2D SIGMA MODELS ON SUPER TARGETS AND RANDOM GEOMETRICAL PROBLEMS

Transcription

1 2D SIGMA MODELS ON SUPER TARGETS AND RANDOM GEOMETRICAL PROBLEMS H. Saleur Based on works with C. Candu, S. Caracciolo, Y. Ikhlef, J. Jacobsen, N. Read, V. Schomerus, A. Sokal and A. Sportiello. 1

2 SIGMA MODELS ON SUPER TARGETS: BACKGROUND Supergroups: symmetry groups mixing bosons and fermions. We will encounter U(n m), OSp(n 2m). Simplest example is GL(1 1) whose elements read ( ) a α g = β b where a, b are real numbers and α, β Grassmann numbers. The inverse matrix (obeying g 1 g = gg 1 = 1) exists iff ab 0 reads g 1 = b ab αβ β ab ab α a ab αβ We will encounter cosets such as the superspheres OSp(n 2m)/OSp(n 1 2m). In condensed matter physics super-manifold target spaces arise in the study of non interacting disordered systems. For instance the solution of the transition between plateaux in the integer quantum Hall effect boils down to identifying the strong coupling limit of the U(1, 1 2)/U(1 1) U(1 1) sigma model at θ = π. 2

3 Supergroups and super-coset targets appear in the description of strings in anti de Sitter space, which play a major role on the string theory side of the AdS/CFT duality. For instance strings on AdS 3 S 3 are related with P SL(2 2) sigma model. Supergroups in the context of geometrical problems appeared in the pioneering paper of Parisi and Sourlas as early as Supersymmetry/disorder/random geometry problems have a long tangled history. No confusion should arise with space time supersymmetry of particle physics. 3

4 WHY THESE MODELS ARE DIFFICULT Theories are violently non unitary: paths integrals are naively divergent, S matrices cannot be unitary ( p i = 1 but some p i > 1!). Underlying representation theory involves disgusting problems of indecomposability and wilderness. The space of CFTs itself is hard to map out. In the case of ordinary compact groups, conformal invariance and global group symmetry imply Kac Moody symmetry. WZW models are easily manageable. In the case of supergroups, WZW models form only a small class of possible CFTs. Other classes include: 1. PCM on OSp(2n + 2 2n), P SL(n, n) which are conformal with or without a WZW term and provide lines of CFTs 2. Supersphere sigma models OSp(2n+2 2n)/OSp(2n+ 1 2n) which also provide lines of CFTs 3. Strong coupling limits of super-projective sigma models U(n + m n)/u(1) U(n + m 1 n) at θ = π 4

5 4. Weak coupling limits of supersphere sigma models in (Goldstone) spontaneously broken symmetry phases In these other classes local Noether currents obey logarithmically deformed current OPEs, and or involve null states. There is, to this day,no known classification principle. 5

6 A QUICK TOUR OF THE FIELD: DENSE POLYMERS REVISITED Dense polymers are one of the best understood universality classes of 2D geometrical problems. They are obtained by considering a finite number of SAWs (or SALs) on a lattice such that, in the thermodynamic limit, the density of monomers remains finite. Forcing the density to be maximal on finite lattices (ie, considering hamiltonian walks) can change the universality class to those of compact polymers which are non generic and unstable under any small decreases of density. 6

7 Nevertheless loop coverings of the surrounding lattice of the square lattice are known to be in the universality class of dense polymers. RELATIONS WITH SUPER SIGMA MODELS? Here is the plan: (Super) spin chains for dense polymers 6 leg crossings and the conformal CP n 1 n sigma model (unpublished!) 4 leg crossings and the S n 1 n sigma model Duals: trees and forests 7

8 SUPER SPIN CHAINS FOR DENSE POLYMERS AND THE SUPER PROJECTIVE SIGMA MODEL 8

9 Since we allow two and only two possible splittings at every vertex of the lattice, we can consider that the edges carry a fixed orientation (this is reminiscent of the Chalker Coddington model): We can think of transfer matrices propagating say vertically x y Take 2L sites labelled i = 0,..., 2L 1. With odd sites associate the of SU(m) and with even sites. Use a bosonic representation with vector space V i = C m at each site. 9

