On the shape of fundamental domains in GL(n,R)∕O(n)

Grenier, Douglas

doi:10.2140/pjm.1993.160.53

Cited by 19 publications

(24 citation statements)

References 7 publications

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“…We must now compute F rk=1,2 de f E 6 with respect to unfolded fundamental domains F rk=1,2 . The SL(3) fundamental domain F is known [33,34], a complete description is given in Appendix B.2, see in particular equations (B.17) and (B.19). Its construction is in terms of a height function given by the maximal abelian torus coordinate L 3 in (4.2) along with the actions of an (overcomplete) set of SL(3, Z) generators S 1,...,5 , T 1,2 and U 1,2 given in (B.16,B.18).…”

Section: Rank 1 and 2 Winding Modesmentioning

confidence: 99%

The Automorphic Membrane

Pioline

Waldron

2004

J. High Energy Phys.

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We present a 1-loop toroidal membrane winding sum reproducing the conjectured M -theory, four-graviton, eight derivative, R 4 amplitude. The U -duality and toroidal membrane world-volume modular groups appear as a Howe dual pair in a larger, exceptional, group. A detailed analysis is carried out for M -theory compactified on a 3-torus, where the target-space SL(3, Z) × SL(2, Z) U -duality and SL(3, Z) world-volume modular groups are embedded in E 6(6) (Z). Unlike previous semi-classical expansions, U -duality is built in manifestly and realized at the quantum level thanks to Fourier invariance of cubic characters. In addition to winding modes, a pair of new discrete, flux-like, quantum numbers are necessary to ensure invariance under the larger group. The action for these modes is of Born-Infeld type, interpolating between standard Polyakov and Nambu-Goto membrane actions. After integration over the membrane moduli, we recover the known R 4 amplitude, including membrane instantons. Divergences are disposed of by trading the non-compact volume integration for a compact integral over the two variables conjugate to the fluxes -a constant term computation in mathematical parlance. As byproducts, we suggest that, in line with membrane/fivebrane duality, the E 6 theta series also describes five-branes wrapped on T 6 in a manifestly U-duality invariant way. In addition we uncover a new action of E 6 on ten dimensional pure spinors, which may have implications for ten dimensional super Yang-Mills theory. An extensive review of SL(3) automorphic forms is included in an Appendix.

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Section: Rank 1 and 2 Winding Modesmentioning

confidence: 99%

The Automorphic Membrane

Pioline

Waldron

2004

J. High Energy Phys.

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“…Note that our bound is stronger than the result for SL 3 (Z) from [Mi96] where it was proved that λ 1 ≥ 1. In general, our method applies to G = SL n (R) with Γ = SL n (Z) to give the bound λ 1 > 3π 2 /M where M is the number of fundamental domains which intersect a Siegel set containing the fundamental domain constructed in [Gr93]; however, for n ≥ 4, this bound is rather weak. Finally, let us remark that our theorem is indeed a consequence of the Ramanujan conjecture, which asserts that all nontrivial automorphic representations come from tempered representations.…”

Section: Discussionmentioning

confidence: 99%

“…To this end, we introduce a fundamental domain for Γ\H. Specifically, computations on page 56 of [Gr93] show that a fundamental domain D is described through the following set of inequalities:…”

Section: §1 Statement Of the Main Theoremmentioning

confidence: 99%

“…The set S contains the fundamental domain D. Further, results from page 61 of [Gr93] show the existence of elements γ 1 , ..., γ 10 ∈ SL 3 (Z) such that S ⊂ (we have used the notation Dγ to denote the image of the fundamental domain D under left multiplication by γ). The main aspects of the above points which we shall use are the assertions that for any τ ∈ S we have y 1 (τ ) > √ 3/2 and that S is contained in ten translates of D.…”

Section: §1 Statement Of the Main Theoremmentioning

confidence: 99%

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On an analogue of Selberg’s eigenvalue conjecture for 𝑆𝐿₃(𝐙)

Catto

Huntley

Jorgenson

et al. 1998

Proc. Amer. Math. Soc.

