# Dr Martin Wolf

+44 (0)1483 684770
25A AA 04

Department of Mathematics, Fields, Strings, and Geometry Group.

### University roles and responsibilities

• Maths IT Liaison Officer
• Newton Institute Correspondent

### Publications

Hyungrok Kim, Branislav Jurčo, TOMMASO MACRELLI, Christian Saemann, MARTIN WOLF (2021)Double Copy from Homotopy Algebras, In: Fortschritte der Physik2100075 Wiley

We show that the BRST Lagrangian double copy construction of N = 0 supergravity as the ‘square’ of Yang–Mills theory finds a natural interpretation in terms of homotopy algebras. We significantly expand on our previous work arguing the validity of the double copy at the loop level, and we give a detailed derivation of the double-copied Lagrangian and BRST operator. Our constructions are very general and can be applied to a vast set of examples.

Branislav Jurčo, Lorenzo Raspollini, Christian Sämann, Martin Wolf (2019)L‐Algebras of Classical Field Theories and the Batalin–Vilkovisky Formalism, In: Fortschritte der Physik67(7)1900025pp. 1-60 Wiley

We review in detail the Batalin–Vilkovisky formalism for Lagrangian field theories and its mathematical foundations with an emphasis on higher algebraic structures and classical field theories. In particular, we show how a field theory gives rise to an L∞‐algebra and how quasi‐isomorphisms between L∞‐algebras correspond to classical equivalences of field theories. A few experts may be familiar with parts of our discussion, however, the material is presented from the perspective of a very general notion of a gauge theory. We also make a number of new observations and present some new results. Most importantly, we discuss in great detail higher (categorified) Chern–Simons theories and give some useful shortcuts in usually rather involved computations.

Branislav Jurčo, Tommaso Macrelli, Lorenzo Raspollini, Christian Sämann, Martin Wolf (2019)L-Algebras, the BV Formalism, and Classical Fields, In: Fortschritte der Physik - Special Issue: Proceedings of the LMS/EPSRC Durham Symposium on Higher Structures in M‐Theory (Durham University (UK) 12–18 August 2018)67(8-9) Wiley-VCH Verlag

We summarise some of our recent works on L∞‐algebras and quasi‐groups with regard to higher principal bundles and their applications in twistor theory and gauge theory. In particular, after a lightning review of L∞‐algebras, we discuss their Maurer–Cartan theory and explain that any classical field theory admitting an action can be reformulated in this context with the help of the Batalin–Vilkovisky formalism. As examples, we explore higher Chern–Simons theory and Yang–Mills theory. We also explain how these ideas can be combined with those of twistor theory to formulate maximally superconformal gauge theories in four and six dimensions by means of L∞‐quasi‐isomorphisms, and we propose a twistor space action.

A Pittelli, Alessandro Torrielli, Martin Wolf (2014)Secret symmetries of type IIB superstring theory on AdS(3) x S-3 x M-4, In: JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL47(45)455402 IOP PUBLISHING LTD

We establish features of so-called Yangian secret symmetries for AdS3 type IIB superstring backgrounds, thus verifying the persistence of such symmetries to this new instance of the AdS/CFT correspondence. Specifically, we find two a priori different classes of secret symmetry generators. One class of generators, anticipated from the previous literature, is more naturally embedded in the algebra governing the integrable scattering problem. The other class of generators is more elusive and somewhat closer in its form to its higher-dimensional AdS5 counterpart. All of these symmetries respect left-right crossing. In addition, by considering the interplay between left and right representations, we gain a new perspective on the AdS5 case. We also study the $R____mathcal{T}____mathcal{T}$-realisation of the Yangian in AdS3 backgrounds, thus establishing a new incarnation of the Beisert–de Leeuw construction.

