# On Fields over Fields

## DOI:

https://doi.org/10.56725/instemm.v1iS1.9## Abstract

We investigate certain arithmetic properties of field theories. In particular, we study the vacuum structure of supersymmetric gauge theories as algebraic varieties over number fields of finite characteristic. Parallel to the Plethystic Programme of counting the spectrum of operators from the syzygies of the complex geometry, we construct, based on the zeros of the vacuum moduli space over finite fields, the local and global Hasse-Weil zeta functions, as well as develop the associated Dirichlet expansions. We find curious dualities wherein the geometrical properties and asymptotic behaviour of one gauge theory is governed by the number theoretic nature of another.

## References

P. Candelas, X. de la Ossa, and F. Rodriguez-Villegas, “Calabi-Yau manifolds over finite fields, I,” arXiv:hepth/0012233, 2000.

P. Candelas, X. de la Ossa, and F. Rodriguez-Villegas, “Calabi-Yau manifolds over finite fields, II,” arXiv:hepth/0402133, 2004.

C. Meyer, Modular Calabi-Yau Threefolds, vol. 22 of Fields Institute Monographs. Providence, RI: American Mathematical Society, 1st ed., 2005.

J. Bryan and N. Leung, “The enumerative geometry of K3 surfaces and modular forms,” Journal of the American Mathematical Society, vol. 13, pp. 371–410, 1999.

D. Maulik, R. Pandharipande, and R. P. Thomas, “Curves on K3 surfaces and modular forms,” Journal of Topology, vol. 3, no. 4, pp. 937–996, 2010.

M. Aganagic, V. Bouchard, and A. Klemm, “Topological strings and (almost) modular forms,” Communications in Mathematical Physics, vol. 277, no. 3, pp. 771–819, 2007.

Y.-H. He and V. Jejjala, “Modular matrix models,” arXiv:hep-th/0307293, 2003.

A. Kapustin and E. Witten, “Electric-magnetic duality and the geometric Langlands program,” arXiv:hepth/0604151, 2006.

M. A. Luty and W. Taylor, “Varieties of vacua in classical supersymmetric gauge theories,” Physical Review D, vol. 53, no. 6, pp. 3399–3405, 1996.

J. Gray, Y.-H. He, V. Jejjala, and B. D. Nelson, “Vacuum geometry and the search for new physics,” Physics Letters B, vol. 638, no. 2-3, pp. 253–257, 2006.

D. Berenstein, “Reverse geometric engineering of singularities,” Journal of High Energy Physics, vol. 2002, no. 04, p. 052, 2002.

F. Ferrari, “On the geometry of super Yang–Mills theories: phases and irreducible polynomials,” Journal of High Energy Physics, vol. 2009, no. 01, p. 026, 2009.

J. Gray, Y.-H. He, V. Jejjala, and B. D. Nelson, “Exploring the vacuum geometry of gauge theories,” Nuclear Physics B, vol. 750, no. 1-2, pp. 1–27, 2006.

J. Gray, Y.-H. He, A. Hanany, N. Mekareeya, and V. Jejjala, “SQCD: a geometric aperçu,” Journal of High Energy Physics, vol. 2008, no. 05, p. 099, 2008.

S. Benvenuti, B. Feng, A. Hanany, and Y.-H. He, “Counting BPS operators in gauge theories: quivers, syzygies and plethystics,” Journal of High Energy Physics, vol. 2007, no. 11, p. 050, 2007.

B. Feng, A. Hanany, and Y.-H. He, “Counting gauge invariants: the plethystic program,” Journal of High Energy Physics, vol. 2007, no. 03, p. 090, 2007.

F. Buccella, J. P. Derendinger, S. Ferrara, and C. A. Savoy, “Patterns of symmetry breaking in supersymmetric gauge theories,” Physics Letters B, vol. 115, pp. 375– 379, Sept. 1982.

R. Gatto and G. Sartori, “Consequences of the complex character of the internal symmetry in supersymmetric theories,” Communications in Mathematical Physics, vol. 109, p. 327, 1987.

C. Procesi and G. W. Schwarz, “The geometry of orbit spaces and gauge symmetry breaking in supersymmetric gauge theories,” Physics Letters B, vol. 161, pp. 117– 121, 1985.

E. Witten, “Phases of N = 2 theories in two dimensions,” Nuclear Physics B, vol. 403, no. 1-2, pp. 159–222, 1993.

D. Forcella, A. Hanany, Y.-H. He, and A. Zaffaroni, “The master space of N=1 gauge theories,” Journal of High Energy Physics, vol. 2008, no. 08, p. 012, 2008.

D. Forcella, A. Hanany, Y.-H. He, and A. Zaffaroni, “Mastering the master space,” Letters in Mathematical Physics, vol. 85, no. 2-3, pp. 163–171, 2008.

D. Forcella, “Master space and Hilbert series for N=1 field theories,” arXiv:0902.2109, 2009.

D. Forcella, A. Hanany, and A. Zaffaroni, “Baryonic generating functions,” Journal of High Energy Physics, vol. 2007, no. 12, p. 022, 2007.

A. Butti, D. Forcella, A. Hanany, D. Vegh, and A. Zaffaroni, “Counting chiral operators in quiver gauge theories,” Journal of High Energy Physics, vol. 2007, no. 11, p. 092, 2007.

