### Fermions

Paper 2, of Dan Freed’s, Greg Moore’s and my series of papers on Orientifolds is out. This one focusses on the worldsheet formulation — particularly the worldsheet fermions. Because it’s for a volume dedicated to Is Singer, it’s written in a somewhat mathematical style. So, while it’s more accessible than our telegraphic Précis, it’s maybe a little tough-going for some of our physics audience. There will be some more physics-oriented papers in the series, but I don’t think any of them will be specifically devoted to the worldsheet. So, contrary to my usual practice, I’m going to try to distill some of the salient points of our current paper, here.

If you wish, we address two questions

- What is the NS B-field in the Type-II string?
- What is the role of spin structures, both on the worldsheet and in spacetime, and how do those notions generalize in an orientifold background?

Obviously, any theory with spacetime fermions must invoke a spin structure on spacetime, $X$. Type-II string theory invokes a *pair* of spin structures, because the R-NS and NS-R sectors are a-priori distinct. Let’s call those spin structures, respectively, $\kappa_l$ and $\kappa_r$. In Type-IIB, they have the same chirality; in IIA, they have the opposite chirality. The difference between the two is a *pair*, $(t,a)\in H^0(X,\mathbb{Z}/2)\oplus H^1(X,\mathbb{Z}/2)$, where “$t$” measures the difference in chirality, and “$a$” measures the difference in boundary conditions for the fermions around any 1-cycle in $X$.

In an oriented Type-II background, one can think, interchangeably, about a pair of spin structures, $(\kappa_l,\kappa_r)$, or about a single spin structure, $\kappa\equiv\kappa_l$, and the cohomology classes, $(t,a)$. An obvious question is: which picture generalizes most straightforwardly to an orientifold background?

We argue that it’s the latter picture which generalizes: instead of a spin structure on $X$, one has a *twisted spin structure*, $\kappa$, and the pair $(t,a)$ lies in the Borel-equivariant cohomology of $X$. The precise definition is in section 6 of our paper, but the salient point is that it *depends* both on the orientifold double cover (which we can think of as specifying a class $w\in H^1(X,\mathbb{Z}/2)$) and on the pair $(t,a)$. In fact, the condition for the *existence* of a twisted spin structure is
$\begin{split}
w_1(X) = t w \\
w_2(X) = t w^2 + a w
\end{split}$
(which reduces to the usual conditions for the existence of a spin structure, in the non-orientifold case, where $w=0$).

The Ramond-Ramond fields are bispinors. Hence *their* geometrical nature is also controlled by $(t,a)$. $t$ determines whether the bispinor decomposes into even or odd differential forms. When $a\in H^1(X,\mathbb{Z}/2)$ is nonzero, those differential forms are twisted by a real line bundle, $L(a)$. Changing the nature of the RR fields is the role-in-life of the NS B-field (or, at least, of its familiar 3-form field strength); the fancy phrase is that it twists K-theory. So it’s natural to combine $(t,a)$ with what would have been the B-field of the bosonic string. The latter is a twisted differential form (twisted by $w$), and fits into an exact sequence
$0 \to \check{H}^{3+w}(X)\to \check{R}^{-1}(X)\to H^0(X,\mathbb{Z}/2)\oplus H^1(X,\mathbb{Z}/2)\to 0$
for a certain very interesting generalized (differential) cohomology theory, $\check{R}^{\bullet}(X)$, which is where — we would like to argue — the Type-II NS B-field “lives.” As an exact sequence of Abelian groups, the above sequence doesn’t split, but it does split on the level of sets. So it does make sense (if you’re suitably careful) to think of our NS B-field as a triple $(h,t,a)$, with $h\in \check{H}^{3+w}(X)$, which is where (for instance) the B-field of the bosonic string (orientifold, when $w\neq 0$) lives.

On the worldsheet of the oriented Type-II string, the worldsheet fermions couple to spin-structures, $\alpha_l,\alpha_r$ of opposite chirality. This spin-structure data can be thought of as a (discrete) worldsheet field, which is to be integrated (summed) over, in the worldsheet path integral.

In the orientifold case, there are still local spin-structures, $\alpha_l,\alpha_r$, but they don’t fit together to global structures, because $\Sigma$ need not even be orientable, let alone spin. Instead, $\alpha_l,\alpha_r$ fit together to a (single) spin structure on the orientation double-cover $\hat{\Sigma}\to\Sigma$. (I have to say that there’s some confusion on this point, in the literature. You’ll find mention – when the issue is discussed at all – of a $Pin^-$ structure on $\Sigma$, which is not at all the same thing.)