Tommaso Macrelli, Christian Sämann, Martin Wolf (2019)Scattering amplitude recursion relations in Batalin-Vilkovisky–quantizable theories, In: Physical Review D100(4)045017pp. 045017-1 American Physical Society

Tree-level scattering amplitudes in Yang-Mills theory satisfy a recursion relation due to Berends and Giele which yields e.g., the famous Parke-Taylor formula for maximally helicity violating amplitudes. We show that the origin of this recursion relation becomes clear in the Batalin-Vilkovisky (BV) formalism, which encodes a field theory in an L∞-algebra. The recursion relation is obtained in the transition to a smallest representative in the quasi-isomorphism class of that L∞-algebra, known as a minimal model. In fact, the quasi-isomorphism contains all the information about the scattering theory. As we explain, the computation of such a minimal model is readily performed in any BV quantizable theory, which, in turn, produces recursion relations for its tree-level scattering amplitudes.

M Wolf (2017)Twistors and aspects of integrability of self-dual SYM theory, In: Proc Intern Workshop on Supersymmetries and Quantum Symmetries 1 (2005) 44
M Wolf, Z Horvath, O Lechtenfeld (2003)Non-commutative instantons via dressing and splitting approaches, In: JHEP 0212 (2002) 060
M Wolf, AD Popov, AG Sergeev (2003)Seiberg-Witten monopole equations on non-commutative R^4, In: J Math Phys 44 (2003) 4527
Christian Saemann, Branislav Jurco, Hyungrok Kim, Tommaso Macrelli, Martin Wolf (2020)Perturbative Quantum Field Theory and Homotopy Algebras, In: Proceedings of Corfu Summer Institute 2019 "School and Workshops on Elementary Particle Physics and Gravity" — PoS(CORFU2019) Proceedings Of Science

We review the homotopy algebraic perspective on perturbative quantum field theory: classical field theories correspond to homotopy algebras such as A∞- and L∞-algebras. Furthermore, their scattering amplitudes are encoded in minimal models of these homotopy algebras at tree level and their quantum relatives at loop level. The translation between Lagrangian field theories and homotopy algebras is provided by the Batalin-Vilkovisky formalism. The minimal models are computed recursively using the homological perturbation lemma, which induces useful recursion relations for the computation of scattering amplitudes. After explaining how the homolcogical perturbation lemma produces the usual Feynman diagram expansion, we use our techniques to verify an identity for the Berends-Giele currents which implies the Kleiss-Kuijf relations.

M Wolf (2007)Self-dual supergravity and twistor theory, In: Class Quant Grav 24 (2007) 6287
M Wolf, AD Popov, AG Sergeev (2005)Non-trivial solutions of the Seiberg-Witten equations on the non-commutative four-dimensional Euclidean space, In: Proc Steklov Inst 251 (2005) 127
M Wolf, J Schmiegel, M Greiner (2000)Artificiality of multifractal phase transitions, In: Phys Lett A 266 (2000) 276
Branislav Jurčo, Tommaso Macrelli, Christian Sämann, Martin Wolf (2020)Loop Amplitudes and Quantum Homotopy Algebras, In: Journal of High Energy Physics20203 Springer Verlag

We derive a recursion relation for loop-level scattering amplitudes of La- grangian field theories that generalises the tree-level Berends-Giele recursion relation in Yang-Mills theory. The origin of this recursion relation is the homological perturbation lemma, which allows us to compute scattering amplitudes from minimal models of quantum homotopy algebras in a recursive way. As an application of our techniques, we give an alternative proof of the relation between non-planar and planar colour-stripped scattering amplitudes.

Martin Wolf, B Jurco, C Saemann (2016)Higher Groupoid Bundles, Higher Spaces, and Self-Dual Tensor Field Equations, In: Fortschritte der Physik / Progress of Physics64(8-9)pp. 674-717 Wiley

We develop a description of higher gauge theory with higher groupoids as gauge structure from first principles. This approach captures ordinary gauge theories and gauged sigma models as well as their categorifications on a very general class of (higher) spaces comprising presentable differentiable stacks, as e.g. orbifolds. We start off with a self-contained review on simplicial sets as models of (∞,1)-categories. We then discuss principal bundles in terms of simplicial maps and their homotopies. We explain in detail a differentiation procedure, suggested by Severa, that maps higher groupoids to L∞-algebroids. Generalising this procedure, we define connections for higher groupoid bundles. As an application, we obtain six-dimensional superconformal field theories via a Penrose-Ward transform of higher groupoid bundles over a twistor space. This construction reduces the search for non-Abelian self-dual tensor field equations in six dimensions to a search for the appropriate (higher) gauge structure. The treatment aims to be accessible to theoretical physicists.

