Carlos Perez-Sanchez math-phys

Main areas of work and research contributions:

Path integrals in noncommutative geometry and random matrices

Noncommutative geometry on graphs and lattice gauge-Higgs theory

Random tensors

Functional renormalisation

Bootstrap (non-conformal, beginner)

Before sketching these five points in the illustrated tour that follows, let me briefly describe in plain terms what they have in common. Several physical theories care about a collection of mathematical objects \(a_0,a_1,a_2,\ldots,\) from which measurable quantities can be extracted. Independent of this last meaning, a common approach by some physicists, mathematicians and computer scientists is to study the generating series \( \sum_k a_k z^k\)\( =a_0+a_1z + a_2 z^2+\ldots,\) instead of looking at \(a_0,a_1,a_2,\ldots,\) as isolated entities. For instance, on the square lattice on the plane, or for concreteness the corners in a grid-planned city, fix a point, \(0\), and consider the numbers \(a_0,a_1,\ldots,\) where \(a_r\) is the number of points at a distance \(r\) from \(0\) (below, the displayed numbers are the radius of such \(L^1\)-spheres around \(0\)). Their generating series can be expressed as a known function:

HarerZagier N=2

One advantage of the generating function on the right is the generalisations it promptly motivates: [Sorry. Maybe in other browser you could see an animation of the
Harer-Zagier function for d=3]

[Sorry. Maybe in other browser you could see an animation of the
Harer-Zagier function for d=3]

Indeed, it can be proven that \( \mathrm{HZ}_d(z)=[(1+z)/(1-z)]^d\) is the generating series of the volumes of the \( L^1 \)-spheres in a \(d\)-dimensional square lattice. (The 'HZ' is due to this being the Harer-Zagier function that solves a true mathematical problem that concerns the Euler characteristic of \(\mathcal M_g^1\); but, as you see above, their function also enumerates earthly objects.)

These initial remarks are for sake of vocabulary. Below, physically motivated quantities are implicit in partition functions––a special kind of generating series of non-integer numbers (whose coefficients can themselves be series in some cases, and which can even be indexed not by integers, but by richer geometry-encoding combinatorial data, in others). Path integrals will be the method to obtain them.

landscape of one of my research interests

Without being a probabilist, roughly speaking, I work with random geometry. In particular, in noncommutative geometry (NCG, as depicted in the chart below), physics is encoded in the spectrum of a preferred operator called Dirac. Accepting this principle implies the Connes-Chamseddine spectral action, our classical action functional, and a reason to choose the Dirac operator as our 'field content' is that, under the hood, it accommodates gravity, gauge interactions and Higgs fields simultaneously. Further, the spectral action yields known physics:

landscape of one of my research interests

Path integrals over finite-rank Dirac operators become then a random matrix problem. One example of this class, due to Barrett, is known as matrix or fuzzy geometries, which, interestingly, feature interaction polynomials on multiple matrices and products of multitraces, as proven here in general. The artist's impression could be:

fuzzy sphere and torus, the latter the artist's interpretation

fuzzy sphere and torus, the latter the artist's interpretation

Gauge and gauge-Higgs theories on such spaces are not meant to be depicted, but they are described here.

In a related but independent approach, another type of matrix ensemble emerges when one considers gauge theories on graphs, also in the formalism of noncommutative geometry. When additional data decorate the vertices (nodes) and the edges (links) of a graph one often refers to it as 'quiver'. In our case vertices \(v\) carry certain C*-algebras \(A_v\) and Hilbert spaces \(H_v\), while paths \(\gamma: u\to v\) on the graph are labelled by maps \(\phi_\gamma\) between the algebras at the endpoints \(\phi_\gamma: A_u \to A_v\), along with unitarities \(\mathrm{hol}_\gamma: H_u\to H_v\) called holonomies, as exposed here (my work relies on Baez' and on that of Marcolli-van Suijlekom, who first used quivers in this context, although a different target category). These matrix ensembles are no longer hermitian but unitary––as in lattice field theory to which this theory reduces when it ought to. Such unitary matrices are the ingredient to construct a Dirac operator, which together with the pair \( \oplus_v A_v, \oplus_v H_v\) yields a spectral triple –– that is, a noncommutative geometry. To understand its dynamics, the spectral action was computed (cf. details here) as a certain expansion over the holonomy of loops on the graph, which generalise Wilson-plaquettes.

fuzzyspaces

This way, the Dirac operator path integral is transformed into an ensemble of unitary matrices for which loop equations or Makeenko-Migdal-like equations were obtained.

A third topic I'm interested in deals with simplicial or (singular) PL-approximations of manifolds, which can be generated by the Feynman expansion of tensor integrals, whose large-N was an essential finding by Gurău. Under precise rules (e.g. index contraction = face identification), tensors represent simplices and thereby Feynman graphs of invariant tensor polynomials yield simplicial gluings. In this context, I am interested in topological and geometric problems.

tensor model interpretation

I also used quantum field theoretical techniques for random tensors, like the loop (or Dyson-Schwinger) equations proven here and this Ward-Takahashi Identity, as well as on the geometric interpretation of their correlation functions, as sketched below.

