Seminar talk, 18 November 2020
Speaker: Andrey Losev
Title: Tau theory, d=10 N=1 SUSY and BV
1. David Gross and Edward Witten said, "Everyone in string theory is convinced...that spacetime is doomed. But we don't know what it's replaced by." Nathan Seiberg said, "I am almost certain that space and time are illusions. These are primitive notions that will be replaced by something more sophisticated."
Tau-theory (Tau stands for Tensor Algebra Universe) is an attempt to introduce such replacement. In the contrast with the expectations of abovementioned scientists the replacement is not that sophisticated, it is rather simple: the space of solutions to A-infinity equations embedded into tensor algebra. Being space of solutions it forms what I call a tau-landscape.
Most of points of tau-landscape are not interesting for physics like most of exoplanets are not interesting for biology. However, we may look at commutative associative (super)algebras, concentrating on those whose spectrum is smooth. These are like planets suitable for life but with no life yet.
Things become more interesting if we study tau-landscape in the neighborhood of such points. The first order deformations of A-infinity algebra are Hochschild cohomology of the A-infinity algebra, and for smooth scheme they are polyvector fields on these scheme (due to Hochschild-Kostant-Rosenberg theorem) that would form fields of the emerging QFT. The arising world would be made out of "fluctuations of the multiplication table".
Having spacetime and fields we would need equations of motion of these fields. We do not need to invent them since they naturally come from the obstruction for the first order deformations to be deformations of A-infinity algebra to the second order. And this obstruction is given by Schouten bracket on polyvectors. Thus we have universal equations, this is like to find life on a planet. Biologists would be satisfied at that point (like mathematical physicists that like to play with model theories) but it would be much interesting to find intelligent life. For physicists this would mean to find something like a gauge theory with matter fields, moreover, not a topological theory like Chern-Simons but rather some version of Yang-Mills theory.
The main point of my talk is that it can be done if we "condensate" the very specific homological vector field on a very specific manifold and consider fluctuations around this condensate. It would be the pure spinor construction for N=1 d=10 supersymmeric Yang-Mills theory.
2. In the second part of my talk I would explain how supersymmetry of N=1 d=10 SYM is realized from the point of view of pure spinor construction.
2a. Symmetry from BV point of view.
Those, who love BV think that BV is a language, and all interesting statements may be turned into a statement that some action solves master equation and all constructions are some kinds of BV integral. In BV language the statement that a function on a manifold (action) is invariant under the action of certain Lie algebra is just a statement that the extended action (involving ghosts) solves classical master equation. However, this form of the action is not invariant under the BV integral, so we should generalize. Thus, we get the notion of the infinity-representation of the Lie algebra, where zero-th power of the Lie algebra is mapped to a function on a manifold (invariant action), k-th power - to k-polyvector fields, that satisfy quadratic consistency equations. If the bivector part of this map is nonzero, the infinity-representation is no longer representation, rather it is what is called (in physics literature) "the action of the Lie algebra that closes on-shell", and I would like to add "completed by bivectors". I will describe appearance of such infinity-representation of so(n) acting on , , using BV integral as a prototype of the explanation of "on-shell" supersymmetry of N=1 d=10 SYM.
I will recall the projection that makes from homological vector field (L-infinity algebra) solution to master equation on cotangent bundle to manifold M (with the inversed parity of the fiber) a solution to master equation on M (if M has a BV symplectic form preserved by the homological vector field). This solution is a "Hamiltonian" for the vector field. In this way CS theory may be obtained from BF theory.
2c. Induction of SYM.
I will recall how N=1 d=10 SYM appears from BF-like theory (sometimes called Witten Open String Field theory) constructed from supercommutative DGA with values in End(V), i.e. matrices). The DGA would be a ring , where are even while are odd, and is the ideal generated by ten -quadrics (equations for pure spinor manifold).
Differential in this DGA is a sum of two differentials .
Cohomology of turn out to be cotangent bundle over the space of fields and gauge symmetries of N=1 d=10 SYM. Induction of homological vector field to cohomology of followed by projection gives the action of SYM.
2d. Induction of infinity=BV=refined on-shell SUSY of d=10 N=1 SYM.
Original DGA is classically (not infinity) supersymmetric, that is realized by the differential depending on the superghosts that anticommute with D. Induction to cohomology followed by projection take this supersymmetry into the infinity-symmetry ("on-shell") of the N=1 D=10 SYM.
Based on old but not well known papers arXiv:0705.2191 "On Pure Spinor Superfield Formalism" JHEP 0710:074,2007 and arXiv:0707.1906 "Materializing Superghosts" JETP Lett.86:439-443,2007 by Victor Alexandrov, Dmitry Krotov, Andrei Losev, Vyacheslav Lysov.