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Abstract. Let H be the homogeneous space associated to the group PGL 3 (R). Let X = Γ\H where Γ = SL 3 (Z) and consider the first nontrivial eigenvalue λ 1 of the Laplacian on L 2 (X). Using geometric considerations, we prove the inequality λ 1 > 3π 2 /10. Since the continuous spectrum is represented by the band [1, ∞), our bound on λ 1 can be viewed as an analogue of Selberg's eigenvalue conjecture for quotients of the hyperbolic half space. §1. Statement of the main theorem A fundamental question in the spectral theory of automorphic forms is whether small eigenvalues exist. More specifically, let G be a noncompact reductive group with finite center, Γ a nonuniform lattice, K a maximal compact subgroup of G, and set X = Γ\G/K. It is well known from the theory of Eisenstein series that L 2 (X) has continuous spectrum for the ring of invariant differential operators, and in particular for the positive Laplacian, ∆. The continuous spectrum will be, in cases of interest such as PGL n (R), an interval [a, ∞) with a > 0. The question we referred to above is: Do nonconstant square integrable eigenforms exist with eigenvalue λ < a? This problem is important for various considerations in number theory. In the case G = PGL 2 (R) and Γ is a congruence subgroup, Selberg conjectured that no such nontrivial small eigenvalues exist.In this paper, we consider the case when G = PGL 3 (R) and Γ = SL 3 (Z).Our main result is the following.Theorem. Let λ 1 be the eigenvalue for the first nontrivial eigenform on L 2 (X). Then λ 1 > 3π 2 /10 > 2.96088.

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“…The trivial V 3 dependence ofF L is always implicit 9. This simply follows from the structure of divergences of the Mercedes diagrams[22] which arise from the boundary of the fundamental domain of SL(3, Z)[38][39][40]. Our auxiliary geometry is a subspace of the T 3 geometry with the same boundary structure 10.…”

mentioning

confidence: 99%

Some finite terms from ladder diagrams in three and four loop maximal supergravity

Basu¹

2015

Class. Quantum Grav.

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We consider the finite part of the leading local interactions in the low energy expansion of the four graviton amplitude from the ladder skeleton diagrams in maximal supergravity on T 2 , at three and four loops. At three loops, we express the D 8 R 4 and D 10 R 4 amplitudes as integrals over the moduli space of an underlying auxiliary geometry. These amplitudes are evaluated exactly for special values of the the moduli of the auxiliary geometry, where the integrand simplifies. We also perform a similar analysis for the D 8 R 4 amplitude at four loops that arise from the ladder skeleton diagrams for a special value of a parameter in the moduli space of the auxiliary geometry. While the dependence of the amplitudes on the volume of the T 2 is very simple, the dependence on the complex structure of the T 2 is quite intricate. In some of the cases, the amplitude consists of terms each of which factorizes into a product of two SL(2, Z) invariant modular forms. While one of the factors is a non-holomorphic Eisenstein series, the other factor splits into a sum of modular forms each of which satisfies a Poisson equation on moduli space with source terms that are bilinear in the Eisenstein series. This leads to several possible perturbative contributions unto genus 5 in type II string theory on S 1 . Unlike the one and two loop supergravity analysis, these amplitudes also receive non-perturbative contributions from bound states of three D-(anti)instantons in the IIB theory. 1 email address: anirbanbasu@hri.res.in 2 For theories with self-dual field strengths, what we mean are covariant equations of motion.

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On the shape of fundamental domains in GL(n,R)∕O(n)

Cited by 19 publications

References 7 publications

The Automorphic Membrane

The Automorphic Membrane

On an analogue of Selberg’s eigenvalue conjecture for 𝑆𝐿₃(𝐙)

Some finite terms from ladder diagrams in three and four loop maximal supergravity

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