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Kevin J. Costello

Publications and source records attributed to Kevin J. Costello.

13 recordsLinked to original sources
arXiv2025

Bootstrapping two-loop QCD amplitudes

Form factors of self-dual gauge theory are equal to correlators of an (extended) celestial chiral algebra. This suggests that these form factors can be computed using the "bootstrap" method familiar from 2d CFTs. The method can also be applied to certain QCD amplitudes, which are built from form-factors of self-dual gauge theory. In this paper this bootstrap method is applied to compute two-loop all-plus QCD amplitudes, for $SU(N)$ gauge theory with certain special matter content. A closed formula is presented for all single-trace amplitudes.

hep-th↗
arXiv2021

Quantizing local holomorphic field theories on twistor space

This paper studies a class of four-dimensional quantum field theories which arise by quantizing local holomorphic field theories on twistor space. These theories have some remarkable properties: in particular, all correlation functions are rational functions. The two main examples are the $WZW_4$ model of Donaldson and Losev, Moore, Nekrasov and Shatashvili, and self-dual Yang-Mills theory. In each case, anomalies on twistor space must be cancelled by a Green-Schwarz mechanism, which introduces additional fields. For $WZW_4$, this only works for $G = SO(8)$ and the additional field is gravitational. For self-dual Yang-Mills, this works for $SU(2)$, $SU(3)$, $SO(8)$ and the exceptional groups, and the additional field is an axion.

hep-th↗
arXiv2013

Integrable lattice models from four-dimensional field theories

This note gives a general construction of an integrable lattice model (and a solution of the Yang-Baxter equation with spectral parameter) from a four-dimensional field theory which is a mixture of topological and holomorphic. Spin-chain models arise in this way from a twisted, deformed version of N=1 gauge theory.

hep-th↗
arXiv2013

Notes on supersymmetric and holomorphic field theories in dimensions 2 and 4

These notes explore some aspects of formal derived geometry related to classical field theory. One goal is to explain how many important classical field theories in physics -- such as supersymmetric gauge theories and supersymmetric sigma-models -- can be described very cleanly using derived geometry. In particular, I describe a mathematically natural construction of Kapustin-Witten's P^1 of twisted supersymmetric gauge theories.

math.QA↗
arXiv2012

Quantum BCOV theory on Calabi-Yau manifolds and the higher genus B-model

Bershadsky-Cecotti-Ooguri-Vafa (BCOV) proposed that the B-model of mirror symmetry should be described by a quantum field theory on a Calabi-Yau variety, which they called the Kodaira-Spenser theory (we call it the BCOV theory). This is the first of three papers in which we construct and analyze the quantum BCOV theory. In this paper, we construct the classical field theory on a Calabi-Yau variety of arbitrary dimension; define what it means to give a quantization; analyze the relation Givental's symplectic formalism for Gromov-Witten theory; prove uniqueness of the quantization on an elliptic curve; and prove the Virasoro constraints on an elliptic curve. The second paper (arXiv:1112.4063) proves that the partition function of the quantum BCOV theory on the elliptic curve is equivalent to the Gromov-Witten theory of the mirror elliptic curve. The third paper, in progress, constructs the quantum BCOV theory on a general Calabi-Yau.

math.QA↗
arXiv2011

A geometric construction of the Witten genus, II

I give a rigorous construction of a 2-dimensional quantum field theory of maps from an elliptic curve to a compact complex manifold X, and I show that the partition function of this theory is the Witten genus of X. The results proved here were announced (in a slightly different form) in arXiv:1006.5422.

math.QA↗
arXiv2008

Closed String TCFT for Hermitian Calabi-Yau Elliptic Spaces

We describe an explicit action of the prop of the chains on the moduli space of Riemann surfaces on the Hochschild complex of a Calabi-Yau elliptic space. One example of such an elliptic space extends the known string topology operations, for all compact simply-connected manifolds, to a collection indexed by the de Rham currents on the moduli space. Another example pertains to the B-model at all genera.

math.QA↗
arXiv2007

Renormalisation and the Batalin-Vilkovisky formalism

This paper gives a way to renormalise certain quantum field theories on compact manifolds. Examples include Yang-Mills theory (in dimension 4 only), Chern-Simons theory and holomorphic Chern-Simons theory. The method is within the framework of the Batalin-Vilkovisky formalism. Chern-Simons theory is renormalised in a way respecting all symmetries (up to homotopy). This yields an invariant of smooth manifolds: a certain algebraic structure on the cohomology of the manifold tensored with a Lie algebra, which is a "higher loop" enrichment of the natural Lie-infinity structure.

math.QA↗
arXiv2006

Topological conformal field theories and gauge theories

This paper gives a construction, using heat kernels, of differential forms on the moduli space of metrised ribbon graphs, or equivalently on the moduli space of Riemann surfaces with boundary. The construction depends on a manifold with a bundle of Frobenius algebras, satisfying various conditions. These forms satisfy gluing conditions which mean they form an open topological conformal field theory, i.e. a kind of open string theory. If the integral of these forms converged, it would yield the purely quantum part of the partition function of a Chern-Simons type gauge theory. Yang-Mills theory on a four manifold arises as one of these Chern-Simons type gauge theories.

math.QA↗
arXiv2006

Topological conformal field theories and Calabi-Yau categories

This is the first of two papers which construct a purely algebraic counterpart to the theory of Gromov-Witten invariants (at all genera). These Gromov-Witten type invariants depend on a Calabi-Yau A-infinity category, which plays the role of the target in ordinary Gromov-Witten theory. When we use an appropriate A-infinity version of the derived category of coherent sheaves on a Calabi-Yau variety, this constructs the B model at all genera. When the Fukaya category of a compact symplectic manifold X is used, it is shown, under certain assumptions, that the usual Gromov-Witten invariants are recovered. The assumptions are that a good theory of open-closed Gromov-Witten invariants exists for X, and that the natural map from the Hochschild homology of the Fukaya category of X to the ordinary homology of X is an isomorphism.

math.QA↗
arXiv2006

A dual point of view on the ribbon graph decomposition of moduli space

In this note, I discuss in some detail the dual version of the ribbon graph decomposition of the moduli spaces of Riemann surfaces with boundary and marked points, which I introduced in math.AG/0402015, and used in math.QA/0412149 to construct open-closed topological conformal field theories. This dual version of the ribbon graph decomposition is a compact orbi-cell complex with a natural weak homotopy equivalence to the moduli space.

math.GT↗
arXiv2005

The Gromov-Witten potential associated to a TCFT

This is the sequel to my preprint "TCFTs and Calabi-Yau categories", math.QA/0412149. Here we extend the results of that paper to construct, for certain Calabi-Yau A-infinity categories, something playing the role of the Gromov-Witten potential. This is a state in the Fock space associated to periodic cyclic homology, which is a symplectic vector space. Applying this to a suitable A-infinity version of the derived category of sheaves on a Calabi-Yau yields the B model potential, at all genera. The construction doesn't go via the Deligne-Mumford spaces, but instead uses the Batalin-Vilkovisky algebra constructed from the uncompactified moduli spaces of curves by Sen and Zwiebach. The fundamental class of Deligne-Mumford space is replaced here by a certain solution of the quantum master equation, essentially the "string vertices" of Zwiebach. On the field theory side, the BV operator has an interpretation as the quantised differential on the Fock space for periodic cyclic chains. Passing to homology, something satisfying the master equation yields an element of the Fock space.

math.QA↗