Supersymmetric Gauge Theories, Matrix Models and Geometric Transitions Frank FERRARI Université Libre de Bruxelles and International Solvay Institutes XVth Oporto meeting on Geometry, Topology and Physics: Mathematical aspects of supersymmetry Oporto, 20/23 July 2006 1
The aim of this series of lecture is to explain general ideas and techniques that allow to derive exact results in strongly coupled supersymmetric gauge theories. This includes the famous Seiberg- Witten solution of N=2, and also many other results in N=1 theories. Many approaches have been used in the literature: S-duality, integrable systems, singularity theory and mirror symmetry, brane constructions, etc... We shall use a unified approach that can be justified from first principles. Its origin lies in the open/closed string duality, but this will become clear only in the third lecture... On the physics side, I have tried to include several interesting applications, for N=4, N=2 and N=1 (in the second lecture). On the math side, we shall discover a fascinating relation between matrix models and some problems in algebraic geometry (in the third lecture). These are mainly open questions. 2
1. Gauge theory set-up 1.1. Basics of super Yang-Mills 1.2. Anomaly equations 1.3. The gauge theory/matrix model correspondence 1.4. Gauge invariance and quantum equations of motion 2. Applications 3. String theory set-up 5
1.1. Basics of super Yang-Mills Gauge group (compact Lie group) Vector multiplet (in the adjoint) L YM = 1 [ ] 4π Im τ 0 d 2 θ tr F W α W α G = U(N) W α = (A µ, λ α ) τ 0 = ϑ 2π + 4iπ g 2 0 Chiral multiplet(s) in some anomaly-free representation φ = (ϕ, ψ α ) Adjoints X k = (X, Y,...) Fundamentals and anti-fundamentals Q f, Qf Other representations (symmetric, antisymmetric) L kin = N d 2 θd 2 θ φ k e2v φ k k Tree-level superpotential L W = 2N Re W m d 2 θ W m (φ k ) W m = Tr F W tree (X k ) + T Qf m f f (X k )Q f 6
Exact!-functions with canonical kinetic terms (physical; includes Z-factors) β c (g c ) = g3 c 3N 1 2 k (1 γ k)i Rk 16π 2 1 N 8π 2 g 2 c Fundamental I F = 1; Adjoint I F = 2N with bare kinetic terms (holomorphic; no Z-factors; one-loop) β(g 0 ) = g3 0 (3N 16π 2 1 ) (1 γ k )I Rk = g3 0 2 16π 2 b or equivalently τ 0 = ib 2π ln Λ 0 Λ The holomorphic coupling is physical when N=2 (Z=1: no wave function renormalization when N=2). In general, however, the physical mass scale M is different from the holomorphic mass scale Λ: M 6N P r I Λ 6N P k I R k Rr = k k ZI R k k 7
F-terms, I: definition An F-term is a term than can be written as an integral over half of superspace and that cannot be written as an integral over the full of superspace at the same time. Example: the superpotential term is an F-term, but the gauge kinetic term is not because it can be rewritten as an integral over the full superspace by using the identity W α D 2 e 2V D α e 2V. THEOREM (non-renormalization): F-terms are not corrected in perturbation theory, i.e. contribution from Feynman diagrams can always be written as integrals over the full of superspace. This can be shown either by using the Feynman rules in superspace, of more conveniently by using holomorphy (more about holomorphy below). VERY IMPORTANT: the non-renormalization theorem does NOT apply to non-perturbative corrections, and we shall see that F- terms *DO* get highly non-trivial and interesting quantum corrections. The miracle is that these quantum corrections can be computed EXACTLY, even though they typically correspond to infinite series of (fractional) instanton corrections. 8
F-terms, II: chiral operators A chiral operator O is a gauge invariant operator that (anti-) commutes with the left-handed supersymmetry charge: [ Q α, O} = 0 The lowest component of a chiral superfield is a chiral operator (the scalar components of chiral multiplets and the gluino are chiral operators). PROPERTIES: The set of chiral operators modulo (anti-)commutators forms a ring called the chiral ring of the theory. Calculations of F-terms are made in the chiral ring. In other words, if O O, d 2 θ O = d 2 θ O + d 2 θd 2 θ D-term 9
MORE PROPERTIES: In any supersymmetric vacuum, O O = O = O. O(x) O(x ) in the chiral ring, for any space-time points x, x. This is proven by using the supersymmetry algebra {Q α, Q α } = 2σ µ α α P µ. Chiral operators expectation values factorize: O K = OK K K This is a fundamental property. Usually, it is a consequence of the existence of a single field configuration dominating the path integral, but in our case this is not true. Rather it follows from space-time independence and cluster decomposition. 10
Effective superpotentials I We define the effective quantum superpotential W low by a path integral in external fields, e i R d 4 x 2N Re R d 2 θ W 0 low (t K)+D-terms = [dv dx] e is(v,x;t K) 0 external chiral superfields vacuum vector, ghosts and matter fields The superpotential W low is a holomorphic function (generally multivalued) of the couplings. t K The fields can couple to any chiral operator in the theory, and we have t K O K 0 O K 0 = W 0 low t K In particular, the chiral operators expectation values are (multivalued) holomorphic functions of the couplings. W low contains all the information about the F-terms, the chiral ring, the vevs of chiral operators. This is the object we want to compute. t K 11
