Self-dual quiver moduli and orientifold Donaldson-Thomas invariants
In string theory, it is of central importance to allow the basic building blocks to be unoriented (one-dimensional manifolds, with or without boundary). For example, the cancellation of ``tadpole-type'' boundaries in the moduli of open strings requires the inclusion of Klein surfaces in the quantization process. Many a hope for eventual contact between string theory and the real world rests on this mechanism for anomaly cancellation. In comparison, triangulated categories with duality, which are the mathematical counterpart of orientifolds, are severely underrepresented in the mathematical literature. This paper contributes to filling this gap in the representation theory of quivers, which is at the heart of many connections between algebra, geometry, and supersymmetric physics. The starting point is a quiver with an involution inverting arrows. Given a choice of duality structure on the fixed nodes, such an involution can be lifted to the abelian category of quiver representations, including the notion of self-dual representations on orthogonal/symplectic vector spaces (Section 2). A compatible stability condition on such a quiver is the choice of a real slope at each vertex that is anti-invariant under the involution. A self-dual representation is semistable with respect to isotropic subobjects if and only if it is semistable as an ordinary representation, and this notion agrees with Mumford's numerical criterion derived from geometric invariant theory for the quotient by the relevant orthogonal/symplectic group (Section 3). The Hall module formalism, introduced by the author in a previous publication, leads to an explicit formula for the number of \({\mathbb F}_q\)-rational points of stacks of semistable self-dual representations. Their generating function is used to give an initial rigorous definition of orientifold Donaldson-Thomas invariants of quiver representations. This is the central result of the paper (Section 4). The behaviour of these invariants under wall-crossing matches expectations in the physics literature. In a curious departure, the orientifold invariants do not in themselves admit a multiplication, as one would expect from the relationship to physical BPS invariants. Rather, the new invariants form a module over the algebra of oriented BPS invariants of the parent theory, derived, in the present case, from the ordinary representation theory of the quiver. One of the technical difficulties in developping orientifolds mathematically is the oftentimes cumbersome book-keeping of signs and duality structures. This paper does an excellent job at guiding the reader through the formalism, and leaves one looking forward to further work on the subject.
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