Two loop partition function for large N pure Yang-Mills theory on a small three-sphere

We give a direct path-integral calculation of the partition function for pure 3+1 dimensional U(N) Yang-Mills theory at large N on a small three-sphere, up to two-loop order in perturbation theory. From this, we calculate the one-loop shift in the Hagedorn/deconfinement temperature for the theory at small volume, finding that it increases (in units of the inverse sphere radius) as we go to larger coupling (larger volume). Our results also allow us to read off the sum of one-loop anomalous dimensions for all operators with a given engineering dimension in planar Yang-Mills theory on R^4. As checks on our calculation, we reproduce both the Hagedorn shift and some of the anomalous dimension sums by independent methods using the results of hep-th/0412029 and hep-th/0408178. The success of our calculation provides a significant check of methods used in hep-th/0502149 to establish a first order deconfinement transition for pure Yang-Mills theory on a small three-sphere.Comment: 40 pages, 4 figures, harvma

A first order deconfinement transition in large N Yang-Mills theory on a small 3-sphere

We give an analytic demonstration that the 3+1 dimensional large N SU(N) pure Yang-Mills theory, compactified on a small 3-sphere so that the coupling constant at the compactification scale is very small, has a first order deconfinement transition as a function of temperature. We do this by explicitly computing the relevant terms in the canonical partition function up to 3-loop order; this is necessary because the leading (1-loop) result for the phase transition is precisely on the borderline between a first order and a second order transition. Since numerical work strongly suggests that the infinite volume large N theory also has a first order deconfinement transition, we conjecture that the phase structure is independent of the size of the 3-sphere. To deal with divergences in our calculations, we are led to introduce a novel method of regularization useful for nonabelian gauge theory on a 3-sphere.Comment: 63 pages (40 pages + 2 appendices), 6 figures, harvmac. v2: minor correction

The Hagedorn/Deconfinement Phase Transition in Weakly Coupled Large N Gauge Theories

We demonstrate that weakly coupled, large N, d-dimensional SU(N) gauge theories on a class of compact spatial manifolds (including S^{d-1} \times time) undergo deconfinement phase transitions at temperatures proportional to the inverse length scale of the manifold in question. The low temperature phase has a free energy of order one, and is characterized by a stringy (Hagedorn) growth in its density of states. The high temperature phase has a free energy of order N^2. These phases are separated either by a single first order transition that generically occurs below the Hagedorn temperature or by two continuous phase transitions, the first of which occurs at the Hagedorn temperature. These phase transitions could perhaps be continuously connected to the usual flat space deconfinement transition in the case of confining gauge theories, and to the Hawking-Page nucleation of AdS_5 black holes in the case of the N=4 supersymmetric Yang-Mills theory. We suggest that deconfinement transitions may generally be interpreted in terms of black hole formation in a dual string theory. Our analysis proceeds by first reducing the Yang-Mills partition function to a (0+0)-dimensional integral over a unitary matrix U, which is the holonomy (Wilson loop) of the gauge field around the thermal time circle in Euclidean space; deconfinement transitions are large N transitions in this matrix integral.Comment: harvmac, 90 pages, 14 figures, 67 footnotes. V3: added references and minor clarifications. v4: added reference, minor changes. v5: corrected figure captions. v6: small corrections and added footnot

The Phase Structure of Low Dimensional Large N Gauge Theories on Tori

In this paper we continue our study of the thermodynamics of large N gauge theories on compact spaces. We consider toroidal compactifications of pure SU(N) Yang-Mills theories and of maximally supersymmetric Yang-Mills theories dimensionally reduced to 0+1 or 1+1 dimensions, and generalizations of such theories where the adjoint fields are massive. We describe the phase structure of these theories as a function of the gauge coupling, the geometry of the compact space and the mass parameters. In particular, we study the behavior of order parameters associated with the holonomy of the gauge field around the cycles of the torus. Our methods combine analytic analysis, numerical Monte Carlo simulations, and (in the maximally supersymmetric case) information from the dual gravitational theories.Comment: harvmac, 67 pages, 21 figures. v2: minor corrections and clarification

