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Free fermions on a piecewise linear four-manifold. I: Exotic chain complex. (English) Zbl 1391.15081

This work focuses on algebraic realizations of four-dimensional Pachner moves. The paper provides a basis for constructing realizations of all such moves. The key component are Grassmann-Gaussian exponentials of quadratic forms of anticommuting variables.

MSC:

15A66 Clifford algebras, spinors
15A63 Quadratic and bilinear forms, inner products
81T99 Quantum field theory; related classical field theories

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References:

[1] Abłamowicz, R., Lounesto, P.: On Clifford Algebras of a Bilinear Form with an Antisymmetric Part. In: Abłamowicz, R., Lounesto, P., Parra, J.M. (eds.) Clifford Algebras with Numeric and Symbolic Computations. Birkhäuser, Boston (1996) · Zbl 0856.15028
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[4] Fauser, B.: A Treatise on Quantum Clifford Algebras. arXiv:math/0202059 · Zbl 1051.15022
[5] Korepanov, I.G., Sadykov, N.M.: Parameterizing the simplest Grassmann-Gaussian relations for Pachner move 3-3. SIGMA 9, paper 053 (2013). arXiv:1305.3246 · Zbl 1288.15033
[6] Korepanov, I.G.: Two-cocycles give a full nonlinear parameterization of the simplest 3-3 relation. Lett. Math. Phys. 104(10), 1235-1261 (2014). arXiv:1310.4075 · Zbl 1327.15057
[7] Korepanov, I.G.: Multiplicative expression for the coefficient in fermionic 3-3 relation. Mathematics 4(1), paper 3 (2016). doi:10.3390/math4010003. arXiv:1503.02272 · Zbl 1343.57014
[8] Korepanov, I.G.: Free fermions on a piecewise linear four-manifold. II: Pachner moves (2017, work in progress) · Zbl 1409.57025
[9] Lickorish, W.B.R.: Simplicial moves on complexes and manifolds. Geom. Topol. Monogr. 2, 299-320 (1999). arXiv:math/9911256 · Zbl 0963.57013
[10] Pachner, U.: PL homeomorphic manifolds are equivalent by elementary shellings. Eur. J. Comb. 12, 129-145 (1991) · Zbl 0729.52003
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