Introduction
N = 4 Super Yang-Mills Theory And S-Duality
This translation group is isomorphic to R1,9 itself (considered an abelian group), and P is an extension. 2.8) The generators of R1,9 are called the "pulse operators" PI. The bosonic part of the action, in four dimensions, has all three kinds of contributions and becomes.
S-Duality
In fact, a useful string theory realization of the Sp(k)/Spin(2k+1) duality involving orienting trihedra [49] motivates another normalization. The square root means that the group acting on the supersymmetries is a double cover of the duality group Γ.
Topological Field Theory From N = 4 Super Yang-Mills Theory
Twisting N = 4 Super Yang-Mills
This formula uniquely determines sixteen fields (η and η, four components each of ψ and ψ, and six components of e χ) in terms of the sixteen components of λ. After some Γ-matrix gymnastics, one finds the transformation of the four-dimensional gauge field under the topological supersymmetry:.
A Family Of A-Models
12 The third equation given here actually derives from vanishing the right-hand side of (3.27). The fields are the bosonic fields A, φ and K = 0; the equations are (3.29); and symmetries are simply gauge symmetries.
Vanishing Theorems
To make the right-hand side positive semidefinite, t must be a minimum of the function f ifR. Fort 6= 0,∞, the second vanishing theorem states that any solution of the supersymmetric equations onM is given by a flat GC-valued connection.
The Topological Lagrangian
However, the problem can be alleviated considerably by introducing an auxiliary field P, a null form with values in the Lie algebra, to complete the algebra for η and ηe (but not χ). We add an auxiliary field H, which must be a two-form with values in the Lie algebra, and postulate.
The Canonical Parameter
After all, the definition of the theory depends on the topological symmetryδt as well as on the action. We will have a symmetry of the topological equations (3.29) if we take T to act on the boson (A, φ).
Compactification And The Geometry Of Hitchin’s Moduli Space
Most of the complex structures on W in this family depend on the complex structure of C. It is possible to give a more "physical" explanation of why the target of the sigma model is the hyper-Kahler quotient of W is.
Complex Structures Of M H
Operation (I) gives the desired modulus space MH, but operation (II) is much easier to understand and almost coincides with it. The reason they almost coincide is that almost every orbit of complex gauge transformations contains a unique orbit of ordinary gauge transformations on which µI = 0.) The easiest way to understand operation (I) is often to first understand operation (II) and then to understand its connection with action (I). Setting the moment map to zero and dividing by gauge transformations gives the same modulus space.
Hitchin’s Fibrations
Using the fact that the fibers of the Hitchin fibration are compact, so a holomorphic function must be a withdrawal from the basis of this fibration, it can be shown that the moving Hamiltonians generate the ring of holomorphic functions in MH. An easy and important consequence of complete integrability is that fibers of the Hitchin fibration are Lagrangian submanifolds on the asymptotic holomorphic structure ΩI or equivalently on the asymptotic real structures ωJ and ωK.
Topological Field Theory In Two Dimensions
Twisted Topological Field Theories
In some cases we can see directly that these models are independent of the complex structure of C. At t = ±i we get the B model in complex structure ±J; this complex structure is independent of the metric of C, as we observed in Section 4.1.
The Role Of Generalized Complex Geometry
The topological field theory associated with IJ is the B-model in the complex structure Jb, and the one associated with Iω is the A-model with the symplectic structure ω.24. At these t values, the model is not actually a B-field transformation of the A-model; is a B-model in the ±J complex structure.
Some Specializations
It must move us out of this family, since the A-model in the J complex structure is not a member of this family. Moreover, we already noticed in the discussion of (5.22) that S (and the whole duality group) leaves an invariant B-model in the complex structure I and likewise an A-model in this complex structure.
S-Duality Of the Hitchin Fibration
But in the complex structure I, a holomorphic function on MH must be constant on the fibers of the Hitchin fibration (which are compact complex submanifolds) and therefore must come from a holomorphic function on the basis. The four-dimensional argument provides more information; will tell us exactly how the duality at the base of the Hitchin fibration works.
Two-Dimensional Interpretation Of S-Duality
If this holds for suitable operatorsOHα, we say that Bis is supported on fiberF in the Hitchin fiber ring. Therefore, Bep is a brane of type (B, A, A) supported on a fiber Fe of the double Hitchin fiber ring.
Branes On M H
That the corresponding fibers in the Hitchin fibers of GandLGer double complex tori have been shown for unitary groups by Hausel and Thaddeus [21]. Other examples of (B, B, B) branes are space-filling branes whose measure is the entire MH, equipped with a Chan–Paton vector bundle that is holomorphic in each of the three complex structures.
Loop and Line Operators
Topological Wilson Operators
The existence of Wilson loop operators possessing the topological symmetry is extremely natural at t = ±i. Just as well, such holonomies are not natural invariants if t 6= ±i, since the topological equations do not necessarily imply flatness.
Topological ’t Hooft Operators
This is in fact not true, because the singularity of the 't Hooft operator causes that Iθ is not invariant under the topological symmetry. Thus Iθ breaks the topological symmetry in the presence of the t Hooft operator, or equivalently, the t Hooft operator spoils the topological symmetry when θ 6= 0.
