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We compute some arithmetic path integrals for BF-theory over the ring of integers of a totally imaginary field, which evaluate to natural arithmetic invariants associated to $mathbb{G}_m$ and abelian varieties.
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Linear wave solutions to the Charney-Hasegawa-Mima partial differential equation with periodic boundary conditions have two physical interpretations: Rossby (atmospheric) waves, and drift (plasma) waves in a tokamak. These waves display resonance in triads. In the case of infinite Rossby deformation radius, the set of resonant triads may be described as the set of integer solutions to a particular homogeneous Diophantine equation, or as the set of rational points on a projective surface. We give a rational parametrization of the smooth points on this surface, answering the question: What are all resonant triads? We also give a fiberwise description, yielding a procedure to answer the question: For fixed $r in mathbb{Q}$, what are all wavevectors $(x,y)$ that resonate with a wavevector $(a,b)$ with $a/b = r$?
This is the text of a series of five lectures given by the author at the Second Annual Spring Institute on Noncommutative Geometry and Operator Algebras held at Vanderbilt University in May 2004. It is meant as an overview of recent results illustrating the interplay between noncommutative geometry and arithmetic geometry/number theory.
We study the two-dimensional topological abelian BF theory in the Lorenz gauge and, surprisingly, we find that the gauged-fixed theory is a free type B twisted N=(2,2) superconformal theory with odd linear target space, with the ghost field c being the pullback of the linear holomorphic coordinate on the target. The BRST operator of the gauge-fixed theory equals the total Q of type B twisted theory. This unexpected identification of two different theories opens a way for nontrivial deformations of both of these theories.
We study two-dimensional non-abelian BF theory in Lorenz gauge and prove that it is a topological conformal field theory. This opens the possibility to compute topological string amplitudes (Gromov-Witten invariants). We found that the theory is exactly solvable in the sense that all correlators are given by finite-dimensional convergent integrals. Surprisingly, this theory turns out to be logarithmic in the sense that there are correlators given by polylogarithms and powers of logarithms. Furthermore, we found fields with logarithmic conformal dimension (elements of a Jordan cell for $L_0$). We also found certain vertex operators with anomalous dimensions that depend on the non-abelian coupling constant. The shift of dimension of composite fields may be understood as arising from the dependence of subtracted singular terms on local coordinates. This generalizes the well-known explanation of anomalous dimensions of vertex operators in the free scalar field theory.
Fix a smooth cubic form $F/mathbb{Q}$ in $6$ variables. For $N_F(X):=#{boldsymbol{x}in[-X,X]^6:F(boldsymbol{x})=0}$, the randomness prediction $N_F(X)=(c_text{HL}+o(1))cdot X^3$ as $Xtoinfty$ of Hardy-Littlewood may fail. Nonetheless, Hooley suggested a modified prediction accounting for special structured loci on the projective variety $V:= V(F)subseteqmathbb{P}^5_mathbb{Q}$. A weighted version of $N_F(X)$ essentially decomposes as a sum of adelic data over hyperplane sections $V_{boldsymbol{c}}subseteq V$, generically with nonzero discriminant $D(boldsymbol{c})$. Assuming standard conjectures for the Hasse-Weil $L$-functions $L(s,V_{boldsymbol{c}})$ over ${boldsymbol{c}inmathbb{Z}^6:D(boldsymbol{c}) eq0}$, Hooley proved the bound $N_F(X) = O_epsilon(X^{3+epsilon})$, essentially for any given diagonal $F$. Now assume (1) standard conjectures for each $L(s,V_{boldsymbol{c}})$, for certain tensor $L$-functions thereof, and for $L(s,V)$; (2) standard predictions (of Random Matrix Theory type) for the mean values of $1/L(s)$ and $1/L(s_1)L(s_2)$ over certain geometric families; (3) a quantitative form of the Square-free Sieve Conjecture for $D$; and (4) an effective bound on the local variation (in $boldsymbol{c}$) of the local factors $L_p(s,V_{boldsymbol{c}})$, in the spirit of Krasners lemma. Under (1)-(4), we establish (away from the Hessian of $F$) a weighted, localized version of Hooleys prediction for diagonal $F$ -- and hence the Hasse principle for $V/mathbb{Q}$. Still under (1)-(4), we conclude that asymptotically $100%$ of integers $a otin{4,5}bmod{9}$ lie in ${x^3+y^3+z^3:x,y,zinmathbb{Z}}$ -- and a positive fraction lie in ${x^3+y^3+z^3:x,y,zinmathbb{Z}_{geq0}}$.
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