Demonstration of the Equivalence of Soft and Zero-Bin Subtractions

Calculations of collinear correlation functions in perturbative QCD and Soft-Collinear Effective Theory (SCET) require a prescription for subtracting soft or zero-bin contributions in order to avoid double counting the contributions from soft modes. At leading order in $\lambda$, where $\lambda$ is the SCET expansion parameter, the zero-bin subtractions have been argued to be equivalent to convolution with soft Wilson lines. We give a proof of the factorization of naive collinear Wilson lines that is crucial for the derivation of the equivalence. We then check the equivalence by computing the non-Abelian two-loop mixed collinear-soft contribution to the jet function in the quark form factor. These results provide strong support for the equivalence, which can be used to give a nonperturbative definition of the zero-bin subtraction at lowest order in $\lambda$.Comment: 14 pages, 3 figure

Matrix factorizations and singularity categories for stacks

We study matrix factorizations of a section W of a line bundle on an algebraic stack. We relate the corresponding derived category (the category of D-branes of type B in the Landau-Ginzburg model with potential W) with the singularity category of the zero locus of W generalizing a theorem of Orlov. We use this result to construct push-forward functors for matrix factorizations with relatively proper support.Comment: 29 page

Hori-mological projective duality

Kuznetsov has conjectured that Pfaffian varieties should admit non-commutative crepant resolutions which satisfy his Homological Projective Duality. We prove half the cases of this conjecture, by interpreting and proving a duality of non-abelian gauged linear sigma models proposed by Hori.Comment: 55 pages. V2: slightly rewritten to take advantage of the `non-commutative Bertini theorem' recently proved by the authors and Van den Bergh. V3: lots of changes in exposition following referees' comments. Section 5 has been mostly cut because it was boring. To appear in Duke Math. J. V3: added funder acknowledgemen

The foam and the matrix factorization sl3 link homologies are equivalent

We prove that the foam and matrix factorization universal rational sl3 link homologies are naturally isomorphic as projective functors from the category of link and link cobordisms to the category of bigraded vector spaces.Comment: We have filled a gap in the proof of Lemma 5.2. 28 page

Derived factorization categories of non-Thom--Sebastiani-type sums of potentials

We first prove semi-orthogonal decompositions of derived factorization categories arising from sums of potentials of gauged Landau-Ginzburg models, where the sums are not necessarily Thom--Sebastiani type. We then apply the result to the category \HMF^{L_f}(f) of maximally graded matrix factorizations of an invertible polynomial $f$ of chain type, and explicitly construct a full strong exceptional collection E_1,\hdots,E_{\mu} in \HMF^{L_f}(f) whose length $\mu$ is the Milnor number of the Berglund--H\"ubsch transpose $\widetilde{f}$ of $f$. This proves a conjecture, which postulates that for an invertible polynomial $f$ the category \HMF^{L_f}(f) admits a tilting object, in the case when $f$ is a chain polynomial. Moreover, by careful analysis of morphisms between the exceptional objects $E_i$, we explicitly determine the quiver with relations $(Q,I)$ which represents the endomorphism ring of the associated tilting object $\oplus_{i=1}^{\mu}E_i$ in \HMF^{L_f}(f), and in particular we obtain an equivalence \HMF^{L_f}(f)\cong \Db(\fmod kQ/I).Comment: Major improvements. The proof of the existence of a tilting object is added, and we compute the associated quiver with relations. 48 page

Quantum-classical equivalence and ground-state factorization

We have performed an analytical study of quantum-classical equivalence for quantum $XY$-spin chains with arbitrary interactions to explore the classical counterpart of the factorizing magnetic fields that drive the system into a separable ground state. We demonstrate that the factorizing line in parameter space of a quantum model is equivalent to the so-called natural boundary that emerges in mapping the quantum $XY$-model onto the two dimensional classical Ising model. As a result, we show that the quantum systems with the non-factorizable ground state could not be mapped onto the classical Ising model. Based on the presented correspondence we suggest a promising method for obtaining the factorizing field of quantum systems through the commutation of the quantum Hamiltonian and the transfer matrix of the classical model.Comment: 5 pages, 2 figure