Contents

cohomology

# Contents

## Idea

For type II superstrings on a global orbifold spacetime $X\sslash G$, D-brane charge is measured by the equivariant K-theory $K_G(X)$ (Witten 98, section 5.1) or at least some subgroup or quotient group thereof (BDHKMMS 01, around (137)). Under the canonical map $K(X/G) \longrightarrow K_G(X)$ from the plain topological K-theory of the quotient space irreducible elements of K-theory in general decompose into direct sums of smaller irreducible elements, hence into “smaller fractions stuck at the orbifold singularity” . The corresponding D-branes are called fractional D-branes in the literature.

In particular, at least for D-branes at an A-type singularity $\mathbb{C}^2\sslash (\mathbb{Z}/n)$, hence for $G = \mathbb{Z}/n$ a cyclic group, the tension (hence mass) of a fractional D-brane is supposed to be the fraction $\tfrac{1}{{\vert G\vert}} = \tfrac{1}{n}$ of that of the unit bulk brane away from the singularity (Douglas-Greene-Morrison 97, p.10).

### At global linear orbifold singularities

If here $X$ is a contractible space (at it is in the majority of examples discussed in the literature!), then its equivariant K-theory is equivalently that of the point

$K_G(X) \;\simeq\; K_G(\ast) \;\simeq\; R_{\mathbb{C}}(G)$

which is isomorphic to the representation ring of $G$.

Under this identification, the (non-fractional) unit brane is identified with the regular representation $k[G/1]$ of $G$:

$\array{ && K(X) &\longrightarrow & K_G(X) \\ && \simeq && \simeq \\ \mathbb{Z} &\simeq& K(\ast) &\longrightarrow& K_G(\ast) &\simeq& R_{\mathbb{C}}(G) \\ 1 && & \mapsto & && k[G/1] }$

The non-fractional nature of the permutation representation is reflected in the fact that its character is still unity on the neutral element $e \in G$ and vanishes on other elements (i.e. in the non-trivial twisted sectors):

$\chi_{k[G/1]} \;\colon\; g \mapsto \left\{ \array{ 1 &\vert& g = e \\ 0 &\vert& \text{otherwise} } \right.$

But, by a standard fact from representation theory, the regular representation is far from irreducible, instead it is a direct sum of all existing irreducible representations $V_i \in R_{\mathbb{C}}(G)$ (each appearing with multiplicity equal to its dimension):

$k[G/1] \;\simeq\; \underset{i}{\bigoplus} d_i V_i \,.$

In this sense, the unit brane on $X$ decays into fractional branes at the orbifold singularities $X\sslash G$.

For a general representation $V \in R(G)$ the general formula (see this Prop.)

$dim\left( V^G \right) \;=\; \frac{1}{{\vert G\vert}} \underset{g \in G}{\sum} \chi_V(g)$

shows that the total charge of the brane corresponding to $V$, summed over all twisted sectors, is equal to the dimension of the fixed point space $V^G$.

### In terms of twisted sector boundary states

In the worldsheet-description of D-branes via boundary conformal field theory, fractional D-branes are reflected by boundary states in “twisted sectors”. (e.g. Diaconescu-Gomis 99, Recknagel-Schomerus 13, around p. 173)

### D-branes on resolutions of orbifold singularities

In parts of the string theory literature, fractional D-branes are identified in a “dual” formulation of the situation:

At least for $X \simeq \mathbb{C}^2$ and $G$ a finite subgroup of SU(2) acting in the canonical way, hence for $X\sslash G$ an ADE-singularity, the K-theoretic McKay correspondence (Gonzalez-Sprinberg & Verdier 83) identifies the equivariant K-theory of $X$ with the plain K-theory of a nice blow-up resolution $\tilde X$:

(1)$\array{ && && K( X \times_{X/G} \tilde S ) \\ && & {}^{\mathllap{ p_1^\ast }}\nearrow && \searrow^{\mathrlap{Inv \circ (p_2)_\ast}} \\ R(G) \simeq K_G(\ast) &\simeq& K_G(X) && \overset{\simeq}{\longrightarrow} && K(\tilde X) }$

Under this equivalence (isomorphism of K-theory groups), fractional D-branes on the orbifold are identified with D-branes on $\tilde X$ which are wrapped around some of the cycles in $\tilde X$ that appear through the blow-up of the ADE-singularity (in physics jargon these are the “vanishing cycles”).

