We review the use of the matrix model in studying the flow of renormalization groups and the space of conformal field theories in three dimensions. Finally, quite surprisingly, a localization calculation on three spheres was used to provide insight into the flow of the renormalization group and the space of conformal field theories in three dimensions.
- Spinor Conventions
- The N = 2 Supersymmetry Algebra
- Multiplets
- Components
- Supersymmetric Actions
- Global Symmetries
- Abelian Duality
A real measure for the U(1)B symmetry can be shifted by shifting the origin of the Coulomb branch. A triplet of Fayet-Iliopoulos terms in the (1,3) representation exists for every factor U(1) of the gauge set.
Dynamical Aspects
- Vacua
- Renormalization group flow
- Anomalies
- Instantons
- Vortices
- Monopole operators
- Chern-Simons Theory
- Brane Constructions
The theory is topological and the observables are Wilson loops in a representation of the gauge group G. When one of the branes is of type (1, k), the segments on the left and right additionally get a Chern-Simons term of respectively the levels k and-k [21].
Supersymmetric Lagrangians on S 3
- Coupling to Supergravity
- Coordinates on S 3
- The Spectrum
- Killing Spinors
- Gauge Theories on S 3
- The Duistermaat-Heckman Theorem
- The Atiyah-Bott-Berline-Vergne theorem
Note that all the dualities described in 3.1 are valid at the IR fixed points of the theories. Then one rotates the time direction associated with R to obtain a Euclidean version of the theory. The resulting action can then be evaluated in the limit where the magnitude of the S1 goes to zero and the theory becomes three-dimensional.
Q is the pullback and e(NKQ) is the equivalent Euler class of the normal bundle of KQ in M. In terms of the objects considered above, we can identify α with eiSdV and e(NKQ) with psdet(Hess(S(p) Note that in the definition of α we can include any Q-closed observable part of the theory, without changing the conclusions.
Localizing 3D Gauge Theories
- The Supercharge
- The Functional
- Zero Modes
- Fluctuations
The zero locus of the supercharge used to localize the action coincides with the set of bosonic configurations for which the gaugino variation vanishes. This is a quite simple null locus, reminiscent of that obtained for theories on S4 in [25]. Note that the remaining gauge freedom of the theory allows us to diagonalize the zero state σ0.
The superpotential limits the choice of global symmetries and thus indirectly affects the effect of the theory on S3 (see 1.1.6 and 1.3.1). This is a free operation for which the evaluation of the path integral reduces to the calculation of a determinant. Below is a list of the relevant operators for the localization function given in 2.2.2 after accounting for the space of zero states in 2.2.3.
The Matrix Model
The Integration Measure
The analysis in 2.2.3 implies that for the class of N= 2 theories under consideration, the zero locus of the supersymmetry generator Q is the set of constant modes on S3 of the scalar and auxiliary scalar,σ0 =−D0, in every vector multiple. If the vector multiple in question is dynamic, the constant matrix must be integrated over. Denote the measure group, which can be a direct product, for the dynamic vector multiplication by G.
We can use the residual gauge symmetry, the freedom to perform constant gauge transformations on S3, to twist into the Cartan subalgebra.
Classical Contributions
One-Loop Contributions
A matrix model for the pure Chern-Simons theory was written in [38] and agrees with our results. Note that this means that the chiral multiplet N= 2 with ∆ = 1 does not contribute to the matrix model.
Observables
This system can be solved to give a 1/2 BPS loop by choosing η= 0 andγan integral curve for the vector field†γµ corresponding to the Killing spinor given in 1.3.4. Note that the formal product of any number of such operators Wi is invariant when supported on γi, all of which are integral curves corresponding to the same spinor.
Convergence
The condition that this asymptotic behavior describes the exponential decay of the integrand in all directions (i.e. a > 0) was further shown to be equivalent to the Gaiotto-Witten criterion for N = 4 gauge theories that are 'good' or 'good'. We also show an example of calculating the expectation value of an observable, a supersymmetric Wilson loop, using localization and the matrix model. We describe how the matrix model can be used to calculate exact coefficients for the R-symmetry mixing phenomenon described in 1.1.6.
We show how a partition function on three spheres computed by localization can be used to limit the flow of the renormalization group and explore the space of conformal field theories. Duality can exchange weakly coupled theories with strongly coupled ones, associate theories with different gauge groups and different matter contents, and create a map between common observables built from fundamental fields and perturbation operators. In this section, we show how localization and the matrix model can be used to test some duality assumptions involving three-dimensional theories.
Mirror Symmetry
The two partition functions are properly related due to the basic Fourier identity. Parameter matching in dual theories is more complicated and is given explicitly in [49]. The mirror symmetry for these theories can be derived from the SL(2,Z) duality of the type IIB string theory used in the brane construction.
