On the volume conjecture for classical spin networks

Abstract

We prove an upper bound for the evaluation of all classical SU(2) spin networks conjectured by Garoufalidis and van der Veen. This implies one half of the analogue of the volume conjecture which they proposed for classical spin networks. We are also able to obtain the other half, namely, an exact determination of the spectral radius, for the special class of generalized drum graphs. Our proof uses a version of Feynman diagram calculus which we developed as a tool for the interpretation of the symbolic method of classical invariant theory, in a manner which is rigorous yet true to the spirit of the classical literature. Comment: More than 200 diagrams, pdf is better for download. Minor corrections, references added

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arXiv:0904.1734v2 [math.GT] 28 May 2009
On the volume conjecture for classical spin networks
Abdelmalek Abdesselam
May 28, 2009
Department of Mathematics, P. O. Box 400137, University of Virginia,
Charlottesville, VA 22904-4137, USA
email: malek@virginia.edu
In memoriam Pierre Leroux
Abstract. We prove an upper bound for the evaluation of all classical SU2
spin networks conjectured by Garoufalidis and van der Veen. This implies
one half of the analogue of the volume conjecture which they proposed for
classical spin networks. We are also able to obtain the other half, namely, an
exact determination of the spectral radius, for the special class of generalized
drum graphs. Our proof uses a version of Feynman diagram calculus which we
developed as a tool for the interpretation of the symbolic method of classical
invariant theory, in a manner which is rigorous yet true to the spirit of the
classical literature.
Mathematics Subject Classification (2000): 13A50; 22E70; 57M15; 57M25;
57N10;
Keywords: spin networks, ribbon graphs, 6-j symbols, angular momentum,
asymptotics, volume conjecture.
Contents
1 Introduction
1.1 Motivation
1.2History
1.3 Definitions and statement of results . . . . . . . . . . . . . . .
2
2
3
4
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Clebsch-Gordan networks and a brief tour of classical in-
variant theory10
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3 Clebsch-Gordan networks with external legs and SL2invari-
ance26
4 The negative dimensionality theorem32
5The existence of smooth orientations 40
6 The bridge reduction42
7 Estimates on special spin networks
7.1 The tetrahedron or 6-j symbol . . . . . . . . . . . . . . . . . .
7.1.1Preparation . . . . . . . . . . . . . . . . . . . . . . . .
7.1.2The upper bound . . . . . . . . . . . . . . . . . . . . .
7.1.3 The lower bound . . . . . . . . . . . . . . . . . . . . .
7.2The generalized drum . . . . . . . . . . . . . . . . . . . . . .
7.2.1 The upper bound . . . . . . . . . . . . . . . . . . . . .
7.2.2 The lower bound . . . . . . . . . . . . . . . . . . . . .
46
46
46
47
49
50
50
52
8 Proof of the main theorem56
1 Introduction
1.1 Motivation
The volume conjecture [55, 67] is one of the most important open problems
in low-dimensional topology, with vast ramifications in many active areas
of mathematics and theoretical physics (see, e.g., [77, 33, 34] and references
therein). The conjecture relates the exponential growth rate of the colored
Jones polynomial of a hyperbolic knot evaluated at a root of unity to the
hyperbolic volume of the knot complement. There is a close connection
between this problem and that of analysing asymptotics of q-deformed or
quantum spin networks [77, 68, 84, 26, 27, 86]. Perhaps as a simpler setting
for studying such asymptotics, a vigorous program for the systematic study
of large angular momentum asymptotics of classical spin networks (CSN),
with precisely formulated conjectures, was presented in [44]. It is the main
source of inspiration for the present article. As a further piece of motiva-
tion for the study of CSN’s, one can note that spin networks feature in a
great variety of topics, e.g., quantum computation [64], Tyurin’s approach
to nonabelian theta functions [85], shell models of turbulence [38], to only
cite some of the perhaps less well known.
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1.2 History
The introduction of CSN’s is usually attributed to Roger Penrose [69, 70]
who used them in an attempt to combinatorially quantize gravity. This is in
similar spirit to Regge’s calculus [75]. Indeed, by analysing the asymtotics
of CSN’s when all angular momenta are large, Ponzano and Regge [73] made
a strong case for the emergence of the Regge action of 3d gravity from this
semiclassical regime. This is based on a precise asymptotic formula for the
6-j symbol conjectured in [73] but proved much later by Roberts in [76].
Recently spin networks and their generalizations such as the Barrett-Crane