C Saemann, R Wimmer, M Wolf (2012)A twistor description of six-dimensional N = (1,1) super Yang-Mills theory, In: JOURNAL OF HIGH ENERGY PHYSICS(5)ARTN 020 SPRINGER
Branislav Jurčo, Christian Sämann, Urs Schreiber, Martin Wolf (2019)Higher Structures in M‐Theory, In: Fortschritte der Physik - Special Issue: Proceedings of the LMS/EPSRC Durham Symposium on Higher Structures in M‐Theory (Durham University (UK) 12–18 August 2018)67(8-9) Wiley-VCH Verlag

The key open problem of string theory remains its non-perturbative completion to M-theory. A decisive hint to its inner workings comes from numerous appearances of higher structures in the limits of M-theory that are already understood, such as higher degree flux fields and their dualities, or the higher algebraic structures governing closed string field theory. These are all controlled by the higher homotopy theory of derived categories, generalised cohomology theories, and L∞-algebras. This is the introductory chapter to the proceedings of the LMS/EPSRC Durham Symposium on Higher Structures in M-Theory. We first review higher structures as well as their motivation in string theory and beyond. Then we list the contributions in this volume, putting them into context.

C Samann, M Wolf (2013)On twistors and conformal field theories from six dimensions, In: JOURNAL OF MATHEMATICAL PHYSICS54(1)ARTN 01350 AMER INST PHYSICS
M Wolf, N Akerblom, C Saemann (2010)Marginal deformations and 3-algebra structures, In: Nuclear Physics, Section B826(3)pp. 456-489
M Wolf, R Ricci, AA Tseytlin (2007)On T-duality and integrability for strings on AdS backgrounds, In: JHEP 0712 (2007) 082
M Wolf, DH Correa (2010)Shaping up BPS states with matrix model saddle points, In: Journal of Physics A: Mathematical and Theoretical43(46)465402
M Wolf, LJ Mason (2009)Twistor actions for self-dual supergravities, In: Communications in Mathematical Physics288(1)pp. 97-123 Springer
M Wolf (2009)A connection between twistors and superstring sigma models on coset superspaces, In: The Journal of High Energy PhysicsJHEP09(071)

We consider superstring sigma models that are based on coset superspaces G/H in which H arises as the fixed point set of an order-4 automorphism of G. We show by means of twistor theory that the corresponding first-order system, consisting of the Maurer-Cartan equations and the equations of motion, arises from a dimensional reduction of some generalised self-dual Yang-Mills equations in eight dimensions. Such a relationship might help shed light on the explicit construction of solutions to the superstring equations including their hidden symmetry structures and thus on the properties of their gauge theory duals.

M Wolf (2010)A first course on twistors, integrability and gluon scattering amplitudes, In: Journal of Physics A: Mathematical and Theoretical43393001

These notes accompany an introductory lecture course on the twistor approach to supersymmetric gauge theories aimed at early stage PhD students. It was held by the author at the University of Cambridge during the Michaelmas term in 2009. The lectures assume a working knowledge of differential geometry and quantum field theory. No prior knowledge of twistor theory is required.

M Wolf, N Beisert, R Ricci, AA Tseytlin (2008)Dual superconformal symmetry from AdS5xS5 superstring integrability, In: Physical Review D78(126004)pp. 1-21
Leron Borsten, Branislav Jurco, Hyungrok Kim, Tommaso Macrelli, Christian Saemann, Martin Wolf (2021)BRST-Lagrangian Double Copy of Yang-Mills Theory, In: Physical Review Letters126191601pp. 191601-1-191601-7

We show that the double copy of gauge theory amplitudes to $\mathcal{N}=0$ supergravity amplitudes extends from tree level to loop level. We first explain that color-kinematic duality is a condition for the Becchi-Rouet-Stora-Tyutin operator and the action of a field theory with cubic interaction terms to double copy to a consistent gauge theory. We then apply this argument to Yang-Mills theory, where color-kinematic duality is known to be satisfied onshell at tree level. Finally, we show that the latter restriction can only lead to terms that can be absorbed in a sequence of field redefinitions, rendering the double copied action equivalent to $\mathcal{N}=0$ supergravity.

