Discrete field theory: symmetries and conservation laws

We present a general algorithm constructing a discretization of a classical field theory from a Lagrangian. We prove a new discrete Noether theorem relating symmetries to conservation laws and an energy conservation theorem not based on any symmetry. This gives exact conservation laws for several discrete field theories: electrodynamics, gauge theory, Klein-Gordon and Dirac ones. In particular, we construct a conserved discrete energy-momentum tensor, approximating the continuum one at least for free fields. The theory is stated in topological terms, such as coboundary and products of cochains.Comment: 40 pages, 7 figures; exposition improve

Packing a cake into a box

Given a cake in form of a triangle and a box that fits the mirror image of the cake, how to cut the cake into a minimal number of pieces so that it can be put into the box? The cake has an icing, so that we are not allowed to put it into the box upside down. V.G. Boltyansky asked this question in 1977 and showed that three pieces always suffice. In this paper we provide examples of cakes that cannot be cut into two pieces to put into the box. This shows that three is the answer to V.G. Boltyansky's question. Also we give examples of cakes which can be cut into two pieces.Comment: 9 pages, 13 figure

Suspension theorems for links and link maps

We present a new short proof of the explicit formula for the group of links (and also link maps) in the 'quadruple point free' dimension. Denote by $L^m_{p,q}$ (respectively, $C^{m-p}_p$) the group of smooth embeddings $S^p\sqcup S^q\to S^m$ (respectively, $S^p\to S^m$) up to smooth isotopy. Denote by $LM^m_{p,q}$ the group of link maps $S^p\sqcup S^q\to S^m$ up to link homotopy. Theorem 1. If $p\le q\le m-3$ and $2p+2q\le 3m-6$ then \begin{equation*} L^m_{p,q}\cong \pi_p(S^{m-q-1})\oplus\pi_{p+q+2-m}(SO/SO_{m-p-1})\oplus C^{m-p}_p\oplus C^{m-q}_q. \end{equation*} Theorem 2. If $p, q\le m-3$ and $2p+2q\le 3m-5$ then $LM^m_{p,q}\cong \pi^S_{p+q+1-m}$. Our approach is based on the use of the suspension operation for links and link maps, and suspension theorems for them.Comment: in English and in Russian, 12 pages, 3 figures; minor correction in the definition of the vertical homomorphisms in Theorem 3.

Lattice gauge theory and a random-medium Ising model

We study linearization of lattice gauge theory. Linearized theory approximates lattice gauge theory in the same manner as the loop O(n)-model approximates the spin O(n)-model. Under mild assumptions, we show that the expectation of an observable in linearized Abelian gauge theory coincides with the expectation in the Ising model with random edge-weights. We find a similar relation between Yang-Mills theory and 4-state Potts model. For the latter, we introduce a new observable.Comment: 10 pages, 2 figure

The rational classification of links of codimension >2

Fix an integer m and a multi-index p = (p_1, ..., p_r) of integers p_i < m-2. The set of links of codimension > 2, with multi-index p, E(p, m), is the set of smooth isotopy classes of smooth embeddings of the disjoint union of the p_i-spheres into the m-sphere. Haefliger showed that E(p, m) is a finitely generated abelian group with respect to embedded connected summation and computed its rank in the case of knots, i.e. r=1. For r > 1 and for restrictions on p the rank of this group can be computed using results of Haefliger or Nezhinsky. Our main result determines the rank of the group E(p, m) in general. In particular we determine precisely when E(p,m) is finite. We also accomplish these tasks for framed links. Our proofs are based on the Haefliger exact sequence for groups of links and the theory of Lie algebras.Comment: 16 page