We propose a nonperturbative approach to nonabelian two-form gauge theory. We formulate the theory on a lattice in terms of plaquette as fundamental dynamical variable, and assign U(N) Chan-Paton colors at each boundary link. We show that, on hypercubic lattices, such colored plaquette variables constitute Yang-Baxter maps, where holonomy is characterized by certain dynamical deformation of quantum Yang-Baxter equations. Consistent dimensional reduction to Wilson's lattice gauge theory singles out unique compactness condition. We study a class of theories where the compactness condition is solved by Lax pair ansatz. We find that, in naive classical continuum limit, these theories recover Lorentz invariance but have degrees of freedom that scales as N^2 at large N. This implies that nontrivial quantum continuum limit must be sought for. We demonstrate that, after dimensional reduction, these theories are reduced to Wilson's lattice gauge theory. We also show that Wilson surfaces are well-defined physical observables without ordering ambiguity. Utilizing lattice strong coupling expansion, we compute partition function and correlation functions of the Wilson surfaces. We discover that, at large $N$ limit, the character expansion coefficients exhibit large-order behavior growing faster than exponential, in striking contrast to Wilson's lattice gauge theory. This hints a hidden, weakly coupled theory dual to the proposed tensor gauge theory. We finally discuss relevance of our study to topological quantum order in strongly correlated systems.
Deep Dive into A Nonperturbative Proposal for Nonabelian Tensor Gauge Theory and Dynamical Quantum Yang-Baxter Maps.
We propose a nonperturbative approach to nonabelian two-form gauge theory. We formulate the theory on a lattice in terms of plaquette as fundamental dynamical variable, and assign U(N) Chan-Paton colors at each boundary link. We show that, on hypercubic lattices, such colored plaquette variables constitute Yang-Baxter maps, where holonomy is characterized by certain dynamical deformation of quantum Yang-Baxter equations. Consistent dimensional reduction to Wilson’s lattice gauge theory singles out unique compactness condition. We study a class of theories where the compactness condition is solved by Lax pair ansatz. We find that, in naive classical continuum limit, these theories recover Lorentz invariance but have degrees of freedom that scales as N^2 at large N. This implies that nontrivial quantum continuum limit must be sought for. We demonstrate that, after dimensional reduction, these theories are reduced to Wilson’s lattice gauge theory. We also show that Wilson surfaces are wel
"What is the use" thought Alice "of a book if it has no picture or conversation?" -Louis Carroll An outstanding problem in theoretical physics is a constructive definition of p-form gauge theories, especially, nonabelian and self-interacting ones. Variants of p-form gauge theory arise in diverse contexts, ranging from string or M theories [1,2] and higher-dimensional integrable systems [3] to topological order and phases in strongly correlated systems [4] and to quantum error correction codes [5] in quantum information sciences. Of particular interest is whether a nonabelian p-form gauge theory exists and, if so, what sort of self-interactions are allowed by the gauge invariance.
To the problem posed, one’s first guess is that the fundamental degrees of freedom are some sort of nonabelian extension of the abelian p-form gauge theory, but then the question is now more to “what types of nonabelian extensions can be endowed to abelian p-form gauge theory?” and to “what types of self-interaction are possible for a given nonabelian extension?”.
The Z 2 Ising model provides the simplest situation of all. Consider the model defined on a d-dimensional hypercubic lattice. In dual formulation, the Ising model is mapped to a gauge theory, where the gauge potential is a p = (d -2)-form and takes a value in G = Z 2 . As is firmly established, the Ising model does not exhibits any nontrivial renormalization group fixed point for d ≥ 4. It implies that the dual p-form gauge theory in d ≥ 4 ought to be free in the continuum limit, yielding no obvious self-interaction among the p-form gauge potentials * .
In string and M theories, variety of the p-form gauge potentials is rich and complex. Foremost, all known string theories, whether supersymmetric or not, contain universally the Kalb-Ramond two-form potential B 2 = 1 2! B mn x . m ∧x . n , and the fundamental string couples minimally to it [1]. Type I and II superstring theories contain in addition R-R(Ramond-Ramond) p-form potentials [2]. The D(p -1)-branes are the charged objects coupled minimally to these R-R potentials. Because of different chirality projection, type IIA superstring gives rise to p = 1, 3, • • •, 9 only, while Type IIB superstring does so for p = 0, 2, • • •, 10 only. These NS-NS and R-R fields exhaust all possible pform potentials permeating through the ten-dimensional bulk spacetime. To the extent understood so far, they are essentially abelian. There are, in addition, p-form potentials residing only on the worldvolume of string solitons.
when N parallel D-branes stack on top of one another, combinatorially, there are N × N possible open strings and the lowest excitation of them forms U(N) matrix-valued 1-form potential [8]. Thus, for D-branes, N 2 variety of open strings constitutes the microscopic field and particle degrees of freedom of the D-brane worldvolume. Consider next the NS5-brane, the magnetic dual to the fundamental string. Worldvolume dynamics of an NS5-brane in Type IIA string theory, equivalently, an M5-brane in M-theory is described at low-energy by the six-dimensional N = (2, 0)
superconformal theory [18], and the theory is known to contain a selfdual two-form (p = 2) potential of gauge group U (1). Again, a novelty is that, when N parallel NS5-branes or M5-branes stack on top of one another, the microscopic degrees of freedom constitute tensionless strings that arise in M-theory from the variety of open M2-branes connecting all possible combinatoric pairs of the N M5-branes [10]. Similar to the D-branes, one might anticipate that N 2 variety of open M2-branes connecting all possible pairs of M5-brane constitute the microscopic field and string degrees of freedom of the M5-brane worldvolume. However, in stark contrast, various considerations ranging thermodynamic free energy [11] and gravitational anomaly cancellation [12] all indicate a peculiarity that the degrees of freedom are intrinsically quantum-mechanical and scales in N → ∞ limit as N 3 , in stark contrast to N 2 behavior [13] observed for the D3-brane worldvolume dynamics described by the N = 4 super Yang-Mills theory.
In this paper, we study a viable extension of nonabelian gauge invariance to tensor gauge field and put forward a nonperturbative approach for tensor gauge theory in d ≥ 4 dimensions † by putting the theory on a d-dimensional hypercubic lattice. The lattice formulation allows a tranparent and concerete description for origin of nonabelian gauge symmetries and self-interactions among the p-form gauge fields ‡ . By putting the theory on a lattice, we are sidestepping from other structures that can be endowed to the tensor gauge theory such as (extended) supersymmetry or (anti)self-duality. Though these structures are desirable for making contact with those arising in string theory, in this paper, we focus primarily on the issue of nonabelian gauge symmetry and self-interactions thereof. This paper is organized as follows. We begin in section 2 with etiol
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