10 The states can be represented using b a i, b ia for i even, b ia, b a i for i odd, with [b a i, b jb ] = δ ijδb a (a, b = 1,..., m), and similarly for i odd. The destruction operators b a i, b ia destroy the vacuum state, the daggers indicate the adjoint, and the spaces V i are defined by the constraints b ia ba i = 1 (i even), b a i b ia = 1 (i odd) of one boson per site. Generators of U(m) (or in fact of gl m ) acting in the spaces V i are Jia b = b ia bb i for i even, Jia b = bb i b ia for i odd, and the commutation relations among the J i s (for each i) are i-independent. Hence the global gl m algebra, defined by Ja b = i Jia b, acts in the tensor product V = ( ) L. SU(m)-invariant nearest-neighbor coupling in the chain is unique, up to additive and multiplicative constants. It is the usual Heisenberg coupling of magnetism, and can be written in terms of operators E i E i = b a i+1 b ia bb i b i+1,b, i even, b a i b i+1,a bb i+1 b ib, i odd. The E i s are Hermitian, E i = E i. Acting in the constrained space V, they satisfy Ei 2 = me i, E i E i±1 E i = E i, E i E j = E j E i (j i, i ± 1). 10

11 which define the Temperley Lieb algebra T L N (m). Relations have well known graphical interpretation: Transfer matrices propagating along the (1, 1) direction of the square lattice can be written in terms of elementary vertex interactions T t 1 t 3 t 2L 3 t 0 t 2 t 2L 2, By taking either of the two terms in t i for each vertex in the graph, the expansion in space-filling loops mentioned above is obtained with corresponding coefficients for each vertex, and a factor m for each loop. It is possible to generalize this to the closed (periodic) version of the SU(m) spin chain models by invoking more complicated algebraic objects (periodic Temperley Lieb algebras).

12 To now get m = 0 we generalize the construction to the SU(m + n n) case. Now each site carries a Z 2 - graded vector space of dimensions m + n for the even (bosonic), n for the odd (fermionic), subspace (n 0 is an integer). This space is the fundamental of the Lie superalgebra gl(m + n n) for i even, and its dual for i odd. The chain is the graded tensor product of these V i. It may be constructed using fermion operators f ia, f a i for a = m + n + 1,..., m + 2n, while a = 1,..., m + n corresponds to boson operators as in the n = 0 special case. For i even we have boson operators b a i, b ia, [ba i, b jb ] = δ ij δ a b (a, b = 1,..., n + m), and fermion operators f α i, f iα, {f i α, f jβ } = δ ijδβ α (α, β = 1,..., n); here labels like α on the fermion operators stand for α = a (m+n) for a corresponding a index. For i odd, we have similarly boson operators b ia, b a i, [b ia, b b j ] = δ ijδa b (a, b = 1,..., n + m), and fermion operators f iα, f α i, {f iα, f β j } = δ ij δα β (α, β = 1,..., n). Notice the minus sign in the last anticommutator; since our convention is that the stands for the adjoint, this minus sign implies that the norm-square of any two states that are mapped onto each other by the action of a single f iα or f α i have opposite signs, and 11

13 the Hilbert space has an indefinite inner product. The space V is now defined as the subspace of states that obey the constraints a a b ia ba i + α b a i b ia α f iα f α i = 1 (i even), f α i f iα = 1 (i odd). The generators of the Lie superalgebra gl(m + n n) acting on each site of the chain are the bilinear forms Jia b = b ia bb i, f iα f β i, b ia f β i, f iα bb i for i even, and similarly for i odd. The TL generators are constructed as follows. First, we note that for any two sites i (even), j (odd), the combinations a b ja b a i + α f jα f α i, a b ia ba j + α f iα f α j are invariant under gl(n+m n). The introduce for each pair of neighbors i, i + 1 d Vi+1 = a b i+1,a b a i + α f i+1,α f α i (i even), d Vi+1 = a b ia b a i+1 + α f iα f α i+1 (i odd), b Vi = a b Vi = a b ia ba i+1 + α b i+1,a ba i + α f iα f α i+1 (i even), f i+1,α f α i (i odd). 12