Now for a puzzle. We can even dispense with the orientifold case, and ask the question for the oriented Type-II string:

How does the (pair of) spacetime spin structure(s) affect the worldsheet path integral?

Indeed, the obvious conundrum of the NSR formalism is that the worldsheet theory, at least classically, does not seem to make any mention of the spacetime spin struncture at all. That, of course, is an illusion. The quantum theory does know about the spacetime spin structure; the quantization of the workdsheet fermions depends on it. In fancy words, to trivialize the fermion Pfaffian line bundle (and, thus, to interpret the fermion Pfaffian as a *function* on the bosonic parameter space) *requires* a spacetime spin structure — more precisely, a pair $(\kappa_l,\kappa_r)$ or, equivalently, the pair $\kappa\equiv\kappa_l$ and $(t,a)$.

We can make this more concrete, after a little disgression on integration in differential $\check{R}^\bullet$ theory.

The B-field in the oriented bosonic string, takes values $\check{\beta}\in\check{H}^3(X)$. Integration, in that theory, requires an orientation on $\Sigma$ (which we have), and lands us $\int_\Sigma \phi^\ast\check{\beta} \in \check{H}^1(pt) = \mathbb{R}/\mathbb{Z}$. In a bosonic orientifold, $\Sigma$ is unoriented, but a twisted form, $\check{\beta}\in\check{H}^{3+w}(X)$, can be integrated *without* need of an orientation (using that $\phi^\astw = w_1(\Sigma)$).

Integration in $R$-theory requires more than an orientation; it requires a spin structure, $\alpha$. The integral of the triple $(t(\check{\beta}),a(\check{\beta}),h(\check{\beta}))$ is

where $q(\cdot)$ is known, variously, as

- the quadratic refinement of the intersection pairing, $\langle\cdot,\cdot\rangle$, on $H^1(\Sigma,\mathbb{Z}/2)$
- the mod-2 Dirac Index
- the Kervaire invariant in $d=2$.

$q(\alpha)$ is 1 for $\alpha$ an odd spin structures, and 0 for $\alpha$ an even spin structures^{1}.

Say we change the pair of spacetime spin structures, $(\kappa_l,\kappa_r)$, or equivalently, the pair $(\kappa, a )$, to $(\kappa', a')$. The claim is that the integrand of the worldsheet path integral changes by a factor of

This formula has a number of interesting properties.

- It obeys the obvious cocycle condition. To see that requires the identity $q(\alpha + a + b) = q(\alpha + a) + q(\alpha + b) - q(\alpha) + \langle a, b\rangle$
- When $a'=a$, which is to say when the nature of the RR field is unchanged by the change in spin structure(s), the expression is holomorphically-factorized, i.e. $\langle\alpha_l-\alpha_r, \phi^*(\kappa'-\kappa)\rangle = q(\alpha_l +\phi^*(\kappa'-\kappa)) - q(\alpha_l) + q(\alpha_r +\phi^*((\kappa'+a)-(\kappa+a))) - q(\alpha_r)$
- When $a'\neq a$ — ie, when the nature of the RR fields is altered by the change in spin structures — the formula is
*not*holomorphically-factorized. Nonetheless, it invariant under exchanging the roles of left- and right-movers $\begin{gathered}\alpha_l \leftrightarrow \alpha_r \\ \kappa \to \kappa + a \\ \kappa'\to \kappa'+ a' \end{gathered}$ - When the spin structures for the R-NS and NS-R sectors are the same, i.e. when $a'=a=0$, it agrees with known formulæ about “Scherk-Schwarz” compactifications.

So, already for the oriented Type-II string, we have an interesting generalization of Scherk-Schwarz. For instance, one might break half of the spacetime supersymmetries using this mechanism. In the orientifold case, the story is considerably more subtle; whereas here I could write down the B-field integral (1) and the fermion Pfaffian *separately* as functions on the bosonic parameter space, in the orientifold case, only the product of the two makes sense as a function.

But that’s the subject of another paper in this series …

^{1} Note that the $t$-dependence of (1) is exactly the difference between the sum over worldsheet spin structures in IIB ($t=0$) and in IIA ($t=1)$. In that sense we like to say that IIA and IIB (even in 10 dimensions) are different backgrounds of the *same* theory (with different vacuum expectation values of the NS B-field). The $a$-dependence is a little more exotic; it corresponds to what we like to call “generalized Scherk-Schwarz”.