Modular Forms on the Double Half-Plane

We formulate a notion of modular form on the double half-plane for half-integral weights and explain its relationship to the usual notion of modular form. The construction we provide is compatible with certain physical considerations due to the second author.Comment: 17 pages: Minor corrections in text (due to a helpful referee), updated affiliations. Accepted for publication in the International Journal for Number Theory (IJNT

Spectral sum rules for confining large-N theories

We consider asymptotically-free four-dimensional large-$N$ gauge theories with massive fermionic and bosonic adjoint matter fields, compactified on squashed three-spheres, and examine their regularized large-$N$ confined-phase spectral sums. The analysis is done in the limit of vanishing 't Hooft coupling, which is justified by taking the size of the compactification manifold to be small compared to the inverse strong scale $\Lambda^{-1}$. Our results motivate us to conjecture some universal spectral sum rules for these large $N$ gauge theories.Comment: 25 pages, 2 figures. v2: fixed typos, added references, and some minor improvements to discussio

Bose-Fermi Degeneracies in Large NN Adjoint QCD

We analyze the large $N$ limit of adjoint QCD, an $SU(N)$ gauge theory with $N_f$ flavors of massless adjoint Majorana fermions, compactified on $S^3 \times S^1$. We focus on the weakly-coupled confining small-$S^3$ regime. If the fermions are given periodic boundary conditions on $S^1$, we show that there are large cancellations between bosonic and fermionic contributions to the twisted partition function. These cancellations follow a pattern previously seen in the context of misaligned supersymmetry, and lead to the absence of Hagedorn instabilities for any $S^1$ size $L$, even though the bosonic and fermionic densities of states both have Hagedorn growth. Adjoint QCD stays in the confining phase for any $L \sim N^0$, explaining how it is able to enjoy large $N$ volume independence for any $L$. The large $N$ boson-fermion cancellations take place in a setting where adjoint QCD is manifestly non-supersymmetric at any finite $N$, and are consistent with the recent conjecture that adjoint QCD has emergent fermionic symmetries in the large $N$ limit.Comment: 35 pages, 5 figures. v3: further minor correction

The chiral condensate in a constant electromagnetic field

We study the shift of the chiral condensate in a constant electromagnetic field in the context of chiral perturbation theory. Using the Schwinger proper-time formalism, we derive a one-loop expression correct to all orders in $m_{\pi}^2 / eH$. Our result correctly reproduces a previously derived ``low-energy theorem'' for $m_\pi = 0$. We show that it is essential to include corrections due to non-vanishing $m_\pi$ in order for a low energy theorem to have any approximate regime of validity in the physical universe. We generalize these results to systems containing electric fields, and discuss the regime of validity for the results. In particular, we discuss the circumstances in which the method formally breaks down due to pair creation in an electric field.Comment: 9 pages, 6 figures, LaTeX; removed extraneous section + minor revision