14 Then the TL generators can be written as for all i. e i = b Vi d Vi+1 It is important to stress that in the loop model formulation, a factor (n + m) m = str 1 = m is obtained for each loop (in all the models considered in this paper, we will consider only m, n integer, with n + m, n 0). The latter is equal to the supertrace in the fundamental representation (denoted str), of 1, because the evaluation of contributions for each loop can be viewed in terms of states in V flowing around the loop. This holds true for loops homotopic to a point as well as for topologically nontrivial loops. Superalgebra invariance requires that non contractible loops (in the space direction) for a periodic system have the same weight as contractible ones.changing their weight requires breaking the supersymmetry. This kind of formalism is useful to properly build the CFT of dense polymers or hulls of percolation, It is what gave rise to the discovery that dense polymers were related with symplectic fermions. More on this later. More to the point here, the fact that eg dense polymers are related with SL(n n) Heisenberg chain allows to use profitably sigma model ideas. 13

15 Let us recall what happens in the usual SU(2) case. The general strategy is to use geometrical or coherent state quantization to describe individual spins via a path integral. Taking (antiferromagnetic) interactions into account, in the large spin limit one gets the O(3) = SU(2)/U(1) sigma model at θ = 2πs. One also finds that the bare coupling constant is gσ 2 1 s and that there is a flow to large coupling so physics at s = 2 1 is described by the XXX antiferromagnetic spin chain, and thus the SU(2) 1 WZW model. Extending this to SU(m) and alternating reps (so as to have antiferromagnetic physics) can be done using coherent state representation of the states on every site of the spin chain. This leads to matrices living in the group modulo the isotropy group of the highest weight state, Q = g hw hw g 1. For reps... and their conjugates this gives SU(m)/SU(m 1) U(1). One can show similarly that in the case at hand, large distance physics for the hamiltonian H = e i is described by the U(n + m n)/u(1) U(n + m 1 n) or CP n+m 1 n model at θ = π. The fields can be represented by complex components z a (a = 1,..., n + m), ζ α (α = 1,..., n), where z a 14

16 is commuting, ζ α is anticommuting. In these coordinates, at each point in spacetime, the solutions to the constraint z az a +ζ αζ α = 1 (conjugation obeys (ηξ) = ξ η for any η, ξ), modulo U(1) phase transformations z a e ib z a, ζ α e ib ζ α, parametrize CP n+m 1 n. The Lagrangian density in two-dimensional Euclidean spacetime is L = 1 [ ( µ 2gσ 2 ia µ )z a( µ + ia µ )z a + ( µ ia µ )ζ α( µ + ia µ ) where a µ (µ = 1, 2) stands for a µ = i 2 [z a µz a + ζ α µζ α ( z a )za ( ζ α )ζα ], Fields are subject to the constraint, and under the U(1) gauge invariance, a µ transforms as a gauge potential; a gauge must be fixed in any calculation. This setup is similar to the nonsupersymmetric CP m 1 model The coupling constants are g 2 σ, (there is only one such coupling, because the target supermanifold is a supersymmetric space), and θ, the coefficient of the topological term, so θ is defined modulo 2π.

17 Now we finally get to the important points. the β function of the model obeys First, dg 2 σ = β(g 2 dl σ) = mgσ 4 + O(gσ) 6 For m = 0 is vanishes to leading order. In fact the β function is independent of n. Now for n = 1 we have a theory of symplectic fermions L = 1 2gσ 2 µ ζ µ ζ For which it is clear that g 2 σ is redundant. Thus β = 0 to all orders. We have a conformal sigma model (more about topological angle later). Strictly speaking, the most general SU(n n) hamiltonian allows for more interactions. The simplest one is given by permutations of (next to nearest neighbour) reps or. In the hamiltonian language this corresponds to considering H = e i + w P i,i+2 15

18 (the Wall Brauer algebra), that is allowing 6 leg crossings - something do-able elegantly on the triangular lattice. So it is natural to expect that the continuum limit of this more general H(w) corresponds to the sigma model with gσ 2 a function of w (one can argue that gσ 2 0 as w ). Now the second point is that the L legs polymer operators for L > 2 do not belong to the psl(1/1) case, only the n > 1 one. So we expect: That as 6 leg crossings are allowed the L > 2 exponents change while the symplectic fermions psl(1/1) subsector does not. Recall the formula for usual dense polymers h L = L So in particular h 2 = 0, h 4 = 12 32, h 6 = 1, h 8 = In the weak coupling limit gσ 2 small meanwhile, we can solve the sigma model using the minisuperspace (particle limit) approach. The general spectrum of the Laplacian on CP m is given by E l = 4l(l + m) which leads, after setting m = 1 to a prediction for the exponents (2l L then) h L = gσ 2 [ (L 1) 2 1 ] 16