### In terms the worldvolume gauge theories

In terms of the worldvolume gauge field theory the equivalence (1) between

1. fractional D-branes stuck at orbifold singularities

2. and wrapped branes on the blow-up resolution

is supposed to be exhibited by passage from the Higgs branch to the Coulomb branch:

The first key insight is due to Kronheimer 89. He showed that the (resolutions of) the orbifold quotients $\mathbb{C}^2/\Gamma$ for $\Gamma$ a finite subgroup of SU(2) are precisely the generic form of the gauge orbits of the direct product of $U(n_i)$-s acting in the evident way on the direct sum of $Hom(C^{n_i}, C^{n_j})$-s, where $i$ and $j$ range over the vertices of the Dynkin diagram, and $(i,j)$ over its edges.

This becomes more illuminating when interpreted in terms of Yang-Mills gauge theory: in a “quiver gauge theory” the gauge group is a direct product group of circle group $U(n_i)$ factors associated with vertices of a quiver, and the particles which are charged under this gauge group arrange, as a linear representation, into a direct sum of $Hom(C^{n_i}, C^{n_j})$-s, for each edge of the quiver.

Pick one such particle, and follow it around as the gauge group transforms it. The space swept out is its gauge orbit, and Kronheimer says that if the quiver is a Dynkin diagram, then this gauge orbit looks like $\mathbb{C}^2/\Gamma$.

On the other extreme, gauge theories are of interest whose gauge group is not a big direct product, but is a simple Lie group, in the technical sense, such as the special unitary group $SU(N)$ or the exceptional Lie group $E_8$. The mechanism that relates the two classes of examples is spontaneous symmetry breaking (“Higgsing”): the ground state energy of the field theory may happen to be achieved by putting the fields at any one point in a higher dimensional space of field configurations, acted on by the gauge group, and fixing any one such point “spontaneously” singles out the corresponding stabilizer subgroup.

Now here is the final ingredient: it is N=2 super Yang-Mills theories (“Seiberg-Witten theory”) which have a potential that is such that its vacua break a simple gauge group such as $SU(N)$ down to a Dynkin diagram quiver gauge theory. One place where this is reviewed, physics style, is Albertsson 03, section 2.3.4.

More precisely, these theories have two different kinds of vacua, those on the “Coulomb branch” and those on the “Higgs branch” depending on whether the scalars of the “vector multiplets” (the gauge field sector) or of the “hypermultiplet” (the matter field sector) vanish. The statement above is for the Higgs branch, but the Coulomb branch is supposed to behave “dually”. (see e.g. Diaconescu-Gomis 99)

So that then finally is the relation, in the ADE classification, between the simple Lie groups and the finite subgroups of SU(2): start with an N=2 super Yang-Mills theory with gauge group a simple Lie group. Let it spontaneously find its vacuum and consider the orbit space of the remaining spontaneously broken symmetry group. That is (a resolution of) the orbifold quotient of $\mathbb{C}^2$ by a finite subgroup of SU(2).

### Fractional M-branes

An analogous McKay correspondence for (fractional) M-branes

graphics grabbed from HSS18

is considered informally in the string theory literature (for instance in discussion of M-theory on G2-manifolds) but has not been given a correspondingly precise cohomological formulation yet.

## Properties

### RR-Charge

Under the identification (above)

$KU_G^0(\ast) \simeq R_{\mathbb{C}}(G)$

of the fractional D-brane charges at a $G$-orbifold singularity with of the equivariant K-theory of the point and hence with the representation ring of $G$, a character

$\chi_V \coloneqq tr_V(-) \;\colon\; ConjCl(G) \to \mathbb{C}$

of a representation $V \in R(G)$ correspondes to the RR-field charge

(2)$Q_V(g) \;=\; \frac{ \chi_V(g) }{ {\vert G \vert} }$

of the corresponding fractional D-brane in the $g$-twisted sector.

See at RR-field tadpole cancellation the section For fractional D-branes

## References

The concept originates with

based on the analysis of perturbative string theory on (global) orbifold backgrounds in

The proposal that D-brane charge on orbifolds is given by equivariant K-theory goes back to

but it was pointed out that only a subgroup or quotient group of equivariant K-theory can be physically relevant, in

Further discussion in terms of equivariant K-theory:

• Hugo Garcia-Compean, D-branes in Orbifold Singularities and Equivariant K-Theory, Nucl.Phys. B557 (1999) 480-504 (arXiv:hep-th/9812226)

Survey with an eye towards string phenomenology is in

The McKay correspondence as an integral transform (Fourier-Mukai transform) in (equivariant) K-theory, and hence in terms of fractional D-brane charge is due to

Detailed mathematical discussion of fractional D-branes in their incarnation as Ext-groups of coherent sheaves is in

and with relation to Bridgeland stability conditions in

Also on stability: