On exact superpotentials in confining vacua

We consider the N=1 super Yang-Mills theory with gauge group U(Nc) or SU(Nc) and one adjoint Higgs field with an arbitrary polynomial superpotential. We provide a purely field theoretic derivation of the exact effective superpotential W(S) for the glueball superfield S in the confining vacua. We show that the result matches with the Dijkgraaf-Vafa matrix model proposal. The proof brings to light a deep relationship between non-renormalization theorems first discussed by Intriligator, Leigh and Seiberg, and the fact that W(S) is given by a sum over planar diagrams.Comment: 15 pages including one appendix; v2: minor changes and one reference added; v3: typos correcte

A note on theta dependence

The dependence on the topological theta angle term in quantum field theory is usually discussed in the context of instanton calculus. There the observables are 2 pi periodic, analytic functions of theta. However, in strongly coupled theories, the semi-classical instanton approximation can break down due to infrared divergences. Instances are indeed known where analyticity in theta can be lost, while the 2 pi periodicity is preserved. In this short note we exhibit a simple two dimensional example where the 2 pi periodicity is lost. The observables remain periodic under the transformation theta -> theta + 2 k pi for some k >= 2. We also briefly discuss the case of four dimensional N=2 supersymmetric gauge theories.Comment: 6 pages; v2: a couple of clarifying sentences added; v3: references adde

Quantum parameter space and double scaling limits in N=1 super Yang-Mills theory

We study the physics of N=1 super Yang-Mills theory with gauge group U(Nc) and one adjoint Higgs field, by using the recently derived exact effective superpotentials. Interesting phenomena occur for some special values of the Higgs potential couplings. We find critical points with massless glueballs and/or massless monopoles, confinement without a mass gap, and tensionless domain walls. We describe the transitions between regimes with different patterns of gauge symmetry breaking, or, in the matrix model language, between solutions with a different number of cuts. The standard large Nc expansion is singular near the critical points, with domain walls tensions scaling as a fractional power of Nc. We argue that the critical points are four dimensional analogues of the Kazakov critical points that are commonly found in low dimensional matrix integrals. We define a double scaling limit that yields the exact tension of BPS two-branes in the resulting N=1, four dimensional non-critical string theory. D-brane states can be deformed continuously into closed string solitonic states and vice-versa along paths that go over regions where the string coupling is strong.Comment: 32 pages, 4 figures, 1 appendix; v2: typos corrected and the physical distinction between the fields z and S made clearer in section 4.4; v3: more typos correcte

Black Hole Horizons and Bose-Einstein Condensation

Consider a particle sitting at a fixed position outside of a stable black hole. If the system is heated up, the black hole horizon grows and there should exist a critical temperature above which the particle enters the black hole interior. We solve a simple model describing exactly this situation: a large N matrix quantum mechanics modeling a fixed D-particle in a black hole background. We show that indeed a striking phenomenon occurs: above some critical temperature, there is a non-perturbative Bose-Einstein condensation of massless strings. The transition, even though precisely defined by the presence of the condensate, cannot be sharply detected by measurements made in a finite amount of time. The order parameter is fundamentally non-local in time and corresponds to infinite-time correlations.Comment: 11 pages, 1 figur

The Analytic Renormalization Group

Finite temperature Euclidean two-point functions in quantum mechanics or quantum field theory are characterized by a discrete set of Fourier coefficients $G_{k}$, $k\in\mathbb Z$, associated with the Matsubara frequencies $\nu_{k}=2\pi k/\beta$. We show that analyticity implies that the coefficients $G_{k}$ must satisfy an infinite number of model-independent linear equations that we write down explicitly. In particular, we construct "Analytic Renormalization Group" linear maps $\mathsf A_{\mu}$ which, for any choice of cut-off $\mu$, allow to express the low energy Fourier coefficients for $|\nu_{k}|<\mu$ (with the possible exception of the zero mode $G_{0}$), together with the real-time correlators and spectral functions, in terms of the high energy Fourier coefficients for $|\nu_{k}|\geq\mu$. Operating a simple numerical algorithm, we show that the exact universal linear constraints on $G_{k}$ can be used to systematically improve any random approximate data set obtained, for example, from Monte-Carlo simulations. Our results are illustrated on several explicit examples.Comment: 52 pages, 25 figures; v2: a few comments and explanations adde