19 so h 2 = 0, h 4 = 8g 2 σ, h 6 = 24g 2 σ, h 8 = 48g 2 σ... We can conveniently compare ratios of exponents: h 6 /h 4 = 32/12 (h 8 /h 4 = 5) for usual dense polymers but equal to 3 (6) in the small coupling limit of the sigma model h 2 h 4 where L 10 2L 12 2L 14 2L 16 2L w

20 It is not clear what the exact values of the exponents might be. There are expectations that the minisuperspace formula is exact, and that there is a singularity at w = 0 (region w < 0 not clear either). The spin chain formalism is crucial to derive fusion rules for the polymer theory. It shows the role of projective representations of the Virasoro algebra. Other values of m are usueful as well: m = 1 corresponds to percolation and spin quantum Hall effect. This model is a close cousin to the P SL(n n) PCM which has been proposed to describe the large coupling limit of the quantum Hall sigma model (usually believed not to be a correct guess). 17

21 FOUR LEGS CROSSINGS IN DENSE POLYMERS AND THE SUPER SUPER SPHERE SIGMA MODEL 18

22 Although SAWs are usually referred to as O(n), n 0, it is important to realize that our favorite lattice models have not O(n) but U(n) symmetry. The best way to see this is to think of the dense case again and of a spin chain built now with the fundamental (vector) representation of OSP (m+2n 2n) on every site. Recall = 1 + Adj SU(m) V V = 1 + Sym + Antisym O(m) Going over steps similar to SU(m) we now have three possible interactions in the chain, which can be represented, in terms of a loop model as = w t I E P Hence configurations will look generically as allowing now four leg crossings. 19

23 Another way to think of this is that when four leg crossings are allowed, links cannot be oriented any longer Now in the dilute case the four leg operator is irrelevant. But it becomes relevant h = in the dense case. We thus expect a flow to another CFT. Long distance properties should be described by the supersphere sigma model OSp(2n 2n)/OSp(2n 1 2n). Indeed the Mermin Wagner theorem which forbids the spontaneous breaking of a continuous symmetry in two dimensions does not hold for supergroups (because of the lack of unitarity)! Models with orthosymplectic OSp(2n + m 2n) symmetry do exhibit a low temperature phase with spontaneous broken symmetry provided m < 2. Use as coordinates a real scalar field : φ (φ 1,..., φ m, ψ 1,..., ψ 2n ) 20

24 and the invariant bilinear form φ φ = φ a φ a + J αβ ψ α ψ β where J αβ is the symplectic form which we take consisting of diagonal blocks :. The unit su- ( ) persphere is defined by the constraint : φ.φ = 1 Action of the sigma model (conventions are that the Boltzmann weight is e S ) S = 1 2gσ 2 d 2 x µ φ. µ φ The perturbative β function depends only on m to all orders : β(g 2 σ ) = (m 2)g4 σ + O(g6 σ ) The model for g σ positive thus flows to strong coupling for m > 2. Like in the ordinary sigma models case, the symmetry is restored at large length scales, and the field theory is massive. For m < 2 meanwhile, the model flows to weak coupling, and the symmetry is spontaneously broken. One expects this scenario to work for g σ small enough, and the Goldstone phase to be separated from a non perturbative strong coupling phase by a critical point. 21

25 Stating the behaviour of the beta function might not be enough to convince one of the invalidity of the Mermin Wagner theorem. The point in usual O(m) models is that fluctuations of the transverse degrees of freedom diminish the value of the two point function of the order parameter in the searched for broken symmetry phase by logarithmic terms that grow at large distances. To lowest order in g 2 σ n(r) n(0) = [ 1 cstgσ(m 2 2) ln R ]m 1 m 2 a where we have not made explicit the cst term which is normalization dependent (but positive and independent of m). For m > 2 we see indeed that fluctuations diminish the expectation value. But if m < 2 we see that they in fact increase this value. In the Goldstone phase, the supersphere sigma models turn out in fact to have correlations that diverge (logarithmically) at large distances. This is because the symmetry being spontaneously broken the fundamental field has non vanishing expectation value, and thus the fields φ a do exist in the conformal field theory, unlike in the case of a compact boson. 22