V. Balasubramanian, B. Czech, Y.-H. He, K. Larjo, and J. Simón, “Typicality, black hole microstates and superconformal field theories,” Journal of High Energy Physics, vol. 2008, no. 03, p. 008, 2008.

A. Hanany, N. Mekareeya, and A. Zaffaroni, “Partition functions for membrane theories,” Journal of High Energy Physics, vol. 2008, no. 09, p. 090, 2008.

D. J. Anick, “Non-commutative graded algebras and their Hilbert series,” Journal of Algebra, vol. 78, no. 1, pp. 120–140, 1982.

J.-P. Serre, Cours d’arithmétique. Paris: Presses Universitaires de France, 1st ed., 1970.

R. Hartshorne, Algebraic Geometry. Graduate Texts in Mathematics, New York: Springer-Verlag, 1st ed., 1977.

T. L. Apostol, Introduction to Analytic Number Theory. New York: Springer-Verlag, 1st ed., 1976.

S. Mandelbrojt, Dirichlet Series: Principles and Methods. Dordrecht: Springer Netherlands, 1st ed., 1971.

Y. Nambu, Quantum field theory and Quantum Statistics, vol. 1, ch. Field Theory Of Galois Fields, pp. 625–636. Bristol: IOP Publishing, 1st ed., 1987.

A. Connes and M. Marcolli, Noncommutative Geometry, Quantum Fields and Motives. Providence, RI: American Mathematical Society, 1st ed., 2008.

H. Nakajima and K. Yoshioka, “Instanton counting on blowup. I. 4-dimensional pure gauge theory,” Inventiones mathematicae, vol. 162, no. 2, pp. 313–355, 2005.

D. R. Grayson and M. E. Stillman, “Macaulay2, a software system for research in algebraic geometry.” Available at http://www.math.uiuc.edu/Macaulay2/.

I. R. Klebanov and E. Witten, “Superconformal field theory on threebranes at a Calabi–Yau singularity,” Nuclear Physics B, vol. 536, no. 1-2, pp. 199–218, 1998.

J. Hewlett and Y.-H. He, “Probing the space of toric quiver theories,” Journal of High Energy Physics, vol. 2010, no. 3, 2010.

B. Szendrői, “Non-commutative Donaldson–Thomas invariants and the conifold,” Geometry & Topology, vol. 12, no. 2, pp. 1171–1202, 2008.

K. Oh and R. Tatar, “Branes at orbifolded conifold singularities and supersymmetric gauge field theories,” Journal of High Energy Physics, vol. 1999, no. 10, p. 031, 1999.

M. Aganagic, A. Karch, D. L¨ust, and A. Miemiec, “Mirror symmetries for brane configurations and branes at singularities,” Nuclear Physics B, vol. 569, no. 1-3, pp. 277–302, 2000.

B. Feng, A. Hanany, and Y.-H. He, “D-brane gauge theories from toric singularities and toric duality,” Nuclear Physics B, vol. 595, no. 1-2, pp. 165–200, 2001.

A. Hanany and A. Zaffaroni, “On the realization of chiral four-dimensional gauge theories using branes,” Journal of High Energy Physics, vol. 98, no. 05, p. 001, 1998.

M. Wijnholt, “Large volume perspective on branes at singularities,” arXiv:hep-th/0212021, 2002.

V. Gorbounov, “Homological algebra and divergent series,” Symmetry, Integrability and Geometry: Methods and Applications, 2009.

V. Golyshev, “The canonical strip, I,” arXiv:0903.2076, 2009.

R. P. Stanley, “Hilbert functions of graded algebras,” Advances in Mathematics, vol. 28, no. 1, pp. 57–83, 1978.

I. Gradshteyn and I. Ryzhik, Tables of Integrals, Series, and Products. Cambridge MA: Elsevier Academic Press, 8th ed., 2014.

A. Okounkov, N. Reshetikhin, and C. Vafa, “Quantum Calabi-Yau and classical crystals,” arXiv:hepth/0309208, 2003.

A. Hanany and Y.-H. He, “Non-Abelian finite gauge theories,” Journal of High Energy Physics, vol. 1999, no. 02, pp. 013–013, 1999.

A. Hanany and Y.-H. He, “M2-branes and quiver Chern-Simons: A taxonomic study,” arXiv:0811.4044, 2008.

G. Hardy and M. Riesz, The General Theory of Dirichlet’s Series. Mineola NY: Dover Publications, 2005.

A. Hanany, D. Orlando, and S. Reffert, “Sublattice counting and orbifolds,” Journal of High Energy Physics, vol. 2010, no. 6, 2010.

N. I. Koblitz, Introduction to Elliptic Curves and Modular Forms. New York: Springer-Verlag, 2nd ed., 1993.

J. Davey, A. Hanany, N. Mekareeya, and G. Torri, “Brane tilings, M2-Branes and Chern-Simons theories,” arXiv:0910.4962, 2009.

W. Bosma, J. Cannon, and C. Playoust, “The MAGMA algebra system I: The user language,” Journal of Symbolic Computation, vol. 24, no. 3, pp. 235–265, 1997.

S. R. Finch, “Primitive cusp forms.” Available at https://oeis.org/A001616/a001616.pdf, 2009.

G. Meinardus, “Asymptotische aussagen über partitionen,” Mathematische Zeitschrift, vol. 59, pp. 388–398, 1954.

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