Duality Symmetries for N=2 Supersymmetric QCD with Vanishing beta-Functions

We construct the duality groups for N=2 Supersymmetric QCD with gauge group SU(2n+1) and N_f=4n+2 hypermultiplets in the fundamental representation. The groups are generated by two elements $S$ and $T$ that satisfy a relation $(STS^{-1}T)^{2n+1}=1$. Thus, the groups are not subgroups of $SL(2,Z)$. We also construct automorphic functions that map the fundamental region of the group generated by $T$ and $STS$ to the Riemann sphere. These automorphic functions faithfully represent the duality symmetry in the Seiberg-Witten curve.Comment: 20 pages, 3 figures, harvmac (b); v2, typos corrected, statement about curves of marginal stability is correcte

Argyres-Seiberg duality and the Higgs branch

We demonstrate the agreement between the Higgs branches of two N=2 theories proposed by Argyres and Seiberg to be S-dual, namely the SU(3) gauge theory with six quarks, and the SU(2) gauge theory with one pair of quarks coupled to the superconformal theory with E_6 flavor symmetry. In mathematical terms, we demonstrate the equivalence between a hyperkaehler quotient of a linear space and another hyperkaehler quotient involving the minimal nilpotent orbit of E_6, modulo the identification of the twistor lines.Comment: 27 pages; v2: published versio

An Instanton Toolbox for F-Theory Model Building

Several dimensionful parameters needed for model building can be engineered in a certain class of SU(5) F-theory GUTs by adding extra singlet fields which are localized along pairwise intersections of D7-branes. The values of these parameters, however, depend on dynamics external to the GUT which causes the singlets to acquire suitable masses or expectation values. In this note, we demonstrate that D3-instantons which wrap the same 4-cycle as one of the intersecting D7's can provide precisely the needed dynamics to generate several important scales, including the supersymmetry-breaking scale and the right-handed neutrino mass. Furthermore, these instantons seem unable to directly generate the \mu term suggesting that, at least in this class of models, it should perhaps be tied to one of the other scales in the problem. More specifically, we study the simple system consisting of a pair of D7-branes wrapping del Pezzo surfaces which intersect along a curve $\Sigma$ of genus 0 or 1 and classify all instanton configurations which can potentially contribute to the superpotential. This allows one to formulate topological conditions which must be imposed on \Sigma for various model-building applications. Along the way, we also observe that the construction of arXiv:0808.1286 which engineers a linear superpotential in fact realizes an O'Raifeartaigh model at the KK scale whose 1-loop Coleman-Weinberg potential generically leads to a metastable, long-lived SUSY-breaking vacuum.Comment: 18 pages, 2 figures; v2: updated to reflect corrections in v2 of 0808.128

N=3 Warped Compactifications

Orientifolds with three-form flux provide some of the simplest string examples of warped compactification. In this paper we show that some models of this type have the unusual feature of D=4, N=3 spacetime supersymmetry. We discuss their construction and low energy physics. Although the local form of the moduli space is fully determined by supersymmetry, to find its global form requires a careful study of the BPS spectrum.Comment: 27 pages, v2: 32pp., RevTeX4, fixed factors, slightly improved sections 3D and 4B, v3: added referenc

The Pomeron and Gauge/String Duality

The traditional description of high-energy small-angle scattering in QCD has two components -- a soft Pomeron Regge pole for the tensor glueball, and a hard BFKL Pomeron in leading order at weak coupling. On the basis of gauge/string duality, we present a coherent treatment of the Pomeron. In large-N QCD-like theories, we use curved-space string-theory to describe simultaneously both the BFKL regime and the classic Regge regime. The problem reduces to finding the spectrum of a single j-plane Schrodinger operator. For ultraviolet-conformal theories, the spectrum exhibits a set of Regge trajectories at positive t, and a leading j-plane cut for negative t, the cross-over point being model-dependent. For theories with logarithmically-running couplings, one instead finds a discrete spectrum of poles at all t, where the Regge trajectories at positive t continuously become a set of slowly-varying and closely-spaced poles at negative t. Our results agree with expectations for the BFKL Pomeron at negative t, and with the expected glueball spectrum at positive t, but provide a framework in which they are unified. Effects beyond the single Pomeron exchange are briefly discussed.Comment: 68 pages, uses JHEP3.cls, utphys.bst; references added, typos corrected, and clarifying remarks adde