26 The central charge in the w > 0 theory should become c = 1 (recall it is c = 2 at w = 0!) and the fuseau exponents should all scale to zero logarithmically 23

27 We now specialize to the case of the OSp(2 2) model. The UV limit is easy to understand. We can parametrize the supersphere by setting φ φ ψ 1ψ 2 = 1 φ 1 = cos ϕ (1 ψ 1 ψ 2 ) φ 2 = sin ϕ (1 ψ 1 ψ 2 ), ϕ ϕ + 2π The action then reads S = 1 2g 2 σ d 2 x [ ( µ ϕ) 2 (1 2ψ 1 ψ 2 ) + 2 µ ψ 1 µ ψ 2 4ψ 1 ψ 2 µ ψ 1 µ ψ 2 ] The coupling gσ 2 > 0 flows to zero at large distances. On the other hand, we can absorb it by rescaling all fields so the action reads S = 1 2 d 2 x [ ( µ ϕ) 2 (1 2g 2 σ ψ 1ψ 2 ) + 2 µ ψ 1 µ ψ 2 +4g 2 σψ 1 ψ 2 µ ψ 1 µ ψ 2 ] where now ϕ has a different radius, ϕ ϕ + 2π g σ. We see that as g σ 0 all interaction terms disappear and we get a free boson φ together with a pair of free symplectic fermions ψ 1,2. Moreover the radius of compactification goes to infinity in that limit, so the boson ϕ appears as non compact. 24

28 This holds in the true large distance limit. At intermediate scales, we can use the RG equation for the coupling (now with precise normalizations) dg 2 σ d log l = m 2n 2 gσ 4 = 1 2π π g4 σ Writing more generally dgσ 2 d log l = αg4 σ we see that g σ approaches its vanishing large distance value as 1 = 1 (gσ 0 + α log(l/l 0) α log(l/l 0 ) )2 g 2 σ Here, l is a characteristic dimensionless scale ratio, roughly of the order of the ratio of the scale at which one is observing the physics to the lattice cut-off. On the cylinder, l can be identified with the width in lattice units, l = N. In the limit of large l, we can estimate more precisely the contribution to the spectrum coming from the boson φ. Recall that for a free bosonic theory where the action is normalized as S = 8π 1 ( µ X) 2 and the field compactified as X X + 2πR, the spectrum of dimensions is x = e2 R 2 + m2 R

29 Matching the normalization gives R 2 = 4 π gσ 2 in our case, and thus we expect the scaled gaps coming from the bosonic degrees of freedom to read, at large distances : x = e 2 4πα log(l/l 0 ) + m2 πα log(l/l 0 ) In the limit l the dimensions become degenerate and the spectrum can be considered as a continuum starting above + = 0. To emphasize the latter point, consider the contribution to the partition function coming from the φ degrees of freedom: Z ϕ = = R 1 η η e,m q (e/r+mr/2)2 /2 q (e/r mr/2)2 /2 R 2 1 Im τη η R 1 2 Im τη η m,m exp ( πr2 mτ m 2 2Im τ where η(q) = q 1/24 n=1 (1 q n ) = q 1/24 P (q). Observe now that one can write 1 = 4 Im τη η 0 s2 qs2 q which can be interpreted as an integral over a continuum of critical exponents = = s 2. η η )

30 In the partition function R plays the role of the density of levels, and is proportional to the (diverging) size of the target space. We thus get a non compact target starting from a model with finite dimensional representations on every site. Going back to the specific case of the OSp(2 2)/OSp(1 2 model, we have α = 1/π. We get the radius R 2 = 4 log(l/l 0 ), and the contribution of the free boson ϕ to the spectrum is : x = (e 2) 2 4 log(l/l 0 ) + (m 2) 2 log(l/l 0 ) with e 2, m 2 arbitrary integers. Hence for any w 0, we expect that for large systems the spectrum of the hamiltonian will be made of a discrete symplectic fermion component, and a continuum component coming from the boson. While this is very hard to establish numerically in general, it turns out that for the special value w = 2 1 the model is integrable. That such a value exists is expected using the general quantum inverse scattering construction for the algebra OSp(2 2) with the four dimensional fundamental representation on every site. 26

