Volumes of Classical Supermanifolds

Voronov, Theodore

doi:10.1007/978-3-319-63594-1_29

Cited by 4 publications

(5 citation statements)

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“…We conclude, that volume of SU(−N) isn't given by analytically continued volume of SU(N). This is in correspondence with the fact, that volume of SU(N|M) doesn't analytically depend on N − M [30]. However, N ↔ −N remains symmetry of the theory, realized in more complicated way -volume function give rise to two analytical functions, which combine into the doublet of this symmetry.…”

Section: Resultssupporting

confidence: 54%

“…Various formulae for volume of compact simple Lie groups are given by Macdonald [15] (see also [8]), Marinov [17], Kac and Peterson [10], and Fegan [7]. For few series of (super)groups volume formulae are given by Voronov [30]. In this section we derive the universal expression for volume of compact Lie groups, which generalizes these formulae for an arbitrary points on Vogel's plane, and which presents them in a uniform way.…”

Section: Universal Invariant Volume Of Simple Lie Groupsmentioning

confidence: 96%

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Universal volume of groups and anomaly of Vogel’s symmetry

Khudaverdian

Mkrtchyan

2017

Lett Math Phys

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We show that integral representation of universal volume function of compact simple Lie groups gives rise to six analytic functions on CP 2 , which transform as two triplets under group of permutations of Vogel's projective parameters. This substitutes expected invariance under permutations of universal parameters by more complicated covariance.We provide an analytical continuation of these functions and particularly calculate their change under permutations of parameters. This last relation is universal generalization, for an arbitrary simple Lie group and an arbitrary point in Vogel's plane, of the Kinkelin's reflection relation on Barnes' G(1 + N ) function. Kinkelin's relation gives asymmetry of the G(1 + N ) function (which is essentially the volume function for SU (N ) groups) under N ↔ −N transformation (which is equivalent of the permutation of parameters, for SU (N ) groups), and coincides with universal relation on permutations at the SU (N ) line on Vogel's plane. These results are also applicable to universal partition function of Chern-Simons theory on three-dimensional sphere.This effect is analogous to modular covariance, instead of invariance, of partition functions of appropriate gauge theories under modular transformation of couplings.

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Section: Resultssupporting

confidence: 54%

Section: Universal Invariant Volume Of Simple Lie Groupsmentioning

confidence: 96%

Universal volume of groups and anomaly of Vogel’s symmetry

Khudaverdian

Mkrtchyan

2017

Lett Math Phys

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“…Thus we have achieved our goal of expressing the volume of a symplectic supermanifold M purely in terms of bosonic geometry. The result can be compared to explicit computations in examples, such as those in [97].…”

Section: A Volumes Of Symplectic Supermanifoldsmentioning

confidence: 99%

JT Gravity and the Ensembles of Random Matrix Theory

Stanford¹,

Witten²

2019

Preprint

112

356

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We generalize the recently discovered relationship between JT gravity and double-scaled random matrix theory to the case that the boundary theory may have time-reversal symmetry and may have fermions with or without supersymmetry. The matching between variants of JT gravity and matrix ensembles depends on the assumed symmetries. Time-reversal symmetry in the boundary theory means that unorientable spacetimes must be considered in the bulk. In such a case, the partition function of JT gravity is still related to the volume of the moduli space of conformal structures, but this volume has a quantum correction and has to be computed using Reidemeister-Ray-Singer "torsion." Presence of fermions in the boundary theory (and thus a symmetry (−1) F ) means that the bulk has a spin or pin structure. Supersymmetry in the boundary means that the bulk theory is associated to JT supergravity and is related to the volume of the moduli space of super Riemann surfaces rather than of ordinary Riemann surfaces. In all cases we match JT gravity or supergravity with an appropriate random matrix ensemble. All ten standard random matrix ensembles make an appearance -the three Dyson ensembles and the seven Altland-Zirnbauer ensembles. To facilitate the analysis, we extend to the other ensembles techniques that are most familiar in the case of the original Wigner-Dyson ensemble of hermitian matrices. We also generalize Mirzakhani's recursion for the volumes of ordinary moduli space to the case of super Riemann surfaces.

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“…The same holds for the odd variables θ. As clearly explained for example in the appendix of the paper [26] if one introduces also the natural concept of even differential (in order to make more contact with the standard definition of cotangent bundle of a manifold) our cotangent bundle (that we consider as the bundle of one-forms) should, more appropriately, be denoted by ΠT * .…”

Section: Super Hodge Dual and Super Convolutionsmentioning

confidence: 99%