Effective superpotentials II We can define other quantum effective superpotentials from, by taking the Legendre transform with respect to some of the couplings ( integrating in the corresponding fields), The ) ( W eff (O k ; t K ) = W low (ˆt k (O j ; t α ), t β + tk ˆt k (O j ; t α ) ) O k. ˆt k are obtained by solving the equations W ( ) low t k = ˆt k = O k. t k The resulting W eff is holomorphic and gives a description equivalent to W low, as long as the Legendre transform is welldefined (otherwise some branches or phases of the theory may be missed). The couplings of the fields that are integrated in appear linearly in. W eff k W low the couplings appear linearly 12
The expectation values are obtained by solving the equations W eff depends on the vacuum, but unlike it can describe several vacua at the same time (the above equation can have several solutions). W eff W ( ) eff O k = O k O k = 0. W low is NOT a low energy effective superpotential in general (one can integrate in any field, independently of its mass). The non-renormalization theorem for F-terms implies that the effective superpotentials are not corrected in perturbation theory (except for the one-loop contribution to the gauge coupling constant). But they do get very interesting non-perturbative corrections. 13
EXAMPLE Let s consider the theory with one adjoint X (this is the basic example on which we will mainly focus; other cases amounts to rather straightforward generalizations). It is convenient to parametrize the tree level superpotential as W tree(x) = d g k X k = g d k=0 d (X a I ). The classical vacua N are labeled by integers N 1 ;... ; N d I, d with I=1 N I = N, corresponding to the number of eigenvalues of X that are equal to a I. The pattern of gauge symmetry breaking in such a vacuum is U(N) U(N 1 ) U(N d ), through a Higgs mechanism that is similar, for example, to the SU(5) GUT. The following operators can be shown to generate the chiral ring of the theory: u k = Tr F X k, u α k = 1 4π Tr F W α X k, v k = 1 16π 2 Tr F W α W α X k. Note that only single-trace operators need to be considered because of the factorization property of expectation values. I=1 14
To each of these operators we can associate a coupling u k t k, u α k t kα, v k τ k such that u k = W low t k, u α k = W low t kα, v k = W low τ k Clearly, t Moreover, v 0 = (1/16π 2 ) Tr F W α k = g k 1 /k. W α = NS is the glueball superfield and correspond to the gauge kinetic term in the lagrangian. This shows that Nτ 0 = 2iπτ = ln Λ b, and thus in our case ( b = 2N ) the gluino condensate is given by S = W low ln Λ 2N The superpotential obtained by integrating in S from W low is called the glueball superpotential. It will be denoted simply W (S). By construction, it depends on the strong coupling scale Λ only through the linear term S ln Λ 2N. We shall also consider generalized glueball superpotentials, obtained by integrating in the independent generators. v k 15
There are relations between the generators of the chiral ring. These relations are of two types. First type: Identities Let us introduce the characteristic function F (z) = det(z X). Using the representation det(z X) = z N exp Tr F ln(1 X/z) = z N exp ( Tr F X k /(kz k ) ) k 1 we obtain an expansion of the form F (z) = det(z X) = F k (u q )z N k k 0 where the F k (u q ) are elements of the chiral ring. Classically F (z) is a degree N polynomial, and thus there are relations between the that read F k (u q ) = 0 for k > N. u k u k Thus only the for 0 k N are independent. Similarly, only the and v k for 0 k N 1 are independent. u α k 16
The explicit relations between the generators can be derived straightforwardly and take the form u k = P k (u 1,..., u N ) u α k = v k = N 1 q=0 N 1 q=0 u α q P k,q (u 1,..., u N ) v q Q k,q (u 1,..., u N ). The P k, P k,q and Q k,q are suitable homogeneous polynomials of degrees k, k q and k q respectively ( X being of degree one). There are also identities that come from the fermionic nature of the gluino field, for example S N 2 +1 = 0. One actually has the stronger identity in the chiral ring S N 0. 17
Second type: dynamics These identities correspond to the equations of motion W tree(x) = 0. Written in terms of gauge invariant operators, they are equivalent to Tr F X q W tree(x) = 0, Tr F W α X q W tree(x) = 0, Tr F W α W α X q W tree(x) = 0. or g k u k+q = 0, g k u α k+q = 0, g k v k+q = 0, q 0. k 0 k 0 k 0 Introducing a dummy variable z, these equations are equivalent to Tr F W tree(x) z X = 0, Tr F W α W tree(x) z X = 0, Tr F W α W α W tree(x) z X = 0. These relations yield the full solution of the theory in the chiral sector at the classical level. We now want to go to the quantum theory. 18