31 The Bethe equations can be written in various ways. The most elegant is, symbolically, ( ) α i N = α β 2i α + i α β + 2i ( ) N β i = β α 2i β + i β α + 2i with the energy (the scale insures relativistic dispersion relation) E = 4 π α β 2 A study of the different sectors for sizes up to 10,000 shows that the smallest gap vanishes indeed logarithmically with the size x 1,-1-1 a log N + b x 1,-1, gamma=0.495 π, x = 1 / (a log N + b) N 27

32 corresponding to the smallest gap going as x = K ln(l/l 0 ), K in good agreement with the sigma model which predicts K = 1. The Goldstone phase should describe polymers in thin layers. It is fascinating how close to Brownian chains dense walks with four leg crossings can be. 28

33 TREES, FORESTS, FIELDS AND MORE SUPER SIGMA MODELS 29

34 Dense polymers are dual to spanning trees. trees are an interesting topic in combinatorics. Now Consider a loopless undirected graph G = (V, E) with vertex set V and edge set E. The number of vertices is V = n. With each edge e k E associate a formal parameter e k. Define the discrete Laplacian of G as an n n matrix M = {m ij }, with matrix elements, for i j, m ij = connect and for i = j, m ii = + incident e k (sum over edges connecting i and j) e k (sum over edges incident with i) Note that upon setting all e k = 1 this becomes just the standard Laplacian. Kirchhoff s theorem (also known in the mathematics literature as the Matrix Tree Theorem ) is that det M = 0, while the determinant of the matrix M(i) obtained by removing the i th line and the i th column is independent of i, and equal to 30

35 det M(i) = T e k e k T where the sum is over spanning trees T of G. Recall that a tree is a connected graph T G having no cycles; T is said to span G if they have the same vertex set V. Setting {e k } = 1, det M(i) becomes just the number of spanning trees, which is the usual formulation of Kirchhoff s theorem in the physics literature. To generalize the concept of spanning trees, we define spanning p-forests as a set of p disconnected trees, such that each vertex belongs to one, and only one, of the p tree components (an isolated vertex is a tree). An interesting result about forests follows from Kirchhoff s theorem, and is known in the physics literature as well: det M(i 1, i 2,..., i p ) = F p e k. T i F p e k T i Here M(i, j,...) denotes the matrix obtained from M by removing the i th, j th,... lines and columns. On the right-hand side, the sum is over p-forests such that the vertices i 1, i 2,..., i p all belong to different trees {T i } p i=1. 31

36 A simple proof of the result is obtained by observing that det M(i 1, i 2,..., i p ) is the edge product summed over spanning trees on the graph obtained by identifying the vertices i 1, i 2,..., i p. Indeed, represent this contracted graph, and for the edges connecting to i 1, i 2,..., i p which now have a common end point, keep track of which point they were connecting to by giving them an extra label i k (there is an ambiguity if an edge connects two of the i k s, but since these are loops of the contracted graph they cannot appear in the spanning tree anyway). For each tree on the contracted graph one can, by undoing the contraction, obtain a p-forest such that the vertices {i k } p k=1 all belong to different trees. Namely, if two of these vertices were to belong to the same tree, there would have been a cycle in the spanning tree on the contracted graph, which is impossible. Conversely, if one has an n-forest such that the sites i k all belong to different trees, the identification of {i k } p k=1 gives a spanning tree on the contracted graph. 32

37 It is now convenient to introduce a pair of Grassman variables η 1 (i), η 2 (i) per site. By definition, any two of these variables anticommute, η k η l + η l η k = 0, and we have the integration rules dη k 1 = 0 and dη k η k = 1. In particular we have: dη 1 (i)dη 2 (i) η 1 (i)η 2 (i) = 1. i i Introducing the shorthand notation dη 1 dη 2 = i dη 1 (i)dη 2 this allow us to reformulate Kirchhoff s theorem as dη 1 dη 2 e η 1Mη 2 = det M = 0, dη 1 dη 2 η 1 (i)η 2 (i)e η 1Mη 2 = det M(i) = T e k. e k T Note that the factor η 1 (n)η 2 (n) effectively deletes the n th line and column which would otherwise have contributed to the determinant, through the development of the exponential. We then introduce the two objects (n is the number of vertices) Q 1 = Q 2 = 1 2 n i=1 n η 1 (i)η 2 (i), i,j=1 ( m ij )η 1 (i)η 2 (i)η 1 (j)η 2 (j). 33