Compactification To Two Dimensions
Therefore, a loop or line operator can cause a long-range effect; the couplings in the two-dimensional effective theory can be different on both sides. Now in topological field theory there exist for all branes B1,B2 and B3 natural mappings HB1,B2 ⊗ HB2,B3 → HB1,B3 defined by the union of Riemann surfaces (Fig. In a two-dimensional quantum field theory without topological invariance, one must consider the metric or conformal structures of surfaces, for which an analogous discussion leads to the extension of the product of the operator.) These maps obey the obvious associativity relation, which states that when the three sets are combined (Fig. .
Line Operator Near A Boundary
We can be more specific in the case of weirs that are built from geometry. In the same way, we can reason for special 't Hooft operators, i.e. 't Hooft operators supported on curves of the same type.
Fluxes and S-Duality
Review
Naively, we just define E so that its limit top×C is "the" bundle over C determined by p. Let M(Gad, C)0 be the component of M(Gad, C) that parametrizes topologically trivial bundles – those that when restricted top×C, forp∈M(Gad, C), can be lifted to G-bundles .
Compactification To Two Dimensions
Our next goal is to interpret 0, e1, m0 and m1 in the effective two-dimensional sigma- model with MH(Gad) objective. As we have seen, the MH(Gad) target sigma model is equipped with a flat B-field e0(ζ).
Electric Eigenbranes
How Wilson Operators Act On Branes
In the limit that γ approaches the limit Q of Σ, it has the same form as the term that comes anyway from the Chan-Paton bundle U of the original brane B:38. We can now see explicitly how this is reflected in the action of the Wilson operator on branes.
Zerobranes As Electric Eigenbranes
In this section we first analyze static 't Hooft operators in the context of the standard Bogomolny equations (9.5). The action of the corresponding ’t Hooft operator T(Lw) on a SU(N) bundle E is as before: with respect to some decomposition E =⊕Ni=1Li nearp, we have Li → Li⊗ O(p0)mi.
The Space Of Hecke Modifications
First of all, the space of Hecke modifications of type (m1+c, m2+c) is the same as that of type (m1, m2), since adding (c, c) only has the effect of changing the output of the Hecke -transformation with O(p0)c. Some of what we have explained will be more transparent to most physicists if we shift from Lw = (a,0) to Lw = (a/2,−a/2). The space of Hecke modifications, of course, remains unchanged in this shift.) As we explained in Section 9.1, weights (a/2, −a/2) make sense for gauge group G = P SU(2) = SO(3) , LG = SU(2), and in addition the 't Hooft operator with these weights is S-dual to a Wilson loop of SU(2) in the representation of spin a.
The Affine Grassmannian
For U(N) (or SU(N), or P U(N)), let us define the dimension of the space of Hecke modifications of an arbitrary type. We conclude that the space of Hecke modifications Y(Lw;p0) is the orbit of the point.
The Bogomolny Equations And The Space Of Hecke Modifications
Boundary Conditions
Let the module space be the solution of the Bogomolny equations with elliptic boundary conditions in Fig. Lw = (a/2,−a/2) with a positive integer a shifts the apparent (complex) dimension of the space of modules Z of the solutions of the Bogomolny equations for a.
Monopole Bubbling
Since the group F preserves the hyper-Kahler structure of R4, the space of F-equivariant instantons is hyper-Kahler (and again, the ADHM construction makes this obvious). Indeed, the modulo space of solutions of the Bogomolny equations in R3 is hyper-Kahler [105].45 The spaces of Z(Lw1,.
Kahler Structure And The Moment Map
Now we want to find the moment mapping µ for the operation of the measuring group. The definition of Symplectic structure C used the planar form of C, but not its complex structure.
Operator Product Expansion Of ’t Hooft Operators
In the A-model, in the absence of instant uncorrections (they are absent here for reasons discussed at the beginning of Section 9), the space of physical states is the cohomology of the space of time-independent supersymmetric configurations. In the case of the spaces Y(Lw) of Hecke modifications, the submanifolds of interest are the spaces Y(Lwα), where Lwα is a weight associated with Lw.
The Extended Bogomolny Equations
This is the same as the dimension of the space of Hecke modifications of beams as determined in Eq. In the ordinary semi-simple case, this means that u must be one of the two eigenvectors of ϕ.
A-Branes and D-Modules
The Canonical Coisotropic A-Brane
In this case, the set of algebras of open strings is the chain of differential operators acting on sections of the line bundle L. The opposite of the algebra of differential operators on a line bundle. differential operators onL−1⊗KX, where KX is the canonical line bundle of X (that is, the bundle of holomorphic forms of top degree).
D-modules Corresponding to A-Branes
In this approach, open-string states with (Bc.c.,B′) boundary conditions are sections of the tensor product with N−1 of a vector bundle obtained by quantizing the space of fermion null modes. The Chan-Paton line bundle L → Fp of the brane BFp has a flat unit connection which can be easily represented as.
Generalizations of the c.c. Brane and Twisted Differential Operators
For non-integer Ψ, the objects of study will therefore not be rank-one branes supported on a Hitchin fibration fiber. For rational Ψ, there are A-branes of finite rank supported on a fiber of the Hitchin fibration; we can find what they are by following the duality starting from Ψ = 0.
Branes From Gauge Theory
General Properties Of Boundary Conditions
Branes of Type (B, B, B)
Branes Of Type (B, A, A)
Branes Of Type (A, B, A)