A model for gauge theories with Higgs fields

We discuss in details a simple, purely bosonic, quantum field theory belonging to larger class of models with the following properties: a) They are asymptotically free, with a dynamically generated mass scale. b) They have a space of parameters which gets quantum corrections drastically modifying the classical singularity structure. The quantum theory can have massless solitons, Argyres-Douglas-like CFTs, exhibit confinement, etc... c) The physics can, to a large extent, be worked out in models with a large number of supersymmetries as well as in purely bosonic ones. In the former case, exact BPS mass formulas can be derived, brane constructions and embedding in M theory do exist. d) The models have an interesting 1/N expansion, and it is possible to define a double scaling limit in the sense of the ``old'' matrix models when approaching the singularities in parameter space. These properties make these theories very good toy models for four dimensional gauge theories with Higgs fields, and provide a framework where the effects of breaking supersymmetry can be explicitly studied. In our model, we work out in details the quantum space of parameters. We obtain the non-local lagrangian description of the Argyres-Douglas-like CFT, and show that it admits a strongly coupled fixed point. We also explicitly demonstrate property d). The possibility of defining such double scaling limits was not anticipated on the gauge theory side, and could be of interest to understand the gauge theory/string theory correspondence.Comment: 74 pages, 8 figures, 3 appendice

Non-perturbative double scaling limits

Recently, the author has proposed a generalization of the matrix and vector models approach to the theory of random surfaces and polymers. The idea is to replace the simple matrix or vector (path) integrals by gauge theory or non-linear sigma model (path) integrals. We explain how this solves one of the most fundamental limitation of the classic approach: we automatically obtain non-perturbative definitions in non-Borel summable cases. This is exemplified on the simplest possible examples involving O(N) symmetric non-linear sigma models with N-dimensional target spaces, for which we construct (multi)critical metrics. The non-perturbative definitions of the double scaled, manifestly positive, partition functions rely on remarkable identities involving (path) integrals.Comment: 18 pages, one figur

The BPS Spectra and Superconformal Points in Massive N=2 Supersymmetric QCD

We present a detailed study of the analytic structure, BPS spectra and superconformal points of the $N=2$ susy $SU(2)$ gauge theories with $N_f=1,2,3$ massive quark hypermultiplets. We compute the curves of marginal stability with the help of the explicit solutions for the low energy effective actions in terms of standard elliptic functions. We show that only a few of these curves are relevant. As a generic example, the case of $N_f=2$ with two equal bare masses is studied in depth. We determine the precise existence domains for each BPS state, and show how they are compatible with the RG flows. At the superconformal point, where two singularities coincide, we prove that (for $N_f=2$) the massless spectrum consists of four distinct BPS states and is S-invariant. This is due to the monodromy around the superconformal point being S, providing strong evidence for exact S-duality of the SCFT. For all $N_f$, we compute the slopes $\omega$ of the $\beta$-functions at the fixed point couplings and show that they are related to the anomalous dimensions $\alpha$ of $u=$ by $\omega= 2 (\alpha -1)$.Comment: 58 pages, 27 figures, uses phyzzx, speculative discussion of UV behaviour remove

Multi-Loop Zeta Function Regularization and Spectral Cutoff in Curved Spacetime

We emphasize the close relationship between zeta function methods and arbitrary spectral cutoff regularizations in curved spacetime. This yields, on the one hand, a physically sound and mathematically rigorous justification of the standard zeta function regularization at one loop and, on the other hand, a natural generalization of this method to higher loops. In particular, to any Feynman diagram is associated a generalized meromorphic zeta function. For the one-loop vacuum diagram, it is directly related to the usual spectral zeta function. To any loop order, the renormalized amplitudes can be read off from the pole structure of the generalized zeta functions. We focus on scalar field theories and illustrate the general formalism by explicit calculations at one-loop and two-loop orders, including a two-loop evaluation of the conformal anomaly.Comment: 85 pages, including 17 pages of technical appendices; 4 figures; v2: typos and refs correcte

Exact Multiplets of Spontaneously Broken Discrete Global Symmetries: the Example of N=2N=2 Susy QCD

In these notes, we emphasize the r\^ole of spontaneous broken global discrete symmetries acting on the moduli space of $N=2$ susy Yang-Mills theories and show how they can be used, together with the BPS condition, as a spectrum generating symmetry. In particular, in the strong-coupling region, all BPS states come in multiplets of this broken symmetry. This played a key r\^ole in the determination of the strong-coupling spectra.Comment: 6 pages, uses PHYZZX, to appear in the Proceedings of the Second International Sakharov Conference, Moscow, May 1996, based on a talk given by A.