38 Observe now that dη 1 dη 2 Q p 1 eη 1Mη 2 = F p p! e k e k F p T i F p V i. Here, V i is the number of vertices in the tree T i. To prove this statement we note that each term η 1 (i 1 )η 2 (i 1 ) in the expansion of Q p 1 gives a p-forest with vertices i 1,..., i p all belonging to different trees. The sum over all the terms weighs each tree by its number of vertices, and the p! is a symmetry factor. Similarly, one has dη 1 dη 2 Q p 2 eη 1Mη 2 = F p p! e k e k F p T i F p E i, where E i is the number of edges in the tree T i. see this, we first consider the quantity p k=1 ( m ik j k ) det M(i 1, j 1,..., i p, j p ), To which will appear in the expansion of Q p 2. Here for each k = 1,..., p, the vertices i k and j k are supposed to be connected by an edge in E (otherwise, one would have m ik j k = 0). The product p k=1 m i k j k turns the 2p-forest represented by det M(i 1, j 1,..., i p, j p ) into a p-forest, since each factor of m ik j k joins two tree components. 34

39 In this p-forest, i k and j k belong to the same tree T k, and T k is further required to contain the edge joining i k and j k. Summing over all vertices i k, j k, each tree component gets weighed by its number of edges,. Note that the factor 2 p coming from the different assignments of end-points to the edges (i k j k ) cancels the factor 2 p in the development of Q p 2 ; the symmetry factor p! for permuting the tree components among themselves is the same as before. The main use of theses result is in the grand generating function, dη 1 dη 2 e a 1Q 1 +a 2 Q 2 e η 1Mη 2 = e k p=1 F p e k F p T i F p (a 1 V i + a 2 E i ). A particular case of this identity is when {e k } = 1 and a 2 = 0. One then gets det(m + a 1 I) = a p 1 p=1 F p T i F p V i, which is a result of David and Duplantier. 35

40 Another use is obtained with a 1 = a 2. One then gets, using that V i = E i + 1 for each component tree, dη 1 dη 2 e a 1(Q 1 Q 2 ) e η 1Mη 2 = a p 1 p=1 F p 1 = p=1 a p 1 N(F p), where N(F p ) is the number of p-forests on G. Hence actions with four fermion terms also have a geometrical interpretation. Now what about super sigma models? 36

41 It turns out that the generating functions of spanning forests - the functional integral with a 1 = a 2 a possesses a non linearly realized OSP (1/2) symmetry, for any graph. To see this, let us introduce on each vertex an auxiliary field x(i) subject to the constraint that x 2 (i) + 2aη 1 (i)η 2 (i) = 1. Solving for x gives two solutions, one whose body is positive, the other one whose body is negative. We will for now restrict to the first choice, indicated by + sign in the integrals below. Using the basic rule that δ(x 2 + 2a 1 η 1 η 2 1) = 1 2x δ(x + a 1η 1 η 2 1), together with 1 x = ea 1 η 1 η 2 2, one can entirely eliminate the a 1 term in the action and write the functional integral as Z 2 N = + exp i i,j dx(i)dη 1 (i)dη 2 (i)δ[x 2 (i) + 2a 1 η 1 (i)η 2 (i) 1] η 1 (i)m ij η 2 (j) + a 2 <ij> η 1 (i)η 2 (i)η 1 (j)η 2 (j) We now introduce the vector u = (x, η 1, η 2 ) with the scalar product u(i) u(j) = x(i)x(j) + a 1 η 1 (i)η 2 (j) + aη 1 (j)η 2 (i). By explicitely writing down the discrete Laplacian terms and regrouping a bit, one finds that the action can be rewritten as A = 1 a 1 (a 1 + a 2 ) <ij> <ij> u(i) u(j) 1 + η 1 (i)η 2 (i)η 1 (j)η 2 (j) 37

