps between sums of two squares and level spacings for toral point scatterers
- 2017
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Ingår i: Communications in Number Theory and Physics. - 1931-4523 .- 1931-4531. ; 11:4, s. 837-877
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Tidskriftsartikel (refereegranskat)abstract
- We investigate the level spacing distribution for the quantum spectrum of the square billiard. Extending work of Connors-Keating, and Smilansky, we formulate an analog of the Hardy-Littlewood prime k-tuple conjecture for sums of two squares, and show that it implies that the spectral gaps, after removing degeneracies and rescaling, are Poisson distributed. Consequently, by work of Rud-nick and Ueberschar, the level spacings of arithmetic toral point scatterers, in the weak coupling limit, are also Poisson distributed. We also give numerical evidence for the conjecture and its implications.
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| 8. |
- Gaberdiel, Matthias R., et al.
(författare)
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Generalized Mathieu Moonshine
- 2013
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Ingår i: Communications in Number Theory and Physics. - 1931-4531 .- 1931-4523. ; 7:1, s. 145-223
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Tidskriftsartikel (refereegranskat)abstract
- The Mathieu twisted twining genera, i.e., the analogues of Norton's generalized Moonshine functions, are constructed for the elliptic genus of K3. It is shown that they satisfy the expected consistency conditions, and that their behaviour under modular transformations is controlled by a 3-cocycle in H-3(M-24, U(1)), just as for the case of holomorphic orbifolds. This suggests that a holomorphic VOA may be underlying Mathieu Moonshine.
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| 9. |
- Nilsson, Bengt E W, 1952, et al.
(författare)
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Instanton Corrections to the Universal Hypermultiplet and Automorphic Forms on SU(2,1).
- 2010
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Ingår i: Communications in Number Theory and Physics. - 1931-4531 .- 1931-4523. ; 4:1, s. 187-266
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Tidskriftsartikel (refereegranskat)abstract
- Abstract: The hypermultiplet moduli space in Type IIA string theory compactified on a rigid Calabi-Yau threefold X , corresponding to the “universal hypermultiplet”, is described at tree-level by the symmetric space SU(2,1)/(SU(2)×U(1)). To determine the quantum corrections to this metric, we posit that a discrete subgroup of the continuous tree-level isometry group SU(2,1), namely the Picard modular group SU(2,1;Z[i]), must remain un- broken in the exact metric – including all perturbative and non-perturbative quantum cor- rections. This assumption is expected to be valid when X admits complex multiplication by Z[i]. Based on this hypothesis, we construct an SU(2,1;Z[i])-invariant, non-holomorphic Eisenstein series, and tentatively propose that this Eisenstein series provides the exact contact potential on the twistor space over the universal hypermultiplet moduli space. We analyze its non-Abelian Fourier expansion, and show that the Abelian and non-Abelian Fourier coefficients take the required form for instanton corrections due to Euclidean D2- branes wrapping special Lagrangian submanifolds, and to Euclidean NS5-branes wrapping the entire Calabi-Yau threefold, respectively. While this tentative proposal fails to repro- duce the correct one-loop correction, the consistency of the Fourier expansion with physics expectations provides strong support for the usefulness of the Picard modular group in constraining the quantum moduli space.
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| 10. |
- Paquette, N. M., et al.
(författare)
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Monstrous BPS-algebras and the superstring origin of moonshine
- 2016
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Ingår i: Communications in Number Theory and Physics. - 1931-4531 .- 1931-4523. ; 10:3, s. 433-526
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Tidskriftsartikel (refereegranskat)abstract
- We provide a physics derivation of Monstrous moonshine. We show that the McKay-Thompson series T-g, g epsilon M, can be interpreted as supersymmetric indices counting spacetime BPS-states in certain heterotic string models. The invariance groups of these series arise naturally as spacetime T-duality groups and their genus zero property descends from the behaviour of these heterotic models in suitable decompactification limits. We also show that the space of BPS-states forms a module for the Monstrous Lie algebras m(g), constructed by Borcherds and Carnahan. We argue that m(g) arise in the heterotic models as algebras of spontaneously broken gauge symmetries, whose generators are in exact correspondence with BPS-states. This gives mg an interpretation as a kind of BPS-algebra.
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| 11. |
- Persson, Daniel, 1978, et al.
(författare)
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Second-quantized Mathieu moonshine
- 2014
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Ingår i: Communications in Number Theory and Physics. - 1931-4531 .- 1931-4523. ; 8:3, s. 403-509
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Tidskriftsartikel (refereegranskat)abstract
- We study the second-quantized version of the twisted twining genera of generalized Mathieu moonshine, and prove that they give rise to Siegel modular forms with infinite product representations. Most of these forms are expected to have an interpretation as twisted partition functions counting 1/4 BPS dyons in type II superstring theory on K3 x T-2 or in heterotic CHL-models. We show that all these Siegel modular forms, independently of their possible physical interpretation, satisfy an "S-duality" transformation and a "wall-crossing formula". The latter reproduces all the eta-products of an older version of generalized Mathieu moonshine proposed by Mason in the 1990s. Surprisingly, some of the Siegel modular forms we find coincide with the multiplicative (Borcherds) lifts of Jacobi forms in umbral moonshine.
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| 12. |
- Pioline, B., et al.
(författare)
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The automorphic NS5-brane
- 2009
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Ingår i: Communications in Number Theory and Physics. - : International Press of Boston. - 1931-4531 .- 1931-4523. ; 3:4, s. 697-754
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Tidskriftsartikel (övrigt vetenskapligt/konstnärligt)abstract
- Understanding the implications of SL(2, Z) S-duality for the hyper-multiplet moduli space of type II string theories has led to much progress recently in uncovering D-instanton contributions. In this work, we suggest that the extended duality group SL(3, Z), which includes both S-duality and Ehlers symmetry, may determine the contributions of D5 and NS5-branes. We support this claim by automorphizing the perturbative corrections to the "extended universal hypermultiplet," a five-dimensional universal SO(3)\SL(3, R) subspace which includes the string coupling, overall volume, Ramond zero-form and six-form and NS axion. Using the non-Abelian Fourier expansion of the Eisenstein series attached to the principal series of SL(3, R), worked out many years ago by Vinogradov, Takhtajan and Bump, we extract the contributions of D(-1)-D5 and NS5-brane instantons, corresponding to the Abelian and non-Abelian coefficients, respectively. In particular, the contributions of k NS5-branes can be summarized into a vector of wave functions Psi(k,l), l = 0,..., k - 1, as expected on general grounds. We also point out that for more general models with a symmetric moduli space K\G, the minimal theta series of G generates an infinite series of exponential corrections of the form required for "small" D(-1)-D1-D3-D5-NS5 instanton bound states. As a mathematical spin-off, we make contact with earlier results in the literature about the spherical vectors for the principal series of SL(3, R) and for minimal representations.
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