Specifically, the S generator exchanges D5 and NS5 branes and leaves D3 branes invariant, thus changing the gauge group and matter content of the theory in the manner described above. These are indeed induced in the mirror transform, although it is difficult to see without evaluating the partition functions as was done above. We then replace the two NS5 branes, and change the number of suspended D3 branes in the interval.
Seiberg-Like Dualities
- Naive Duality
- Giveon-Kutasov Duality
- Fractional M-Brane (ABJ) Duality
The integrals involved in the calculation of partition functions deformed by FI parameters and real mass terms can be done exactly in this case [17]. The deformations of the theory necessary to go from (b) to (a) and from (d) to (e) are identified. To compare the partition functions, we use a version of duality that preserves the N = 3 supersymmetry by adding an appropriate superpotential to the electrical theory (1).
The naive version of the duality described in the previous section is the "k= 0" version of this proposition (assuming we start with N= 3 gauge theory with the Yang-Mills and Chern-Simons expressions). However, we will find that the calculation of the partition function without modification supports the dualities proposed in [60]. This is nothing more than the duality examined in the last section performed on only one of the factors in the measurement group.
V± are mapped to first-theory monopole operators under duality, while Mab is mapped to QaQ˜b. Note that the possibility of mixing between the UV R-symmetry and the global U(1)asymmetry, which could change the dimensions of the chiral superfields, is actually fully accounted for by promoting µ into a complex parameter. This is not surprising given the supergravity construction of act 1.3.1.
The comparison of the partition functions for the dual theories is insensitive to this mixing, being a similarity between two functions which depend on µ in a holomorphic way. 2+ma+µ λ=−2η (3.1.48) This also defines the analytic continuation of the partition functions for complex values of the background parameters needed when R-symmetry mixing is possible. The superfield ˜Y parametrizes the Coulomb branch of the dual theory, and Y is identified with the Coulomb branch of the original theory.
Maximally Supersymmetric Gauge Theory
The ABJM Partition Function
The mirror dual is identical to N = 8 SYM, with the addition of one hypermultiplet in the basic representation. This is doubled to N = 8 SYM, because the latter can be built using an NS5 single brane elliptical sleeve. An additional hypermultiplet breaks supersymmetry at N= 4, but maximum supersymmetry is restored in the IR limit.
The supersymmetric deformations of the partition function for this theory are a Fayet-Iliopoulos term η and a real mass term ω for the adjacent hypermultiplet. This partition function was shown in [52] to be equal to the partition function forN= 8 SYM, with the additional. This is a powerful check of the IR equivalence of these two theories, and thus of the identification of ABJM theory with the low-energy theory on coincident M2 branes.
A Supersymmetric Wilson Loop
Note that the version of ABJM considered above, which describes M2-branes in flat space, is strongly coupled. Calculations in the large N limit are described in 3.2.3. and for the Wilson loop in the fundamental representation. Using the trivial framing, the expression for the Wilson loop above can be expanded into large N and small N/kto yield [19].
Holography
In addition, the expected value of Wilson loops, such as that described in 3.2.2, can be calculated and compared with the gravity results. The Wilson loop in 3.2.2 is 1/2 BPS in terms of N = 2, but there is an operator that preserves 6 supercharges (and 6 conformal supercharges) and is therefore 1/2 BPS in terms of N = 6 [ 70 ]. Its expected value under strong coupling, in fact the entire interpolation function, can be calculated using the same methods as for the partition function.
The result for the 1/2 BPS Wilson loop in the fundamental representation (denoted by a single box) under strong coupling is [71]. Note that this operator preserves the right amount of supersymmetry to be dual for a fundamental type IIA sequence (or an M2-brane ending in the ABJM stack of branes in the M-theory description) [70].
Superconformal Fixed Points
R-Symmetry Mixing
It is the conformal dimension of the scalar in the chiral multiplet that the commutator acts on. The utility of the above observation is that in the CFT the quantity is given by . Examining XYZ theory 3.1.3 in light of the above criterion trivially yields the correct dimensions for the chiral multiplets (which can be deduced from symmetry considerations).
The free energy is derived from the finite part of the partition function, which is unambiguous and can be calculated using localization. Finally, explicit tests of the F-theorem performed in [78], comparing the free energy in UV and IR, confirm the general statement. For the operators in question, this can be evaluated in terms of the Hurwitz zeta function (25.11 in [85]).
Figures B.1 and B.2 summarize the comparison of the absolute value and phase of the partition functions for the dualities in 3.1.2.2. Heckman, On the variation in the cohomology of the symplectic form of the reduced phase space., Inventiones Mathematicae.