42 The case a 1 = a 2 a is obviously very special since then the additional four fermions term disappears from the action, which exhibits global OSP (1/2) symmetry. The generating function in fact looks exactly like the one of a discretized sigma model on the target manifold OSP (1/2)/OSP (0/2) UOSP (1/2)/SU(2). It is important to notice that the parameter a does not affect the symmetry. Indeed, it can always be absorbed in a rescaling of the fermions. Thus the problem is entirely equivalent to a N Z 2 N = + dxdη 1dη 2 δ(x 2 + 2η 1 η 2 1) exp 1 a <ij> ( u i u j 1) An infinitesimal OSP (1/2) transformation reads δx = δξ 1 η 1 + δξ 2 η 2 δη 1 = δξ 2 x + δaη 1 + δcη 2 δη 2 = δξ 1 x + δbη 1 δaη 2 where δξ 1, δξ 2 are small fermionic deformation parameters, δa, δc small bosonic parameters. By definition, this change leaves x 2 + 2η 1 η 2 invariant. 38

43 In terms of the fermion variables, the symmetry is realized non linearly: δη 1 = δξ 2 (1 η 1 η 2 ) + δaη 1 + δcη 2 δη 2 = δξ 1 (1 η 1 η 2 ) + δaη 1 δaη 2 The issue of the sign ambiguity in solving the constraint is a bit delicate. Strictly speaking, what we have seen is that the arboreal gas model maps onto the hemi-supersphere sigma model. The correspondence is fine perturbatively, but it breaks down non perturbatively. This is not expected to affect the physics in the broken symmetry phase, where the field is slowly varying - but it might have an incidence on the critical point for instance. We now specialize to the square lattice, and take the continuum limit by introducing slowly varying fermionic fields η 1 (x, y) and η 2 (x, y). The euclidean action (with Boltzmannn weight e S ) reads S = 1 g 2 σ dxdy [ ( µ x) µ η 1 µ η 2 ]. with coupling gσ 2 = 2a. Note that no lattice spacing appears in this action as the a 2 0 from the expansion of the fields is compensated by the 1/a 2 0 from the transformation of the discrete sum over lattice sites into an integral. 39

44 We can then read the phase diagram of the forest model from the known results about the sigma model dgσ 2 d log l = m 2n 2 2π g 4 σ = 1 π g4 σ For gσ 2 < 0 (a > 0), the model flows to weak coupling in the IR, where the symmetry is restored and excitations are massive. For gσ 2 > 0 (a < 0), the model is massless in the IR, and described by a symplectic fermions theory. This works up to the critical coupling a c where, by analogy with what is known to happen for models based on ordinary groups in higher dimension, the symmetry is restored. These conclusions are compatible with our knowledge of the phase diagram of the Q-state Potts model, using the fact that the generating function of spanning forests is identical with the Q 0 limit of the Q state Potts model partition function, provided that e K 1 Q 1 a. One finds as a bonus the value of the critical coupling, which corresponds with the antiferromagnetic critical point, a c = 4. 40

45 Indeed write the partition function as Z Potts = = G e Kδ σ i σ j σ i =1,...,Q <ij> ( e K 1 ) B Q Cl ( ) Q S a G a Cl Cy Q Cy where G are subgraphs of the lattice made of B bonds and Cl clusters (connected components), S is the total number of sites, Cy the number of cycles on the clusters. We have used Euler s relation (valid in the plane or on an annulus, a geometry we will consider below), S = B + Cl Cy. In the last equation Q 0, e K 1 Q a. As Q 0, only graphs without cycles survive, giving the result for the arboreal gas after a renormalization of Z. This is true irrespective of the lattice. A lot more is known for the square lattice: 41

46 1 e K K e > 1-Q/4 : Massive region K e = 1-Q/4 : Critical point e K< 1-Q/4 : Massless region; flows to c=-2 Q Of course the questions of the universality class at the critical value, of the geometrical exponents in the massless phase and at the critical point etc are all interesting!

47 CONCLUSIONS: One of the most fascinating aspects is the emergence of non compact target spaces, having to do with a certain amount of relaxing of the SAW constraint.. While this is useful for applications (from CFT to quantum Hall effect) one can wonder about SLE. Can it predict the new exponents? Another aspect is the algebraic machinery required to properly build CFTs for geometrical problems. Dense polymers have even more ramifications: an example is the TSP. So here is a map of Finland (a country with 4461 cities rather homogeneously distributed cities 42

48 43