A result with a combinatorial component is the Tutte-like recursion satisfied by the correlation functions or moments of tensor integrals, as proven at the end of this article. Briefly, the original recursion by Tutte is a statement in two dimensions that describes how generating series \(\mathcal{T}_{\ell_1,\ldots,\ell_n} \) of polygonal gluings with boundaries of prescribed perimeters \( \ell_1,\ldots, \ell_n \) are related among themselves. This can be proven with loop equations for matrix integrals or, combinatorially deduced, by edge-removals. In the case of the pictures below for planar rooted quadrangulations, one has \( \mathcal{T}_{\ell+1} = \lambda \mathcal{T}_{\ell+3} +\sum_{\ell_1 + \ell_2 = \ell -1} \mathcal{T}_{\ell_1} \mathcal{T}_{\ell_2} \), as interpreted in the picture below, where the coefficient \( [\lambda^q]\mathcal{T}_{\ell} \) of \( \lambda^q\) in the generating series \(\mathcal{T}_{\ell} \) yields the number gluings with \(q\) quadrangles with one boundary of perimeter \(\ell\).

Tutte recursion, matrices

The tensor analogue of Tutte's recursion describes the three-dimensional situation for simplices generated by quadratic-quartic tensor integrals.

Tutte-like equations for tensors

For given boundary data \(\mathcal{D}=(\mathcal{R},\mathcal{Q})\) that can be grasped as the higher-dimensional analogue of the perimeters in Tutte's two-dimensional case (hence \(\mathcal{R}\) and \(\mathcal{Q} \) represent discrete closed surfaces, since we are in one dimension higher), the obtained equations are sketched above. These describe several generating series \( \mathfrak f_{\mathcal R},\varsigma({\mathcal D}),\ldots,\) etc. (blobs in cyan) that yield, after contraction with the quartic tensor invariant (depicted in form of a pillow), the initial boundary data. The proof of this claim is not combinatoric, but uses this tensor Ward Identity, motivated by the work of Dissertori-Gurău-Magnen-Rivasseau for matrix integrals, along with the loop equations. (This combination was a key strategy for the Grosse-Wulkenhaar model.)

Working with functional renormalisation, particularly with Wetterich equation, I focused first (in this article) on matrix models motivated by noncommutative geometry as mentioned above. The parameter of the RG-flow is (the logarithm of) a threshold on the Dirac spectrum, which in the two-sphere is related to the azimuthal number \(\ell\) and can be depicted as:

A truncation by N

The interactions (or equivalently, the probability measure) governing these models contain certain products of traces of polynomials in several matrices like \(\mathrm{Tr}(A^2) \times \mathrm{Tr}(B^2) \) or \(\mathrm{Tr} (A BC+CBA).\) The Wetterich equation in functional renormalisation traces powers of the Hessian of such expressions and, particularly due to the double traces, obeys an algebra I found interesting and described here. Indeed, using (a bigger 'cousin' of) the free algebra speeds up computations and has a transparent Feynman diagrammatics interpretation. The need for an algebraic structure, as opposed to graph combinatorics or to a matrix entries computation, can be motivated by the next ribbon-graphical example, where white-filled strands represent a matrix, while those in solid colour a derived matrix.

Hessian

(Definitions for \( \boxtimes \) and \(\otimes \), to wit how this algebra works, are explained here.) These latter strands in solid colour form a loop when Hessians are multiplied and traced, and the uncontracted white-filled edges––read off in the proper order the words \( w_1\) and \( w_2\) around the two boundaries of that loop––yield the effective interactions, e.g. \(\mathrm{Tr} (w_1) \times \mathrm{Tr}(w_2)\) in the case below that illustrates a generic model:

1-loop structure

Lately I'm interested in the bootstrap approach to random tensors and random matrix integrals. Bootstrap usually evokes a conformal field theory context, but in the present case it just means to exploit the (easy to prove, albeit powerful) fact that a certain matrix of expectation values, typically Hankel- or Toeplitz-shaped, is positive. As the size of its upper-left block submatrices grows, one gets tighter and tighter constraints from the sign of their determinant, which eventually yield the solution to the model (in the sense that a family of expectation values is determined in terms of the coupling and thus, by loop equations, the whole correlation functions).

Applying the bootstrap to the simplest interesting quiver (details here), whose partition function boils down to the Gross-Witten-Wadia integral, yields the following. The planes represent the coupling \(x\) and the first moment \(y\), and specifically, the red region denotes the negativity of the determinant of said \(2\times 2\) submatrix, the orange one that of the \(3\times 3,\ldots, \) the blue region of the \(6\times 6\) block, etc. Hence the solution \(y(x)\) must lie in all the transparent regions simultaneously. The exact solution in the rightmost plot shows neat agreement with the bootstrap:

Effective interaction gets two traces

Without having published on the next topics, I'm also interested in JT-gravity (short handout from our STRUCTURES-seminar), Topological Recursion (handout from a talk) and lately in free probability, the BV-formalism and Weingarten calculus. I'm also interested in the mathematical theory of generative linguistics developed in 2023 by Marcolli-Chomsky-Berwick, which exhibits the precise algebraic structure of the Merge (Chomsy et al.).

I work in the group of Răzvan Gurău at the Institute of Theoretical Physics, University of Heidelberg. Before, I was a postdoc in the group of Piotr Sułkowski at the Institute of Theoretical Physics, University of Warsaw, Poland. Prior to that, I did my Ph.D. in mathematics at the University of Münster, under supervision of Raimar Wulkenhaar.

This term I am teaching Theoretical Statistical Physics in the MSc programme (Tutorium WiSe 24/25).

Thanks for visiting!

Kontakt:
Carlos Perez-Sanchez
Institut für Theoretische Physik,
Philosophenweg 19
69120, Heidelberg, Deutschland.
Tel. +49-6221-54-9103
My e-mail is my first surname (perez) with domain ITP-Heidelberg
(i.e. @uni-heidelberg.de is insufficient, use thphys.uni-heidelberg.de):

emails

Carlos I. Perez-Sanchez