model [12] and spin foams [71] have become a staple food at the table of
loop quantum gravity (see [78] and references therein). A central issue in
this approach to quantum gravity is the understanding of this semiclassical
regime (see, e.g., [9, 13, 40, 50]).
Earlier, CSN’s essentially appeared in the works of the Lituanian School
of quantum angular momentum theory (QAMT) [88]. Indeed some sort of
graphical notation becomes indispensable when calculating complicated 3n-j
symbols. However, the beginnings of QAMT and the theory of CSN’s go
much further back to the classical invariant theory (CIT) of binary forms,
although there is some effort in mathematical translation needed in order to
see the connection. The ideal tool for this translation is Feynman diagram
calculus (FDC), namely a diagrammatic representation of contractions of
tensors as in the remarkable book [30]. Enough elements of such a trans-
lation were presented in [3, 4, 5] to suit the needs of these articles. More
is required here in order to translate the modern CSN formalism into the
framework of CIT, and this is the object of §2. A piece of data which ap-
pears naturally in the CIT/FDC picture is the notion of smooth orientations
of a cubic graph, according to the terminology of [60]. This is key to our
solution of some of the questions raised in [44].
There is a huge physical literature on QAMT where such objects as
Clebsch-Gordan, Clebsch-Gordon (sic), Clebsh-Gordon (sic),...coefficients
are ubiquitous. Yet, with only a few exceptions such as [32], this literature
shows almost no sign of awareness or aknowledgement of the work of Alfred
Clebsch and Paul Gordan. Aside from introducing some foundational mate-
rial needed in the subsequent proofs, we attemp to correct this injustice in
§2. Our hope is that by allowing the users of QAMT to tap into the vast and
most often perfectly rigorous 19th century literature on CIT, and through
cross-fertilization with modern theories in mathematics and physics, new
ideas and unexpected connections will emerge.
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1.3 Definitions and statement of results
We will follow the definitions of [44] where a CSN is defined as a pair (Γ,γ)
consisting of an abstract cubic ribbon graph Γ together with a decoration γ
of the edges by natural integers. By cubic ribbon graph we mean a trivalent
regular graph equipped with a cyclic ordering of the edges at each vertex.
These are also called rotation systems, or fat graphs, although one has to
be careful as definitions may vary in the literature [16, §2.1]. The notion we
use here is that of pure rotation systems as in [49, Chap. 3]. It is well known
that such a ribbon graph defines (up to orientation preserving equivalence
of imbeddings) a unique imbedding of the underlying abstract graph into a
compact orientable Riemann surface [49, Thm. 3.2.3]. Note that we allow
multiple edges and loops, i.e., edges with both ends attached to the same
vertex. We also allow Γ to be disconnected. Finally, although this means
a slight arm twisting on the usual definition of a graph, we allow trivial
components without vertices which are made of a single edge closing upon
itself.
a loop
,
a trivial
component
Note that the decorations γ = (γ(e))γ∈E(Γ)in NE(Γ)where E(Γ) is the edge
set of Γ must satisfy the following admissibility conditions. For every vertex,
the three integers a,b,c associated to the incident edges are such that a+b+c
is even and the triangle inquality |a − b| ≤ c ≤ a + b holds. Note that for a
loop vertex
a
b
the decoration a is counted twice so the constraint reduces to: b is even and
0 ≤ b ≤ 2a. For a trivial component
a
the decoration a can be any nonnegative integer.
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We can now define as in [44] the Penrose evaluation ?Γ,γ?Pof such a
spin network:
1. Use the imbedding into the surface Σ to thicken the vertices into discs
and the edges into bands.
2. On an edge carrying the decoration a, draw a parallel strands and a
perpendicular bar
P
which we call a Penrose bar:
a=3
−→
P
.
3. Connect the strands at each vertex as in the picture
a=6
b=7
c=5
−→
seen from outside Σ. This connection is made possible by the admis-
sibility constraints. It is also essentially unique since the number of
strands connecting the a edge to the b edge isa+b−c
2
, etc.
4. Replace each Penrose bar
tions σ ∈ Saas in
P
a
by an alternating sum over permuta-
P
a=3
−→
−−−++
.
5. Finally each term in the sum over permutations will produce a collec-
tion of closed curves drawn on the surface Σ. One associates a factor
(-2) to each such curve.
6. The Penrose evaluation ?Γ,γ?Pis the result of summing the corre-
sponding (-2) to the power of the number of curves times (-1) to the
power of the number of crossings, over all possible states or connection
schemes specified by the permutations σ at each edge.
5