Christian Samann, Martin Wolf (2017)Supersymmetric Yang–Mills Theory as Higher Chern–Simons Theory, In: Journal of High Energy Physics111 Springer Verlag

We observe that the string field theory actions for the topological sigma models describe higher or categorified Chern–Simons theories. These theories yield dynamical equations for connective structures on higher principal bundles. As a special case, we consider holomorphic higher Chern–Simons theory on the ambitwistor space of four-dimensional space-time. In particular, we propose a higher ambitwistor space action functional for maximally supersymmetric Yang–Mills theory.

MC Abbott, J Murugan, S Penati, A Pittelli, D Sorokin, P Sundin, J Tarrant, M Wolf, L Wulff (2015)T-duality of Green-Schwarz superstrings on AdS(d) x S d x M 10-2d, In: JOURNAL OF HIGH ENERGY PHYSICS(12)ARTN 104 SPRINGER
C Sämann, Martin Wolf (2014)Non-Abelian Tensor Multiplet Equations from Twistor Space, In: Communications in Mathematical Physics328(2)pp. 527-544 Springer

We establish a Penrose-Ward transform yielding a bijection between holomorphic principal 2-bundles over a twistor space and non-Abelian self-dual tensor fields on six-dimensional flat space-time. Extending the twistor space to supertwistor space, we derive sets of manifestly N = (1,0) and N = (2,0) supersymmetric non-Abelian constraint equations containing the tensor multiplet. We also demonstrate how this construction leads to constraint equations for non-Abelian supersymmetric self-dual strings. © 2014 Springer-Verlag Berlin Heidelberg.

M Wolf (2012)Contact manifolds, contact instantons, and twistor geometry, In: JOURNAL OF HIGH ENERGY PHYSICS(7)ARTN 074 SPRINGER
M Wolf (2017)Twistor geometry and gauge theory, In: Proc Modave Summer School Math Phys 1 (2005) 248
B Jurčo, C Sämann, M Wolf (2015)Semistrict higher gauge theory, In: Journal of High Energy Physics2015(4)

We develop semistrict higher gauge theory from first principles. In particular, we describe the differential Deligne cohomology underlying semistrict principal 2-bundles with connective structures. Principal 2-bundles are obtained in terms of weak 2-functors from the Cech groupoid to weak Lie 2-groups. As is demonstrated, some of these Lie 2-groups can be differentiated to semistrict Lie 2-algebras by a method due to Severa. We further derive the full description of connective structures on semistrict principal 2-bundles including the non-linear gauge transformations. As an application, we use a twistor construction to derive superconformal constraint equations in six dimensions for a non-Abelian N=(2,0) tensor multiplet taking values in a semistrict Lie 2-algebra.

C Sämann, Martin Wolf (2014)Six-Dimensional Superconformal Field Theories from Principal 3-Bundles over Twistor Space, In: Letters in Mathematical Physics104(9)pp. 1147-1188

We construct manifestly superconformal field theories in six dimensions which contain a non-Abelian tensor multiplet. In particular, we show how principal 3-bundles over a suitable twistor space encode solutions to these self-dual tensor field theories via a Penrose-Ward transform. The resulting higher or categorified gauge theories significantly generalise those obtained previously from principal 2-bundles in that the so-called Peiffer identity is relaxed in a systematic fashion. This transform also exposes various unexplored structures of higher gauge theories modelled on principal 3-bundles such as the relevant gauge transformations. We thus arrive at the non-Abelian differential cohomology that describes principal 3-bundles with connective structure. © 2014 Springer Science+Business Media Dordrecht.