Research
I am generally interested in homotopical aspects of quantum field theory, as well as applications of QFT to representation theory and complex geometry through the lens of factorization algebras. Currently, my current work can be organized into three buckets:
Holomorphic QFT. Holomorphic field theories generalize the notion of a chiral CFT on Riemann surfaces, and exist in any dimension. They are strictly more abundant than topological field theories, yet are still described by attainable mathematical structures of algebraic (vertex algebras in CFT, say) and geometric (conformal blocks, for instance) flavors.
(Higher) gauge theories and the BV formalism.
Dualities in string theory and QFT.
Check out some of my collaborators: Kevin Costello, Chris Elliott, Vassily Gorbounov, Ryan Grady, Owen Gwilliam, Natalie Paquette, Eugene Rabinovich, Ingmar Saberi, Pavel Safronov, Matt Szczesny, Philsang Yoo.
Holomorphic quantum field theory
Quantization of topological-holomorphic field theories: local aspects.
We study one-loop renormalization and quantization for theories which are partially topological and holomorphic. Our main result states that for theories with at least one topological direction there is no one-loop anomaly.
With Owen Gwilliam and Eugene Rabinovich.
Twisting pure spinor superfields with applications to supergravity
We develop a twisted version of the pure spinor formalism with applications to supergravity. Along the way we resolve several conjectures of Costello-Li on twists of supergravity in dimensions 10 and 11.
With Ingmar Saberi.
Constraints in the BV formalism: six-dimensional SUSY and its twists
The theory of constraints in the BV formalism is developed with an eye towards the ubiquitous six-dimensional superconformal theory. We additionally compute the holomorphic and partially topological twists of the theory.
With Ingmar Saberi.
Holomorphic Poisson field theories
A family of quantum field theories which exist in the world of holomorphic Poisson geometry is considered. In the symplectic case such theories give rise to examples of ``weakly" topological factorization algebras. We provide evidence of a relationship of such theories to twists of supergravity.
With Chris Elliott.
Superconformal algebras and holomorphic field theories
We compute the holomorphic twist of superconformal algebras in dimension four, and use them to study deformations of holomorphic factorization algebras. As an application we provide a mathematical model for the localization of 4d N=2 theories to chiral algebras.
With Ingmar Saberi.
Twisted characters and holomorphic symmetries
We formulate and axiomatize the theory of characters for observables of holomorphic field theories in arbitrary dimensions, generalizing the notion of a ``q-character" of a vertex algebra. We compute several physically meaningful examples.
With Ingmar Saberi.
Higher Kac-Moody algebras and symmetries of holomorphic field theories
We introduce the higher Kac-Moody algebra using the language of factorization algebras. In parallel with the story in 2d CFT, these algebras appear as symmetries of holomorphic QFT in arbitrary dimensions. As an application, we give a free field realization of the algebra.
With Owen Gwilliam.
Renormalization for holomorphic field theories
Inspired by the work of Costello--Li on BCOV theory, we formulate a general notion of a holomorphic field theory, and study its one-loop renormalization.
Dualities in QFT and string theory
Twisted heterotic / type I duality
We formulate a twisted version of the conjectured duality between heterotic and type I string theories. We provide a non-trivial check of this duality by showing that certain infinite-dimensional Lie algebras of global gauge transformations built from each theory are isomorphic.
With Kevin Costello.
Koszul duality in QFT
We introduce basic aspects of the algebraic notion of Koszul duality for a physics audience. We then review its appearance in the physical problem of coupling QFTs to topological line defects, and illustrate the concept with some examples drawn from twists of various simple supersymmetric theories.
With Natalie Paquette.
Gauge theories and the BV formalism
Higher deformation quantization for KW theories
We pursue a uniform quantization of all twists of 4-dimensional N = 4 supersymmetric Yang-Mills theory, using the BV formalism, and we explore consequences for factorization algebras of observables. Our central result is the construction of a one-loop exact quantization on flat space for all such twists and for every point in a moduli of vacua.
With Chris Elliott and Owen Gwilliam.
Batalin-Vilkovisky quantization and supersymmetric twists
We show that a family of topological twists of SUSY mechanics with Kähler target exhibits a BV quantization. Using this we make a general proposal for the Hilbert space of states in terms of the cohomology of a certain perverse sheaf. We give several examples including DT invariants, Haydys-Witten theory, and the 3d A-model.
With Pavel Safronov.
A taxonomy of twists of supersymmetric Yang--Mills theory
We characterize all twists of supersymmetric Yang--Mills theory in any dimension using the BV formalism. Central to our construction is the use of L-infinity algebras to provide an off-shell homotopical action of SUSY.
With Chris Elliott and Pavel Safronov.
Factorization algebras and abelian CS/WZW correspondences
We develop a method of quantization for free field theories on manifolds with boundary. Our main application is to realize the CS/WZW correspondence in terms of stratified factorization algebras.
With Owen Gwilliam and Eugene Rabinovich.
A one-loop exact quantization of Chern--Simons theory
We examine Chern--Simons theory as a deformation of a partially topological version of BF theory. We extract various consequences of the resulting one-loop exact quantization.
With Owen Gwilliam.
Asymptotic freedom in the BV formalism
We provide a mathematical definition of the beta-function in the context of perturbative QFT. As an application, we prove that four-dimensional Yang--Mills theory is asymptotically free.
With Chris Elliott and Philsang Yoo.
Quantization of sigma-models
CDO's via the quantization of the holomorphic sigma-model
We show that the local observables of the curved beta-gamma system encode the sheaf of chiral differential operators using the machinery of factorization algebras and QFT.
With Vassily Gorbounov and Owen Gwilliam.
The holomorphic bosonic string
We analyze a holomorphic version of the bosonic string in the formalism of QFT developed by Costello and collaborators, which provides a powerful combination of renormalization theory and the BV formalism.
With Owen Gwilliam.
Homotopy RG flow and the non-linear sigma-model
Using the setup from "Asymptotic freedom in the BV formalism", we compute the beta-function for the 2d Riemannian sigma-model.
With Ryan Grady.
The Virasoro vertex algebra and factorization algebras on Riemann surfaces
We construct a factorization algebra on any Riemann surface, which is locally equivalent to the Virasoro vertex algebra. We compute conformal blocks and correlation functions, as well as give a model for free field realization using the BV formalism.
Toroidal prefactorization algebras
Toroidal prefactorization algebras associated to holomorphic fibrations and a relationship to vertex algebras. We introduce a factorization algebra associated to any locally trivial holomorphic fibration. In the case that the base manifold is a Riemann surface, this gives a geometric description of toroidal vertex algebras.
With Matt Szczesny and Jackson Walters